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\begin{document}
\title[Stable families and exceptional extensions]{Stable algebraic families\\of polarized dynamical systems\\and exceptional extensions}
\date{}
\begin{abstract}
The notion of exceptional
maps in arbitrary dimension was recently introduced by Ji, Song, and Xie.  In this paper, we introduce exceptional extensions of polarized dynamical systems:
a family of systems admitting a positive-dimensional exceptional factor whose generic fiber dynamics have simultaneous local
product coordinates with an isotrivial system. 
We then  prove that an algebraic family of polarized endomorphisms of
normal projective varieties over a quasi-projective
curve is $J$-stable on a nonempty Zariski open subset if and only if
it is isotrivial or it is an exceptional
extension after a finite base change. This extends McMullen's theorem for stable algebraic families of rational maps to higher dimension.
One ingredient is inspired by Ji and Xie's proof of McMullen's
theorem: the use of homoclinic orbits and the multipliers of adjoint
periodic points to control local return maps. The new ingredients are a higher-dimensional algebraization method and an inductive rigidity argument for the fiber dynamics, which together yield the exceptional extension structure. 
\end{abstract}
\maketitle

\section{Introduction and definitions}
\label{sec:main-definitions}

McMullen's rigidity theorem asserts that a stable irreducible
algebraic family of rational maps of degree at least two is
either isotrivial or consists of flexible Latt\`es maps; see
\cite[Theorem~2.2]{McM87} and the modern formulation
\cite[Theorem~1.2]{JX23}. Ji and Xie give a new proof using
homoclinic orbits and the multipliers of adjoint periodic
points \cite[Sections~2.2--2.3 and~6]{JX23}. Their viewpoint
motivates the local return arguments used here to study generic
$J$-stability in higher phase dimension, while allowing the
normal projective fibers to vary.

The case of $\mathbf P^2$ has recently been treated in the Hexagon
preprint contributed by Jit-Wu Yap
\cite[Theorem~1.2]{Yap26}. Its main theorem classifies
equilibrium-stable algebraic families as projectively isotrivial
families, homogeneous lifts of flexible Latt\`es families on
$\mathbf P^1$, or families of plane Latt\`es maps, with the relevant
presentations obtained after finite base change and restriction
to a nonempty algebraic open. The present paper formulates a
structural characterization in arbitrary phase dimension and
allows varying normal projective phase spaces.

We introduce the notion of an \emph{exceptional extension}
(Definition~\ref{def:exceptional-extension}) to formulate the
structure of stable families in higher dimension. It combines an
exceptional factor of abelian variety type with a fixed polarized
complex fiber system through simultaneous local product coordinates.
Theorem~\ref{thm:introduction-normal-reduction} characterizes
generically $J$-stable algebraic families as isotrivial families
or families whose generic systems become exceptional extensions
after a finite parameter extension. The extension condition records
the compatibility of the fiber dynamics along all finite forward
paths over the exceptional base.

Throughout, varieties are integral and the ground fields have
characteristic zero. The parameter space $S$ is a smooth connected
complex algebraic curve, and $K=\mathbf C(S)$.
A finite parameter extension means a finite extension $K'/K$,
realized by the normalization of $S$ in $K'$ and restriction to
a nonempty open subset. The resulting parameter curve is again
smooth and connected. For a complex variety $X$, write
$X_{K'}=X\times_{\mathbf C}\operatorname{Spec}K'$.
For a family $p:\mathcal X\to S$, write
$X_\eta=\mathcal X\times_S\operatorname{Spec}K$ and
$X_{\eta,K'}=X_\eta\times_K\operatorname{Spec}K'$.

\begin{definition}[Polarized families]
\label{def:polarized-family-stability}
A \emph{$q$-polarized endomorphism} of a normal projective
variety $V$ is a finite surjective morphism $u:V\to V$ together
with an ample line bundle $L$ such that
$u^*L\simeq L^{\otimes q}$, where $q>1$ is an integer.
The triple $(V,u,L)$ is a \emph{$q$-polarized dynamical system}.
An \emph{algebraic family} consists of a flat
projective morphism $p:\mathcal X\to S$ with geometrically
integral normal fibers, an $S$-endomorphism $f$ of $\mathcal X$,
and a relatively ample line bundle $L$ satisfying
$f^*L\simeq L^{\otimes q}$. A family on a fixed $X$ has
$\mathcal X=S\times X$.

\end{definition}

\begin{definition}[$J$-stability]
\label{def:J-stability}
Let $(p:\mathcal X\to S,f,L)$ be an algebraic family as above,
with fibers of dimension $N$. For $s\in S(\mathbf C)$, let $T_s$
be the Green current of $f_s$ and set
\[
 \mu_s=\frac{T_s^N}{(L_s^N)},\qquad
 J_s=\operatorname{supp}\mu_s.
\]
Here $T_s=\lim_{m\to\infty}q^{-m}(f_s^m)^*\omega_s$ for a smooth
positive representative $\omega_s$ of $c_1(L_s)$. The continuous
local potentials of $T_s$ define its Monge--Amp\`ere measure;
on a normal fiber this can equivalently be computed on a resolution
and pushed down. For $n\geq1$, write
\[
 R_n(s)=\left\{x\in X_{s,\mathrm{reg}}\cap J_s:
 \begin{array}{l}
 f_s^n(x)=x,\\[-2pt]
 |\lambda|>1\text{ for every eigenvalue }\lambda
       \text{ of }D_xf_s^n
 \end{array}\right\}.
\]
Thus $R_n(s)$ consists of the smooth repelling Julia points fixed
by $f_s^n$; their exact periods may divide $n$.

The family is \emph{$J$-stable} if, for every $s_0\in S(\mathbf C)$,
there is an analytic disk $B\subset S^{\mathrm{an}}$ centered
at $s_0$ such that, for every $n\geq1$
and every $x\in R_n(s_0)$, there is a holomorphic section
\[
 \gamma_{n,x}:B\longrightarrow\mathcal X_B,
 \qquad p\circ\gamma_{n,x}=\operatorname{id}_B,
 \qquad \gamma_{n,x}(s_0)=x,
\]
and, for every $s\in B$, evaluation induces a bijection
\[
 R_n(s_0)\longrightarrow R_n(s),\qquad
                     x\longmapsto\gamma_{n,x}(s).
\]
For each $n$, these sections are disjoint, remain smooth and
repelling, and exhaust the smooth repelling Julia points fixed
by $f_s^n$ in every fiber. The same disk $B$ works for all
$n$. This is also called \emph{strong $J$-stability}.

The family is \emph{generically $J$-stable} if its restriction
to $S\setminus E$ is $J$-stable in this sense for some finite
subset $E\subset S$. Equivalently, it is $J$-stable on a nonempty
Zariski open subset of $S$.
\end{definition}

\begin{definition}[Isotriviality and rational factors]
\label{def:iso-main}
A family is \emph{isotrivial} if, after a finite extension of $K$,
its generic dynamical system is isomorphic to a dynamical system
defined over $\mathbf C$. A \emph{rational fibration}
$\pi:V\dashrightarrow Z$ is a dominant rational map whose
geometric generic fiber is integral; equivalently, $k(Z)$ is
relatively algebraically closed in $k(V)$. It is a
\emph{dynamical factor} of $(V,u)$ if $Z$ is normal projective
and has an endomorphism $g$ satisfying
\[
                         \pi u=g\pi.
\]
Equalities of rational maps are understood on their common
domain of definition. The \emph{join} of finitely many rational
factors is the normalization of their joint image. Its connected
version is obtained by taking the relative algebraic closure of
its function field in $k(V)$.
\end{definition}

\begin{definition}[Exceptional maps of abelian variety type]
\label{def:abelian-type-exceptional}
An \emph{abelian variety} is a connected projective algebraic
group. An \emph{isogeny} is a surjective homomorphism of abelian
varieties with finite kernel. It is \emph{$Q$-polarized} if it
pulls an ample line bundle back to its $Q$th tensor power.

A $q$-polarized system $(Z,g)$ is \emph{exceptional of abelian variety type} if
there exist an abelian variety $A$, a finite surjective morphism
$\eta:A\to Z$, an integer $e\geq1$, and a $q^e$-polarized
isogeny $\alpha:A\to A$ such that
\begin{equation}\label{eq:abelian-type-presentation}
                         \eta\alpha=g^e\eta.
\end{equation}
We call $(A,\alpha,\eta,e)$ an \emph{abelian presentation}.
These are the abelian variety type among the exceptional systems
of \cite[Definitions~1.2--1.4]{JSX26}. The presentation data are
defined after a finite extension of the ground field.

\end{definition}

\begin{definition}[Simultaneous local product coordinates]
\label{def:simultaneous-fiber-coordinates}
A \emph{phase open} is a Zariski open subset of a dynamical
variety over the parameter function field. Let $K=\mathbf C(S)$, let $\pi:X\dashrightarrow Z$ be a dominant
rational map of normal projective $K$-varieties with geometrically
integral generic fiber, and suppose that
\[
                       \pi\circ f=g\circ\pi.
\]
Let $\Gamma$ be the normalization of the closure of the graph of
$\pi$ in $X\times Z$, let $\rho:\Gamma\to Z$ be its projection,
and let $F$ be the endomorphism induced by $f\times g$.
Choose an ample line bundle $L$ on $\Gamma$ satisfying
$F^*L\simeq L^{\otimes q}$.  Write $\Gamma_z$ for the fiber of
$\rho$ over a geometric point $z$ of $Z$.  The \emph{fiber path
of length $n$} starting at $z$ is
\[
 \Gamma_z\longrightarrow\Gamma_{g(z)}
       \longrightarrow\cdots\longrightarrow\Gamma_{g^n(z)},
\]
whose arrows are induced by $F$ and whose polarizations are the
restrictions of $L$.  A \emph{generic fiber path} is this diagram
at a geometric generic point of $Z$.

The generic fiber arrows have \emph{simultaneous local product
coordinates with one fixed polarized fiber system over
$\mathbf C$} if there are a normal projective complex variety $W$,
an ample line bundle $L_W$, and a finite endomorphism $h:W\to W$
satisfying $h^*L_W\simeq L_W^{\otimes q}$ such that every generic
fiber path is geometrically polarized isomorphic to
\[
                      W\xrightarrow{h}W\xrightarrow{h}\cdots
                                        \xrightarrow{h}W.
\]
A \emph{frame} is an isomorphism from the fixed chain to the
actual path, given by polarized isomorphisms at the vertices
commuting with every arrow.

In geometric coordinates, choose a dense open $V\subset Z$ over
which $\rho$ has normal geometrically integral fibers.  For each
$n\geq1$, after restricting to a dense open
\[
                  V_n\subseteq\bigcap_{i=0}^n g^{-i}(V),
\]
there are an \emph{\'etale} surjection $T_n\to V_n$ and polarized
isomorphisms
\[
 c_i:T_n\times W\xrightarrow{\sim}
             \bigl((g^i)^*\Gamma\bigr)|_{T_n},
                       \qquad 0\leq i\leq n.
\]
Here $(g^i)^*\Gamma=Z\times_{g^i,Z,\rho}\Gamma$, and the target
of $c_i$ is its base change to $T_n$; its fiber over a point lying
above $z\in V_n$ is $\Gamma_{g^i(z)}$.  The coordinates satisfy
\[
               F_i\circ c_i
                   =c_{i+1}\circ(\operatorname{id}_{T_n}\times h).
\]
Here $F_i$ is the induced $i$th fiber arrow, and all products with
$W$ include its base change from $\mathbf C$.  By the stabilization
argument in the proof of Lemma~\ref{lem:stationary-constant-path},
the frames can be chosen by restriction of one finite-length
frame torsor.
After spreading over a parameter open, its analytic local sections
give holomorphic product coordinates.  Intersecting these charts
with the locus where $\Gamma\dashrightarrow X$ is an isomorphism
gives the corresponding coordinates on dense phase opens of $X$.
For a zero-dimensional fiber we take $W=\operatorname{Spec}\mathbf C$
and $h=\operatorname{id}_W$.
\end{definition}

\begin{definition}[Exceptional extensions]
\label{def:exceptional-extension}
Let $K'$ be a finite extension of $K=\mathbf C(S)$, and let
$(X,f,L_X)$ be a $q$-polarized dynamical system on a normal
projective $K'$-variety. We call $(X,f)$ an
\emph{exceptional extension} if there exist a normal projective
$K'$-variety $Z$, a $q$-polarized endomorphism $g:Z\to Z$
exceptional of abelian variety type, and a rational fibration
\[
 \pi:X\dashrightarrow Z,\qquad \pi f=g\pi,\qquad
                  1\leq r:=\dim Z\leq\dim X,
\]
whose generic fiber arrows have simultaneous local product
coordinates with one fixed polarized complex fiber system
$(W,L_W,h)$ of dimension $\dim X-r$, in the sense of
Definition~\ref{def:simultaneous-fiber-coordinates} applied over $K'$.
We call $\pi$ an \emph{exceptional extension structure}, $(Z,g)$
its \emph{exceptional base}, and $(W,L_W,h)$ its
\emph{fixed fiber system}. A zero-dimensional fiber is allowed,
with the point system specified in
Definition~\ref{def:simultaneous-fiber-coordinates}.
\end{definition}

\begin{theorem}[Stable families and exceptional extensions]
\label{thm:introduction-normal-reduction}
Let $f:\mathcal X\to\mathcal X$ be an algebraic family of
$q$-polarized endomorphisms of relative dimension $N\geq1$
over a smooth connected complex algebraic curve $S$.
Put $K=\mathbf C(S)$, and write $(X_\eta,f_\eta)$ for its
generic dynamical system.
Then $f$ is generically $J$-stable if and only if one of the
following holds:
\begin{enumerate}
\item The family $f$ is isotrivial.
\item After a finite extension $K'/K$, the generic system
$(X_{\eta,K'},f_{\eta,K'})$ is an exceptional extension in the
sense of Definition~\ref{def:exceptional-extension}.
\end{enumerate}
\end{theorem}

In the second alternative, write
$\pi:X_{\eta,K'}\dashrightarrow Z$ for the exceptional extension
structure, with exceptional base $(Z,g)$.
Put $r=\dim Z$. Then $1\leq r\leq N$, and its fixed fiber
system has dimension $N-r$.
On a dense open $U$ where $\pi$ is regular, equivariance gives the map to the
dynamical pullback
\[
 U\cap f^{-1}(U)\longrightarrow
 Z\mathbin{\times}_{g,Z,\pi}U,\qquad
 x\longmapsto(\pi(x),f(x)).
\]
Its fiber over $z$ is the arrow from the fiber over $z$ to the
fiber over $g(z)$.

\begin{corollary}[Families on projective space]
\label{cor:projective-space-stability}
Let $S$ be a smooth connected complex algebraic curve and let
$f:S\times\mathbf P^N\to S\times\mathbf P^N$ be an algebraic
family of endomorphisms of degree $d\geq2$. Put $K=\mathbf C(S)$.
Then $f$ is $J$-stable if and only if, after a finite extension
$K'/K$, one of the following holds:
\begin{enumerate}
\item There are $f_0\in\operatorname{End}_d(\mathbf P^N)(\mathbf C)$
and $M\in\operatorname{PGL}_{N+1}(K')$ such that
$f_{K'}=M^{-1}f_0M$.
\item The system $(\mathbf P^N_{K'},f_{K'})$ is an exceptional
extension in the sense of Definition~\ref{def:exceptional-extension},
with extension structure $\pi:\mathbf P^N_{K'}\dashrightarrow Z$
and geometrically unirational exceptional base $(Z,g)$.
\end{enumerate}
Here the polarization on the source is $\mathcal O_{\mathbf P^N}(1)$
and $q=d$. If $\dim Z=1$, then
$Z_{\overline K}\simeq\mathbf P^1_{\overline K}$ and $g$ is a
Latt\`es map. If $\dim Z=N$, then $f_{\overline K}$ is itself
a Latt\`es map.
\end{corollary}

Here \emph{geometrically unirational} means dominated by a projective
space after passage to an algebraic closure. A \emph{Latt\`es map}
on $\mathbf P^r$ is an endomorphism admitting a finite surjective
semiconjugacy from an affine isogeny of an abelian variety of
dimension $r$.

Constant repelling spectra give local return structures, whose
algebraization produces polarized rational factors. Uniformly
bounded covers of periodic fibers close the induction on the
phase dimension. A maximal varying abelian factor has constant
finite fiber-path classes, and a stationary path representative
gives the fixed system $(W,L_W,h)$, completing the exceptional
extension structure. For the converse, compatible frames of an
exceptional extension extend over the abelian phase base. Periodic sections of
the resulting bundle give an equilibrium web, and the relative
web criterion proves $J$-stability on a nonempty parameter open.


\section{Equilibrium webs and removable parameters}
\label{web:sec:web-criterion}

The main result of this section is
Theorem~\ref{thm:web-criterion-normal}: for a polarized family
on a fixed normal projective variety, the existence of an
arbitrary equilibrium web, the holomorphic motion of all
repelling Julia cycles, and the vanishing of the Lyapunov
bifurcation current $dd^cL$ are equivalent. Here $L$ is the sum
of the Lyapunov exponents of the equilibrium measure.
We distinguish the new statements from the existing projective-space
theory. Equilibrium webs were introduced by
Berteloot--Bianchi--Dupont \cite{BBD18}. Their Theorem~1.1 proves
this equivalence for arbitrary families on $\Pj^2$ over simply
connected parameter spaces, and for simply connected open subsets
of the full parameter space in every dimension. Berteloot--Buff
\cite[Proposition~1.2 and Theorem~4.1]{BB26} prove in every
dimension that $\mu$-stability, defined by the existence of an
equilibrium lamination, implies the holomorphic motion of all
repelling Julia cycles. The passage from acritical ergodic webs
to laminations is established in \cite[Theorem~4.1]{BBD18}; see
also \cite[Theorem~2.2]{BB26}. In particular, the equivalence
between $dd^cL=0$, an equilibrium lamination, and the motion of
all repelling Julia cycles on $\Pj^N$ in arbitrary dimension
is already contained in \cite[Theorem~4.1]{BB26}; the Lyapunov
criterion itself is not a new assertion of this paper.

The new implication in Theorem~\ref{thm:web-criterion-normal}
starts from an arbitrary equilibrium web, without assuming
acriticity, ergodicity, or a pre-existing equilibrium lamination.
This strengthens the cited results even for arbitrary
one-parameter holomorphic families on $\Pj^N$ with $N\geq3$:
the implication from such a web to the motion of every repelling
Julia cycle was not previously available in that generality.
Consequently, an arbitrary equilibrium web can also be added to
the equivalent Lyapunov and lamination criteria in this setting.
The key input is the jet rigidity of
Lemma~\ref{web:jet}, which gives uniqueness of a section through
almost every point at each fixed parameter for any equilibrium
web. This is a pointwise-in-parameter uniqueness statement for
each given web, stronger than merely constructing one acritical
web. Lemma~\ref{web:lifts} then supplies the inverse histories
used in the argument of \cite[Proposition~1.2]{BB26}. This yields
the criterion on normal projective varieties, including repelling
cycles at singular points. The normal-space Lyapunov identity
and its connection with this criterion are proved in
Subsection~\ref{web:lyapunov-subsection}, rather than inferred
from results stated only for $\Pj^N$. Further extensions here are
the removability result for a fixed normal phase variety
(Lemma~\ref{lem:stability-removable-parameters}) and the relative
criterion for varying normal fibers on a nonempty parameter open
(Proposition~\ref{normal:relative-web-criterion}).

Work over a coordinate disk $B$ in a smooth complex parameter curve, with
fixed normal projective phase variety $X$ of dimension $k$.
We first treat this fixed phase space; the final subsection gives
the relative criterion for varying normal phase families.
Put $J_\lambda=\operatorname{supp}\mu_\lambda$. For this section,
a periodic point is called repelling when the eigenvalues on its
Zariski tangent space all have modulus greater than one. Write
$R_n(\lambda)$ for these points in $J_\lambda$ fixed by
$f_\lambda^n$, and $R_n^{\reg}(\lambda)=R_n(\lambda)\cap X_{\reg}$.
The notation $\sO(B,X)$ means the space of holomorphic sections,
with the topology of locally uniform convergence. Set
\[
 (\sF\gamma)(\lambda)=f_\lambda(\gamma(\lambda)),\qquad
 e_\lambda(\gamma)=\gamma(\lambda).
\]

\begin{definition}[Equilibrium web]
An equilibrium web is a compactly supported Borel probability
measure $\mathfrak m$ on $\sO(B,X)$ such that
\[
 \sF_*\mathfrak m=\mathfrak m,\qquad
 (e_\lambda)_*\mathfrak m=\mu_\lambda\quad(\lambda\in B).
\]
\end{definition}

\begin{definition}[Lyapunov stability]\label{def:lyapunov-stability}
Let $\chi_1(\lambda),\ldots,\chi_k(\lambda)$ be the Lyapunov
exponents of $(f_\lambda,\mu_\lambda)$ on the smooth locus,
counted with multiplicity, and put
\[
 L(\lambda)=\sum_{j=1}^k\chi_j(\lambda)
 =\int_X\log|\operatorname{Jac}f_\lambda|\,d\mu_\lambda.
\]
The Jacobian norm uses the Hermitian metric induced by a fixed
projective embedding; Proposition~\ref{web:lyapunov-current}
justifies the integral and metric independence. The family is
\emph{Lyapunov stable} on $B$ if $dd^cL=0$ on $B$, in the
sense of currents. Since the parameter is a curve, this says
that $L$ is harmonic. In phase dimension greater than one the
definition uses the sum of the exponents, not one individual
exponent. The positive current $dd^cL$ is called the
\emph{bifurcation current}.
\end{definition}

\begin{theorem}[Equilibrium webs, $J$-stability, and Lyapunov stability]\label{thm:web-criterion-normal}
The following are equivalent:
\begin{enumerate}
\item an equilibrium web exists over $B$;
\item every $R_n(\lambda)$ is represented by disjoint holomorphic
sections over $B$, exhausting $R_n(\lambda)$ at every parameter;
\item the same assertion holds for $R_n^{\reg}(\lambda)$;
\item $dd^cL=0$ on $B$.
\end{enumerate}
Thus $J$-stability and Lyapunov stability give equivalent
definitions of stability on $B$.
\end{theorem}

\subsection{Polarized dynamics and periodic sections}

We record the dynamical properties and the compactness of periodic sections used in the web criterion.

\begin{prop}\label{web:dynamics}
Let $h:X\to X$ satisfy $h^*L\simeq L^{\otimes q}$. Its equilibrium measure $\mu$ has the following properties.
\begin{enumerate}
\item It is mixing and satisfies $h_*\mu=\mu$ and $h^*\mu=q^k\mu$. It gives no mass to proper local analytic subsets of $X$.
\item If $R_n$ is the set of repelling points of $h^n$ in $X_\reg\cap\supp\mu$, then
\begin{equation}\label{web:periodic-equid}
 q^{-kn}\sum_{x\in R_n}\delta_x\longrightarrow\mu,
 \qquad \#R_n=q^{kn}+o(q^{kn}).
\end{equation}
\item Fix a projective embedding of $X$ and its induced Hermitian norm on $TX_\reg$. There is a completely invariant Borel set $E\subset\supp\mu\cap X_\reg$ of full $\mu$-measure such that $h$ is unramified along every orbit in $E$, every point of $E$ has $q^k$ distinct preimages in $E$, and
\begin{equation}\label{web:unbounded-derivative}
 \sup_{n\geq0}\|D(h^n)_xv\|=+\infty
 \quad(x\in E,\ 0\ne v\in T_xX).
\end{equation}
\end{enumerate}
\end{prop}

\begin{proof}
Let $\pi:Y\to X$ be a projective resolution which is an isomorphism over $X_\reg$, and put $g:=\pi^{-1}\circ h\circ\pi$. The dynamical degrees of $g$ are
\[
 d_j(g)=q^j,\qquad 0\leq j\leq k.
\]
Indeed, on $X$ the degree of order $j$ of $h^n$, computed with $L$, is $q^{nj}(L^k)$, and dynamical degrees are birationally invariant \cite[Theorem~1]{Dan20}. Thus $g$ has dominant topological degree $q^k$.

Let $\nu$ be the equilibrium measure of $g$. The construction and properties of this measure are given in \cite[Theorem~2.1]{Gue05} and \cite[Section~1]{DNT15}. It is balanced and mixing, and gives no mass to pluripolar sets. Its pushforward $\pi_*\nu$ is the equilibrium measure $\mu$ of $h$. This description is independent of the resolution: outside a countable union of proper analytic sets, the normalized preimages of a point correspond under $\pi$, and their limits are given by \cite[Theorem~1.2]{DNT15}. The balance and mixing identities descend to $X$. The inverse image of a proper local analytic subset is a proper local analytic subset of $Y$, so its $\mu$-mass is zero.

By \cite[Theorem~1.1]{DNT15}, the repelling periodic points of $g$, including those in $\supp\nu$, are equidistributed. The exceptional locus of $\pi$ is closed and has zero $\nu$-mass. Its contribution to these normalized counting measures therefore tends to zero. A repelling periodic point in $X_\reg$ has its entire orbit in $X_\reg$: the return germ has local degree one, hence every finite germ along the orbit has local degree one and is an isomorphism, by normality. Such cycles correspond exactly to the cycles of $g$ disjoint from the exceptional locus. Away from that locus, supports also correspond under $\pi$. This proves \eqref{web:periodic-equid}.

The Lyapunov exponents of $\nu$ are at least $\frac12\log q$ by \cite[Theorem~5.4]{DNT15}. Choose a compact set $Q\subset Y\setminus\pi^{-1}(\Sing X)$ of positive $\nu$-measure. For $\nu$-almost every $y$, all Lyapunov exponents are positive, the orbit is defined and avoids the exceptional and critical loci, and there are arbitrarily large $n$ with $g^n(y)\in Q$. On $Q$, the differential $D\pi$ has a uniformly positive conorm. Hence, for $x=\pi(y)$ and $0\ne v\in T_xX$,
\[
 D(h^n)_xv
 =D\pi_{g^n(y)}D(g^n)_y(D\pi_y)^{-1}v
\]
is unbounded along these return times. This proves \eqref{web:unbounded-derivative} on a set of full measure.

Both direct and inverse images of null sets are null. The inverse-image assertion follows from invariance; for the direct image one uses
\[
 \mu(h(A))\leq q^k\mu(A).
\]
This inequality follows from balance by counting preimages. Delete the grand orbit of the exceptional null set just obtained, together with the singular and ramification loci and their images. Intersect the remainder with the support of $\mu$. Balance and the local inverse theorem give complete invariance of this full-measure set. Its points have $q^k$ distinct preimages, and all of them belong to the same set. This gives $E$.
\end{proof}

\begin{lemma}\label{web:normality}
Let $B$ be a coordinate disk. The set of all holomorphic periodic sections $\gamma:B\to X$ is relatively compact in $\sO(B,X)$.
\end{lemma}

\begin{proof}
After shrinking the parameter disk, replace the relative polarization
by one sufficiently high power. Its direct image is a holomorphic vector
bundle, and a local trivialization gives a relative projectively normal
embedding $B\times X\hookrightarrow B\times\Pj^s$; see
\cite[Chapter~1]{Laz04}. Denote the affine cone of its fiber at
$\lambda$ by $C_\lambda\subset\C^{s+1}$. The fiber embeddings may vary
with $\lambda$.

The relative polarization identity, together with a local
trivialization on $B$, lifts the family to homogeneous maps
$F_\lambda:\C^{s+1}\to\C^{s+1}$ of degree $q$. Relative projective
normality and a holomorphic splitting of the restriction map on sections
give lifts depending holomorphically on $\lambda$. They satisfy
\[
 F_\lambda(C_\lambda)\subset C_\lambda,\qquad
 F_\lambda^{-1}(0)\cap C_\lambda=\{0\},\qquad
 [F_\lambda(Z)]=f_\lambda([Z]).
\]
For every $U\Subset B$, there is $C_U\geq1$ with
\begin{equation}\label{web:cone-estimate}
 C_U^{-1}\|Z\|^q\leq\|F_\lambda(Z)\|
       \leq C_U\|Z\|^q
 \quad(\lambda\in U,\ Z\in C_\lambda).
\end{equation}

Let $\gamma$ have period $n$. Since $B$ is contractible and Stein, the pullback of the tautological line bundle is trivial. Thus $\gamma$ has a nowhere zero holomorphic lift $Z_\gamma:B\to\C^{s+1},\quad Z_\gamma(\lambda)\in C_\lambda$, and
\[
 F_\lambda^n(Z_\gamma(\lambda))
       =a_\gamma(\lambda)Z_\gamma(\lambda)
\]
for some $a_\gamma\in\sO(B)^*$. Choose a holomorphic logarithm of $a_\gamma$ and set
\[
 H_\gamma:=a_\gamma^{-1/(q^n-1)}Z_\gamma.
\]
Then $F_\lambda^n(H_\gamma)=H_\gamma$. Iterating \eqref{web:cone-estimate} gives
\begin{equation}\label{web:annulus}
 C_U^{-1/(q-1)}\leq\|H_\gamma(\lambda)\|
       \leq C_U^{1/(q-1)}.
\end{equation}
These bounds are independent of $n$ and $\gamma$. Montel's theorem and a diagonal argument give a locally uniformly convergent subsequence of the lifts of any sequence of periodic sections. Its limit belongs fiberwise to $C_\lambda\setminus\{0\}$ by \eqref{web:annulus}. Projectivizing proves local normality. Applying the construction
locally on $B$ and taking a diagonal subsequence proves the assertion
on the original disk.
\end{proof}

\subsection{Rigidity and inverse histories}

The main results of this subsection are Lemma~\ref{web:jet},
which gives uniqueness of sections through expanding points,
and Lemma~\ref{web:strong}, which describes exponentially
converging inverse histories near a fixed point. For acritical
ergodic equilibrium webs, almost-everywhere unbranching is part
of the lamination theory of \cite[Theorem~4.1]{BBD18}.
The jet argument below obtains the pointwise uniqueness needed
here directly from expansion in any compact forward-invariant
family of sections. We continue to use the metric induced by a
projective embedding of $X$.

\begin{lemma}\label{web:jet}
Let $B$ be connected and let $\sK\subset\sO(B,X)$ be compact with $\sF(\sK)\subset\sK$. Fix $\lambda_0\in B$ and $x\in X$. Suppose that, on the Zariski tangent space,
\[
 \sup_n\|D(f_{\lambda_0}^n)_xv\|=+\infty
       \quad\text{for every }0\ne v\in T_xX.
\]
Then $e_{\lambda_0}^{-1}(x)\cap\sK$ contains at most one element.
\end{lemma}

\begin{proof}
Suppose that distinct $\gamma,\eta\in\sK$ agree at $\lambda_0$. Choose a local parameter $t$ centered at $\lambda_0$. Their first different Taylor coefficient has some order $r\geq1$. In a local embedding at $x$,
\[
 \gamma(t)-\eta(t)=t^rv+O(t^{r+1}),\qquad
       0\ne v\in T_xX.
\]
The coefficient belongs to $T_xX$ because both maps take values in $X$ and their lower-order coefficients agree.

Put $x_n=f_0^n(x)$. Choose projective unitary coordinates $\psi_n$ centered at $x_n$. The order-$r$ coefficient of
\[
 \psi_n(f_t^n(\gamma(t)))-\psi_n(f_t^n(\eta(t)))
\]
is
\begin{equation}\label{web:jet-coefficient}
 D\psi_n(x_n)D(f_0^n)_xv.
\end{equation}
This follows by composing the two Taylor expansions: at the first differing order only the differential in the phase variable contributes.

The sections $\sF^n\gamma$ and $\sF^n\eta$ belong to $\sK$. Equicontinuity therefore gives a parameter disk, independent of $n$, on which the above coordinate maps are uniformly bounded. Cauchy's estimates uniformly bound \eqref{web:jet-coefficient}. The unitary coordinates compare their norms with the fixed projective metric by constants independent of $n$. This contradicts the assumption.
\end{proof}

\begin{lemma}\label{web:lifts}
Let $\mathfrak m$ be an equilibrium web over a connected $B$, and put $\sK:=\supp\mathfrak m$. Fix $\lambda_0\in B$ and let $E$ be as in Proposition~\ref{web:dynamics} for $f_{\lambda_0}$. Set
\[
 \sK_E:=\{\gamma\in\sK:\gamma(\lambda_0)\in E\}.
\]
Then $\mathfrak m(\sK_E)=1$, evaluation induces a bijection $\sK_E\to E$, and $\sF:\sK_E\to\sK_E$ is $q^k$-to-one. More precisely, each preimage of $\gamma(\lambda_0)$ determines a unique preimage of $\gamma$ in $\sK_E$.
\end{lemma}

\begin{proof}
Compactness and the defining identities of a web give
\begin{equation}\label{web:support-identities}
 \sF(\sK)=\sK,
 \qquad e_\lambda(\sK)=J_\lambda
       \quad(\lambda\in B).
\end{equation}
Thus evaluation onto $E$ is surjective, and Lemma~\ref{web:jet} gives injectivity. The full-measure assertion follows from $(e_{\lambda_0})_*\mathfrak m=\mu_{\lambda_0}$.

For $x\in f_{\lambda_0}^{-1}(\gamma(\lambda_0))$, choose the unique $\eta\in\sK_E$ with $\eta(\lambda_0)=x$. Then $\sF\eta$ and $\gamma$ agree at $\lambda_0$, so Lemma~\ref{web:jet} gives $\sF\eta=\gamma$. The $q^k$ preimages of $\gamma(\lambda_0)$ are distinct and belong to $E$. Conversely, two preimage sections with the same value at $\lambda_0$ coincide. This proves the claim.
\end{proof}

\begin{lemma}\label{web:propagation}
Let $B$ be a connected Riemann surface, let $\sK\subset\sO(B,X)$ be compact, and let $p\in\sK$. For each $n\geq0$, let $\mathscr H_n\subset\sK$. Suppose that on some nonempty open subset $V\subset B$ there are $C>0$ and $0<a<1$ such that
\[
 \sup_{\gamma\in\mathscr H_n}\sup_{\lambda\in V}
       \dist(\gamma(\lambda),p(\lambda))\leq Ca^n.
\]
For every compact $Q\subset B$, there exist $C_Q>0$ and $0<a_Q<1$ such that
\[
 \sup_{\gamma\in\mathscr H_n}\sup_{\lambda\in Q}
       \dist(\gamma(\lambda),p(\lambda))\leq C_Qa_Q^n.
\]
\end{lemma}

\begin{proof}
First, convergence to $p$ is locally uniform, uniformly over $\mathscr H_n$. Otherwise, a sequence of violating sections with $n\to\infty$ would have a limit in $\sK$ which equals $p$ on $V$ and differs from it elsewhere. This contradicts the identity theorem.

Join a relatively compact disk in $V$ to a point of $Q$ by a finite chain of parameter coordinate disks. Choose these disks small enough that the image of $p$ over each slightly larger disk lies in one projective coordinate chart. The uniform convergence just proved puts every section in $\mathscr H_n$ in the same chart for all sufficiently large $n$.

We use the following consequence of the three-circle theorem. If $0<r<s<R$ and a scalar holomorphic function $u$ on a disk of radius $R$ satisfies $\sup|u|\leq M$ and $\sup_{|z|\leq r}|u|\leq\epsilon$, then
\[
 \sup_{|z|\leq s}|u|
 \leq M^{1-\theta}\epsilon^\theta,
 \qquad \theta=\frac{\log(R/s)}{\log(R/r)}>0.
\]
Apply this estimate to each coordinate of $\gamma-p$. A bound $Ca^n$ on a smaller disk gives a bound $C'(a^\theta)^n$ on a larger disk. Successive overlapping disks propagate an exponential bound along the chain. A finite number of chains covers $Q$, and finitely many initial values of $n$ are absorbed into $C_Q$.
\end{proof}

\begin{lemma}\label{web:strong}
Let $H:(\C^e,0)\to(\C^e,0)$ be a holomorphic germ. Fix $r>1$ which is not the modulus of an eigenvalue of $A:=DH_0$, and let $E^u$ be the sum of the generalized eigenspaces with eigenvalues of modulus greater than $r$. There is a local complex submanifold $W$ through $0$, tangent to $E^u$, with the following property. If
\[
 H(z_{n+1})=z_n,
 \qquad \|z_n\|=O(b^n),\qquad 0<b<r^{-1},
\]
then $z_n\in W$ for all sufficiently large $n$.
\end{lemma}

\begin{proof}
Write $\C^e=E^u\oplus E^s$ and $H(z)=Az+R(z)$, where $R(0)=DR(0)=0$. Choose an adapted norm with
\[
 \|A_u^{-1}\|<r^{-1},\qquad \|A_s\|<r.
\]
Use the Banach space of sequences $z=(z_n)_{n\geq0}$, with $z_n=(u_n,s_n)$ and norm
\[
 \|z\|_r:=\sup_{n\geq0}r^n\|z_n\|.
\]
For a sufficiently small prescribed $u_0$, consider the equations
\begin{align}
 u_n&=A_u^{-n}u_0-
       \sum_{j=1}^n A_u^{-(n-j+1)}R_u(z_j),\label{web:unstable-u}\\
 s_n&=\sum_{j\geq1}A_s^{j-1}R_s(z_{n+j}).\label{web:unstable-s}
\end{align}
Both convolution operators are bounded in the weighted norm. On a sufficiently small ball, the Lipschitz constant of $R$ makes the right-hand side a contraction. Its fixed point depends holomorphically on $u_0$. The values at $n=0$ form a holomorphic graph $s_0=\varphi(u_0)$ with $\varphi(0)=D\varphi(0)=0$; denote this graph by $W$.

Any sufficiently small inverse history with finite weighted norm satisfies \eqref{web:unstable-u} by iteration. It satisfies \eqref{web:unstable-s} because $\|A_s\|<r$ makes the remainder at infinity vanish. Uniqueness of the fixed point places its initial value in $W$. For the history in the statement, every sufficiently late tail has arbitrarily small weighted norm, since $rb<1$. Apply the preceding assertion to each such tail.
\end{proof}

\subsection{Proof of Theorem~\ref{thm:web-criterion-normal}}

We first prove that an equilibrium web makes every repelling $J$-cycle move holomorphically. The argument also proves that a section through a regular point remains in $X_\reg$. We then construct a web from moving regular periodic points.

\begin{prop}\label{web:web-to-motion}
Suppose that $B$ is connected and that $f$ admits an equilibrium web over $B$. For every $\lambda_0\in B$ and every $p_0\in R_n(\lambda_0)$, there is a unique $p\in\supp\mathfrak m$ such that
\[
 p(\lambda_0)=p_0,\qquad p(\lambda)\in R_n(\lambda)
       \quad\text{for every }\lambda\in B.
\]
\end{prop}

\begin{proof}
Put $\sK:=\supp\mathfrak m$. Replace the family by its $n$th iterate, and continue to denote it by $f$, writing $q$ for its polarization factor. Its equilibrium measures and the web are unchanged. Thus $p_0$ is fixed and repelling.

By \eqref{web:support-identities}, some $p\in\sK$ satisfies $p(\lambda_0)=p_0$. Lemma~\ref{web:jet}, applied to this repelling fixed point, gives uniqueness. Applying it to $\sF p$ and $p$ yields
\begin{equation}\label{web:periodic-section}
 f_\lambda(p(\lambda))=p(\lambda)
       \quad(\lambda\in B).
\end{equation}

We will use the local inverse theorem also at singular points. In a minimal embedding $(X,p_0)\subset(\C^e,0)$, the germ of $f_{\lambda_0}$ extends to an ambient germ with invertible derivative. The ambient inverse theorem applies, and the resulting biholomorphism sends the irreducible germ of $X$ onto itself. Its inverse restricts to an inverse germ on $X$. Since all inverse eigenvalues have modulus less than one, this inverse is contracting in an adapted ambient norm. The same construction applies to the nearby family and its fixed section $p$.

\medskip
\noindent\emph{Construction of inverse histories.}
Choose a parameter neighborhood $V$ of $\lambda_0$ and a neighborhood of $p|_V$ in a local embedding on which the inverse branch $\phi_\lambda$ fixing $p(\lambda)$ is defined and uniformly contracting. An adapted ambient norm gives a contraction constant $0<a<1$. Let $E$ and $\sK_E$ be as in Lemma~\ref{web:lifts}, for the iterated family now under consideration.

Equicontinuity of $\sK$ allows us to choose a small relative neighborhood $\Omega$ of $p_0$ and shrink $V$ so that every $\gamma\in\sK$ with $\gamma(\lambda_0)\in\Omega$ lies in this inverse-branch neighborhood over $V$. We choose $\Omega$ invariant under $\phi_{\lambda_0}$. Set
\[
 \mathcal A:=\{\gamma\in\sK_E:\gamma(\lambda_0)\in\Omega\}.
\]
Then
\begin{equation}\label{web:positive-mass}
 \mathfrak m(\mathcal A)=\mu_{\lambda_0}(\Omega)>0.
\end{equation}
Lemma~\ref{web:lifts} supplies, for every $\gamma\in\mathcal A$, a unique sequence $\gamma_{-j}\in\sK_E$, with $\gamma_0=\gamma$, such that
\[
 \sF\gamma_{-j-1}=\gamma_{-j},\qquad
 \gamma_{-j}(\lambda_0)=\phi_{\lambda_0}^j(\gamma(\lambda_0)).
\]
All these sections belong to $\mathcal A$. The local inverse theorem and the identity theorem give
\[
 \gamma_{-j}(\lambda)=\phi_\lambda^j(\gamma(\lambda))
       \quad(\lambda\in V).
\]
In particular, for a constant independent of $\gamma\in\mathcal A$,
\begin{equation}\label{web:base-contraction}
 \sup_{\lambda\in V}\dist(\gamma_{-j}(\lambda),p(\lambda))
       \leq Ca^j.
\end{equation}

\medskip
\noindent\emph{Repulsion on the Zariski tangent space.}
Fix an arbitrary parameter $\lambda_1\in B$ and put $p_1:=p(\lambda_1)$. Lemma~\ref{web:propagation}, applied to all the sections $\gamma_{-j}$ above, gives constants $C_1>0$ and $0<b<1$ such that
\begin{equation}\label{web:target-contraction}
 \dist(\gamma_{-j}(\lambda_1),p_1)\leq C_1b^j
       \quad(\gamma\in\mathcal A,\ j\geq0).
\end{equation}

Choose a minimal local embedding $(X,p_1)\hookrightarrow(\C^e,0)$, where $e=\dim_\C T_{p_1}X$. In these coordinates $T_{p_1}X=\C^e$. Extend the germ of $f_{\lambda_1}$ to a holomorphic germ $H:(\C^e,0)\to(\C^e,0)$. The derivative $DH_0$ is the action of $Df_{\lambda_1}$ on the Zariski tangent space. Local holomorphic embedding coordinates are Lipschitz with respect to a fixed projective chart after shrinking, so \eqref{web:target-contraction} remains an exponential bound in these coordinates.

Choose $1<r<b^{-1}$ avoiding the moduli of the eigenvalues of $DH_0$. Let $W$ be the submanifold supplied by Lemma~\ref{web:strong}. Suppose that $DH_0$ has an eigenvalue of modulus at most $1$. Then $T_0W\subsetneq\C^e$. The germ of $X$ is irreducible, and it cannot be contained in $W$, since this would give
\[
 T_{p_1}X\subseteq T_0W\subsetneq\C^e.
\]
Thus $W\cap X$ is a proper local analytic subset of $X$.

For every $\gamma\in\mathcal A$, the values $\gamma_{-j}(\lambda_1)$ form an exponentially converging inverse history for $H$. Lemma~\ref{web:strong} therefore gives
\[
 \gamma(\lambda_1)\in
       \bigcup_{j\geq0}f_{\lambda_1}^j(W\cap X).
\]
Proposition~\ref{web:dynamics} gives $\mu_{\lambda_1}(W\cap X)=0$. The inequality $\mu_{\lambda_1}(f_{\lambda_1}(A))\leq q^k\mu_{\lambda_1}(A)$ then gives zero mass to the union above. On the other hand, its inverse image under evaluation contains $\mathcal A$, contradicting \eqref{web:positive-mass}. We have proved
\begin{equation}\label{web:tangent-repulsion}
 \text{every eigenvalue of }Df_{\lambda_1}|_{T_{p_1}X}
       \text{ has modulus greater than }1.
\end{equation}
The parameter $\lambda_1$ was arbitrary.

\medskip
\noindent\emph{Persistence of smoothness.}
We show that $p^{-1}(\Sing X)$ is open. Suppose that $p_1\in\Sing X$. In the minimal embedding just used, extend the nearby family to holomorphic maps $H_\lambda$ of an ambient neighborhood. By \eqref{web:tangent-repulsion}, the total map
\[
 (\lambda,z)\longmapsto(\lambda,H_\lambda(z))
\]
is a local biholomorphism at $(\lambda_1,0)$. It maps the irreducible germ $B\times X$ into itself, hence onto the same germ. Its inverse therefore preserves $B\times X$ and its singular locus.

The inverse germs fixing $p(\lambda)$ are uniformly contracting after shrinking the parameter and phase neighborhoods and choosing an adapted norm. For every nearby $\lambda$, start the inverse iteration at the point $p_1\in\Sing X$, chosen independently of $\lambda$. All iterates belong to $\Sing X$, and they converge to $p(\lambda)$. Hence $p(\lambda)\in\Sing X$. This proves openness. The set $p^{-1}(\Sing X)$ is also closed. Connectedness of $B$ shows that it is either empty or all of $B$. In particular, $p_0\in X_\reg$ implies $p(B)\subset X_\reg$.

Together with \eqref{web:periodic-section}, \eqref{web:tangent-repulsion}, and $p\in\sK$, this proves the proposition.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:web-criterion-normal}]
Assume first that an equilibrium web exists. Fix $\lambda_0\in B$. For each $n$, apply Proposition~\ref{web:web-to-motion} to all points of $R_n(\lambda_0)$, obtaining a finite set $\sR_n$ of sections. Distinct sections cannot meet: at a common value, which is a repelling periodic point, Lemma~\ref{web:jet} would make them equal. To see that their values exhaust $R_n(\lambda)$ at another parameter, apply Proposition~\ref{web:web-to-motion} with that parameter as basepoint. The resulting section is repelling at $\lambda_0$ and therefore is one of the sections already constructed. This proves $(1)\Rightarrow(2)$.

The persistence-of-smoothness argument in Proposition~\ref{web:web-to-motion} uses only that the section is periodic and repelling on the Zariski tangent space at every parameter. Hence every section in $(2)$ is either everywhere regular or everywhere singular. Keeping the regular sections proves $(2)\Rightarrow(3)$.

Finally, suppose that the regular repelling $J$-cycles move holomorphically. Denote their sections by $\sR_n^\reg$, write $N_n:=\#\sR_n^\reg$, and, for all sufficiently large $n$, set
\[
 \mathfrak m_n:=\frac1{N_n}\sum_{\gamma\in\sR_n^\reg}\delta_\gamma.
\]
Lemma~\ref{web:normality} places the supports of all these probability measures in one compact subset of $\sO(B,X)$. Take a weakly convergent subsequence with limit $\mathfrak m$. The map $\sF$ permutes $\sR_n^\reg$, as follows from the uniqueness of the local continuation of a regular repelling periodic point. Thus $\sF_*\mathfrak m_n=\mathfrak m_n$ and $\sF_*\mathfrak m=\mathfrak m$.

For every $\lambda\in B$, Proposition~\ref{web:dynamics} gives
\[
 (e_\lambda)_*\mathfrak m_n
       =\frac1{N_n}\sum_{x\in R_n^\reg(\lambda)}\delta_x
       \longrightarrow\mu_\lambda.
\]
Evaluation is continuous, so $(e_\lambda)_*\mathfrak m=\mu_\lambda$ for every $\lambda$. Hence $\mathfrak m$ is an equilibrium web, proving $(3)\Rightarrow(1)$.
The equivalence with $(4)$ is proved in
Proposition~\ref{web:lyapunov-equivalence} below, using only
the equivalence of $(1)$--$(3)$ just established.
\end{proof}


\begin{proposition}[Periodic sections give a web]
\label{prop:periodic-sections-equilibrium-web}
Let $\mathcal M_j$ be probability measures on holomorphic periodic
sections of a polarized family on a fixed normal projective $X$.
Suppose they are invariant under the same iterate $\sF^e$ and that
$(e_\lambda)_*\mathcal M_j\to\mu_\lambda$ for every $\lambda$.
Then the family is $J$-stable.
\end{proposition}
\begin{proof}
Lemma~\ref{web:normality} puts their supports in one compact set,
locally on the parameter disk. A weak limit $\mathcal M$ is
$\sF^e$-invariant and has evaluations $\mu_\lambda$.
The average $e^{-1}\sum_{i=0}^{e-1}\sF_*^i\mathcal M$ is an
equilibrium web. Apply Theorem~\ref{thm:web-criterion-normal}.
\end{proof}

\subsection{The Lyapunov criterion}
\label{web:lyapunov-subsection}

We prove the equivalence with assertion~(4) of
Theorem~\ref{thm:web-criterion-normal}. On projective space the
Lyapunov criterion and the critical-mass argument are classical;
see \cite[Theorem~1.6 and Proposition~3.12]{BBD18} and
\cite[Theorem~4.1]{BB26}. We give the additional arguments needed
for a normal phase variety. In this subsection we denote the
polarizing line bundle by $\mathscr L$, reserving $L$ for the
Lyapunov function, and put $v=(\mathscr L^k)$ and $d=q^k$.

\begin{proposition}[The bifurcation current on a normal variety]
\label{web:lyapunov-current}
Let $F(\lambda,x)=(\lambda,f_\lambda(x))$ and
$p:B\times X\to B$. Let $\mathcal T$ be the relative Green
current, normalized by $F^*\mathcal T=q\mathcal T$ and
$\mathcal T|_{X_\lambda}=T_\lambda$, and let $R_F$ be the
effective ramification Weil divisor of the finite map $F$.
The Lyapunov function is continuous and subharmonic, and
\begin{equation}\label{web:lyapunov-current-formula}
 dd^cL=\frac1v p_*(\mathcal T^k\wedge[R_F]).
\end{equation}
Here $dd^c$ is normalized by $dd^c\log|z|=\delta_0$ on
$\C$. The formula does not require $X$ to be $\Q$-Gorenstein.
\end{proposition}

\begin{proof}
We first specify the products and integrability used below.
In a projective embedding the polarization series gives
$\mathcal T=\omega+dd^cG$, with locally uniformly H\"older
continuous potentials. Indeed, on a compact parameter disk the
maps $f_\lambda$ and the initial metric potential have uniform
Lipschitz bounds. Splitting the series with coefficients $q^{-j}$
at the scale where the Lipschitz growth reaches the inverse
distance gives a positive H\"older exponent, uniformly on that
disk. This argument applies on the embedded normal space.
Pulling back to a fixed resolution $Y\to X$ gives the same local
H\"older bounds on $B\times Y$.

The exponential integrability theorem for Monge--Amp\`ere
currents with H\"older potentials
\cite[Theorem~1.1 and Corollary~1.2]{DNS10} therefore applies
both to these currents and to their fiber measures. On compact
parameter sets it gives uniform integrability of the logarithms
of meromorphic functions whose zeros and poles have no fiber
component. To check the uniformity, use finitely many charts
of the fixed resolution: the local logarithms of the holomorphic
numerators and denominators form compact families of
plurisubharmonic functions in $L^1$, and the H\"older constants
of the Green potentials are uniform. Truncating these logarithms
and then using the exponential tail estimate justifies slicing,
passage to limits, and integration by parts. For a rational
differential form, its norm in the induced projective metric
has at most logarithmic growth on the resolution: its
coefficients and the minors of the differential of the resolution
are algebraic, so this follows from \L{}ojasiewicz inequalities.
The same integrability therefore applies to that norm.

For a Weil divisor $A$, the product $\mathcal T^k\wedge[A]$
means the Bedford--Taylor product on a resolution of each
component of $A$, pushed down to $B\times X$. This definition
agrees with the usual one on the regular locus. A divisor
exceptional over the singular locus of $B\times X$ has image
of dimension at most $k-1$. The restriction of
$(\pi^*\mathcal T)^k$ to that divisor is zero: on the open
strata this follows from the dimension of the image, and the
remaining analytic subsets have zero mass for the locally
bounded-potential product. Thus exceptional divisors contribute
nothing, and all ensuing divisor identities can be checked in
codimension one. The same observation proves the projection
formula for these products with Weil divisors.

Choose a nonzero rational $k$-form $\eta$ on $X$, viewed as a
relative meromorphic form on $B\times X$, and set
$D=\operatorname{div}\eta$. Write
$F^*\eta=J\eta$, where $J$ is a meromorphic function on the
total space. At the generic point of every prime divisor the
source and target are smooth, so the usual Jacobian identity gives
\begin{equation}\label{web:meromorphic-jacobian}
 \operatorname{div}J=R_F+F^*D-D.
\end{equation}
The pullback here is the valuation-theoretic pullback of Weil
divisors under a finite map; no Cartier assumption is used.
None of these divisors has a fiber component. If
$b=\log\|\eta\|$ for the metric induced by a projective
embedding, then, away from a set of measure zero,
\[
 \log|J_\lambda|
 =\log|\operatorname{Jac}f_\lambda|
   +b\circ f_\lambda-b.
\]
The integrability just established and invariance of $\mu_\lambda$
show that
\begin{equation}\label{web:lyapunov-meromorphic-integral}
 L(\lambda)=\int_X\log|J_\lambda|\,d\mu_\lambda.
\end{equation}
They also justify Oseledets' formula for the sum of the exponents
and its independence of the projective metric or of a smooth
metric on a resolution:
the logarithms of the changes of metric are integrable and hence
give tempered changes of coordinates. Weak continuity of the
fiber Green measures, together with the uniform logarithmic
tail estimate, proves continuity in
\eqref{web:lyapunov-meromorphic-integral}.

Apply Poincar\'e--Lelong on the resolution and push down.
Equation~\eqref{web:meromorphic-jacobian} gives
\[
 dd^cL=\frac1v p_*\bigl(
 \mathcal T^k\wedge([R_F]+F^*[D]-[D])\bigr).
\]
Since $F$ has degree $d=q^k$, the identity
$F^*\mathcal T^k=d\mathcal T^k$ implies
$F_*\mathcal T^k=\mathcal T^k$. The projection formula and
$p\circ F=p$ cancel the last two terms. This proves
\eqref{web:lyapunov-current-formula}; its right side is positive.
\end{proof}

\begin{lemma}[Periodic approximation of the Lyapunov sum]
\label{web:lyapunov-approximation}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective $k$-fold, put $d=q^k$, and let $L(f)$ be the sum of
its equilibrium Lyapunov exponents.  If $R_n^{\reg}$ denotes
the smooth repelling Julia points fixed by $f^n$, then
\begin{equation}\label{web:periodic-lyapunov-limit}
 L(f)=\lim_{n\to\infty}\frac{1}{n d^n}
       \sum_{p\in R_n^{\reg}}\log|\det D_pf^n|.
\end{equation}
\end{lemma}

\begin{proof}
We give the determinant version of the argument of
\cite[Theorem~1.5 and Section~4.2]{BDM08}, including the
inverse-branch input on a normal phase variety.  Choose a
resolution $\pi:Y\to X$ and put $g=\pi^{-1}f\pi$.  Its
equilibrium measure $\nu$ integrates quasi-plurisubharmonic
functions \cite[Theorem~2.1]{Gue05}, and its Lyapunov exponents
are positive \cite[Theorem~5.4]{DNT15}.  Work on the full-measure
set of inverse histories avoiding the critical, indeterminacy,
and exceptional loci.  The changes of tangent metric induced
by $\pi$ have logarithmically integrable norms and inverse
norms, so the exponents computed there agree with those of $f$.

Here are the required local estimates.  Fix a finite coordinate
atlas on $Y$.  Algebraic \L{}ojasiewicz inequalities and the
quantitative inverse function theorem give a proper algebraic
set $A$, constants $a,C>0$, and the following bounds at every
unramified point $y\notin A$: an inverse germ at $g(y)$ is
defined on a ball of radius at least
$C^{-1}\dist(y,A)^a$, and its first two derivatives, in the
chosen charts, are bounded by $C\dist(y,A)^{-a}$ after reducing
that radius by a fixed factor.  Enlarge $A$ to include the
exceptional loci used above.  The function $\log\dist(\cdot,A)$
is $\nu$-integrable: locally an ideal norm for $A$ has
plurisubharmonic logarithm, and \L{}ojasiewicz inequalities
compare this norm with a power of the distance.  Consequently
all these chart bounds are tempered along almost every
inverse history.

For clarity, this produces contracting branches before any
normalization theorem is applied.  Oseledets' theorem gives
tempered linear coordinates in which every one-step inverse
derivative has norm at most $e^{-\chi+\epsilon}$, where
$\chi>0$ is the smallest exponent and $0<3\epsilon<\chi$.
The preceding second-derivative bounds allow us to choose
positive tempered radii on which the full inverse maps have
derivative norm at most $e^{-\chi+2\epsilon}$.  Replace these
radii by an $\epsilon$-slow minorant: if $r_0$ is a tempered
radius and $\tau$ is the backward shift, take
$r(\hat x)=\inf_{j\in\Z}e^{\epsilon|j|}r_0(\tau^j\hat x)$.
It is positive almost everywhere and
$r(\tau\hat x)\ge e^{-\epsilon}r(\hat x)$.
Thus each inverse map sends its radius ball into the next one,
and all inverse iterates are defined.  Returning to the fixed
atlas gives exponentially contracting inverse branches on a
ball of positive history-dependent radius.  This is also the
inverse-bundle construction in \cite[Section~4.1]{BDM08}.

Given $\epsilon>0$, restriction to a measurable history block
$H$ of measure greater than $1-\epsilon/2$, whose zeroth
coordinates lie in a compact subset above $X^{\reg}$, makes the radius
at least a fixed $r>0$ and, for every sufficiently large $n$,
makes the Lipschitz constant of its $n$th inverse branches
$h_{\hat x,n}$ on the full radius-$r$ ball at most $e^{-cn}$
for some $c>0$.
Birkhoff's theorem and Egorov's theorem allow the same block
to satisfy, uniformly,
\[
 -\frac1n\log|\det Dh_{\hat x,n}(x_0)|=L(f)+o(1).
\]
The coordinate and metric determinants differ by uniformly
bounded endpoint terms in the fixed atlas.  For a branch
returning to one small regular coordinate neighborhood,
$u_n=-\log|\det Dh_{\hat x,n}|$ is positive pluriharmonic
on the radius-$r$ ball for all sufficiently large $n$.
Harnack's inequality shows that its values within distance
$\delta r$ of $x_0$ differ from $u_n(x_0)$ by at most
$C_k\delta u_n(x_0)$.  Choosing $\delta$ sufficiently small
therefore gives
\begin{equation}\label{web:good-periodic-determinants}
 \left|\frac1n\log|\det D_pf^n|-L(f)\right|<2\epsilon
\end{equation}
at the fixed point of every such returning branch.  All
constructions may be made over the regular locus of $X$.

We recall the counting step to specify how many points satisfy
\eqref{web:good-periodic-determinants}.  Choose finitely many
pairwise disjoint convex coordinate boxes $B_i^+\Subset X^{\reg}$,
with inner boxes $B_i\Subset B_i^+$, diameter less than
$\delta r$, and
\[
 \mu\Bigl(\bigcup_i B_i\Bigr)>1-\epsilon/2,
 \qquad \mu(B_i)>(1-\epsilon)\mu(B_i^+).
\]
Write $\widehat B_i$ for their inverse images in the natural
extension.  Mixing gives, for all sufficiently large $n$,
\[
 \widehat\mu\bigl(\widehat f^{-n}
       (H\cap\widehat B_i)\cap\widehat B_i\bigr)
 >(1-\epsilon)\widehat\mu(H\cap\widehat B_i)\mu(B_i).
\]
Each associated inverse branch is defined on $B_i^+$; its image
meets $B_i$ and, by contraction, lies relatively compactly
inside $B_i^+$.  Different branch images are disjoint and each
has mass $d^{-n}\mu(B_i^+)$ by balance.  Each contains one
repelling fixed point in the Julia set, since it is obtained
by iterating an inverse branch from a Julia point.  The displayed
inequality therefore supplies at least
\[
 d^n(1-\epsilon)^2\widehat\mu(H\cap\widehat B_i)
\]
such points in $B_i^+$.  Summing gives at least
$d^n(1-\epsilon)^3$ points satisfying
\eqref{web:good-periodic-determinants}, exactly as in
\cite[Lemma~4.5]{BDM08}.

Finally, in a fixed projective embedding $\|Df\|\le C$ on
$X^{\reg}$: local ambient holomorphic extensions and a finite
cover of $X$ give a uniform bound.  Hence every smooth
repelling fixed point of $f^n$ satisfies
\[
 0\le \frac1n\log|\det D_pf^n|\le k\log\max\{C,1\}.
\]
Together with $\#R_n^{\reg}=d^n+o(d^n)$, the preceding count
proves \eqref{web:periodic-lyapunov-limit} by letting
$\epsilon$ tend to zero.
\end{proof}

\begin{corollary}[Stability implies harmonicity]
\label{web:stable-lyapunov-harmonic}
On a parameter disk where the smooth repelling Julia cycles
move holomorphically and exhaust the corresponding cycles in
each fiber, their Lyapunov sum $L(s)$ is harmonic.  This holds
both for a fixed normal phase variety and for the relative
families considered in this section.
\end{corollary}

\begin{proof}
Put $d=q^k$.  For the finite collection $\mathcal R_n^{\reg}$
of moving sections define
\[
 L_n(s)=\frac1{n d^n}\sum_{\gamma\in\mathcal R_n^{\reg}}
       \log\left|\det D_{\gamma(s)}f_s^n\right|.
\]
The return derivative is an endomorphism of the relative
tangent bundle along $\gamma$; its determinant is a nowhere
zero holomorphic function.  Thus $L_n$ is nonnegative and
harmonic.  Relative projective embeddings and local ambient
extensions give a uniform bound for the vertical derivatives
on compact parameter disks.  Since
$\#\mathcal R_n^{\reg}=d^n+o(d^n)$, the functions $L_n$ are
locally uniformly bounded.  Lemma~\ref{web:lyapunov-approximation}
gives $L_n(s)\to L(s)$ at every parameter.  Compactness of
locally bounded harmonic functions then gives locally uniform
convergence to $L$.  In particular $dd^cL=0$.
\end{proof}

\begin{lemma}[Critical mass estimate]
\label{lem:critical-mass-estimate-normal}
Let $\mathcal T$ be the relative Green current of a polarized family
over a disk, let $R_f$ be its ramification divisor, and suppose that
\[
 p_*(\mathcal T^k\wedge[R_f])=0.
\]
For each relatively compact parameter disk $B_0$, one has
\[
 \big\|(f^n)_*[R_f]\big\|_{p^{-1}(B_0)}
 =O\big(q^{(k-1)n}\big).
\]
In particular the ramification series in
Lemma~\ref{lem:summable-ramification-web} has locally finite mass.
\end{lemma}

\begin{proof}
Choose a smooth relatively positive representative $\omega$ of
$c_1(\mathscr L)$ and a positive form $\beta$ on the parameter disk.
Enlarging the coefficient of $p^*\beta$, we may use
$\omega+p^*\beta$ to compute masses on a fixed relatively compact
parameter neighborhood. Put
\[
 \omega_n=q^{-n}(f^n)^*\omega.
\]
The Green-potential series gives
\[
 \omega_n-\mathcal T=dd^c u_n,
 \qquad \|u_n\|_\infty=O(q^{-n})
\]
on that neighborhood. Choose a smooth cutoff $\chi$ on the parameter
disk, identically one near $\overline{B_0}$. Integration by parts on
the ramification cycle gives
\[
 \begin{split}
 &\int p^*\chi\,[R_f]\wedge
       (\omega_n^k-\mathcal T^k)\\
 &\quad=\int u_n\,p^*(dd^c\chi)\wedge[R_f]\wedge
       \sum_{j=0}^{k-1}\omega_n^{k-1-j}\wedge\mathcal T^j
       =O(q^{-n}).
 \end{split}
\]
The last bound is uniform: after the base form is inserted, the
remaining intersections are fiberwise intersections of fixed
cohomology classes with the ramification cycle. These identities
can be computed on a resolution of that cycle; all potentials being
continuous, Bedford--Taylor products and integration by parts apply.
The hypothesis makes the integral containing $\mathcal T^k$ zero.
Multiplication by $q^{kn}$ therefore bounds the contribution of
$(f^n)^*\omega^k$ by $O(q^{(k-1)n})$.

The remaining term in the mass is
\[
 k\int p^*(\chi\beta)\wedge[R_f]\wedge
       (f^n)^*\omega^{k-1},
\]
which has the same bound by the fiberwise polarization identity.
There are no terms containing two base forms. This proves the
estimate. Finally $[Y]\leq f_*[R_f]$, so the $j$th term of the
ramification series has mass $O(q^{-j})$ on every such neighborhood.
\end{proof}

\begin{lemma}[Summable ramification and equilibrium webs]
\label{lem:summable-ramification-web}
Let $p:\mathcal X\to B$ be a flat projective family with normal
geometrically integral fibers of dimension $k$ over a disk. Let
$f:\mathcal X\to\mathcal X$ be a polarized endomorphism over $B$, of
fiberwise topological degree $d=q^k$. Write
$\Sigma=\mathcal X\setminus\operatorname{Sm}(p)$, let $R_f$ be the
ramification Weil divisor of $f$, and put
$Y=(f(\operatorname{Supp}R_f))_{\mathrm{red}}$.
Suppose that
\[
 \mathscr R=\sum_{j\geq0}d^{-j}(f^j)_*[Y]
\]
has locally finite mass on $\mathcal X$. Then every parameter admits
a neighborhood carrying an equilibrium web.
\end{lemma}

\begin{proof}
The inverse-branch counting argument is that of
\cite[Lemma~3.1]{BBr23}. We give its application to transverse disks,
which also accounts for the singular fibers.

Fix $s_0\in B$, and use a coordinate $t$ with $t(s_0)=0$. Choose
$a\in\mathcal X_{s_0,\mathrm{reg}}$ outside the exceptional set for
backward equidistribution and outside
\[
 \bigcup_{j\geq0}f^j(Y\cup\Sigma).
\]
Such a point exists: the divisors $f^j(Y)$ have no vertical
components, and $f^j(\Sigma)$ has fiberwise codimension at least two.
The backward-equidistribution exceptional set is a countable union
of proper analytic sets; use Proposition~\ref{web:dynamics} and
\cite[Theorem~1.2]{DNT15} on a resolution.
Take relative smooth coordinates $(t,z)$ centered at $a=(0,0)$.
We first check that $\nu(\mathscr R,a)=0$. Every finite partial
sum has Lelong number zero at $a$, by its choice. For a fixed ball
$Q$ centered at $a$, the Lelong number of the positive tail is at
most a constant depending on $Q$ times its mass on $Q$. The latter
tends to zero by local mass summability. Additivity of Lelong
numbers for the sum of the finite partial sum and its tail proves
the assertion.

Fix $0<\epsilon<1/4$. The radial Crofton formula, or the slicing
argument in \cite[Lemma~3.1]{BBr23}, now gives a radius and a set of
complex line directions of arbitrarily large measure on which
\[
 \sum_{j\geq0}d^{-j}
 \big\|(f^j)_*[Y]\wedge[\Delta]\big\|\leq\epsilon.
 \tag{*}
\]
For completeness, apply the radial slicing formula to every
divisorial summand, and then sum by monotone convergence. The
average of the intersection masses on disks of radius $r$ is,
up to the fixed normalization, the Lelong quotient of
$\mathscr R$ at radius $r$, because no summand contains $a$.
It tends to zero as $r\to0$. Markov's inequality proves (*).
We also shrink the radius so that the ball avoids $Y$.
Restrict to directions transverse to $t=0$ and bounded
away from the vertical directions. After a common restriction of the
parameter disk, these lines are graphs
\[
 a_v(t)=(t,tv),\qquad |t|<\rho,
\]
where $v$ belongs to a bounded measurable set $V\subset\mathbb C^k$
of positive Lebesgue measure, and (*) holds on every graph.
Discarding a null set of directions ensures that these graphs avoid
every $f^j(\Sigma)$. Indeed, radial projection of each analytic set
$f^j(\Sigma)$ to the space of line directions has measure zero,
since its codimension in the total space is at least two and it
does not contain $a$.

For each $v\in V$, construct the inverse branches along $a_v$ in
the backward tree, retaining only those for which every intermediate
branch avoids $Y$. Let $\Gamma_n(v)$ be the retained sections
$\gamma$ satisfying $f^n\circ\gamma=a_v$.
Off $Y\cup f(\Sigma)\cup\Sigma$, the map $f$ is an unramified
cover of degree $d$. This follows from purity of the branch locus
on the smooth target, applied to the finite map with normal source.
Thus every retained branch at level $n$ has $d$ lifts at level
$n+1$ before branches meeting $Y$ are removed. The number removed
is at most the intersection number with $(f^{n+1})_*[Y]$.
More explicitly, with
$I_n(v)=\| (f^n)_*[Y]\wedge[a_v]\|$ and
$c_n(v)=d^{-n}\#\Gamma_n(v)$, the recurrence is
\[
 \#\Gamma_n(v)\geq d\#\Gamma_{n-1}(v)-I_n(v),\qquad
 c_n(v)\geq c_{n-1}(v)-d^{-n}I_n(v).
\]
Since $c_0(v)=1$, (*) gives, as in
\cite[Lemma~3.1]{BBr23},
\[
 \#\Gamma_n(v)\geq(1-\epsilon)d^n,\qquad
 f\Gamma_{n+1}(v)\subseteq\Gamma_n(v),
 \tag{**}
\]
and each section has at most $d$ predecessors.
All these sections remain in the relative smooth locus, because
their images under $f^n$ lie on graphs avoiding every $f^j(\Sigma)$.

The union of these sections is relatively compact on every smaller
parameter disk. To see this, use a relative projectively normal
embedding and the homogeneous lifts $F_t$ constructed in
Lemma~\ref{web:normality} (and, for varying fibers, in the proof
of Lemma~\ref{normal:periodic-normality}). The target sections $a_v$ have homogeneous
lifts $A_v$ with uniformly bounded, nowhere zero values on compact
parameter sets. A lift of a section in $\Gamma_n(v)$ may be
multiplied by a holomorphic scalar so that
\[
 F_t^n(H_\gamma(t))=A_v(t).
\]
The required $q^n$-th root exists on the parameter disk. Iterating
the homogeneous norm bounds in Lemma~\ref{web:normality} bounds
$\|H_\gamma\|$ both above and away from zero, independently of
$n$, $v$, and $\gamma$. Montel's theorem proves the assertion.

Normalize Lebesgue measure on $V$ to a probability $\ell$, and set
\[
 m_n=\int_V d^{-n}\sum_{\gamma\in\Gamma_n(v)}\delta_\gamma\,
 d\ell(v),\qquad
 M_N=\frac1N\sum_{n=1}^N m_n.
\]
The inverse branches and the condition that a branch avoids $Y$
are Borel in the slope; the integrals can equivalently be defined
on the finite analytic inverse-cover spaces and then pushed to the
space of sections. By (**), the masses $b_n=\|m_n\|$ decrease and
belong to $[1-\epsilon,1]$, while
\[
 m_{n-1}-\mathcal F_*m_n\geq0,\qquad
 \|m_{n-1}-\mathcal F_*m_n\|=b_{n-1}-b_n.
\]
Consequently $\|M_N-\mathcal F_*M_N\|\leq3/N$.
Compactness gives a weak limit $M$ with
$\mathcal F_*M=M$ and mass $b\geq1-\epsilon$.

For each $t$, evaluation is dominated by the full inverse-image
measures:
\[
 (e_t)_*m_n\leq
 \int_V d^{-n}(f_t^n)^*\delta_{a_v(t)}\,d\ell(v).
\]
At $t=0$ the measures on the right converge to $\mu_0$ by the
choice of $a$. At any fixed $t\ne0$, the map $v\mapsto tv$ is
invertible and complex linear. Its image measure is absolutely
continuous in the relative smooth coordinate chart, so backward
equidistribution and dominated convergence again give the limit
$\mu_t$. Hence $(e_t)_*M\leq\mu_t$ for every $t$.
The left side is $f_t$-invariant and has mass $b$; ergodicity of
$\mu_t$ therefore gives $(e_t)_*M=b\mu_t$. The probability
$b^{-1}M$ is an equilibrium web.
\end{proof}

\begin{proposition}[Equivalence of the Lyapunov criterion]
\label{web:lyapunov-equivalence}
For a polarized holomorphic family on a fixed normal projective
variety over a disk, $dd^cL=0$ if and only if the three equivalent
conditions $(1)$--$(3)$ of
Theorem~\ref{thm:web-criterion-normal} hold.
\end{proposition}
\begin{proof}
Condition $(3)$ implies $dd^cL=0$ by
Corollary~\ref{web:stable-lyapunov-harmonic}. Conversely,
suppose $dd^cL=0$. Formula~\eqref{web:lyapunov-current-formula}
and Lemma~\ref{lem:critical-mass-estimate-normal} give locally
finite mass of the ramification series. By
Lemma~\ref{lem:summable-ramification-web}, every parameter has
a neighborhood carrying an equilibrium web. The already proved
implication $(1)\Rightarrow(3)$ gives holomorphic motion of
all smooth repelling Julia cycles on each such neighborhood.
For each period these local continuations are unique, disjoint,
and exhaustive. They therefore define a finite covering of $B$.
Since $B$ is simply connected, this covering is trivial, so
condition $(3)$ holds over all of $B$. The equivalence of
$(1)$--$(3)$ completes the proof.
\end{proof}

\begin{remark}[The bifurcation locus]\label{web:bifurcation-locus}
It follows that the complement of the stability locus is
exactly $\supp(dd^cL)$. The equation $dd^cL=0$ is a local
equation of currents on an open parameter set; it is not a
pointwise assertion about one chosen parameter. On a parameter
curve, harmonicity need not mean that $L$ is constant.
\end{remark}

\subsection{Removable parameters}
\begin{lemma}[Persistence of a repelling Julia point]
\label{lem:repelling-julia-persistence}
Let $f_s$ be a holomorphic polarized family on a fixed normal projective
variety $X$. A smooth repelling periodic point in $J_{s_0}$ has a
holomorphic continuation which remains smooth, repelling, and in $J_s$
near $s_0$.
\end{lemma}

\begin{proof}
The implicit function theorem gives the continuation and preserves
smoothness and repulsion. Write $d=q^{\dim X}$. A sufficiently high
iterate of the return has inverse branches $\psi_s:B\to B$, defined
on one ball in the smooth locus, whose images contract uniformly to
the continued point. The polarization series converges locally uniformly and gives
jointly continuous Green potentials. Bedford--Taylor continuity
on a fixed resolution \cite[Proposition~2.3]{BT76} gives weak
continuity of the equilibrium measures. Together with
$\mu_{s_0}(B)>0$, this gives $\mu_s(B)>0$ after restriction. The balance
identity implies
\[
 \mu_s\bigl(\psi_s^j(B)\bigr)=d^{-\ell j}\mu_s(B)>0,
\]
where $\ell$ is the chosen return time. These images eventually lie
in any prescribed neighborhood of the continued point. Hence that
point belongs to $\operatorname{supp}\mu_s$.
\end{proof}

\begin{lemma}[Removable parameters for stability]
\label{lem:stability-removable-parameters}
Let $f_s$ be a holomorphic family of $q$-polarized endomorphisms of a
fixed normal projective variety $X$, parametrized by a disk $D$.
If the family is $J$-stable on $D\setminus\{0\}$, it is $J$-stable
on $D$.
\end{lemma}

\begin{proof}
Put $d=q^{\dim X}$. Let $R_n$ be the smooth repelling Julia points
of $f_0$ fixed by $f_0^n$. Periodic-point equidistribution, applied
on a resolution and then restricted to the smooth locus, gives
\[
 \#R_n=d^n+o(d^n),\qquad
 d^{-n}\sum_{x\in R_n}\delta_x\longrightarrow\mu_0;
\]
see \cite[Theorem~1.1]{DNT15}. By
Lemma~\ref{lem:repelling-julia-persistence}, each $x\in R_n$ defines
a smooth repelling Julia section near $0$. Stability on the
punctured disk continues this section along every path there.
Its monodromy around $0$ is trivial, since the section is already
defined near $0$. We obtain a holomorphic section
$\gamma_x:D\to X$, smooth repelling and in $J_s$ for every $s$.
Different points of $R_n$ give disjoint sections. Moreover, $f$
permutes these sections, because it does so at $0$ and continuation
is unique.

Lemma~\ref{web:normality} makes these periodic sections a relatively
compact family on every smaller parameter disk.

For large $n$ set
\[
                  \mathcal M_n=
                  \frac1{\#R_n}\sum_{x\in R_n}\delta_{\gamma_x}.
\]
These are invariant probability measures on the space of
holomorphic sections. Their supports lie in one compact set after
restriction to a smaller disk. For each $s$, their evaluations are
uniform measures on $d^n+o(d^n)$ distinct smooth repelling Julia
points of $f_s$. The upper bound and equidistribution in
\cite[Theorem~1.1]{DNT15} show that these evaluations converge to
$\mu_s$. A weak limit of $\mathcal M_n$ is therefore an equilibrium
web. The equilibrium-web criterion
(Theorem~\ref{thm:web-criterion-normal}) proves stability near $0$.
\end{proof}

\begin{corollary}[Extension across finitely many parameters]
\label{cor:stability-zariski-open-extension}
Let $S$ be a smooth connected complex algebraic curve and let
$f$ be an algebraic family of polarized endomorphisms of a fixed
normal projective variety $X$. If $f$ is $J$-stable on a nonempty
Zariski open subset $S^\circ$, then it is $J$-stable on $S$.
\end{corollary}
\begin{proof}
The complement $S\setminus S^\circ$ is finite. For each omitted
parameter $s$, choose an analytic disk $D$ centered at $s$ such
that $D\setminus\{s\}\subset S^\circ$. The restricted family
is stable on this punctured disk, so
Lemma~\ref{lem:stability-removable-parameters} makes it stable
on $D$. Together with stability on $S^\circ$, this proves the
assertion.
\end{proof}


\subsection{Equilibrium webs for varying normal fibers}
\label{normal:relative-subsection}

Let $S$ be a smooth connected complex algebraic curve and let
$p:\mathcal X\to S$ be flat and projective, with geometrically
integral normal fibers of dimension $k$. Let $f:\mathcal X\to\mathcal X$ be an
$S$-endomorphism satisfying $f^*L\simeq L^{\otimes q}$ for a
relatively ample line bundle $L$ and $q>1$. On a parameter disk
$B\subset S^{\mathrm{an}}$, write $\sO_B(\mathcal X)$ for the
space of holomorphic sections, with its compact-open topology,
and put $\sF\gamma=f\circ\gamma$ and $e_s(\gamma)=\gamma(s)$.
An equilibrium web is a compactly supported probability measure
$\mathfrak m$ on this space with
\[
 \sF_*\mathfrak m=\mathfrak m,
 \qquad (e_s)_*\mathfrak m=\mu_s\quad(s\in B).
\]
The construction by periodic sections follows
\cite[Lemma~2.1 and Proposition~2.2(2)]{BBD18}. We establish
the relative criterion needed on a nonempty parameter open.

\begin{lemma}[Compactness of relative periodic sections]
\label{normal:periodic-normality}
The holomorphic periodic sections over a parameter disk $B$
form a relatively compact subset of $\sO_B(\mathcal X)$.
\end{lemma}
\begin{proof}
Locally on $B$, a sufficiently high power of $L$ gives a
relative projectively normal embedding
$\mathcal X\hookrightarrow B\times\Pj^M$. Relative Serre
vanishing and cohomology and base change
\cite[Chapter~III, Sections~5 and~12]{Har77} provide holomorphic
splittings of the restriction maps for homogeneous polynomials.
The polarization therefore lifts $f$ to homogeneous degree-$q$
maps $F_s$ preserving the varying affine cones $C_s$ and having
no zero on $C_s\setminus\{0\}$.

Compactness of their unit links over compact parameter sets
gives the uniform estimate \eqref{web:cone-estimate}. The
normalization of periodic lifts in Lemma~\ref{web:normality}
then gives \eqref{web:annulus}, independently of the period.
Montel's theorem produces limits in $C_s\setminus\{0\}$,
since the relative cone is closed. Projectivization and a
diagonal argument over parameter disks prove the assertion.
\end{proof}

\begin{proposition}[The web criterion on a parameter open]
\label{normal:relative-web-criterion}
There is a nonempty Zariski open $S^\circ\subset S$
such that, on every parameter disk $B\subset(S^\circ)^{\mathrm{an}}$, the
following conditions are equivalent: the existence of an
equilibrium web, $J$-stability, and $dd^cL=0$, where
$L(s)=\sum_{j=1}^k\chi_j(s)$ is the equilibrium Lyapunov sum.
\end{proposition}
\begin{proof}
Let
\[
 \Sigma=(\mathcal X\setminus\operatorname{Sm}(p))_{\mathrm{red}}.
\]
Its intersection with $X_s$ is set-theoretically
$\operatorname{Sing}(X_s)$. By generic flatness \cite[Tag~052A]{Stacks}, choose
$S^\circ\subset S$ so that
$\Sigma|_{S^\circ}\to S^\circ$ is flat. Its analytification
is then open.
Choose also a resolution $\pi:\mathcal Y\to\mathcal X$
which is an isomorphism over the relative smooth locus, and
a nonzero rational relative $k$-form on $\mathcal X$.
After deleting finitely many further parameters, the map
$\mathcal Y\to S^\circ$ is smooth, each $\pi_s$ is a
resolution of $X_s$, and the chosen form and its pullback
under $f$ restrict to nonzero rational forms on every fiber.
These restrictions follow from generic smoothness and the
finiteness of the vertical components of their divisors.

We explain the changes to the fixed-variety argument. In a
relative projective embedding, the first differing coefficient
of two sections with equal lower jets is in the vertical
Zariski tangent space. Subtracting their defining equations
cancels the parameter terms; after composition with $f^n$,
the coefficient becomes $D(f_s^n)v$. Thus projective Cauchy
estimates prove Lemma~\ref{web:jet} for relative sections.
The coordinate differences are also holomorphic functions of
the parameter, so the three-circle argument of
Lemma~\ref{web:propagation} is unchanged. These two facts,
the support identities, and the fiberwise dynamics in
Proposition~\ref{web:dynamics} give the relative lift lemma
\ref{web:lifts}.

Suppose a web exists. Its support and jet uniqueness give a
unique periodic section $\gamma$ through each smooth repelling
Julia point at $s_0$. The inverse-history construction in
Proposition~\ref{web:web-to-motion} starts in relative smooth
coordinates near that point. The lift lemma provides a
positive-measure set of section histories which approach
$\gamma$ exponentially near $s_0$; propagation gives an
exponential bound at every parameter $s\in B$.

In a minimal embedding of $(X_s,\gamma(s))$, apply
Lemma~\ref{web:strong} to an ambient extension of the return.
If a tangent eigenvalue had modulus at most one, the strong
unstable manifold would have tangent space smaller than
$T_{\gamma(s)}X_s$. The irreducible fiber germ cannot be
contained in this manifold. Its intersection with that
manifold is consequently a proper local analytic subset.
The inverse histories would place a positive-measure set in
the union of its forward images, contradicting
Proposition~\ref{web:dynamics}. Hence the section is repelling
on its full Zariski tangent space at every parameter. This is
the inverse-history argument underlying
\cite[Proposition~1.2]{BB26}, with minimal embeddings allowing
singular fibers.

We must also show that $\gamma$ remains smooth. Suppose
$\gamma(s_1)\in\Sigma$. Eliminate the independent linear
fiber equations in a relative embedding at this point. This
gives a relative embedding into $B\times\C^e$, minimal on
the fiber at $s_1$. Translate $\gamma$ to zero and extend
the return to ambient maps $H_s$ fixing zero. The vertical
derivative at $(s_1,0)$ is expanding, so
$(s,z)\mapsto(s,H_s(z))$ is a local biholomorphism. The
total space is normal, because the base is smooth and the
fibers are geometrically normal. Its irreducible local germ
is mapped into itself and hence onto itself. The ambient
inverse therefore preserves the family and $\Sigma$.

In an adapted norm these inverse branches are uniformly
contracting on a small neighborhood of the section.
Openness of $\Sigma\to B$ ensures that each nearby fiber
contains a singular point in that neighborhood. Its inverse
iterates stay singular and converge to $\gamma(s)$.
Thus $\gamma^{-1}(\Sigma)$ is open; it is also closed and
does not contain $s_0$, so it is empty. Jet uniqueness makes
the resulting smooth repelling Julia sections disjoint.
Starting at any parameter proves that they exhaust all
such points in every fiber. This proves $J$-stability.

Conversely, take the uniform measures on the moving smooth
repelling Julia points fixed by $f^n$. They are invariant,
their evaluations converge to $\mu_s$ by
Proposition~\ref{web:dynamics}, and
Lemma~\ref{normal:periodic-normality} gives compact support
for their union. Any weak limit is an equilibrium web.

It remains to include the Lyapunov criterion. On each disk in
$S^\circ$, the relative polarization series and the smooth
family of resolutions just chosen give uniform H\"older
potentials and uniform logarithmic integrability exactly as
in Proposition~\ref{web:lyapunov-current}. A rational relative
top form replaces the fixed form there. The identity
$\operatorname{div}J=R_f+f^*D-D$ holds in codimension one;
exceptional divisors again map to sets of dimension at most
$k-1$. Thus, with $v=(L_s^k)$, which is independent of $s$,
\[
 dd^cL=\frac1v p_*(\mathcal T^k\wedge[R_f])
\]
holds on the disk, with continuous subharmonic $L$.
If it vanishes, Lemmas~\ref{lem:critical-mass-estimate-normal}
and~\ref{lem:summable-ramification-web} give a web locally
at each parameter. The relative web argument just proved
then gives local $J$-stability, and uniqueness of continuation
and simple connectedness of the disk give $J$-stability on
the whole disk. Conversely,
Corollary~\ref{web:stable-lyapunov-harmonic} applies fiberwise
and proves that $J$-stability implies $dd^cL=0$.
\end{proof}

\begin{proposition}[Generic periodic-section criterion]
\label{normal:generic-periodic-criterion}
Suppose that, over a nonempty Zariski open $S_1\subset S$,
each sufficiently small parameter disk carries
probability measures $\mathcal M_j$ on holomorphic periodic
sections such that
\[
 \sF_*^e\mathcal M_j=\mathcal M_j,
 \qquad (e_s)_*\mathcal M_j\longrightarrow\mu_s
                         \quad\text{for every }s,
\]
where $e\geq1$ is independent of $j$. Then $f$ is $J$-stable
on a nonempty Zariski open subset of $S$.
\end{proposition}
\begin{proof}
Intersect $S_1$ with the open in
Proposition~\ref{normal:relative-web-criterion}.
Lemma~\ref{normal:periodic-normality} gives a weak limit
$\mathcal M$ with evaluations $\mu_s$ and
$\sF^e$-invariance. As in \cite[Lemma~2.1]{BBD18}, continuity
of evaluation and of $\sF$ passes these identities to the
limit. The measure
\[
                  \frac1e\sum_{i=0}^{e-1}\sF_*^i\mathcal M
\]
is an equilibrium web, so
Proposition~\ref{normal:relative-web-criterion} applies.
\end{proof}


\section{Repelling spectra in arbitrary dimension}


\begin{prop}\label{prop:stable-spectrum-any-dimension}
Let $B$ be a connected smooth algebraic curve over $\mathbb C$, and let
$\pi:\mathcal X\to B$ be flat and projective with normal fibers of dimension
$r$, and let $f:\mathcal X\to\mathcal X$ be a morphism over $B$.
Suppose that a relatively ample line bundle $L$ satisfies
$f^*L\simeq L^{\otimes q}$ for an integer $q>1$.
For each $b\in B$, let $J_b$ be the support of the equilibrium measure.
Assume that, locally on $B$, the smooth repelling Julia periodic points move
holomorphically and that, for every period, these graphs exhaust the smooth
repelling Julia periodic points in every fiber.
Then every local branch $x_b$ of such points satisfying $f_b^n(x_b)=x_b$ has
constant characteristic polynomial
\[
 \det\bigl(T\operatorname{id}-D_{x_b}f_b^n\bigr).
\]
The eigenvalues are counted with their algebraic multiplicities.
\end{prop}

\begin{proof}
Fix $n\geq1$ and let $Z_n=\operatorname{Fix}(f^n)$ be the relative fixed-point
scheme.  It is closed in $\mathcal X$, hence proper over $B$.  Its fibers are
zero-dimensional.  Indeed, a positive-dimensional component of
$(Z_n)_b$ would contain an integral projective curve $C$ on which $f_b^n$ is
the identity.  Restriction of the polarization would give
\[
 \deg(L_b|_C)=q^n\deg(L_b|_C),
\]
contrary to ampleness.  Thus $Z_n\to B$ is proper and quasi-finite, and
therefore finite.

Let $R_n\subset Z_n^{\mathrm{an}}$ consist of the points $x\in X_b^{\mathrm{reg}}$
which lie in $J_b$ and for which every eigenvalue of $D_xf_b^n$ has modulus
greater than one.  We first show that $R_n$ is both open and closed in
$Z_n^{\mathrm{an}}$.

Fix $b_0\in B$.  The motion hypothesis supplies a neighborhood $V$ of $b_0$
and finitely many holomorphic sections $s_1,\ldots,s_N$ whose graphs have union
$R_n\cap\pi^{-1}(V)$.  There are only finitely many sections because
$(Z_n)_{b_0}$ is finite.  Each section has a closed graph in
$\mathcal X_V^{\mathrm{an}}$, since the latter is separated over $V$.
Consequently $R_n\cap\pi^{-1}(V)$ is closed in $Z_n^{\mathrm{an}}|_V$.
Such neighborhoods cover $B$, so $R_n$ is closed.

For openness, take $x_0\in R_n$ above $b_0$.  Flatness and smoothness of the
fiber at $x_0$ imply that $\pi$ is smooth near $x_0$.  In relative local
coordinates the fixed-point equations are $f_b^n(z)-z=0$, and their derivative
in $z$ at $(b_0,x_0)$ is $D_{x_0}f_{b_0}^n-\operatorname{id}$.  This matrix is
invertible.  The holomorphic implicit function theorem identifies a
neighborhood of $x_0$ in $Z_n^{\mathrm{an}}$ with a single section over a
neighborhood of $b_0$.  It is the section provided by the motion hypothesis,
by uniqueness, and is contained in $R_n$.  Hence $R_n$ is open.

Let $C$ be the irreducible component of $(Z_n)_{\mathrm{red}}$ containing
the given local branch.  The preceding local description shows that $C$
dominates $B$.  Let $\nu:S\to C$ be its normalization.  The curve $S$ is
connected, and $S\to B$ is finite and surjective.  The inverse image
$\nu^{-1}(R_n)$ is nonempty, open and closed in $S^{\mathrm{an}}$.
Since $S^{\mathrm{an}}$ is connected, it equals $S^{\mathrm{an}}$.
Thus every point on this entire algebraic periodic cover is a smooth
repelling Julia point.  Moreover, the implicit-function description applies
at every point of $C$: the normalization is an isomorphism there and
$S\to B$ is finite \textup{\'{e}tale}.

Let $\iota:S\to\mathcal X$ be the tautological map.  Along $\iota(S)$ the
relative tangent sheaf is locally free, so
\[
 E=\iota^*T_{\mathcal X/B}
\]
is a rank-$r$ vector bundle on $S$.  The identity $f^n\iota=\iota$ makes the
relative derivative an algebraic bundle endomorphism $A:E\to E$.  Therefore
\[
 P(T)=\det(T\operatorname{id}-A)
\]
is a monic polynomial with regular coefficients on $S$.

Take a finite splitting field of $P$ over $\mathbb C(S)$ and let
$\tau:S'\to S$ be the normalization in that field.  This is a finite
surjective morphism of smooth connected algebraic curves.  The roots
$\lambda_1,\ldots,\lambda_r$ of $P$, repeated according to multiplicity,
are regular on $S'$: they are integral over the coordinate ring of every
affine open of $S$, and $S'$ is its normalization in the splitting field.
At every $u\in S'$ they specialize to eigenvalues of $A_{\tau(u)}$.
The preceding paragraph therefore gives
\[
 |\lambda_j(u)|>1\qquad(u\in S',\ 1\leq j\leq r).
\]
In particular $1/\lambda_j$ is a bounded holomorphic function on $S'$.

Let $\overline{S'}$ be the smooth projective completion of $S'$.  Its
complement is finite.  The removable-singularity theorem extends
$1/\lambda_j$ across every point of this complement.  A holomorphic function
on the compact connected Riemann surface $\overline{S'}$ is constant.
Each $\lambda_j$ is consequently constant.  Hence
$P(T)=\prod_{j=1}^r(T-\lambda_j)$ has constant coefficients on $S'$, and
therefore on $S$.  Restricting to the original local branch proves the
proposition.
\end{proof}


\section{Horizontal fields and projective rigidity}


\begin{lemma}[Rational horizontal fields on polarized normal families]
\label{lem:rational-horizontal-normal}
Let $\pi:\mathcal X\to S$ be a flat projective family of normal
geometrically integral varieties over a smooth complex curve, and let
$F:\mathcal X\to\mathcal X$ be a finite $S$-morphism.  Suppose an ample
line bundle $H$ on the generic fiber satisfies $F^*H\simeq H^q$ with
$q>1$.  Let $v$ be a nonzero rational vector field on $S$ and let $W$
be a rational vector field on $\mathcal X$ satisfying
\[
                    d\pi(W)=v,\qquad dF(W)=W\circ F.
\]
After shrinking $S$, the field $W$ is holomorphic on all of
$\mathcal X$.  Its local flows trivialize the projective dynamical
family.  In particular the family is algebraically isotrivial after a
finite base change.
\end{lemma}

\begin{proof}
If the fibers have dimension zero, the assertion is immediate after
restriction.  Assume henceforth that their dimension is positive.
The total space is normal.  At the generic point of a horizontal prime
divisor it is smooth, since the generic fiber is normal and the base
has characteristic zero.  Define the polar Weil divisor $P$ of $W$
there by the minimum valuation of its coefficients in a regular local
tangent frame.  Changes of regular frame are invertible over the
corresponding discrete valuation ring, so this definition is intrinsic.

Let $D$ be a horizontal prime and let $D'=F(D)$.  At their generic
points choose regular tangent frames on the total space.  The matrix
of $dF$ has regular coefficients.  Therefore the pole order of
$dF(W)$ at $D$ is no greater than the pole order $P_D$ of $W$.
The pole order of $W\circ F$, regarded as a section of
$F^*T_{\mathcal X}$, is exactly $e_F(D)P_{D'}$.  The transport identity
thus gives
\[
                         F^*P\leq P
\]
on horizontal divisors.  Restrict to the generic fiber $X$ of dimension
$n$.  For an effective Weil divisor and an ample Cartier polarization,
the finite projection formula and $F^*H\simeq H^q$ give
\[
 q(P|_X\cdot H^{n-1})
       =(F^*(P|_X)\cdot H^{n-1})
       \leq P|_X\cdot H^{n-1}.
\]
Ample positivity and $q>1$ force $P|_X=0$.  Hence $W$ has no
horizontal divisorial poles.  Removing finitely many points of $S$
also removes its vertical divisorial poles and the poles and zeros of
$v$.

For any local regular function $a$, the rational function $W(a)$ is
now regular at every codimension-one point.  Normality makes it regular
on the whole open set.  Thus $W$ extends as a regular derivation, and
hence as a global holomorphic vector field, across the remaining
codimension-at-least-two subset.  Horizontality and the transport
identity persist by equality on a dense open.

Local holomorphic derivations integrate also on a singular complex
space: in a local embedding, lift the coefficients to the ambient
space; preservation of the defining ideal makes the ambient flow
preserve the subspace.  Uniqueness glues these flows.  Compactness of
a projective fiber gives a common time interval, so the horizontal
flow identifies nearby whole fibers.  The transport identity makes
these identifications conjugacies of their endomorphisms.

Fix one fiber in the resulting analytic disk.  The space of dynamical
isomorphisms from this fixed fiber to fibers over $S$ is a countable
union of finite-type algebraic spaces: use the graph loci in relative
Hilbert schemes, over their countably many Hilbert polynomials, and
impose the closed conjugacy identity.  Each image in the algebraic
curve $S$ is constructible.  The uncountably many points of the disk
belong to their union, so one finite-type piece has dense image in
$S$.  A closed point of its generic fiber is defined over a finite
extension of $\mathbf C(S)$ and supplies the required algebraic
trivialization after finite base change and restriction.
\end{proof}

\begin{corollary}[An isotrivial iterate]
\label{cor:isotrivial-iterate-normal}
If one fixed positive iterate of a polarized algebraic family is
isotrivial, then the family itself is isotrivial.
\end{corollary}

\begin{proof}
After a finite base change, trivialize that iterate as a fixed
polarized map $h:X\to X$.  Write the resulting original maps as $f_s$,
so $f_s^e=h$.  Each $f_s$ commutes with $h$.  For every
$x\in\operatorname{Fix}(h^n)$, its image $f_s(x)$ belongs to the finite
set $\operatorname{Fix}(h^n)$ and is consequently constant on the
connected parameter curve.  These fixed-point sets are finite because
a positive-dimensional fixed subvariety would have its ample class
both fixed and multiplied by a number greater than one.  Periodic
points of a polarized map are Zariski dense; therefore the morphisms
$f_s$ agree on a dense set and are all equal.
\end{proof}


\section{A spectral class for induction}\label{sec:pointed-spectral}


\subsection{A reference-parameter spectral class}

Fix a countable algebraically closed subfield $k_0\subset\mathbf C$
over which the initial family and all its prescribed algebraic data
are defined. Work with pointed parameter curves $(S,s_0)$ defined
over $k_0$, where $s_0\in S(\mathbf C)$ is generic over $k_0$.
Every nonempty $k_0$-defined Zariski open contains $s_0$. Every lift
of $s_0$ under a finite cover of parameter curves defined over
$k_0$ is again generic over $k_0$. All algebraic operations below
are made over this fixed field.

\begin{definition}[Constant spectra on good reference cycles]
\label{def:reference-good-spectra}
A one-parameter algebraic family $(\mathcal V,F)$ of polarized
endomorphisms of normal projective varieties has the
\emph{good reference-cycle spectral property} if there are a
pointed reference parameter $s_0$ and a dense Zariski open
$U\subset V_{s_0}$ obtained from a $k_0$-defined open of the total
family, such
that the following holds. For every smooth repelling Julia
periodic point whose orbit meets $U$, the characteristic
polynomial of its return is constant as an algebraic function
on the normalization of the periodic parameter component through
that point. Equivalently, it is constant as a germ at the marked
reference point of that component.

Julia membership is specified at $s_0$; each marked cycle is
continued on its own parameter germ. This defines the
\emph{pointed reference spectral class}.
\end{definition}

\begin{proposition}[Closure of the reference spectral class]
\label{prop:reference-spectral-class-closure}
The property in Definition~\ref{def:reference-good-spectra} is
preserved under the following operations, after finite parameter
extension and restriction over $k_0$ whenever required:
\begin{enumerate}
\item finite compatible polarized dynamical covers;
\item connected rational polarized factors and equivariant
      birational polarized models;
\item return families on good marked smooth repelling Julia
      periodic fibers of a connected rational polarized factor.
\end{enumerate}
In the last assertion the good base points are Zariski dense
in the reference phase base. Every strongly $J$-stable algebraic
polarized family has this spectral property after choosing a
countable field of definition and a parameter generic over it.
\end{proposition}

\begin{proof}
All the normality, flatness, smoothness and spreading restrictions
in the proof are nonempty opens defined over $k_0$; thus none
removes the reference parameter. Characteristic-polynomial
coefficients on a smooth marked periodic branch are algebraic functions on its normalized
parameter component. A constant analytic germ is therefore a
constant function on that component. It is enough throughout
to prove constancy on a possibly point-dependent neighborhood
of the reference parameter.

\emph{Finite covers.}
Let $p:\mathcal W\to\mathcal V$ be a finite compatible
polarized cover. Choose the good reference open of $W_{s_0}$
to lie above $U$ and in the locus where both varieties are smooth
and $p_{s_0}$ is \emph{\'etale}. This open is nonempty, since
characteristic zero gives generic separability. A repelling
Julia periodic point in it maps to a smooth repelling Julia
periodic point of $V_{s_0}$. Julia membership follows from
finite equivariance of the equilibrium measures; repulsion and
the equality of return spectra follow from the tangent
isomorphism given by $dp$.

The two local periodic branches exist by the implicit function
theorem. They are compatible with $p$ by uniqueness. If the
image has smaller exact period, take the indicated power of
its constant return polynomial. On a sufficiently small disk,
$dp$ remains invertible, and the return matrices upstairs and
downstairs are conjugate. Their characteristic polynomials are
therefore constant. A periodic orbit merely meeting the good
open has the same return spectrum at every phase, so this
proves the cover assertion.

\emph{Birational models and factors.}
For an equivariant birational polarized model choose an open
on which the birational map is an isomorphism. Equilibrium
measures correspond on that open, and the same conjugacy
argument applies. In particular a rational factor may first be
replaced by its normalized equivariant graph. The tensor product of
the source and base polarizations pulls back to an ample polarization
on that graph normalization.

Let now $\pi:\mathcal V\to\mathcal B$ be a connected factor
morphism. Shrink a dense open $B_0\subset B_{s_0}$ so that its
fibers are normal, geometrically integral and flat, and so that
each fiber meets the original good open in the smooth locus
of $\pi$. Such an open exists by generic normality, generic
smoothness and dominance of the image of the good open.
Choose a smooth repelling Julia periodic point $b\in B_0$,
of period $n$. The polarized return on $Y=\pi^{-1}(b)$ has a
smooth repelling Julia periodic point $x$ in the chosen good
open: its equilibrium measure gives no mass to a proper
algebraic subset, and its repelling Julia periodic points
equidistribute. Write $\ell$ for the period of $x$ under the
fiber return.

Lemma~\ref{lem:continuous-fiber-measures} places $x$ in
$J_{F_{s_0}}$. Smoothness of $\pi$ at $x$ and the tangent exact
sequence show that $x$ is an ambient smooth repeller for
$F^{n\ell}$. Its ambient return polynomial $P(T)$ is constant
by the given reference spectral property. The local motion
$x(s)$ exists by the implicit function theorem, and its
projection is the local period-$n$ motion $b(s)$, by uniqueness
of the repelling base branch. Smoothness of $\pi$ persists at
$x(s)$. Hence
\[
          \det(TI-D_{b(s)}g_s^{n\ell})\quad\text{divides }P(T).
\]
The set of monic divisors of $P$ is finite. Holomorphic
dependence makes this factor constant. Taking $\ell$th roots
of its finite eigenvalue set shows that
$\det(TI-D_{b(s)}g_s^n)$ is constant as well. This proves the
factor property with good open $B_0$. Smooth repelling Julia
periodic points meeting $B_0$ are Zariski dense, again by
repelling equidistribution and the non-pluripolarity of the
equilibrium measure.

\emph{Periodic-fiber returns.}
Fix such a good base point $b$, normalize its algebraic periodic
parameter component, and use its marked section $b(s)$. After
restriction, the corresponding phase fibers form a family of
normal projective varieties, with a polarized return $H_s$.
Use as good reference open in $Y$ the intersection of the
ambient good open with the smooth locus of $\pi$.

Let $x$ be any repelling Julia periodic point of $H_{s_0}$ whose
orbit meets this open, of fiber period $\ell$. The same mixed
measure inclusion and tangent exact sequence make it an ambient
repelling Julia point at the reference parameter. Its local
ambient motion exists and lies over $b(s)$. Both the ambient
polynomial and the base polynomial are constant by the preceding
arguments. The exact tangent sequence gives
\[
 \det(TI-D_{x(s)}F_s^{n\ell})
   =\det(TI-D_{x(s)}H_s^\ell)\,
          \det(TI-D_{b(s)}g_s^{n\ell}).
\]
The fiber characteristic polynomial is their quotient and is
constant. This proves the required property for the return
family. No Julia-persistence assertion away from the reference
parameter is used in this calculation.

Finally, the constant-spectrum theorem for strongly stable
algebraic families, Proposition~\ref{prop:stable-spectrum-any-dimension},
gives the property with the full available smooth reference
locus.
\end{proof}


\subsection{A spectral input stable under the induction operations}

\begin{proposition}[Reference-cycle spectra suffice for the local return construction]
\label{prop:reference-cycle-spectra-local-returns}
Let $\mathfrak f:\mathfrak X\to\mathfrak X$ be an algebraic curve
family of polarized endomorphisms of normal projective varieties.
Fix a parameter $s_0$ and a dense Zariski open
$U\subset X_{s_0}$. Suppose that, for every $m\ge1$, the
characteristic polynomial of $D f_s^m$ is constant on each
algebraic component of the marked $m$-periodic locus whose fibre
at $s_0$ contains a smooth repelling Julia point with an orbit
meeting $U$.
Then all the hypotheses used in the weighted-character and local
return-geometry arguments hold at a suitable reference repeller.
In particular Theorem~\ref{thm:arbitrary-dimensional-local-return-geometry}
applies with this spectral hypothesis in place of $J$-stability.
\end{proposition}

\begin{proof}
Choose a smooth repelling Julia periodic point $p\in U$ satisfying
the return-centre conclusion of
Proposition~\ref{prop:good-repeller-selection}. Replace the family
by its return iterate $F_s=f_s^\ell$ for this construction.
Since $D F_{s_0}(p)-I$ is invertible, the holomorphic implicit
function theorem gives a marked fixed point $p_s$ on one parameter
disc $\Delta$ about $s_0$. The marking belongs to the algebraic
periodic component through $p$. The hypothesis therefore makes
its characteristic polynomial constant. Holomorphic primary
spectral projectors and the parameter Poincar\'e--Dulac theorem
provide coordinates $\sigma_s$ on a common phase neighbourhood,
with
\[
 F_s\sigma_s=\sigma_s\mathcal N_s,
 \qquad \mathcal N_s=S U_s=U_sS,
\]
where $S$ is fixed and expanding. The nilpotent linear and
nonlinear resonant coefficients of $U_s$ may vary. On relatively
compact parameter discs the iterates $\mathcal N_s^{-n}$ contract
bounded phase sets uniformly to $0$, by
Lemma~\ref{lem:resonant-iterate-polynomial-bounds}.
Shrink the phase neighbourhood so that its reference image under
$\sigma_{s_0}$ is contained in $U$.

Construct the entire inverse-return collection first in the single
reference fibre, as in
Proposition~\ref{prop:good-repeller-selection}. Each branch is
unramified along its defining finite inverse orbit. The implicit
function theorem therefore continues that branch as a holomorphic
parameter germ. The continuation can be taken on a fixed relatively
compact phase ball: compactness gives finitely many inverse-function
charts, and uniqueness makes their continuations agree on overlaps.
Compositions and normal-form inverse prefixes continue by the same
identities. No common parameter disc for the infinite collection is
needed.

Fix any finite list of branches $h_1,\ldots,h_r$. Their germs have
a common parameter disc, and there are fixed polydiscs
$D_r\Subset D_R$ such that each $h_{i,s}$ is defined on a
neighbourhood of $\overline{D_r}$ and maps it into $D_R$.
After shrinking that parameter disc, choose $n_0$ so that
$\mathcal N_s^{-n}(D_R)$ is relatively compact in $D_r$ for
all $n\ge n_0$, uniformly in $s$. The same estimate for first
derivatives, or the Cauchy estimate on slightly larger polydiscs,
makes each suitably ordered inverse-prefix composition a strict
contraction. Hence every inverse word with independently chosen
$n_1,\ldots,n_r\ge n_0$ has a unique holomorphic fixed point on
this same parameter disc. Increasing $n_0$ once works for the
whole collection of these integer tuples.

At $s=s_0$, iterate this inverse word starting at the marked Julia
point. Every iterate is a backward image of a Julia point along
an unramified inverse branch, and so lies in the Julia set.
The limit fixed point belongs to the Julia set because that set
is closed. It is smooth and repelling for the forward return,
and lies in $\sigma_{s_0}(D_r)\subset U$. Its holomorphic marking
belongs to an algebraic periodic component covered by the
hypothesis. Consequently its forward return characteristic
polynomial, and hence its inverse fixed-point spectrum, is constant
on the common parameter disc. This proves exactly the simultaneous
spectral assertion needed for the nuclear traces with independently
large integers $n_i$.

The remaining infinite collection is used only through its germs.
Its compressed matrices have entries in the field of meromorphic
parameter germs. The Grassmannian invariance equations are generated
by finitely many polynomial equations, and Burnside reconstruction
uses finitely many words. Thus the construction of the module and
its simultaneous gauge in
Lemma~\ref{lem:irreducible-holomorphic-return-module} takes place on
one parameter germ. Every other branch intertwines that gauge as an
identity of germs. Finally all relative uniqueness assertions use
the reference centres coefficient by coefficient, as established in
Lemma~\ref{lem:relative-uniqueness-individual-germs} below.
These are all the stability inputs in the local return argument.
\end{proof}

\begin{lemma}[Relative uniqueness with individually continued centres]
\label{lem:relative-uniqueness-individual-germs}
Let $\{a_i\}$ be a holomorphic uniqueness set in a phase domain
$D$. For each $i$ let $a_i(s)$ be a holomorphic parameter germ
with $a_i(s_0)=a_i$; these germs need not have a common domain.
Let $G(s,z)$ be holomorphic on a product neighbourhood of
$\{s_0\}\times D$. If $G(s,a_i(s))=0$ as a parameter germ for
every $i$, then $G=0$ as a germ along $\{s_0\}\times D$.
The same assertion holds for a uniqueness set of centres in every
sufficiently small phase neighbourhood.
\end{lemma}

\begin{proof}
Write $G(s,z)=\sum_{j\ge0}(s-s_0)^jG_j(z)$. Evaluation at $s_0$
and uniqueness give $G_0=0$. If $G_0,\ldots,G_{j-1}$ vanish
identically, the coefficient of $(s-s_0)^j$ in
$G(s,a_i(s))$ is $G_j(a_i)$. Hence $G_j=0$ by uniqueness.
Induction annihilates every coefficient. The convergent parameter
expansion proves the assertion. The local version is the same
argument on the chosen phase neighbourhood.
\end{proof}


\begin{lemma}[The one-dimensional spectral base case]
\label{lem:reference-spectral-rational-base}
Let $S$ be a smooth connected algebraic curve and let
$f:S\times\mathbb P^1\to\mathbb P^1$ be an algebraic family of
rational maps of degree $d>1$. Suppose that, at a parameter
$s_0\in S$, there is a dense Zariski open $U\subset\mathbb P^1$
such that the multiplier of every repelling periodic cycle of
$f_{s_0}$ meeting $U$ is constant on its marked algebraic
periodic component. Then the family is isotrivial or a family
of flexible Latt\`es maps.
\end{lemma}

\begin{proof}
Write $Z=\mathbb P^1\setminus U$. The Fatou--Shishikura
inequality gives only finitely many nonrepelling cycles of
$f_{s_0}$; see \cite[Corollary~1]{Shi87}. Choose an integer $M$
larger than $|Z|$, the exact periods of all those cycles, and
every integer $mr$, where $m$ is the exact period of one of
those cycles and its multiplier is a root of unity of order
$r$.

For $n\geq1$ let $\operatorname{Fix}_n$ be the relative fixed
point divisor of $f^n$, with its fixed point multiplicities,
and form the relative dynatomic divisor
\[
       D_n=\sum_{k\mid n}\mu(n/k)\operatorname{Fix}_k,
\]
where $\mu$ denotes the M\"obius function. This is an effective
relative divisor finite and flat over $S$. We recall the local
calculation, including the parabolic case. If $a$ has exact
period $m$ and $h=f^m$ is its return germ, put $\lambda=h'(a)$.
If $\lambda$ is not a root of unity, every germ $h^j(z)-z$
has a simple zero at $a$. If $\lambda$ has order $r$, then
$h^j(z)-z$ has order one for $r\nmid j$, whereas for $j=rk$
its order is a fixed integer $e>1$, independent of $k\geq1$.
Indeed, in a coordinate centered at $a$,
\[
 h^r(z)=z+c z^e+O(z^{e+1}),\quad c\ne0,
 \qquad h^{rk}(z)=z+kc z^e+O(z^{e+1}).
\]
The return germ cannot be the identity because the rational
map has degree greater than one. M\"obius inversion therefore
shows that $a$ occurs in $D_n$ only for $n=m$, or for $n=mr$
when $r>1$. In particular, a parabolic point of multiplier one
occurs only at its actual period, not at arbitrary multiples
of that period. The same calculation over the algebraic
closure of the generic function field proves effectiveness
of the relative divisor. Its components are horizontal,
because the fixed point divisors have no vertical components;
as an effective Cartier divisor on $S\times\mathbb P^1$ with
finite fibers, it is finite and flat over $S$. Restriction of
the Cartier-divisor identity to a fiber gives precisely the
local calculation just made.

It follows that, for every $n>M$, the fiber $(D_n)_{s_0}$
consists entirely of simple points of actual period $n$ which
are repelling. Their cycles meet $U$: a cycle contained in $Z$
has period at most $|Z|$. Let $C$ be any irreducible component
of the support of $D_n$. Since $C\to S$ is finite and
surjective, it meets the reference fiber at one of these
points. The multiplier
\[
             \Lambda_n(s,z)=(f_s^n)'(z)
\]
is an intrinsic regular function on the relative fixed point
scheme: the derivative acts on the relative tangent line at
the fixed point. Near $C\cap(\{s_0\}\times\mathbb P^1)$, the
fixed point scheme is smooth and \emph{\'etale} over $S$, and
$C$ is the marked actual-period-$n$ component. The hypothesis
makes $\Lambda_n$ constant on its normalization, with constant
value of modulus greater than one. The same value holds at
every point of $C$.

Every actual-period-$n$ point of every $f_s$ belongs to $D_n$.
Consequently every periodic point of exact period $n>M$ is
repelling, for every $s\in S$. All attracting cycles in the
whole family have period at most $M$. This is precisely the
stability hypothesis in \cite[Section~2]{McM87}; its
Theorem~2.2 says that the family is trivial or affine. The
classification of affine families immediately following that
theorem identifies every nontrivial algebraic family as a
flexible Latt\`es family. In the trivial case, the generic
M\"obius-conjugacy scheme is nonempty and of finite type over
$\mathbb C(S)$. A closed point supplies a conjugacy after a
finite function-field extension, which is isotriviality.
\end{proof}

\begin{corollary}[Normal curves in the spectral induction class]
\label{cor:reference-spectral-curves-base}
A family of polarized endomorphisms of normal projective
curves with the good reference-cycle spectral property of
Definition~\ref{def:reference-good-spectra} is isotrivial or
exceptional. In genus zero the nonisotrivial alternative is
a family of flexible Latt\`es maps.
\end{corollary}

\begin{proof}
Riemann--Hurwitz gives genus at most one. In genus zero, a
finite extension of the parameter function field identifies
the generic curve with $\mathbb P^1$. Spread the identification
over a nonempty open of the resulting parameter curve. The
fixed-field convention preceding
Definition~\ref{def:reference-good-spectra} retains the marked
reference parameter under this operation. The spectral
property is unchanged by conjugacy and finite parameter
extension. Lemma~\ref{lem:reference-spectral-rational-base}
applies.

In genus one, choose an origin after a finite extension. The
endomorphism is $x\mapsto\alpha(x)+a$, where $\alpha$ is an
isogeny of degree greater than one. The isogeny
$\alpha-\operatorname{id}$ is nonzero, hence surjective. After
a further finite extension choose a fixed point and take it
as origin. The endomorphism becomes the polarized isogeny
$\alpha$, so the family is exceptional.
\end{proof}


\section{Choosing the local returns}

The construction follows the homoclinic-orbit viewpoint of
Ji--Xie \cite[Section~2.2]{JX23}: long returns produce adjoint
periodic points, and their multipliers constrain the return maps.
Their exceptional-map criterion and new proof of McMullen's
theorem appear in \cite[Theorem~2.11 and Section~6]{JX23}.
Here we use repelling spectra to control local returns in
arbitrary phase dimension.

\begin{lemma}[A finite section over a smooth ball]
\label{lem:finite-section-ball}
Let $q:T\to B$ be a finite holomorphic map, where $T$ is normal
and pure dimensional and $B$ is a complex ball of the same
dimension. Every holomorphic section $s:B\to T$ identifies $B$
biholomorphically with an open and closed component of $T$.
\end{lemma}

\begin{proof}
The image of a section is closed: if $s(b_j)\to t$, then
$b_j=q(s(b_j))\to q(t)$ and $t=s(q(t))$. The section is proper
as a map into $T$, and its image is therefore a closed analytic
subset of dimension $\dim T$. Normality makes $T$ locally
irreducible. Thus this image is a union of connected components
of $T$, and is open. The restriction of $q$ and $s$ are mutually
inverse holomorphic maps.
\end{proof}

\begin{proposition}[Selecting a repeller with unique return centres]
\label{prop:good-repeller-selection}
Let $f:X\to X$ be a finite endomorphism of a normal projective
variety. Assume its meromorphic lift to a smooth projective
resolution has dominant topological degree. Let $\mu$ be its
equilibrium measure. In every open set of positive $\mu$-measure
one can choose a smooth repelling periodic point $p$, of period
$r$, with the following property. Put $F=f^r$. There is a
coordinate ball $B$ about $p$ and a composition-closed collection
of biholomorphic inverse branches
\[
          h:B\longrightarrow B,
          \qquad F^{n(h)}h=\operatorname{id}_B,
\]
such that the centres $h(p)$ form a uniqueness set for holomorphic
functions on $B$. Every neighbourhood of $p$ contains the images
of a subcollection whose centres are again a uniqueness set.
The same local uniqueness holds near any return centre, using
compositions with that return. Moreover, the unramified backward
orbit of $p$ under $F$ is dense in $\operatorname{supp}\mu$.
\end{proposition}

\begin{proof}
Let $\pi:Y\to X$ be a resolution which is an isomorphism over
$X_{\rm reg}$, and write $g=\pi^{-1}f\pi$. Let
$d_t$ be its topological degree. The equilibrium measure on $Y$
does not charge pluripolar sets, by Guedj \cite{Gue05},
\emph{Annals of Mathematics} \textbf{161} (2005), Theorem 2.1.
Its pushforward therefore gives every proper local analytic subset
of $X_{\rm reg}$ measure zero. We use Dinh--Nguy\^en--Truong,
\emph{Indiana University Mathematics Journal} \textbf{64} (2015),
Theorem 1.1 and Proposition 3.1. The latter supplies one positive
closed $(1,1)$-current $R$ on $Y$ such that, if
$\nu(R,y)<\varepsilon<1$, a ball about $y$ admits at least
$(1-\varepsilon)d_t^n$ inverse branches of $g^n$, for every $n$,
whose diameters tend uniformly to zero.

Fix, for example, $\varepsilon=1/4$. The Siu level set
$\{\nu(R,\cdot)\ge\varepsilon\}$ is a single proper analytic
set. The equilibrium measure on $Y$ gives it and the exceptional
locus of $\pi$ measure zero. Repelling periodic equidistribution
therefore gives a repeller $\tilde p$ outside these sets, in
the prescribed positive-measure open set. Put $p=\pi(\tilde p)$
and let $r$ be its period. The local conjugacy by $\pi$ makes
$F=f^r$ biholomorphic and expanding at $p$.

The good inverse branches on $Y$ project to genuine inverse
branches on $X$. Indeed, on a ball in the isomorphism locus,
their projections satisfy $f^n s=\operatorname{id}$. This
identity holds on the dense set where the meromorphic conjugacy
is represented by ordinary maps, and hence everywhere by
holomorphic continuation. Lemma~\ref{lem:finite-section-ball}
makes each projected section a biholomorphism onto a smooth open
subset. Distinct branches remain distinct, since $\pi$ is
generically one-to-one. Their diameters still tend uniformly to
zero, by uniform continuity of $\pi$ on compact $Y$.

Restrict to inverse branches of orders $rn$, so that they are
inverse branches of $F^n$. Write $\delta=d_t^r=\deg F$.
Shrink the initial ball $B$ if necessary. We can arrange that
the local inverse branch $v$ of $F$ fixing $p$ maps $B$ relatively
compactly into itself and that $v^\ell(B)$ shrinks to $p$.
All the preceding good branches remain defined after restriction
to this ball.

Choose a smooth probability measure $\sigma$ compactly supported
in $B$. Let $b_{n,j}:B\to X$ be the good inverse branches and
define the two subprobability measures
\[
 \nu_n=\delta^{-n}\sum_j\delta_{b_{n,j}(p)},
 \qquad
 \zeta_n=\delta^{-n}\sum_j(b_{n,j})_*\sigma.
\]
They have the same mass, at least $1-\varepsilon$, and their
difference tends weakly to zero because the branch diameters
tend uniformly to zero. Moreover
\[
                 0\le\zeta_n\le\delta^{-n}(F^n)^*\sigma.
\]
The measures on the right converge to $\mu$. For completeness,
this smooth-measure version also follows by integrating the
pointwise equidistribution theorem, Dinh--Nguy\^en--Truong,
Theorem 1.2: all its exceptional and indeterminacy sets form a
countable union of proper analytic sets and therefore have zero
$\sigma$-measure; bounded test functions permit dominated
convergence. Passing to a subsequence gives
\begin{equation}\label{eq:positive-good-centre-measure}
                   \nu_n\longrightarrow\nu,
                   \qquad 0<\nu\le\mu.
\end{equation}
In particular $\nu$ gives every proper analytic set measure zero.
No pointwise equidistribution assertion at $p$ has been used.

Let $\mathcal B_p$ denote the set of points which map to $p$
under some iterate of $F$ unramified at that point, including $p$
itself. Let $C=\overline{\mathcal B_p}$. Every measure $\nu_n$
is supported in $C$, so \eqref{eq:positive-good-centre-measure}
gives $\mu(C)>0$. The set $C$ is closed and forward invariant:
$F(\mathcal B_p)\subset\mathcal B_p$ and $F$ is continuous.
Invariance of $\mu$ makes $C$ invariant modulo a null set;
ergodicity of $F$ gives $\mu(C)=1$. Hence
\[
                         \operatorname{supp}\mu\subset C.
\]
In particular, $\mathcal B_p\cap B$ is a uniqueness set on $B$.
Indeed, a holomorphic function vanishing there vanishes on
$B\cap\operatorname{supp}\mu$, which has positive $\mu$-measure,
whereas a proper local analytic zero set has zero $\mu$-measure.

For each $z\in\mathcal B_p\cap B$, choose the unramified inverse
germ $b_z$ with $b_z(p)=z$ and $F^{n_z}b_z=\operatorname{id}$.
Its initial domain may depend on $z$. Since $v^\ell(B)$ shrinks
to $p$, for sufficiently large $\ell_z$ the composition
\[
                         h_z=b_zv^{\ell_z}
\]
is defined on the same ball $B$, with image relatively compact
in $B$ and centre still equal to $z$. It is an inverse branch
of $F^{n_z+\ell_z}$, because
$F^{n_z+\ell_z}b_zv^{\ell_z}
 =F^{\ell_z}v^{\ell_z}=\operatorname{id}$.
Thus a common domain has been obtained without discarding any
of the centres in $\mathcal B_p\cap B$.

Adjoin the branch $v$ and all finite compositions. They remain
inverse branches of iterates of $F$ on $B$. For every fixed
$\ell$, the centres of $v^\ell h_z$ are a uniqueness set,
because $v^\ell$ is biholomorphic on $B$.
Their whole images lie in $v^\ell(B)$, which shrinks to $p$.
Composing once more on the left with a fixed return gives the
corresponding uniqueness in every sufficiently small neighbourhood
of its centre. This proves all the assertions.
\end{proof}


\subsection{Weighted characters with varying resonant normal forms}

Fix a parameter disk germ $(B,0)$ and a holomorphically marked smooth
repelling Julia point. Constant spectra and holomorphic
Poincar\'e--Dulac normalization give
\[
 \begin{gathered}
 f_s^\ell\sigma_s=\sigma_s\mathcal N_s,\qquad
 \mathcal N_s=S U_s=U_s S,\\
 S=\operatorname{diag}(\lambda_1,\ldots,\lambda_N),\quad |\lambda_i|>1.
 \end{gathered}
\]
Here $S$ is fixed and $U_s$ is a holomorphic family of unipotent
resonant polynomial automorphisms. The constant-spectrum primary
projectors give holomorphic generalized eigenspace bundles; choose
holomorphic frames for them and keep the varying nilpotent linear
parts in $U_s$. No constancy of their Jordan type or of the
nonlinear resonant coefficients is imposed.
Convergent normalization in the Poincar\'e domain is locally uniform
in the parameter; the homological denominators outside the finitely
many resonant spaces are fixed and bounded away from zero with the
usual degree estimates; see \cite[Appendix]{RR88} and \cite{BDM08}.

For a multiplier $\chi=\lambda^\alpha$, let $V_\chi$ be the span
of the monomials of multiplier $\chi$, and let $P_\chi$ be Taylor
projection. Every $V_\chi$ is finite dimensional. Put
\[
 a=\min_i\log|\lambda_i|>0,\qquad
 b=\max_i\log|\lambda_i|,
 \qquad W_R=\bigoplus_{0<\log|\chi|\le R}V_\chi.
\]
For an actual inverse return $h_s$, set
$T_{h,s}=P_R C_{h_s}|_{W_R}$, where $C_h\varphi=\varphi\circ h$.

\begin{lemma}[Polynomial bounds for resonant iterates]
\label{lem:resonant-iterate-polynomial-bounds}
There are holomorphic nilpotents $L_{\chi,s}$ such that
\[
 C_{\mathcal N_s^{-n}}|_{V_\chi}
       =\chi^{-n}\exp(nL_{\chi,s}).
\]
On any relatively compact parameter disc, there are constants
$C,D$ such that the Taylor coefficients contributed by a monomial
of ordinary degree $m$ are bounded by
\[
                    C^m(1+n)^{Dm}|\chi|^{-n}.
\]
The constants are independent of $m,n$, and $s$.
\end{lemma}

\begin{proof}
Order the variables by $\log|\lambda_i|$. A nonlinear resonant
monomial in coordinate $i$ uses only variables of strictly smaller
absolute weight. The diagonal blocks of the linear part of $U_s$
are unipotent. Induction over these weight blocks shows that every
coordinate of $U_s^n$ is polynomial in $n$: on a linear block the
binomial expansion of a unipotent matrix is finite, and at the next
weight block the additional terms are finite sums of polynomial
sequences in the iterate index. Their degrees admit a bound depending
only on the fixed list of resonant monomials. The same assertion
holds for $U_s^{-n}$.

Substitution by $U_s$ is unipotent on each $V_\chi$. Its logarithm
is therefore a finite sum, and
$L_{\chi,s}=\log(C_{U_s^{-1}}|_{V_\chi})$ is holomorphic and
nilpotent. The first identity follows from commutation with $S$.
The preceding coordinate bounds, multiplied for the $m$ factors of
a degree-$m$ monomial, give the stated estimate, after increasing
$C,D$. All coefficient bounds are uniform on a relatively compact
parameter disc. Notice that $m\le a^{-1}\log|\chi|$.
\end{proof}

\begin{proposition}[Weighted character constancy]
\label{prop:resonant-weighted-character-constancy}
Every trace of a word in
\[
        T_{h,s},\qquad P_\chi|_{W_R},\qquad L_{\chi,s}P_\chi
\]
is independent of $s$. This holds for every finite cutoff $R$.
\end{proposition}

\begin{proof}
Fix the finite collection of returns occurring in a word. Choose
polydiscs $D_r\Subset D_R$ so that these returns send $D_r$
compactly into $D_R$. For all sufficiently large $n$,
$\mathcal N_s^{-n}(D_R)\Subset D_r$, locally uniformly in $s$.
The composition operators between the corresponding Hardy spaces
are bounded, and the inverse-normal-form operators have absolutely
summable Taylor expansions. Consequently the holomorphic
fixed-point trace formula \cite[Theorem~4.2, formula~(11)]{BJ08}
applies to
\begin{equation}\label{eq:resonant-nuclear-word}
 \operatorname{Tr}\bigl(
 C_{h_1,s}C_{\mathcal N_s^{-n_1}}\cdots
 C_{h_r,s}C_{\mathcal N_s^{-n_r}}\bigr).
\end{equation}
Its value is $1/\det(I-Dg_s(p_s))$, where $g_s$ is the
corresponding contracting inverse word. Starting its iterates at
the marked Julia point shows that $p_s$ is a Julia point. The
forward periodic spectra are constant by
Proposition~\ref{prop:stable-spectrum-any-dimension}. Thus
\eqref{eq:resonant-nuclear-word} is constant for all independently
sufficiently large integers $n_1,\ldots,n_r$.

Set $A_{\chi\psi}(h_s)=P_\chi C_{h_s}|_{V_\psi}$.
Expanding \eqref{eq:resonant-nuclear-word} gives exponential
frequencies $\chi_1^{-n_1}\cdots\chi_r^{-n_r}$ and coefficients
\[
 \operatorname{tr}\bigl(
 A_{\chi_r\chi_1}(h_{1,s})e^{n_1L_{\chi_1,s}}
 \cdots
 A_{\chi_{r-1}\chi_r}(h_{r,s})e^{n_rL_{\chi_r,s}}
 \bigr),
\]
which are polynomials in the independent integers $n_i$.
There are only finitely many multiplier weights below a given
modulus. Distinct exponential frequencies of the same modulus,
together with their polynomial coefficients, are linearly
independent; finite-difference annihilators isolate each frequency.
They can be extracted in order of increasing $|\chi|$.

For completeness, the infinite tail does not interfere with this
extraction. Lemma~\ref{lem:resonant-iterate-polynomial-bounds}
and the Cauchy estimates bound the terms of weight $\chi$ by
\[
 \exp\bigl((-n+C_1\log(1+n)+C_2)\log|\chi|\bigr).
\]
The number of monomials with $\log|\chi|\le t$ grows at most
polynomially in $t$. If $\chi_*$ is the next weight modulus after
the finitely many retained ones, the remaining sum is bounded by
$|\chi_*|^{-n}$ times a fixed power of $n$, up to a harmless
constant after increasing the initial $n$. It is exponentially
smaller than every retained frequency. This estimate is locally
uniform in $s$ and is preserved by the other fixed bounded
substitutions. Extraction in each independent integer $n_i$
therefore proves constancy of every coefficient of the displayed
polynomial. These coefficients are exactly the cyclic block traces
with arbitrary inserted powers of the nilpotents. Inserting the
weight projectors and expanding a compressed word proves the
assertion. In the full word \eqref{eq:resonant-nuclear-word} the
constant block contributes the fixed trace $1$; a positive-weight
projector or a nilpotent supported on a positive block annihilates
that block. Passing to $W_R$ therefore gives the asserted traces.
\end{proof}

\begin{lemma}[An irreducible holomorphic summand]
\label{lem:irreducible-holomorphic-return-module}
After a finite change of parameter germ there is a nonzero
holomorphic subbundle $E_s\subset W_R$, invariant under all the
operators in Proposition~\ref{prop:resonant-weighted-character-constancy},
on which they act irreducibly. Their restrictions are simultaneously
constant in a holomorphic frame. In particular there are
$A_s:E_0\xrightarrow{\sim}E_s$, $A_0=I$, intertwining every
compressed return, every weight projector, and every $L_{\chi,s}$.
\end{lemma}

\begin{proof}
Over an algebraic closure of the meromorphic parameter-germ field,
choose a common invariant subspace of least positive dimension.
The invariance equations on its Grassmannian are polynomial and
are generated by finitely many of them. Thus a finite extension
of the parameter-germ field suffices. Properness of the Grassmannian
extends the resulting meromorphic map holomorphically across the
reference point. Invariance extends by closedness. This argument
also accommodates return germs with different parameter domains:
each invariance identity is an identity of meromorphic germs.

For any word, the eigenvalues of its restriction to $E_s$ are
among the constant eigenvalues of its action on $W_R$. The latter
are constant because traces of all powers are constant. Its trace
on $E_s$ is therefore a holomorphic function taking values in a
finite set, hence is constant. Generic irreducibility and Burnside's
theorem give $m^2$ words whose restrictions form a basis of
$\operatorname{End}(E_s)$, where $m=\dim E_s$. Their trace-pairing
Gram determinant is nonzero generically and constant, hence nonzero
at the reference point as well. The multiplication table of this
word basis is determined by the same constant trace pairing.
A fixed system of matrix units consequently yields a holomorphic
trivialization in which every generator is constant. The finite
word basis works for the entire collection of return germs.
\end{proof}


\section{Geometric reconstruction from finite return data}

On a smooth open of a normal variety, a \emph{distribution} is
a subbundle of the tangent bundle. Its rational extension is the
saturated subsheaf of the reflexive tangent sheaf determined at
the generic point. A \emph{foliation} is a distribution closed
under the Lie bracket. Its \emph{algebraic part} is the maximal
subfoliation whose general leaves are algebraic. A foliation is
\emph{purely transcendental} if its algebraic part has rank zero.
A \emph{finite web} is a finite unordered collection of generically
distinct distributions; invariance means that the differential
permutes its members wherever it is invertible. A \emph{Killing
field} of a local geometric structure is a local vector field
whose flow preserves that structure. The structure is of
\emph{finite type} if a fixed finite jet determines each local
infinitesimal symmetry.


\begin{lemma}[An invariant subspace gives an invariant foliation]
\label{lem:truncated-invariant-subspace-foliation}
Let $D\subset\mathbb C^N$ be a connected neighborhood of $0$ and
let $W\subset\mathcal O(D)$ be finite dimensional, with every
element vanishing at $0$.  Suppose a linear Taylor projection
$P_W$ onto $W$ retains all first derivatives:
\[
 d(P_W\varphi)(0)=d\varphi(0).
\]
Let $\mathcal H$ be a collection of local biholomorphisms from
neighborhoods of $0$ into $D$, closed under every composition that
is defined as a germ at $0$.  Assume its centers are a holomorphic
uniqueness set locally at $0$: in every sufficiently small
neighborhood $D'\ni0$, a holomorphic function vanishing at all
$h(0)\in D'$ vanishes identically near $0$.

Write $T_h=P_WC_h|_W$.  If a nonzero subspace $E\subset W$ is
invariant under every $T_h$, set
\[
 E^{(1)}=\{de(0):e\in E\}\subset(\mathbb C^N)^*,
 \qquad d=\dim E^{(1)}.
\]
Then $1\le d\le N$, the differentials of $E$ have constant rank
$d$ near $0$, and their common kernel defines a regular integrable
distribution there.  Its extension as a singular foliation on $D$
is invariant under every return germ $h\in\mathcal H$.
In particular, if $d<N$, it is a nonzero proper invariant foliation.
\end{lemma}

\begin{proof}
Let $a_h=h(0)$.  Invariance of $E$ and the first-jet property of
$P_W$ give
\begin{equation}\label{eq:invariant-space-center-jets}
 dE(a_h)\,Dh(0)\subset E^{(1)}.
\end{equation}
The derivative $Dh(0)$ is invertible.  Hence every $(d+1)$-minor
of a matrix of differentials of a basis of $E$ vanishes at all
centers sufficiently close to $0$.  Uniqueness makes these minors
vanish identically near $0$, and then on $D$ by connectedness.
The rank at $0$ is $d$, so it is exactly $d$ in a neighborhood of
$0$.  If $d=0$, the same argument says $de=0$ on $D$ for all
$e\in E$.  As $e(0)=0$, this would give $E=0$, a contradiction.

Choose $e_1,\ldots,e_d\in E$ with independent differentials at
$0$.  Their joint map is a submersion near $0$, and every
differential from $E$ lies in their span there.  The common kernel
is therefore the tangent distribution to its local fibers and is
integrable.  On $D$ take the saturated foliation given generically
by the same kernel; equivalently it is the kernel of the finite
matrix of differentials of a basis of $E$.

Fix $h\in\mathcal H$.  For every $k\in\mathcal H$ whose center
$a_k$ is sufficiently close to $0$, the composition $h\circ k$
is defined and belongs to $\mathcal H$.  Equation
\eqref{eq:invariant-space-center-jets} for that composition gives
\[
 d(E\circ h)(a_k)\,Dk(0)\subset E^{(1)}.
\]
On the other hand, since the rank of $dE$ is $d$ near $0$,
\eqref{eq:invariant-space-center-jets} for $k$ is an equality:
\[
 dE(a_k)\,Dk(0)=E^{(1)}.
\]
Thus $d(E\circ h)(a_k)\subset dE(a_k)$.  The $(d+1)$-minors
of the combined matrix of $dE$ and $d(E\circ h)$ vanish at all
these centers.  Local uniqueness makes this inclusion an identity
of germs at $0$.  On a dense open subset both sides have rank $d$,
because $h$ is locally biholomorphic.  Their annihilator
distributions agree there, and hence their saturated foliations
agree.  This is the required invariance.
\end{proof}


\begin{proposition}[Compressed invariance yields exact jet geometry]
\label{prop:exact-jet-bundle}
Let $D\subset\mathbb C^N$ be connected, let
$A=\mathbb C1\oplus E$ be a finite-dimensional space of holomorphic
functions on $D$, and suppose that the compressed pullbacks of
a collection $\mathcal H$ of local biholomorphisms preserve $E$.
Assume that the compression retains the complete $k$-jet at $0$
modulo constants. Assume the following composition and uniqueness
conditions: the centres $a_h=h(0)$ determine holomorphic functions
on $D$; for each fixed $h$, the maps $h\circ u$ are among the
tested maps for a locally unique set of sufficiently small centres
$a_u$ in the domain of $h$.

For $0\le d\le k$ put
\[
             L_x^d=j_x^d A\subset J_x^d(\mathcal O_D),
             \qquad r_d=\dim L_0^d.
\]
Then $\dim L_x^d\le r_d$ on $D$. On the common open locus
$D^\circ$ where all these ranks are maximal, the $L^d$ are
holomorphic subbundles and every actual return preserves them:
\begin{equation}\label{eq:actual-jet-bundle-invariance}
           h_x^*:L_{h(x)}^d\xrightarrow{\ \sim\ }L_x^d
           \quad(x,h(x)\in D^\circ).
\end{equation}
Here $h_x^*$ is the ordinary pullback isomorphism of full jet
spaces. If $A$ consists of polynomials, these are algebraic
subbundles on a Zariski open set and the displayed conditions are
algebraic differential equations of order at most $k$ for $h$.
\end{proposition}

\begin{proof}
Choose a basis of $A$. Its derivatives of orders at most $d$
form a holomorphic jet matrix. Compressed invariance gives
\[
                  j_0^d C_h A\subset j_0^d A=L_0^d.
\]
Pullback by the local biholomorphism $h$ is an isomorphism
$J_{a_h}^d\to J_0^d$. Therefore
$\dim L_{a_h}^d\le r_d$. Every minor of size $r_d+1$ of
the jet matrix vanishes at the centres. Centre uniqueness makes
it vanish identically, proving the rank bound. The bound is
attained at $0$, so the maximal-rank locus contains a neighbourhood
of $0$. Intersect these finitely many loci for $d\le k$.

Fix $h$. For all sufficiently small tested $u$ we have
$a_u\in D^\circ$ and hence
\[
          j_0^d C_u A=L_0^d,
          \qquad j_0^d C_{h\circ u}A\subset L_0^d.
\]
The composition convention is
$C_{h\circ u}=C_uC_h$. The invertible jet pullback of $u$
then gives
\[
                       j_{a_u}^d C_hA\subset L_{a_u}^d.
\]
Adjoin the columns of the jet matrix of $C_hA$ to those of $A$.
All minors of size $r_d+1$ vanish at these small centres. Local
centre uniqueness forces them to vanish throughout the domain
of $h$. At any $x,h(x)\in D^\circ$ the inclusion they express
is an equality, since both spaces have dimension $r_d$ and
$h_x^*$ is invertible. This proves
\eqref{eq:actual-jet-bundle-invariance}.

For polynomial $A$, its jet matrices have polynomial entries.
The maximal-rank loci and the corresponding Grassmannian maps
are algebraic. In coordinates, the last inclusion is expressed
by the same finite collection of minors; the chain rule makes
their entries polynomial in $x,h(x)$ and the derivatives of $h$
of orders at most $d$. Thus the equations are algebraic.
\end{proof}

\begin{corollary}[Rank stagnation gives actual finite-dimensional closure]
\label{cor:jet-rank-stagnation}
Under Proposition~\ref{prop:exact-jet-bundle}, suppose that
\[
                          r_d=r_{d-1}
                          \qquad(1\le d\le k).
\]
Then $\dim A=r_{d-1}$ and every tested return satisfies
$C_hA=A$ as a space of holomorphic functions on its domain.
In particular, if $A$ is polynomial, compressed invariance has
upgraded to actual invariance of that polynomial space.
\end{corollary}

\begin{proof}
Work first in a small simply connected open set in $D^\circ$.
Projection of jets induces an isomorphism
$L^d\to L^{d-1}$. It follows by holomorphic linear algebra
that every derivative of order $d$ of an element $f\in A$ is
a holomorphic linear combination of the derivatives of $f$ of
orders at most $d-1$. Choose such coefficient functions and write
the resulting equations as
\begin{equation}\label{eq:finite-type-jet-system}
       \partial^\alpha f
          =\sum_{|\beta|\le d-1}c_{\alpha\beta}(x)
                    \partial^\beta f,
                    \qquad |\alpha|=d.
\end{equation}
Every holomorphic function whose $d$-jets lie in $L^d$ satisfies
these same equations.

Solutions of \eqref{eq:finite-type-jet-system} are determined by
their $(d-1)$-jet at one point. Indeed, collect all derivatives
of orders at most $d-1$ into a vector $Y_f$. Differentiation and
\eqref{eq:finite-type-jet-system} give first-order linear systems
$\partial_iY_f=B_i(x)Y_f$. Along any sufficiently short path
this is a linear ordinary differential equation, so zero initial
data imply $Y_f=0$. Analytic continuation proves uniqueness on
the connected domain; no integrability assertion for arbitrary
initial data is required.

Consequently $j_x^{d-1}:A\to L_x^{d-1}$ is injective, and
therefore is an isomorphism. In particular $\dim A=r_{d-1}$.
For $f\in A$, exact jet-bundle invariance says that $f\circ h$
satisfies the same system wherever source and target lie in
$D^\circ$. At a point there choose $g\in A$ with the same
$(d-1)$-jet as $f\circ h$. Uniqueness gives $f\circ h=g$
near that point and then throughout the connected domain of
$h$. Thus $C_hA\subset A$. Pullback is injective because
$h$ is a local biholomorphism, so equality follows from finite
dimension.
\end{proof}

\begin{corollary}[The first missing jet has an invariant symbol]
\label{cor:first-missing-jet-symbol}
Suppose that $r_{d-1}=\dim J^{d-1}$ but
$r_d<\dim J^d$, with $d\le k$. Then on $D^\circ$
\[
 Q_x:=\ker(L_x^d\longrightarrow J_x^{d-1})
                  \subset\operatorname{Sym}^dT_x^*D
\]
is a proper subbundle satisfying
\[
                         (Dh(x))^*Q_{h(x)}=Q_x.
\]
If $Q=0$, actual closure $C_hA=A$ follows from
Corollary~\ref{cor:jet-rank-stagnation}. If $A$ is polynomial,
the bundle $Q$ is algebraic.
\end{corollary}

\begin{proof}
Projection onto lower jets is surjective by hypothesis and has
constant rank. Thus its kernel is a subbundle of the indicated
symmetric power. Pullback of a jet whose lower terms vanish
acts on its leading term only by the derivative of $h$. The
assertion follows from exact jet-bundle invariance. The case
$Q=0$ is exactly $r_d=r_{d-1}$.
\end{proof}


\begin{lemma}[Integrating a closed tuple of differential operators]
\label{lem:closed-differential-operator-tuple}
Let $N\ge2$. Suppose polynomial-coefficient differential operators
$A_1,\ldots,A_N$ on $\mathbb C^N$, of order at most $r$, satisfy
\[
 \partial_j\circ A_i=\partial_i\circ A_j
 \quad(1\le i,j\le N).
\]
Then there is a unique polynomial-coefficient differential operator
$B$, of order at most $r-1$, such that $A_i=\partial_i\circ B$.
If $r=0$, this means that all $A_i$ vanish.
\end{lemma}

\begin{proof}
Suppose first that $r\ge1$, and let $a_i(x,\xi)$ denote the
degree-$r$ principal symbol of $A_i$, allowing $a_i=0$.
The degree-$(r+1)$ part of the displayed identity gives
\[
 \xi_j a_i=\xi_i a_j.
\]
For each $i$, choose $j\ne i$. Primality of $\xi_i$ shows that
$\xi_i$ divides $a_i$. The quotients $a_i/\xi_i$ coincide, so
$a_i=\xi_i b$ for a polynomial $b(x,\xi)$ homogeneous of degree
$r-1$ in $\xi$. Choose a differential operator $B_{r-1}$ with
symbol $b$. Each $A_i-\partial_i\circ B_{r-1}$ has order at
most $r-1$, and these new operators satisfy the same closedness
identity. Iterating reduces to order zero. At order zero write
$A_i=a_i(x)$. The coefficients of $\partial_j$ in
$\partial_j\circ A_i-\partial_i\circ A_j$ give $a_i=0$ for
$j\ne i$. This proves existence. If $\partial_i\circ B=0$
for every $i$, the highest symbol of any nonzero $B$ would be
annihilated by multiplication by each $\xi_i$, an impossibility.
This proves uniqueness.
\end{proof}


\begin{proposition}[Geometric recovery from jet separation]
\label{prop:jet-spanning-module-recovery}
Let $S=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$ be
expanding, with $N\ge2$, and put
\[
 \begin{gathered}a=\min_i\log|\lambda_i|,\qquad b=\max_i\log|\lambda_i|,\\
d_0=\max\{2,\lfloor b/a\rfloor\},\qquad M=d_0+1.\end{gathered}
\]
Let $E_s\subset W_R$ be a holomorphic family of $S$-invariant
polynomial subspaces, preserved by the compressed returns
$T_{h,s}=P_RC_{h_s}$, with the usual relative uniqueness of the
moving centres. Assume
\[
 j^M_0:\mathbb C1\oplus E_s\longrightarrow
              \mathcal O_{\mathbb C^N,0}/\mathfrak m^{M+1}
 \quad\text{is surjective}.
\]
Suppose there are holomorphic isomorphisms $A_s:E_0\to E_s$,
with $A_0=I$, intertwining all compressed returns and the action of
$S$. Then there are weighted polynomial automorphisms $Q_s$ commuting
with $S$, and nonzero holomorphic scalars $c_s$, such that
\[
                         A_s=c_s^{-1}C_{Q_s}|_{E_0}.
\]
All first derivatives of $Q_sh_sQ_s^{-1}$ at $0$ are constant
in the parameter.
\end{proposition}

\begin{proof}
Every monomial whose multiplier equals one of the $\lambda_i$
has absolute weight at most $b$, hence degree at most
$\lfloor b/a\rfloor<M$. Since the jet map is $S$-equivariant
and surjective, $E_s$ contains each complete multiplier eigenspace
of this kind: no higher-degree monomial with that multiplier can
be invisible to the displayed jet map. In particular $E_s$ contains
all coordinate functions.

Set $Q_s^{(1)}=(A_sx_1,\ldots,A_sx_N)$. Each component has
the same positive multiplier weight as its coordinate. Its
linear part is invertible near the reference parameter. Ordering
the weights makes the nonlinear terms triangular, so successive
substitution gives a weighted polynomial inverse. The normalized spaces
$E'_s=C_{Q_s^{(1)}}^{-1}E_s$ still separate $M$-jets, since
$Q_s^{(1)}$ is a local coordinate change fixing $0$. They still
contain the complete coordinate-multiplier eigenspaces. The normalized
isomorphisms $B_s:E_0\to E'_s$ fix $x_i$. Put
\[
               D_s=\dot B_sB_s^{-1}:E'_s\longrightarrow W_R.
\]
For the normalized returns, differentiation of the intertwining identity
gives
\[
 \dot T_he=D_s(T_he)-T_h(D_se),\qquad e\in E'_s.
\]
Let $\Pi_{\rm c}$ project onto the complete coordinate-multiplier
spaces and define $\ell_D(f)=d(D_s\Pi_{\rm c}f)(0)$.
Extend $D_s$ to constants by $D_s1=0$.
The functional $\ell_D$ has order $m\leq\lfloor b/a\rfloor$
and annihilates constants and linear functions. Put $m=0$ if it vanishes
and $r=\max\{2,m\}$. First derivatives of the commutator give
\begin{equation}\label{eq:jet-separating-centre-locality}
 d(D_se)(a_h)Dh(0)
       =\ell_D(e\circ h)-d\bigl(de(h)\dot h\bigr)(0).
\end{equation}
Thus $d(D_se)(a_h)$ depends on the $r$-jet of $e$ at $a_h$.

Choose finitely many elements of $\mathbb C1\oplus E'_s$ whose
$r$-jets at $0$ form a basis. Their jet matrix remains invertible
on a neighbourhood of $0$. Interpolation therefore gives unique
differential operators $A_i$ of order at most $r$, with holomorphic
coefficients near $0$, agreeing with $\partial_iD_s$ on these
tests. Equation \eqref{eq:jet-separating-centre-locality} and
uniqueness of the small centres give
\[
                      \partial_iD_se=A_ie\qquad(e\in E'_s).
\]
The commutator $\partial_j\circ A_i-\partial_i\circ A_j$
has order at most $r+1\le M$ and annihilates
$\mathbb C1\oplus E'_s$. The $M$-jet surjectivity, which is
open near $0$, forces that differential operator to vanish identically.
The proof of Lemma~\ref{lem:closed-differential-operator-tuple}
works unchanged for holomorphic coefficients. It gives a unique
holomorphic differential operator $G$ of order at most $r-1$ with
$A_i=\partial_i\circ G$.

Testing on the same jet-separating space shows
$A_iC_S=\lambda_iC_SA_i$ as genuine differential-operator
identities. Uniqueness gives $GC_S=C_SG$. Its zeroth coefficient
is therefore constant, and every positive-order coefficient vanishes
at $0$; indeed these coefficients are weighted polynomials. Since
$(D_s-G)e$ is constant and has positive multiplier weight for each
weight eigenvector $e\in E'_s$, it vanishes. Hence $D_s=G$ on
$E'_s$.

Now $\ell_D(e)=d(Ge)(0)$ for every $e\in E'_s$. The difference
between these two functionals has order at most $r$, and
$\mathbb C1\oplus E'_s$ separates all $r$-jets. Consequently
\[
                         \ell_D(f)=d(Gf)(0)
\]
for every holomorphic germ $f$. The right side has order at most
$r-1$ because the positive-order coefficients of $G$ vanish at $0$.
If $m\ge2$ then $r=m$, a contradiction. Thus $m=0$, $G$ has
order at most one. Put $\mathcal E=\sum_i x_i\partial_i$,
the Euler vector field. The identities $D_sx_i=0$ give
\[
                         D_s=\gamma_s(1-\mathcal E)|_{E'_s}.
\]
Integration yields $B_se=c_s^{-1}e(c_sx)$, where
$\dot c_s/c_s=-\gamma_s$ and $c_0=1$. Composing with
$Q_s^{(1)}$ proves the proposition. All interpolations can be made
holomorphically in the parameter, and relative centre uniqueness is
used coefficient by coefficient by
Lemma~\ref{lem:relative-uniqueness-individual-germs}.
\end{proof}


\begin{proposition}[Recovery with a varying resonant normal form]
\label{prop:varying-normal-form-geometric-recovery}
Assume Proposition~\ref{prop:jet-spanning-module-recovery}. In addition,
let $\mathcal N_s$ be holomorphic expanding Poincar\'e--Dulac
polynomial automorphisms with the same semisimple part $S$, and assume
\[
 C_{\mathcal N_s}A_s=A_sC_{\mathcal N_0}\quad\hbox{on }E_0.
\]
The return collection is assumed closed under compositions which are
defined near $0$, and under left composition by sufficiently large
negative powers of $\mathcal N_s$. Then the weighted polynomial
automorphisms supplied by that proposition satisfy
\[
 Q_s\mathcal N_sQ_s^{-1}=\mathcal N_0,
 \qquad Q_sh_sQ_s^{-1}=h_0
\]
as parameter germs for every marked actual return $h$.
\end{proposition}

\begin{proof}
The preceding geometric-recovery proof uses only the fixed semisimple
operator $S$, and supplies $A_s=c_s^{-1}C_{Q_s}|_{E_0}$.
Since $\mathcal N_s$ commutes with $S$, its pullback preserves each
multiplier space and there is no truncation in its action on $W_R$.
Substitution in the additional intertwining identity gives
\[
 e\circ Q_s\circ\mathcal N_s
       =e\circ\mathcal N_0\circ Q_s\qquad(e\in E_0).
\]
Every coordinate function belongs to $E_0$. Testing these functions
proves the first assertion. Henceforth use the coordinates $Q_s$;
thus $\mathcal N_s=\mathcal N_0$, $E_s=E_0=:E$, and every compressed
return on $E$ is constant.

Choose a basis $e_1,\ldots,e_t$ of $E$ and define
\[
 \Phi(z)=\bigl(de_1(z),\ldots,de_t(z)\bigr).
\]
The surjection onto $2$-jets at $0$ implies that $d\Phi_0$ is
injective: if $v\ne0$, a quadratic polynomial with nonzero contraction
of its Hessian against $v$ has the same $2$-jet as an element of
$\mathbb C1\oplus E$. Thus $\Phi$ is an embedding on a neighbourhood
$U$ of $0$, after shrinking $U$.

Let $k$ be a return with reference centre $a_k(0)\in U$. Constancy
of the linear Taylor coefficient of $T_{k,s}e$ gives
\[
 de(a_k(s))Dk_s(0)=de(a_k(0))Dk_0(0)\qquad(e\in E).
\]
Testing coordinate functions first gives $Dk_s(0)=Dk_0(0)$.
This matrix is invertible, so $\Phi(a_k(s))=\Phi(a_k(0))$.
Injectivity on $U$ proves $a_k(s)=a_k(0)$ as a parameter germ.

For an arbitrary return $h$ and a sufficiently small-centred return
$k$, the same first-derivative constancy for $h\circ k$ gives
\[
 Dh_s(a_k(0))Dk_0(0)=D(h_0\circ k_0)(0).
\]
Consequently $Dh_s(a_k(0))=Dh_0(a_k(0))$. Uniqueness of the small
reference centres, applied coefficient by coefficient in the
parameter, gives $Dh_s=Dh_0$. Every small-centred return therefore
has both constant derivative and constant value at $0$, and is
itself constant. Finally choose $m$ with
$\mathcal N_0^{-m}(a_h(0))\in U$. The marked return
$\mathcal N_0^{-m}\circ h_s$ is small-centred and hence constant.
Since $\mathcal N_0$ is constant, so is $h_s$.
\end{proof}


\section{Continuation and algebraization in arbitrary dimension}\label{sec:continuation-rationality}
\label{sec:nd-globalization}

Throughout this section $X$ is a normal complex projective variety of
dimension $N$, and $f:X\to X$ is $q$-polarized. Put $d=q^N$.
We use the inverse-branch and backward-equidistribution theorems of
\cite[Theorem~1.2 and Proposition~3.1]{DNT15}. On a resolution the
meromorphic lift has dynamical degrees $1,q,\ldots,q^N$; in particular
its topological degree dominates the others. The set $\Sigma_0$ in
the construction of \cite[Section~3]{DNT15} may be enlarged by the
exceptional divisors of the resolution. The resulting branches avoid
them at every intermediate step and descend to the smooth locus of
$X$. Thus backward equidistribution holds outside a countable union
of proper algebraic sets. Also, for every $\varepsilon>0$ there is
a proper algebraic set $A$, containing $\operatorname{Sing}X$,
such that every point of $X\setminus A$ has a fixed ball admitting
at least $(1-\varepsilon)d^m$ shrinking unramified inverse branches
of $f^m$ for all sufficiently large $m$. The set $A$ is obtained
from a positive Lelong-number locus of the inverse-branch current;
it is algebraic by Siu's theorem and properness of the resolution.

\begin{proposition}[Majority continuation]
\label{prop:nd-majority}
Let $O\subset X$ be a forward-invariant analytic open set of full
equilibrium measure. Suppose a geometric datum is defined near
$J_f\cap O\cap X^{\mathrm{reg}}$ and is invariant there as a germ.
The datum may be a holomorphic section with holomorphic affine
forward transport, a meromorphic distribution of any fixed rank,
or a meromorphic affine connection. It extends uniquely to
$X^{\mathrm{reg}}\setminus E$ for a proper totally invariant
algebraic set $E$. The extension is holomorphic in the first case
and meromorphic in the other two cases.
\end{proposition}

\begin{proof}
Write $U_0\subset O\cap X^{\mathrm{reg}}$ for the original
domain. The ramification current
\[
                 R=\sum_{j\ge1}d^{-j}(f^j)_*[R_f]
\]
has finite mass. Indeed its $j$th summand has mass
$O(q^{(N-1)j}/q^{Nj})=O(q^{-j})$ against an ample polarization.
The ramification formula gives
\[
                 d^{-m}(f^m)_*[R_{f^m}]\le R.
\]
Choose a generic $a\in J_f\cap O$ with backward equidistribution
and a generic target $x$. A sufficiently ample smooth complete-
intersection curve $C$ through $a,x$ may be chosen to avoid the
forward images of $\operatorname{Sing}X$ and every codimension-two
bad ramification stratum of every iterate, and to meet all remaining
branch divisors transversally. These are countably many dense open
conditions in the parameter space of complete intersections.
Consequently $(f^m)^{-1}C\to C$ is a branched covering between
smooth curves. For $N=1$, take $C=X$.

The restriction $R|_C$ is a finite atomic measure, with neither
endpoint an atom. Choose an arc from $a$ to $x$ and simply connected
strips $\Omega_0\Subset\Omega\subset C$ around it such that
$(R|_C)(\Omega)<\varepsilon$. One obtains such a strip by avoiding
finitely many atoms carrying all but $\varepsilon$ of the mass
and then shrinking around the arc. The total ramification above
$\Omega$ is at most $\varepsilon d^m$. A connected covering of a
disk of degree $e\ge2$ has ramification at least $e-1$, and
$e\le2(e-1)$. Hence at least $(1-2\varepsilon)d^m$ sheets are
unramified sheets of degree one.

The total area of $(f^m)^{-1}C$ is $O(q^{(N-1)m})$.
For example, discarding the sheets with area greater than
$q^{-m/2}$ removes only $O(q^{(N-1/2)m})=o(d^m)$ sheets.
The small-area diameter estimate on the smaller strip, along a
finite chain of disks, makes the diameters of the retained sheets
tend to zero; see \cite[Lemma~3.4]{DNT15}.
Backward equidistribution at $a$, together with a compact core of
$U_0$ of arbitrarily large equilibrium measure, shows that an
arbitrarily large proportion of these branches maps all of
$\Omega_0$ into $U_0$. At each fixed depth they extend to a common
ambient tube around the arc.

Transport the datum along each branch. At $a$ the transported germs
coincide: their source points and intermediate points belong to
$J_f\cap O$, where the invariance identity holds. The identity
theorem identifies their transports throughout the tube. If two
constructions end at $x$ and each retains more than half of the
$d^m$ germs at one sufficiently large depth, they share an inverse
germ. They therefore give the same germ at $x$, independently of
the arc and the starting point.

To obtain a fixed domain, take an inverse-branch ball
$B\subset X\setminus A$. Backward equidistribution at a generic
point and shrinking of the branches show that an arbitrarily large
proportion of its branches maps into $U_0$. At a generic point
$b_0\in B$, more than half of these branches belong to the equality
class furnished by the arc construction. The identity theorem
identifies their transports on all of $B$. It is the unique class
containing more than half of the germs. Comparing with the arc
construction at any other generic point gives the same class.
The resulting data glue on a Zariski open $W\subset X\setminus A$
and satisfy the invariance identity wherever source and target
belong to $W$.

Put $U=\bigcup_{m\ge0}f^m(W)$ and $E=X\setminus U$.
Finite maps to a normal variety are open, so $U$ is Zariski open.
We have $f^{-1}E\subset E$. The descending closed sets
$f^{-m}E$ stabilize, and applying a suitable power of the surjective
map $f$ gives $f^{-1}E=E$.

For $x\in U\cap X^{\mathrm{reg}}$, use a finite proper local
piece of an iterate from a relatively compact subset of $W$ onto
a neighborhood of $x$. Normalized trace extends holomorphic affine
transports across the discriminant. For a connection, trace the
meromorphic Christoffel coefficients; their affine transformation
law commutes with normalized trace. For a rank-$r$ distribution,
use its Pl\"ucker line in $\bigwedge^rT_X$. Trace a local
meromorphic generator with scalar weights chosen by interpolation
on one unramified fiber so that the trace is nonzero. All terms
are generically collinear, so the resulting line is the transported
Pl\"ucker line. Decomposability holds by continuation. These
extensions agree on $W$ and therefore glue on $U\cap X^{\mathrm{reg}}$.
\end{proof}

\begin{lemma}[Depth at a normal boundary]
\label{lem:nd-depth}
Let $E$ be a nonempty reduced totally invariant divisor of $X$.
There are an iterate $F$, an ideal norm $\eta$ of $E$, and a
constant $c$ such that $\rho=-\log\eta\ge1$ is proper on
$X\setminus E$ and
\[
                         \rho\circ F\ge2\rho-c.
\]
On a log resolution of the ideal of $E$, the function $\rho$ is
smooth off the boundary and $dd^c\rho$ extends as a smooth real
form. At a general point of $E$, it equals
$-\log|z_1|+O(1)$.
\end{lemma}

\begin{proof}
Pass to an iterate $F_0$ fixing every prime component $D$ of $E$,
and let $Q$ be its polarization weight. Write $F_0^*D=m_DD$.
The finite projection formula gives
\[
 Q^{N-1}(F_0^*D\cdot H^{N-1})
           =Q^N(D\cdot H^{N-1}),
\]
so $m_D=Q$. For $I=\mathcal I_E$, divisorial valuations on the
normal variety give
\[
                   (F_0^m)^*I\subset I^{(Q^m)}
                                    =\mathcal O_X(-Q^mE).
\]

The symbolic and ordinary topologies of this fixed ideal are
cofinal, as follows from the next ideal-sheaf calculation.
On an affine normal chart, write $I=\bigcap_iP_i$ with $P_i$
the prime component ideals. The rings are excellent, so the
completions of their normal localizations are normal local domains.
For a prime $P_i$ and a strictly larger prime $Q'$, the quotient
of $\widehat{R_{Q'}}$ by $P_i\widehat{R_{Q'}}$ has positive
dimension. Thus Schenzel's criterion
\cite[Theorem~1]{Schenzel85} gives, for each $a$, an integer $b$
such that $P_i^{(b)}\subset P_i^a$.
Write $I^2=\bigcap_jQ_j$ as a primary decomposition and choose
$a$ with $(\sqrt{Q_j})^a\subset Q_j$ for every $j$.
Each $\sqrt{Q_j}$ contains some $P_i$, and therefore, for large $b$,
\[
             I^{(b)}=\bigcap_iP_i^{(b)}
                    \subset\bigcap_iP_i^a\subset I^2.
\]
A finite affine cover makes this choice uniform. Hence an iterate
$F$ satisfies $F^*I\subset I^2$.

Combine squared norms of finite local generating tuples of $I$
using a smooth partition of unity. Their square root is an ideal
norm $\eta$, and the containment gives
$\eta\circ F\le C\eta^2$. On a log resolution the pulled-back
generators have a common divisorial factor and no common zero
after that factor is removed. The norm is therefore the norm of
the resolved defining section times a positive smooth factor.
Rescaling $\eta$ gives all the assertions.
\end{proof}

\begin{proposition}[Rationality of projective cocycles]
\label{prop:nd-projective-cocycle}
Let $E\subset X$ be a proper totally invariant algebraic set.
If a single-valued meromorphic map
$\Phi:X\setminus E\dashrightarrow\mathbb P^M$ satisfies
\[
                   \Phi\circ f=[B]\Phi,
             \qquad B\in\operatorname{GL}_{M+1}(\mathbb C(X)),
\]
then $\Phi$ is rational.
\end{proposition}

\begin{proof}
Meromorphic Hartogs extension on the normal variety removes the
codimension-at-least-two part of $E$. Its reduced divisorial part
is again totally invariant. If that part is empty, projective GAGA
applies. Otherwise use Lemma~\ref{lem:nd-depth} and replace $F$
by a further iterate so that its polarization weight $Q$ is greater
than two. Let $\omega$ be a smooth curvature form of an ample
polarization, and put
\[
                         \omega_j=Q^{-j}(F^j)^*\omega.
\]
On a resolution $\pi:Y\to X$ of the boundary ideal these are
smooth semipositive forms, and
\[
                         \omega_j=\omega_0+dd^cu_j,
                 \qquad \sup_j\|u_j\|_\infty<\infty.
\]
Indeed the potentials are partial sums
$\sum_{i<j}Q^{-i}\psi\circ F^i$ pulled back to $Y$, where
$Q^{-1}F^*\omega=\omega+dd^c\psi$.
We suppress $\pi^*$ in the integrals; no lift of $F$ to $Y$ is used.

For a positive closed $(1,1)$-current $S$ on $X\setminus E$, define
\[
 a_{j,S}(t)=\int_{\{\rho<t\}}S\wedge\omega_j^{N-1},
 \qquad T_{j,S}(s)=\int_0^s a_{j,S}(t)\,dt.
\]
Put $\vartheta=dd^c\rho$, a smooth form on $Y$. At regular levels,
integration by parts gives
\[
 T_{j,dd^cu}(s)=\int u\,d\beta_{j,s}
     -\int_{\{\rho<s\}}u\vartheta\wedge\omega_j^{N-1},
 \quad
 \beta_{j,s}=\delta_{\{\rho=s\}}d\rho\wedge d^c\rho
                                     \wedge\omega_j^{N-1}.
\]
The positive measures $\beta_{j,s}$ have uniformly bounded mass:
Stokes expresses that mass as the integral of
$\vartheta\wedge\omega_j^{N-1}$ over the sublevel, and
$\vartheta$ is bounded by a fixed K\"ahler form on $Y$.

For a nonzero algebraic section $\sigma$ of an ample multiple,
choose $g=C-\log\|\sigma\|\ge0$. For a K\"ahler form $\omega_Y$
on $Y$ one has
\[
                   \sup_j\int_Y g\,\omega_Y
                                      \wedge\omega_j^{N-1}<\infty.
\]
To see this, use the telescoping identity
\[
 \omega_j^{N-1}-\omega_0^{N-1}
   =dd^cu_j\wedge\sum_{a=0}^{N-2}
                   \omega_j^a\wedge\omega_0^{N-2-a}
\]
and integrate by parts. Since
$dd^cg=m\omega_0-[\operatorname{div}\sigma]$, boundedness of
$u_j$ bounds all the resulting positive mixed intersection masses.
For $N=1$ the assertion is immediate. The preceding Stokes identity,
applied to $g$, also gives
$\int g\,d\beta_{j,s}\le C(s+1)$. Consequently
\[
              |T_{j,dd^cu}(s)|\le C(s+1)
                       \quad\text{if }|u|\le g,
\]
uniformly in $j$. These identities may be obtained by truncation
and then used at almost every level.

Take $S=\Phi^*\omega_{\mathrm{FS}}$, defined by the meromorphic
graph. For local reduced homogeneous lifts $P$ of $\Phi$, the
function $u=\log(\|BP\|/\|P\|)$ is independent of the lift.
Bounds for $B$ and $B^{-1}$ give $|u|\le C-\log\|\sigma\|$
for a suitable algebraic section. The curvature identity, allowing
the common factors of $BP$, gives
\[
                        F^*S\le S+dd^cu+[P_B],
\]
where $P_B$ is a fixed effective algebraic divisor. Hence
$T_{j,F^*S}(s)\le T_{j,S}(s)+C(s+1)$ uniformly in $j$.

Now $F^*\omega_j=Q\omega_{j+1}$, whereas $\deg F=Q^N$.
Together with
$F^{-1}\{\rho<2t-c\}\subset\{\rho<t\}$, finite change of
variables therefore gives
\[
 T_{j,S}(R)\le\frac2Q
        T_{j+1,S}\bigl((R+c)/2\bigr)+C(R+1).
\]
The factor is $2/Q$ in every dimension: the $N-1$ source forms
contribute $Q^{N-1}$ and finite degree contributes $Q^N$.
On each compact sublevel, the Chern--Levine--Nirenberg estimate
and the uniformly bounded potentials give a bound independent of
$j$. Iterating the recurrence proves $T_{0,S}(R)=O(R)$.
Monotonicity of $a_{0,S}$ implies
\[
                   \int_{X\setminus E}S\wedge\omega^{N-1}<\infty.
\]

Average linear projections $\mathbb P^M\dashrightarrow\mathbb P^1$
over the unitary group. Their averaged curvature is
$\omega_{\mathrm{FS}}$, so almost every scalar projection of
$\Phi$ has finite spherical mass. Its graph has finite volume:
on a graph in $X\times\mathbb P^1$ the volume form has only
the terms
\[
                  \omega^N+N\omega^{N-1}\wedge\omega_{\mathrm{FS}},
\]
because $\omega_{\mathrm{FS}}^2=0$ on $\mathbb P^1$.
Bishop's finite-volume extension theorem \cite{Bishop64} extends
the graph across $E\times\mathbb P^1$. A generic unitary coordinate
system makes all finitely many coordinate ratios extend
simultaneously. They recover a meromorphic map on $X$, which is
rational by projective GAGA.
\end{proof}

\begin{corollary}\label{cor:nd-rational-data}
The distributions of every rank and the affine connections obtained
in Proposition~\ref{prop:nd-majority} are rational. Integrability,
flatness, and zero torsion persist whenever they hold initially.
\end{corollary}

\begin{proof}
Use a rational tangent frame. A rank-$r$ distribution is represented
by its Pl\"ucker line, with rational cocycle $\bigwedge^rDf$.
The Christoffel coefficients of an affine connection satisfy an
invertible rational affine cocycle; homogenization gives a rational
projective cocycle. Apply Proposition~\ref{prop:nd-projective-cocycle}.
All the stated differential and Pl\"ucker identities extend by
the identity theorem.
\end{proof}

\begin{proposition}[Horizontal fields]
\label{prop:nd-horizontal}
Let a projective algebraic curve family of normal $N$-fold systems
admit a holomorphic horizontal field near its Julia data. Suppose
its fiberwise continuation is single-valued off proper totally
invariant algebraic sets and satisfies the forward differential
transport identity. Then the family is locally holomorphically
trivial and is algebraically isotrivial after finite parameter
base change and restriction.
\end{proposition}

\begin{proof}
Embed the family in a fixed relative projective space. Extend the
ambient components of the field across excluded sets of codimension
at least two on each normal fiber by Hartogs. Tangency and the
transport identity persist. If a reduced divisorial boundary $E$
remains, it is totally invariant. Use the depth function in
Lemma~\ref{lem:nd-depth}.

On a compact fiber the differential of $F$ and the deformation
term of the affine transport have bounded ambient norms. Thus
\[
 M(t):=\sup_{\rho\le t}(\|W\|+1)
          \le C M\bigl((t+c)/2\bigr).
\]
Indeed every target of depth at most $t$ has a preimage of depth
at most $(t+c)/2$. Iteration gives $M(t)=O((1+t)^a)$.
Near a general point of $E$, the growth is polynomial in
$\log(1/|z_1|)$. Every negative Laurent coefficient vanishes,
since its bound is $Cr^k(1+\log(1/r))^a\to0$.
The field therefore extends holomorphically over the general
boundary point. Normal Hartogs extends its ambient components
over the remaining codimension-at-least-two set. The resulting
field remains tangent and horizontal.

After shrinking the parameter curve, cohomology and base change
make the pushforward of the restricted ambient tangent bundle
locally free. Finitely many marked points in the initial common
domain separate the space of its sections on a reference fiber.
A nonvanishing evaluation minor stays nonzero nearby. Solving this
finite linear system shows that the extended fields depend
holomorphically on the parameter. Their flows exist for a uniform
time on a compact fiber, preserve the defining ideal, and commute
with the dynamics. They give local holomorphic trivializations.
The graphs lie in one finite-type relative Hilbert scheme, and
the conjugacy identity is algebraic there. A dominating component
has a closed point over a finite extension of the parameter
function field. This gives algebraic isotriviality after restriction.
\end{proof}

\begin{proposition}[Descent from a normal-form source]
\label{prop:nd-source-descent}
Let $F=f^\ell$ and let $\Pi:\mathbb C^N\to X$ be a Poincar\'e
map at a smooth repelling Julia point $p$, with
\[
                         F\Pi=\Pi\mathcal N,
\]
where $\mathcal N$ is an expanding polynomial normal-form
automorphism. Suppose the good unramified backward centers of $p$
are holomorphic uniqueness sets in the small coordinate neighborhoods
meeting $J_f$, also after deletion of a proper analytic subset.

A regular $\mathcal N$-invariant source distribution, preserved by
all actual inverse returns, descends to a rational $F$-invariant
distribution on $X$. Its rank is arbitrary. If $\mathcal N$ is
linear and all the actual returns are affine, the standard source
connection descends to a rational $F$-invariant flat torsion-free
connection. A holomorphic family of polynomial gauges commuting
with a fixed $\mathcal N$ and identifying all actual returns gives
isotriviality of the family.
\end{proposition}

\begin{proof}
If $\sigma$ is the original injective chart, then near any source
point
\[
                    \Pi=F^m\sigma\mathcal N^{-m}
\]
for sufficiently large $m$. Hence $\Pi$ is locally finite and open,
including over singular target points. Its image $O$ is
$F$-forward invariant and contains a neighborhood of $p$. Ergodicity
of the equilibrium measure for $F$ gives $\mu_f(O)=1$.

Let $g$ be a local deck germ from the original chart to a regular
lift of $p$. Choose $m$ so that $\mathcal N^{-m}g$ takes values
in the original chart, and then choose $\ell$ so that
$h=\mathcal N^{-m}g\mathcal N^{-\ell}$ is defined on the fixed
coordinate ball and maps it into itself. The identity
$F\Pi=\Pi\mathcal N$ makes $h$ an actual inverse return.
Thus $g=\mathcal N^m h\mathcal N^\ell$. Invariance under the
normal form and the actual returns shows that $g$ preserves
the given datum. Let two regular inverse branches of $\Pi$ be
defined over a small smooth ball meeting $J_f$. At a good backward
center $b$ choose $k$ such that $F^k(b)=p$ and $DF^k_b$ is
invertible. Applying $\mathcal N^k$ to both lifts reduces their
comparison to a deck germ over $p$. Pullback by $DF^k_b$ shows
that they give the same datum at $b$. The uniqueness hypothesis,
applied to local Pl\"ucker equations or to connection coefficients,
identifies the two germs on the ball.

The same conclusion holds for finite local pieces over a branch
locus. Indeed, a nonzero holomorphic function vanishing on the
lifted center set would have a nonzero finite norm vanishing on the
center set downstairs. This contradicts uniqueness. Weighted trace
for the Pl\"ucker line, or normalized trace for the connection,
extends the common datum meromorphically across the discriminant.
Proposition~\ref{prop:nd-majority} and
Corollary~\ref{cor:nd-rational-data}, applied to $F$, finish the
descent. Integrability, flatness, and zero torsion are identities
and thus persist.

For a conjugating gauge $\Theta_t$, write $R_t=\Pi_t\Theta_t$.
The actual-return identities identify the branches of $R_t$ over
the good centers. The implicit function theorem for the unramified
forward return, followed by the same uniqueness argument, gives
holomorphic conjugacy germs near the Julia data. Their parameter
derivatives are horizontal fields with holomorphic affine forward
transport. Proposition~\ref{prop:nd-majority} followed by
Proposition~\ref{prop:nd-horizontal} gives isotriviality for
$F_t=f_t^\ell$.

After this trivialization, $F_t=F$ is constant. If $x$ is periodic
for $F$, then $f_t(x)$ lies in the finite set
$\operatorname{Fix}(F^m)$ for a suitable $m$, and hence is constant
in $t$. These fixed-point sets are finite because a positive-
dimensional fixed subvariety contradicts polarization. Periodic
points are Zariski dense by repeller equidistribution. Therefore
$f_t$ itself is constant.
\end{proof}


\section{General symbols and local return geometry}

\subsection{Expansion algebraizes invariant formal geometry}

\begin{lemma}[An invariant formal distribution is polynomial]
\label{lem:expanding-formal-distribution-polynomial}
Let $F$ be an expanding holomorphic germ at $0\in\mathbb C^N$ in
convergent Poincar\'e--Dulac normal form. Write
\[
 F=S\circ U=U\circ S,
 \qquad S=\operatorname{diag}(\lambda_1,\ldots,\lambda_N),
 \qquad |\lambda_i|>1,
\]
where the linear part of $U$ is unipotent and all its monomials
are resonant with $S$. Let $\widehat{\mathcal D}$ be a regular
formal distribution at $0$, invariant under $F$.
Then $\widehat{\mathcal D}$ is the completion of an algebraic
regular distribution on $\mathbb A^N$. If it is formally integrable,
that algebraic distribution is integrable.
\end{lemma}

\begin{proof}
For every $k$, the $k$-jet of a regular distribution is a point of
an algebraic jet space on which the algebraic group of
$(k+1)$-jets of invertible germs acts. Its stabilizer is a closed
algebraic subgroup. Since it contains the jet of $F$, it contains
its semisimple Jordan component. In Poincar\'e--Dulac coordinates
this component is the jet of $S$: the commuting jet of $U$ is
unipotent, as follows either from its action on truncated power
series or from the degree filtration. Thus every finite jet of
$\widehat{\mathcal D}$ is $S$-invariant, and so is the formal
 distribution itself.

Put $V=\widehat{\mathcal D}_0$ and choose an $S$-invariant
complement $W$. A regular formal distribution with this value is
the graph of a formal matrix
\[
                    B(x):V\longrightarrow W,\qquad B(0)=0.
\]
The invariance identity is
\[
                    B(Sx)=S_W B(x)S_V^{-1}.
\]
Use eigenbases on $V$ and $W$. If a coefficient of $x^\alpha$
in the matrix entry from the $\lambda_i$-eigenspace to the
$\lambda_j$-eigenspace is nonzero, then
\[
                         \lambda^\alpha=\lambda_j/\lambda_i.
\]
Let $a=\min_i\log|\lambda_i|>0$ and
$b=\max_i\log|\lambda_i|$. Every such coefficient satisfies
$|\alpha|a\le b-a$. Only finitely many multi-indices occur.
Consequently $B$ is polynomial and its graph defines an algebraic
regular distribution on all of $\mathbb A^N$.

Formal integrability says that the brackets of the polynomial
basis fields of this graph lie in their span to every order at
$0$. The corresponding polynomial Frobenius identities therefore
vanish identically. Hence the distribution is integrable globally.
\end{proof}

\begin{lemma}[Finite invariant formal Lie algebras converge]
\label{lem:expanding-finite-formal-Lie-polynomial}
Let $F$ and $S$ be as in
Lemma~\ref{lem:expanding-formal-distribution-polynomial}.
Let $\mathfrak g$ be a finite-dimensional complex space of formal
vector fields, invariant under $F_*$ and $F_*^{-1}$.
Then every element of $\mathfrak g$ is polynomial.
In particular, if $\mathfrak g$ is a transitive formal Lie algebra,
it is a locally transitive Lie algebra of holomorphic vector fields.
\end{lemma}

\begin{proof}
For every $k$, the space of $k$-jets $j^k\mathfrak g$ is invariant
under the action of the jet of $F$. The stabilizer of this subspace
is algebraic, so it is invariant under the semisimple component,
namely the jet of $S$. A finite-dimensional subspace of formal
series is closed for the adic topology: choose $k_0$ for which its
$k_0$-jet map is injective, and use this map to identify every
compatible inverse system of its jets with a single element of
the subspace. Thus $S_*\mathfrak g=\mathfrak g$.

The induced action on $\mathfrak g$ is diagonalizable, since it
embeds by a sufficiently long jet map into the diagonal action
on polynomial vector fields. Let $Z$ be an eigenvector, with
nonzero eigenvalue $\mu$. A monomial
$x^\alpha\partial/\partial x_j$ can occur in $Z$ only when
\[
                         \lambda_j/\lambda^\alpha=\mu.
\]
Taking absolute values gives
$|\alpha|a\le b-\log|\mu|$. Thus only finitely many monomials
occur in every eigenvector, proving the assertion.
\end{proof}
\begin{lemma}[Equivariant polynomial Frobenius coordinates]
\label{lem:expanding-polynomial-frobenius}
Let $S=\operatorname{diag}(\lambda_1,\ldots,\lambda_N)$,
$|\lambda_i|>1$, preserve a regular integrable holomorphic
distribution $\mathcal D$ near $0$. There is a polynomial
automorphism $Q$ commuting with $S$, fixing $0$, and tangent to
the identity after an $S$-equivariant linear choice of coordinates,
such that $Q_*\mathcal D$ is a constant coordinate distribution.
\end{lemma}

\begin{proof}
Choose an $S$-invariant splitting
$\mathbb C^N=V\oplus W$, with $V=\mathcal D_0$.
The graph calculation in
Lemma~\ref{lem:expanding-formal-distribution-polynomial} shows
that $\mathcal D$ is polynomial and is preserved by the algebraic
diagonal group $T=\overline{\langle S\rangle}^{\mathrm{Zar}}$.
Let $K\subset T$ be its maximal compact subgroup, including the
finitely many components. It acts diagonally and unitarily, and
its Zariski closure is $T$.

Holomorphic Frobenius supplies first integrals
$p_j=x_j+O(|x|^2)$, one for every coordinate $x_j$ of $W$.
Take their common domain to be a sufficiently small $K$-invariant
ball, and let $\chi_j:K\to\mathbb C^*$ be the character on $x_j$.
Put
\[
            P_j(x)=\int_K\chi_j(k)^{-1}p_j(kx)\,dk.
\]
Every integrand is constant along $\mathcal D$, so the same is
true of $P_j$, and $dP_j(0)=dx_j$. Character orthogonality shows
that a Taylor monomial $x^\alpha$ in $P_j$ satisfies
$\lambda^\alpha=\lambda_j$. There are finitely many such monomials
because $|\lambda_i|>1$. Thus $P_j$ is a polynomial and
$P_j\circ S=\lambda_jP_j$.

Set $Q(x)=(x_V,(P_j(x))_{j\in W})$. Its linear part is the
identity. Every nonlinear monomial in its $j$th coordinate uses
variables of strictly smaller absolute multiplier weight. Solving
successively in these weights gives a polynomial inverse. Hence
$Q$ is a polynomial automorphism commuting with $S$. Its last
coordinates are independent first integrals of $\mathcal D$,
so it sends $\mathcal D$ to the constant distribution $V$.
\end{proof}


\subsection{Scalar jet equations exclude primitive infinite symmetry}

\begin{lemma}[The highest scalar jet detects a leading vector field]
\label{lem:scalar-jet-leading-vector}
Let $M$ be a complex manifold, let $p\in M$, and let
$L\subset J^d(\mathcal O_M)$ be a holomorphic subbundle, where $d\ge2$.
Suppose that its projection onto $J^{d-1}(\mathcal O_M)$ is surjective.
Write
\[
 Q_p=L_p\cap\operatorname{Sym}^dT_p^*M.
\]
Let $V$ be an infinitesimal local symmetry of $L$ whose coefficients
vanish to order at least $d$ at $p$, and denote by
$B_d\in T_pM\otimes\operatorname{Sym}^dT_p^*M$ its degree-$d$ term.
Then
\[
                 \ell\circ B_d\in Q_p
                 \qquad(\ell\in T_p^*M).
\]
The same assertion holds for formal symmetries and formal jet bundles.
\end{lemma}

\begin{proof}
Choose local coordinates centered at $p$. The local flow $\phi_t$ of
$V$ fixes $p$ and has $(d-1)$-jet equal to the identity. For a jet
$j_p^d u\in L_p$, invariance gives $j_p^d(u\circ\phi_t)\in L_p$.
Differentiation at $t=0$ gives $j_p^d(Vu)\in L_p$. Its terms of
orders less than $d$ vanish and its degree-$d$ term is
$du(p)\circ B_d$. Every covector occurs as $du(p)$, by the
surjectivity onto the lower jets. This proves the assertion.
The computation involves finitely many coefficients and is unchanged
in the formal category.
\end{proof}

\begin{proposition}[Exclusion of the contact pseudogroup]
\label{prop:contact-scalar-jet-exclusion}
Let $M$ have dimension $2m+1$, where $m\ge1$, and let
$L\subset J^d(\mathcal O_M)$ be as in
Lemma~\ref{lem:scalar-jet-leading-vector}. If $L$ is preserved by all
local contact transformations for a contact structure on $M$, then
$L=J^d(\mathcal O_M)$. The conclusion holds formally as well.
\end{proposition}

\begin{proof}
Use contact Darboux coordinates
$(q_1,\ldots,q_m,p_1,\ldots,p_m,z)$ centered at the point in question,
with contact form
\[
                 \alpha=dz-\sum_{i=1}^m p_i\,dq_i.
\]
For a polynomial $H$, its contact Hamiltonian vector field, with
$\alpha(X_H)=H$, is
\[
 X_H=-\sum_iH_{p_i}\frac{\partial}{\partial q_i}
       +\sum_i(H_{q_i}+p_iH_z)\frac{\partial}{\partial p_i}
       +\left(H-\sum_i p_iH_{p_i}\right)
                         \frac{\partial}{\partial z}.
\]
Indeed, $\mathcal L_{X_H}\alpha=H_z\alpha$.
Let $P$ be an arbitrary homogeneous polynomial of degree $d$ in
all $2m+1$ variables. Choose a homogeneous polynomial $H$ of degree
$d+1$ with $H_{p_1}=-P$. Then $X_H$ vanishes to order at least $d$,
and the $q_1$-component of its degree-$d$ term is $P$.
Lemma~\ref{lem:scalar-jet-leading-vector}, applied with
$\ell=dq_1$, gives $P\in Q_p$. Thus
$Q_p=\operatorname{Sym}^dT_p^*M$. Surjectivity onto the lower jets
now gives $L_p=J_p^d(\mathcal O_M)$ at every point.
The formal argument uses the same polynomial Hamiltonians.
\end{proof}

\begin{proposition}[A primitive scalar-jet symmetry is of finite type]
\label{prop:primitive-scalar-jet-finite}
Let $\Gamma$ be a regular transitive complex analytic Lie pseudogroup
on a connected manifold $M$, generated locally by the flows of its
infinitesimal transformations. Suppose that $\Gamma$ preserves a
proper holomorphic subbundle
\[
       L\subsetneq J^d(\mathcal O_M),\qquad d\ge2,
\]
whose projection onto $J^{d-1}(\mathcal O_M)$ is surjective.
If $\Gamma$ is primitive, then it is of finite type.
Consequently an infinite-type $\Gamma$ as above preserves a regular
foliation of rank strictly between $0$ and $\dim M$.
\end{proposition}

\begin{proof}
The Cartan classification, in its complex analytic form proved by
Guillemin--Quillen--Sternberg \cite{GQS66}, lists six primitive
infinite-type pseudogroups: all local diffeomorphisms; those preserving
a volume form; those preserving a volume form up to a constant;
those preserving a symplectic form; those preserving a symplectic
form up to a constant; and the contact pseudogroup.

Put $Q=L\cap\operatorname{Sym}^dT^*M$. The first five cases have
linear isotropy containing $\operatorname{SL}(T_pM)$ or a standard
symplectic group. Each positive symmetric power of the standard dual
representation is irreducible. Hence their invariant subspace $Q_p$
is either zero or all of $\operatorname{Sym}^dT_p^*M$. The second
possibility contradicts the properness of $L$.

The first possibility is also excluded. In dimension at least two,
a volume-preserving infinitesimal symmetry such as
$x_2^d\partial/\partial x_1$ has a nonzero degree-$d$ term.
In the symplectic cases, a nonconstant homogeneous Hamiltonian of
degree $d+1$ gives such a symmetry. For the full pseudogroup in
dimension one use $x^d\partial/\partial x$. The leading-vector
lemma gives a nonzero element of $Q_p$ in each case. The one-dimensional
volume-preserving and constant-volume-scaling pseudogroups are
already of finite type.

The contact case is excluded by
Proposition~\ref{prop:contact-scalar-jet-exclusion}. Thus $\Gamma$
cannot be primitive and of infinite type. The last assertion is the
negation of primitivity for a transitive pseudogroup: an imprimitive
pseudogroup preserves a nontrivial local foliation.
\end{proof}


\subsection{The regular symmetry pseudogroup of a rational jet equation}

\begin{proposition}[Return centres force regular transitive symmetry]
\label{prop:rational-jet-regular-transitive}
Let $D\subset\mathbb C^N$ be a connected neighbourhood of $0$,
and let $L\subset J^d(\mathcal O_D)$ be a holomorphic subbundle
which is the restriction of a rational subbundle on $\mathbb A^N$.
Let $\mathcal H$ be local biholomorphisms preserving $L$, and
suppose that their centres are a holomorphic uniqueness set in
every sufficiently small neighbourhood of $0$.
Then the full local symmetry pseudogroup of $L$ is regular and
its infinitesimal transformations are locally transitive near $0$.
Its finite-order isotropy may be chosen to be an algebraic group.
\end{proposition}

\begin{proof}
On a Zariski open set where $L$ is an algebraic subbundle,
its invariance equations define an algebraic subgroupoid of the
groupoid of invertible $d$-jets. Their radical differential closure
is an algebraic $\mathcal D$-groupoid; see
\cite[Theorem~1.12]{Cas06}. Its analytic solutions are exactly the
local symmetries of $L$: differentiating invariance adds only
necessary equations, and the initial invariance equations remain
among the defining equations.

We use the generic-base form of Malgrange's regularity theorem,
stated in \cite[Chapter~I, \S\S5.1--5.4]{Mal10}. On one nonempty
Zariski open base $D_*$, all its finite-order truncations $G_k$
are smooth strict algebraic groupoids, their truncation maps are
smooth and surjective, and, from some order onward, the successive
systems are their full prolongations. Prolonging to an involutive
order gives regular analytic Lie equations. Their linearizations
form the corresponding regular equations for vector fields
\cite[Chapter~I, \S5.6]{Mal10}.
After a further generic base restriction and prolongation, the
linearized system is itself involutive and its projections to the
retained lower jets are smooth and surjective. We use this
involutive order to realize infinitesimal jets below.

Choose a return $h$ with centre $a=h(0)\in D_*$. This is possible
by the uniqueness assumption. Conjugate the small return germs by
$h$. Their centres near $a$ remain a holomorphic uniqueness set,
and we may work in a ball contained in $D_*$. The order-zero
groupoid $G_0\subset D_*\times D_*$ is a closed algebraic
equivalence relation. Its fibre at $a$ contains this uniqueness
set, hence equals $D_*$. The equivalence-relation identities give
$G_0=D_*\times D_*$. Smooth surjectivity of the truncation maps
then says that the infinitesimal target map of the source fibre
of $G_\ell$ has rank $N$ at the identity over $a$.

Cartan--K\"ahler applied to the linearized involutive equations
realizes these infinitesimal jets as local infinitesimal symmetries;
compare \cite[Theorem~1.9]{Cas06} and the solution argument in
\cite[Chapter~II, proof of Theorem~2.2]{Mal10}. Thus the local
Killing algebra evaluates surjectively onto $T_aD$. Pulling the
regular local pseudogroup back by $h$ proves regularity and
infinitesimal transitivity near $0$. Composition by a fixed
solution preserves the completed equations and their regularity.

Finally, the isotropy $G_\ell(a,a)$ is an algebraic subgroup of
the algebraic group of invertible $\ell$-jets fixing $a$.
Conjugation by the fixed jet of $h$ identifies it with an algebraic
finite-order isotropy at $0$.
\end{proof}

\begin{proposition}[The local alternative for an arbitrary first missing jet]
\label{prop:arbitrary-symbol-local-alternative}
Assume the hypotheses of
Proposition~\ref{prop:rational-jet-regular-transitive}, and suppose
\[
       L\subsetneq J^d(\mathcal O_D),\qquad d\ge2,
       \qquad L\longrightarrow J^{d-1}(\mathcal O_D)
                   \text{ is surjective}.
\]
Suppose the symmetries include an expanding germ $F$ fixing $0$.
Then one of the following holds in Poincar\'e--Dulac coordinates.
\begin{enumerate}
\item There is a finite-dimensional transitive Lie algebra of
polynomial vector fields normalized by every local symmetry of $L$.
\item There is a finite nonempty collection of proper positive-rank
integrable algebraic distributions, regular near $0$, which is
permuted by every local symmetry of $L$. An iterate of $F$ preserves
each member of this collection.
\end{enumerate}
\end{proposition}

\begin{proof}
Let $\Gamma^0$ be the regular local pseudogroup generated by the
flows of infinitesimal symmetries. It is transitive by
Proposition~\ref{prop:rational-jet-regular-transitive}.
If it is of finite type, its infinitesimal symmetry algebra
$\mathfrak g$ is finite dimensional. Every local symmetry of $L$
normalizes this algebra. Lemma~\ref{lem:expanding-finite-formal-Lie-polynomial}
makes its vector fields polynomial, giving the first case.

Suppose $\Gamma^0$ is of infinite type.
Proposition~\ref{prop:primitive-scalar-jet-finite} gives a proper
positive-rank regular foliation $\mathcal F$ preserved by
$\Gamma^0$. A transitive invariant foliation is determined by its
tangent space $V=T_0\mathcal F$: its tangent spaces elsewhere
are transported by the infinitesimal symmetries. Let $H$ be the
algebraic finite-order isotropy at $0$ furnished by the previous
proposition. Its identity component preserves $V$. Indeed its
Lie algebra consists of isotropy jets of actual infinitesimal
symmetries, by involutivity, and these preserve $\mathcal F$.

The group $H/H^0$ is finite. Therefore the tangent spaces obtained
from $V$ by actual isotropy germs form a finite collection.
Their transported foliations form a finite collection
$\mathcal F_1,\ldots,\mathcal F_s$ preserved by every local
symmetry. To check finiteness for an arbitrary symmetry, first
compose it with a local transformation in $\Gamma^0$ to bring its
centre back to $0$; the remaining transformation is an isotropy
germ, so acts through the stated finite collection. For a centre
outside this regular neighbourhood, first compose with a high
inverse iterate of $F$. The new centre lies in the neighbourhood.
Since $F$ fixes $0$ and acts through the same finite isotropy
orbit, cancellation proves the assertion for the original return.
Each foliation is integrable because it is the image of
$\mathcal F$ by an actual local biholomorphism.

An iterate $F^r$ fixes every member. Apply
Lemma~\ref{lem:expanding-formal-distribution-polynomial} to these
foliations and to $F^r$. Each becomes an algebraic integrable
distribution in the chosen normal-form coordinates. The finite
permutation identities hold as identities of germs, and therefore
hold wherever their algebraic distributions and the local symmetries
are defined. This proves the second case.
\end{proof}


\subsection{The local return alternative in arbitrary dimension}

\begin{theorem}[Local return geometry without a symbol irreducibility assumption]
\label{thm:arbitrary-dimensional-local-return-geometry}
Let $f$ be a $J$-stable algebraic curve family of polarized
endomorphisms of normal projective $N$-folds, with $N\ge2$.
Choose a good smooth Julia repeller as in
Proposition~\ref{prop:good-repeller-selection}.
After restricting the parameter germ and making a finite parameter
change, one of the following occurs.
\begin{enumerate}
\item The marked inverse-return collection is holomorphically
conjugate to its reference collection by a family of weighted
polynomial automorphisms. These automorphisms conjugate the varying
expanding normal form to its reference normal form.
\item At the reference parameter, every marked inverse return
normalizes one finite-dimensional transitive Lie algebra of
polynomial vector fields on $\mathbb C^N$.
\item At the reference parameter, every marked inverse return
permutes a finite nonempty collection of proper positive-rank
integrable algebraic distributions, regular on $\mathbb C^N$.
An iterate of the normal form preserves each member. Each member
admits polynomial Frobenius coordinates commuting with the
semisimple part of that iterate.
\end{enumerate}
\end{theorem}

\begin{proof}
Use the varying normal forms and the notation of
Proposition~\ref{prop:resonant-weighted-character-constancy}.
Let $M=1+\max\{2,\lfloor b/a\rfloor\}$ and fix $R\ge Mb$.
Lemma~\ref{lem:irreducible-holomorphic-return-module} gives a
nonzero irreducible module $E_s\subset W_R$ and a holomorphic
simultaneous gauge. The space $W_R$ consists only of polynomials
vanishing at $0$: it is the canonical zero-constant representative
of the truncated substitution quotient by the constants. Thus
$E_s$ contains a nonconstant polynomial. We use $\mathcal A_s=\mathbb C1\oplus E_s$ for scalar jet
geometry and its truncated substitution action.

At the reference parameter let $r$ be the generic rank of $dE_0$.
Lemma~\ref{lem:truncated-invariant-subspace-foliation}
shows that this rank is attained at $0$, that $r\ge1$, and that
the common kernel of $dE_0$ is preserved by all actual returns.
If $r<N$, this is a proper positive-rank regular integrable
distribution near $0$. Its invariance under the normal form,
Lemma~\ref{lem:expanding-formal-distribution-polynomial}, and
Lemma~\ref{lem:expanding-polynomial-frobenius} give the third
alternative with one member.

Suppose $r=N$. If $\mathcal A_0$ separates $M$-jets, this remains
true after shrinking the parameter germ.
Proposition~\ref{prop:jet-spanning-module-recovery} applies.
The nilpotent intertwiners also identify the actions of
$C_{\mathcal N_s}$, so
Proposition~\ref{prop:varying-normal-form-geometric-recovery}
gives the first alternative.

Otherwise choose the least $d\le M$ for which
$j^d_0\mathcal A_0$ is proper. Since $r=N$, one has $d\ge2$.
Proposition~\ref{prop:exact-jet-bundle} and
Corollary~\ref{cor:first-missing-jet-symbol} give a rational
holomorphic jet subbundle
\[
 L_x=j_x^d\mathcal A_0\subsetneq J_x^d(\mathcal O),
 \qquad L_x\longrightarrow J_x^{d-1}(\mathcal O)
                  \text{ surjective near }0,
\]
preserved by every actual return. The uniqueness of the small
return centres is precisely the hypothesis of
Proposition~\ref{prop:rational-jet-regular-transitive}.
Proposition~\ref{prop:arbitrary-symbol-local-alternative} then
gives either the second alternative or the finite web in the
third. In the latter case apply
Lemma~\ref{lem:expanding-polynomial-frobenius} to each member
after a common iterate. This proves the theorem.
\end{proof}


\begin{corollary}[Global data supplied by the local alternative]
\label{cor:local-alternative-global-data}
In the first alternative of
Theorem~\ref{thm:arbitrary-dimensional-local-return-geometry},
the original family is isotrivial by
Proposition~\ref{prop:nd-source-descent}.
In its second alternative there is an equivariant rational flat
Killing bundle with surjective evaluation onto $TX$, as constructed
in Proposition~\ref{prop:rational-Killing-connection}.
In its third alternative the source-descent and projective-cocycle
arguments give a finite algebraic web of integrable rational
distributions on $X$; a finite component cover selects one member.
\end{corollary}

\begin{proof}
For the first assertion, use the polynomial conjugacy to make the
normal form and all return identifications constant, and apply
the horizontal part of the source-descent proposition.
The second assertion is exactly
Proposition~\ref{prop:rational-Killing-connection}.
For the third, let $\ell_1,\ldots,\ell_t$ be the Pl\"ucker lines
of the members, of common rank $r$. Their unordered product is a
line in $\operatorname{Sym}^t(\bigwedge^rT\mathbb C^N)$.
The natural action is the invertible rational linear cocycle
$\operatorname{Sym}^t(\bigwedge^r Df)$, and unique factorization
of a product of linear forms recovers the unordered set.
The majority-continuation comparison and projective-cocycle
rationality therefore apply to this datum. The resulting algebraic
finite subset of the Grassmannian over $\mathbb C(X)$ splits over
a finite field extension. Normalization in that extension gives
the component cover, on which a selected distribution is rational
and integrable. Equivariance of this cover requires taking the
dynamically compatible component; it is the finite-cover issue
handled separately in the relative construction.
\end{proof}


\subsection{The rational Killing connection}

\begin{lemma}[Continuation of natural jet subspaces]
\label{lem:natural-jet-subspace-globalization}
The majority continuation and rationality statements of
Proposition~\ref{prop:nd-majority} and
Corollary~\ref{cor:nd-rational-data} apply to a fixed-rank subbundle
of $J^k(TX^{\mathrm{reg}})$, transported by the natural jet action of
$f$ on its unramified locus. The source-descent statement of
Proposition~\ref{prop:nd-source-descent} applies to these subbundles
as well.
\end{lemma}

\begin{proof}
On a rational coordinate chart the natural pushforward of a
$k$-jet of a vector field is an invertible linear map whose
coefficients are rational functions of the derivatives of $f$
up to order $k+1$ and of $(Df)^{-1}$. Thus the Grassmannian-valued
map defining a rank-$r$ subspace has the same projective
Pl\"ucker transport law as a distribution, with a rational
invertible cocycle.

The branch comparison in the majority-continuation proof uses
only this invertible transport and equality of germs. Across a
finite discriminant, trace a local meromorphic Pl\"ucker generator
with interpolation weights. The summands are generically collinear,
so their nonzero trace defines the transported Pl\"ucker line.
The Pl\"ucker equations persist, yielding a meromorphic subspace.
Proposition~\ref{prop:nd-projective-cocycle} makes it rational.
The comparison of two regular lifts in the source-descent proof
is unchanged for this natural jet action. This proves all three
assertions.
\end{proof}

\begin{proposition}[Finite local symmetry produces a flat algebraic bundle]
\label{prop:rational-Killing-connection}
Let $\Pi:\mathbb C^N\to X$ and $F$ be as in
Proposition~\ref{prop:nd-source-descent}. Suppose a finite-dimensional
Lie algebra $\mathfrak g$ of holomorphic vector fields near $0$
acts locally transitively there, is normalized by the normal form
$\mathcal N$, and is normalized by every actual inverse return.
Then, on a nonempty Zariski open subset $X^\circ\subset X^{\mathrm{reg}}$,
there are an algebraic vector bundle $\mathcal K$, a flat algebraic
connection $\nabla$, and a surjective algebraic morphism
\[
                         \varepsilon:\mathcal K\longrightarrow TX^\circ.
\]
They have natural rational $F$-equivariance. The horizontal local
sections of $\mathcal K$ evaluate to the local Killing fields.
There is also a parallel algebraic Lie bracket on $\mathcal K$.

The pullback of this flat bundle through $\Pi$ extends over
$\mathbb C^N$ as a trivial flat bundle. In Poincar\'e--Dulac
coordinates, its horizontal sections are the jets of polynomial
vector fields.
\end{proposition}

\begin{proof}
Apply Lemma~\ref{lem:expanding-finite-formal-Lie-polynomial} to
$\mathfrak g$. It consists of polynomial vector fields and is
normalized by the semisimple part $S$ of $\mathcal N$.
Choose $k$ so that $j^k_0:\mathfrak g\to J^k_0(T\mathbb C^N)$ is
injective. Put
\[
 R_j(x)=\{j_x^j Z:Z\in\mathfrak g\}\subset J_x^j(T\mathbb C^N)
                         \qquad(j=k,k+1).
\]
The jet matrices of a polynomial basis of $\mathfrak g$ have full
rank near $0$. Because $S$ normalizes $\mathfrak g$, their ranks
at $x$ and $S^{-n}x$ agree. Since $S^{-n}x\to0$, the rank is
$\dim\mathfrak g$ everywhere on $\mathbb C^N$.
The same argument for evaluation shows that $\mathfrak g$ is
transitive at every source point. Consequently $R_k$ and $R_{k+1}$
are algebraic subbundles, and truncation induces an isomorphism
\[
                         R_{k+1}\xrightarrow{\sim}R_k.
\]
All actual returns preserve both bundles, by the normalization
hypothesis. Lemma~\ref{lem:natural-jet-subspace-globalization}
descends them to rational subbundles of the corresponding jet
bundles on $X$. Shrink $X^\circ$ so that both are algebraic
subbundles, the truncation is an isomorphism, and evaluation of
$R_k$ has rank $N$. Set $\mathcal K=R_k$ on this open set.

Let $\sigma:R_k\to R_{k+1}$ be the inverse of truncation. The
Spencer operator defines
\[
                         \nabla s=D(\sigma s).
\]
This takes values in $\Omega^1\otimes R_k$. Indeed, in a source
chart choose a basis $Z_1,\ldots,Z_m$ of $\mathfrak g$. For
$s=\sum_i a_i j^k Z_i$ one has
\[
 D\left(\sum_i a_i j^{k+1}Z_i\right)
                      =\sum_i da_i\otimes j^kZ_i.
\]
It follows that $\nabla$ is a connection and is flat. Its
coefficients are algebraic on $X^\circ$, because $R_k,R_{k+1}$,
truncation, and the Spencer operator are algebraic. This formula
also shows that the pullback connection extends over all of
$\mathbb C^N$ with the global parallel basis
$j^kZ_1,\ldots,j^kZ_m$.

The zeroth-jet projection gives $\varepsilon$. The $k$-jet of
$[Z,W]$ is a universal bilinear expression in the $(k+1)$-jets
of $Z$ and $W$. Composing this expression with $\sigma\otimes\sigma$
defines an algebraic bracket on $\mathcal K$. In the displayed
parallel basis its coefficients are the constant structure
constants of $\mathfrak g$, so it is parallel. Finally, all these
constructions are natural under local biholomorphisms. They
therefore respect the equivariance of $R_k$ and $R_{k+1}$ under $F$.
\end{proof}


\section{Regularity of invariant rational connections}


\begin{proposition}[Regularity of invariant rational connections]
\label{prop:invariant-connection-regular-singular}
Let $X$ be a normal complex projective variety, and let $f:X\to X$
be a $q$-polarized endomorphism, where $q>1$. Let $(\mathcal E,\nabla)$
be a finite-rank integrable connection over the function field of $X$.
If
\[
                 f^*(\mathcal E,\nabla)\simeq(\mathcal E,\nabla),
\]
then $(\mathcal E,\nabla)$ is regular singular. In particular, an
$f$-invariant rational flat connection on $T_X$ is regular singular.
\end{proposition}

\begin{proof}
Write $n=\dim X$. At the generic point of each prime divisor $D$,
normality gives a discrete valuation ring. Complete its fraction field,
extend the residue coefficient field algebraically, and consider the
formal connection in a transverse parameter. If its slopes are $s_i$
with rank multiplicities $r_i$, set
\[
                  \operatorname{irr}_D(\nabla)=\sum_i r_i s_i.
\]
For an integrable connection this is intrinsic to the divisorial
valuation. It is nonnegative, vanishes precisely when the connection
is regular singular at that valuation, and is zero outside the
finitely many divisorial poles of a rational connection matrix.
Consequently
\[
                  I(\nabla)=\sum_D\operatorname{irr}_D(\nabla)D
\]
is an effective Weil divisor with finite support.

Let $C$ be a prime divisor mapping to $D$, and let $e_C$ be the
ramification index. After extension of the residue coefficient field
and a change of uniformizer, the extension of complete discrete
valuation fields has the form $t=u^{e_C}$. In a
Levelt--Turrittin decomposition, substitution of $u^{e_C}$ for $t$
multiplies every exponential slope by $e_C$ and leaves its rank
multiplicity unchanged. Thus
\[
 \operatorname{irr}_C(f^*\nabla)
       =e_C\operatorname{irr}_D(\nabla),
 \qquad I(f^*\nabla)=f^*I(\nabla).
\]
The given isomorphism therefore implies $I=f^*I$.

Choose an ample Cartier divisor $H$ such that $f^*H\sim qH$.
The projection formula for a finite morphism and Weil cycles gives
\[
\begin{aligned}
 q^{n-1}(f^*I\cdot H^{n-1})
     &=(f^*I\cdot(f^*H)^{n-1})\\
     &=\deg(f)(I\cdot H^{n-1})
       =q^n(I\cdot H^{n-1}).
\end{aligned}
\]
Since $f^*I=I$ and $q>1$, we obtain $(I\cdot H^{n-1})=0$.
Effectivity and ampleness yield $I=0$.

Take a smooth dense open set on which the connection is nonsingular.
Deligne's regularity criterion applies to the fixed normal
compactification $X$: regular singularity at the generic points of
its boundary prime divisors is equivalent to regular singularity on
every algebraic curve; see \cite[II, Proposition~4.4(iii)]{Del70}
and \cite[Theorems~1.1.1 and~1.3.1]{And07}. Hence the connection is regular
singular, including at all divisorial valuations on any resolution.
For a rational tangent connection the differential $Df$ supplies
the required isomorphism over the function field.
\end{proof}


\section{Exceptional factor calculus}

\begin{definition}[Exceptional presentations used in the induction]
We use the exceptional systems of
\cite[Definitions~1.2--1.4]{JSX26}. An \emph{AT-variety}
is a torsor $P\to A$ under an algebraic torus $T$, with $A$
an abelian variety. A torsor is a variety with a group action
which becomes the translation action on the group after
\'etale localization on the base. A \emph{$Q$-lift}
$\Psi:P\to P$ covers a $Q$-polarized isogeny of $A$ and satisfies
$\Psi(tp)=[Q]_T(t)\Psi(p)$, where $[Q]_T$ is multiplication by
$Q$ on the torus. A $q$-polarized system $(Z,g)$ is
\emph{exceptional} if, over an algebraic closure of its ground
field, some iterate has a finite surjective presentation
\[
 \eta:P\longrightarrow Z^\circ,\qquad
 \eta\Psi=g^e\eta,
\]
where $Z^\circ\subset Z$ is dense open and $\Psi$ is a
$q^e$-lift.


An abelian variety over $\overline K$ has \emph{geometric
$\mathbf C$-trace zero} if
\[
 \operatorname{Hom}_{\overline K}(A_0\times_{\mathbf C}\overline K,A)=0
 \quad\text{for every abelian variety }A_0/\mathbf C.
\]
A factor has a \emph{trace-zero abelian presentation} if its
presenting abelian variety has this property. A \emph{maximal}
connected factor of this type contains the function fields of
all other such connected factors.
\end{definition}



\begin{proposition}[A polarized map and an invariant linear system]
\label{prop:polarized-no-automorphic-rational-factor}
Let $F:V\to V$ be a $d$-polarized endomorphism of a normal
projective variety, where $d>1$.  Suppose a rational map
$\gamma:V\dashrightarrow Z$ satisfies
$\gamma F=u\gamma$, where $Z$ is projective and $u$ is an
automorphism preserving an ample line bundle on $Z$.
Then $\gamma$ is constant.
\end{proposition}

\begin{proof}
The assertion is immediate if $\dim V=0$. Otherwise replace $Z$
by the normalization of the image of $\gamma$; the
automorphism $u$ lifts to this normalization. By
\cite[Theorem~1.3]{MZ18},
\[
 (\deg F)^{\dim Z}=(\deg u)^{\dim V}=1.
\]
Since $\deg F>1$, the image has dimension zero.
\end{proof}

\begin{proposition}[Recovering a factor for the original map]
\label{prop:iterate-exceptional-factor-original}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective variety.  Suppose a positive iterate has a dominant rational
factor
\[
                  \pi f^r=h\pi,
\]
where $h:Y\to Y$ is exceptional and $q^r$-polarized.
Then $f$ itself has an exceptional rational factor whose function
field contains $\mathbf C(Y)$.  In particular a positive-dimensional
exceptional factor for an iterate gives one for the original map.
\end{proposition}

\begin{proof}
On $Y^r$ consider the finite cyclic map
\[
 \Phi(y_0,\ldots,y_{r-1})=(y_1,\ldots,y_{r-1},h(y_0)).
\]
Choose an ample line bundle $L$ with $h^*L\simeq L^{q^r}$ and set
\[
              \mathcal L=\bigotimes_{i=0}^{r-1}
                        p_i^*L^{q^{r-1-i}}.
\]
Then $\Phi^*\mathcal L\simeq\mathcal L^q$.  The normalization $Z$ of
the closure of the image of
\[
              \Pi=(\pi,\pi f,\ldots,\pi f^{r-1})
\]
is preserved by $\Phi$ and carries the induced finite $q$-polarized
map $\phi$, with $\Pi f=\phi\Pi$.  Moreover
$\Phi^r=h\times\cdots\times h$.  Take a common AT presentation of
this product and a periodic dominant component over the image of $Z$.
Proposition~\ref{prop:periodic-subvariety-at-lift} makes its
normalization an AT variety with scalar return lift.  It gives an
exceptional presentation for an iterate of $\phi^r$, hence for $\phi$.
The first projection $Z\to Y$ is dominant, proving the containment of
factor fields.
\end{proof}

\begin{proposition}[Periodic subvarieties of an AT lift]
\label{prop:periodic-subvariety-at-lift}
Let $P\to A$ be a complex AT variety with torus $T$, and let
$\psi:P\to P$ be a $d$-lift.  If a closed irreducible subvariety
$Q\subset P$ satisfies $\psi(Q)=Q$, then its normalization
$Q^\nu$ is an AT variety.  The induced endomorphism of $Q^\nu$
is a $d$-lift after choosing the origin of its abelian base.
The same conclusion applies to a periodic subvariety and its
corresponding return.
\end{proposition}

\begin{proof}
Write $T\simeq\mathbf G_m^r$ and
$P=L_1^\times\times_A\cdots\times_A L_r^\times$.
If $\alpha:A\to A$ is the isogeny covered by $\psi$, its
equivariance with $[d]_T$ gives
$\alpha^*L_i\simeq L_i^{\otimes d}$.
Consequently $\psi$ extends by the pure $d$-th powers on the
homogeneous fiber coordinates to a finite endomorphism of
\[
 \overline P=
 \mathbf P(\mathcal O_A\oplus L_1)\times_A\cdots\times_A
 \mathbf P(\mathcal O_A\oplus L_r).
\]
Each relative $\mathcal O(1)$ pulls back to its $d$-th power.
If $H_A$ polarizes $\alpha$, the tensor product of these
relative bundles with $\pi^*H_A^{\otimes N}$ is ample for
$N\gg0$ and is $d$-polarized.  Let $\overline Q$ be the
normalization of the closure of $Q$ in $\overline P$.
It carries a finite $d$-polarized lift $F$, and the projection
$\overline Q\to A$ is a morphism.

Let $C\subset A$ be the image of $\overline Q$. It is invariant
under $\alpha$. Every positive-dimensional invariant abelian
subquotient carries a $d$-polarized induced map, by restriction
and \cite[Theorem~1.3]{MZ18}, so this map cannot have finite order.
Thus $(A,\alpha)$ is strictly mixed in the sense of
\cite[Definition~2.1]{PR04}. By \cite[Theorem~3.4]{PR04},
$C$ is a translate of an abelian subvariety. In particular it is
smooth, and $\alpha_C:C\to C$ is finite \'etale and $d$-polarized.

Take the Stein factorization
\[
 \overline Q\longrightarrow C'\xrightarrow{\rho}C.
\]
Its base $C'$ is normal and projective, and $\rho$ is finite.
The map $F$ induces a finite endomorphism $\alpha'$ of $C'$
with $\rho\alpha'=\alpha_C\rho$.  Pullback of the restricted
polarization on $C$ directly polarizes $\alpha'$ with weight
$d$.  Apply Ji--Song--Xie \cite[Lemma 3.11]{JSX26} with the
whole projective variety $C'$ as its invariant open, and with
the finite map $\rho$ to the smooth variety $C$.
It follows that $\rho$ is \'etale.  Thus $C'$ is an abelian
variety after choosing an origin.  The polarized endomorphism
$\alpha'$ can be made a polarized isogeny by choosing a fixed
point as origin.

Put $k=\dim Q-\dim C$.  The vertical tangent bundle of $P/A$
is canonically $\mathcal O_P\otimes\operatorname{Lie}T$.
On the smooth locus where $Q\to C$ has maximal rank, its
vertical tangent spaces define a rational logarithmic Gauss
map
\[
 \Gamma:Q\dashrightarrow
 \operatorname{Gr}(k,\operatorname{Lie}T).
\]
The vertical differential of $\psi$ is $d\,\mathrm{id}$.
Therefore $\Gamma\psi=\Gamma$, and the first proposition,
applied to $\overline Q$, makes $\Gamma$ constant.
Call its value $W$.

A subvariety of a torus with constant logarithmic tangent
space $W$ is a coset of a subtorus.  Indeed, the invariant
vector fields in $W$ are tangent to the subvariety, so $W$
lies in the Lie algebra of its algebraic translation
stabilizer.  Freeness of the torus action bounds the
stabilizer dimension by the dimension of the subvariety;
the dimensions are equal, and its connected orbit is the
whole irreducible subvariety.  Applied to the geometric
generic fiber over $C'$, this gives a fixed subtorus
$S\subset T$ of dimension $k$, with $\operatorname{Lie}S=W$.
The generic fiber is geometrically integral because $C'$ is
the Stein base of the normal projective variety $\overline Q$.
It is thus a single $S$-coset.

Pull $P|_C$ back to $C'$ and quotient its torus by $S$.
This gives a $(T/S)$-torsor $R\to C'$ with a lift $\theta$
over $\alpha'$ acting by $[d]_{T/S}$.  The generic $S$-coset
defines a rational section $\sigma$ of $R$, and invariance
of $Q$ gives $\theta\sigma=\sigma\alpha'$.
Choose a basis of characters of $T/S$.  In the corresponding
description $R=M_1^\times\times_{C'}\cdots\times_{C'}M_t^\times$,
the rational section gives nonzero rational sections $s_j$
of the $M_j$.  Put $D_j=\operatorname{div}(s_j)$.  The
equivariance gives
\[
 (\alpha')^*D_j=dD_j.
\]
Since $\alpha'$ is finite \'etale, pullback preserves the
maximum of the absolute values of the nonzero coefficients
of a Weil divisor.  Multiplication by $d>1$ strictly increases
that maximum.  Hence $D_j=0$ for every $j$.
The sections $s_j$ therefore extend as nowhere-vanishing
regular sections on the smooth variety $C'$.  Thus $\sigma$
is a global section.

The inverse image of $\sigma(C')$ in $P\times_C C'$ is a
smooth $S$-torsor $Q_S\to C'$.  The normalization $Q^\nu$
maps finitely to $P\times_C C'$: its map to $P$ is finite,
and $P\times_C C'\to P|_C$ is finite.  Its image is closed
and has generic fiber equal to the prescribed $S$-coset,
so that image is $Q_S$.  The resulting finite birational
map $Q^\nu\to Q_S$ is an isomorphism, since $Q_S$ is normal.
The induced endomorphism acts by $[d]_S$ on the torus and
covers the $d$-polarized isogeny $\alpha'$.  It is the
required $d$-lift.
\end{proof}


\begin{lemma}[The exceptional locus of an equivariant birational morphism]
\label{lem:birational-nonfinite-invariant}
Let $p:V\to W$ be a proper birational morphism between normal
projective varieties, and let $F:V\to V$ and $f:W\to W$ be finite
surjective endomorphisms satisfying $pF=fp$.  Let $E\subset V$ be
the non-quasi-finite locus of $p$, and put $Z=p(E)$.  Then
\[
 E=p^{-1}(Z),\qquad F^{-1}(E)=E,\qquad f^{-1}(Z)=Z.
\]
Suppose the endomorphisms are polarized.  If either $V$ or $W$ has
a completely invariant AT open on which its endomorphism is a scalar
lift, $p$ is an isomorphism on that open onto an open of the other
variety.  This other open is also completely invariant.
\end{lemma}

\begin{proof}
A proper birational morphism to a normal variety has connected
fibers.  Every positive-dimensional fiber therefore has positive
local dimension at each of its points: a zero-dimensional irreducible
component would be an isolated component.  This proves $E=p^{-1}(Z)$;
properness also makes $Z$ closed.

If $x\in E$, a positive-dimensional fiber subvariety through $x$
has a positive-dimensional image under the finite map $F$, contained
in a fiber of $p$.  Thus $F(x)\in E$.  Conversely, suppose
$F(x)\in E$.  Choose a positive-dimensional fiber subvariety through
$F(x)$.  The going-down theorem for an integral extension over a
normal domain supplies an irreducible component of its inverse image
through $x$ which dominates that subvariety.  This component has
positive dimension.  Its image under $p$ is contained in the finite
set $f^{-1}(p(F(x)))$, hence is the point $p(x)$.  Thus $x\in E$.
The equality for $Z$ follows using $E=p^{-1}(Z)$ and the surjectivity
of $p$.

Suppose first that $V$ contains the stated AT open $P$.  The closure
of $E\cap P$, if nonempty, is a proper completely invariant closed
subset on whose generic points $F$ is \'{e}tale.  Closure commutes
with inverse image here because finite dominant maps to normal
varieties are open, again by going-down.  Ji--Song--Xie
\cite[Lemma 3.10]{JSX26} rules out this closed subset.  Hence
$E\cap P=\varnothing$.  Connected fibers and Zariski's main theorem
now show that $p$ identifies $P$ with an open $U\subset W$, with
$p^{-1}(U)=P$.  The identity $pF=fp$ and complete invariance of $P$
give $f^{-1}(U)=U$.  If instead the AT open $P$ lies in $W$, apply
the same argument to the closure of $Z\cap P$.  Then $p$ is an
isomorphism over $P$, and its inverse image is completely invariant.
\end{proof}

\begin{proposition}[Birational invariance of exceptionality]
\label{prop:birational-exceptional-invariance}
Two birationally conjugate polarized endomorphisms of normal
projective varieties are simultaneously exceptional.  More generally,
if a generically finite dominant rational map semiconjugates two
polarized endomorphisms of the same weight, either is exceptional
if and only if the other is exceptional.
\end{proposition}

\begin{proof}
First consider a birational conjugacy between $(X,f)$ and $(Y,g)$,
and suppose $g$ is exceptional.  The normalization $G$ of its graph
has a finite lifted endomorphism, polarized by the sum of the pullbacks
of polarizations of $X$ and $Y$.  Take an iterate admitting an AT
presentation $P\to U\subset Y$, and normalize $Y$ in its function
field.  This gives a finite normal projective cover
$\overline P\to Y$ with a polarized lift and completely invariant
AT open $P$.

Normalize the dominant component of $G\times_Y\overline P$, calling
it $G'$.  It is finite over $G$ and birational over $\overline P$;
the lifted dynamics is finite and polarized.  By
Lemma~\ref{lem:birational-nonfinite-invariant}, the latter birational
morphism is an isomorphism over $P$.  Thus $G'$ contains a completely
invariant AT open with scalar dynamics.

The morphism $G'\to X$ is proper and generically finite.  Its Stein
factorization is
\[
                        G'\longrightarrow X'\longrightarrow X,
\]
where the first map is birational and the second finite, with $X'$
normal.  The induced map on $X'$ is finite and polarized by the
pullback of a polarization of $X$.  Apply the same lemma to
$G'\to X'$, this time using its source AT open.  The resulting AT
open on $X'$ is completely invariant, so $X'$ is exceptional.
Ji--Song--Xie \cite[Theorem 3.9]{JSX26} applied to $X'\to X$ makes
$X$ exceptional.  Interchanging $X$ and $Y$ gives the converse.

For a generically finite rational semiconjugacy, take its normalized
graph $G$ and the Stein factorization of $G\to Y$.  The projection
$G\to X$ and the first Stein map are birational, while the second
Stein map is finite.  The sum polarization polarizes the graph, and
the finite Stein target is polarized by pullback from $Y$.  The
birational statement just proved and the finite statement of
\cite[Theorem 3.9]{JSX26} give the equivalence.
\end{proof}


\begin{proposition}[Galois AT presentations]
\label{prop:galois-at-presentation-audited}
Let $f:X\to X$ be a polarized endomorphism of a normal complex
projective variety.  Suppose an iterate $f^\ell$ has an exceptional
presentation
\[
 h:P\longrightarrow U\subset X,\qquad h\psi=f^\ell h,
\]
where $h$ is finite surjective, $P$ is an AT variety, and $\psi$ is
a $q$-lift.  Then $f^\ell$ has such a presentation which is also
Galois.  The new AT lift is again a $q$-lift.  Its deck group $G$
is compatible with the lift: there is an automorphism
$\delta:G\to G$ such that $\psi\gamma=\delta(\gamma)\psi$.
\end{proposition}

\begin{proof}
Replace $f$ by $f^\ell$.  We first note that $f^{-1}(U)=U$.
Indeed, $f(U)\subseteq U$, and finiteness of $h\psi=(f|_U)h$
descends along the finite surjective map $h$, so $f|_U:U\to U$
is finite.  The inclusion $U\hookrightarrow f^{-1}(U)$ is
therefore a finite open immersion.  The latter open is irreducible,
so the inclusion is an equality.

Put $K=\mathbf C(X)$ and $L=\mathbf C(P)$.  Normalize $X$ in $L$,
obtaining a finite map $p:W\to X$ with $p^{-1}(U)=P$.
The field embedding $\sigma=\psi^*:L\hookrightarrow L$ extends
the embedding $f^*$ of $K$.  By the universal property of
normalization it gives a finite endomorphism $\Psi$ of $W$ with
$p\Psi=fp$.  Pullback of a polarization of $f$ polarizes $\Psi$.

The decreasing fields $K\sigma^j(L)$ stabilize, since $L/K$ is
finite.  Write their stable value as $L_0=K\sigma^s(L)$.
Then
\[
 \sigma(L_0)\subseteq L_0,\qquad K\sigma(L_0)=L_0.
\]
Let $p_0:Y\to X$ be the normalization in $L_0$, and let $F_0$
be its polarized lift.  The embedding $\sigma^s:L\hookrightarrow
L_0$ induces a finite morphism $j:Y\to W$ satisfying
\[
 pj=f^sp_0,\qquad jF_0=\Psi j.
\]
Regularity follows from integrality and normality; finiteness
follows because $pj$ is finite.  Complete invariance of $U$ gives
\[
 Y^\circ:=p_0^{-1}(U)=j^{-1}(P).
\]
Apply Ji--Song--Xie \cite[Lemma 3.11]{JSX26} to the normal
projective variety $Y$, its polarized map $F_0$, its completely
invariant open $Y^\circ$, the finite surjection $j:Y^\circ\to P$,
and the finite \'etale endomorphism $\psi$ of the smooth variety
$P$.  The lemma makes $j$ and $F_0|_{Y^\circ}$ \'etale.
Their Proposition 3.7 gives $Y^\circ$ an AT structure for which
$j$ is an AT-isogeny.  Lemma 3.2 and finiteness make the self-map
$F_0|_{Y^\circ}$ an AT-isogeny, and Lemma 3.8 makes it a
$q$-lift after changing the origin of its abelian base.

Let $M/K$ be the normal closure of $L_0/K$ in an algebraic
closure.  We verify that $\sigma$ extends to an embedding
$M\hookrightarrow M$.  Choose a primitive element $\alpha$ of
$L_0/K$, with minimal polynomial $A(T)$.  The equality
$K\sigma(L_0)=L_0$ gives $K(\sigma\alpha)=L_0$.  Thus
$\sigma(A)$ is still irreducible over $K$ and its splitting
field is the same normal closure $M$.  Extend $\sigma$ to an
algebraic closure; images of all roots of $A$ are the roots of
$\sigma(A)$.  Consequently
\[
 \sigma(M)\subseteq M,\qquad K\sigma(M)=M.
\]
Normalize $X$ in $M$, obtaining $p_Z:Z\to X$ and its polarized
lift $F_Z$.  The map $Z\to Y$ is finite and equivariant.  On
$Z^\circ=p_Z^{-1}(U)$, a second application of Lemma 3.11,
followed by Proposition 3.7 and Lemma 3.8, shows that $Z^\circ$
is AT and that $F_Z|_{Z^\circ}$ is a $q$-lift.  The map
$Z^\circ\to U$ is finite Galois, as required.

For the final assertion, write $G=\operatorname{Gal}(M/K)$.
Since $K\sigma(M)=M$ and
$[\sigma(M):\sigma(K)]=[M:K]$, the fields $K$ and $\sigma(M)$
are linearly disjoint over $\sigma(K)$.  Hence conjugation by
$\sigma$ on $\sigma(M)$ extends uniquely to an automorphism
of the group $G$ over the compositum $M$.  This is the required
compatibility $\psi\gamma=\delta(\gamma)\psi$.
\end{proof}

\begin{proposition}[Removing the translation kernel]
\label{prop:bounded-at-galois-degree}
Let $P\to A$ be a torsor under a split torus $T$ of rank $r$, where
$A$ is an abelian variety of dimension $a$.  Let a finite group $G$
act faithfully on $P$.  Put $n=a+r$ and
\[
 M_n=|\operatorname{GL}_{2n}(\mathbf F_3)|.
\]
There is a normal subgroup $N\triangleleft G$ such that $P/N$ is an
AT variety and
\[
 [G:N]\le M_n.
\]
Consequently the quotient $P/G$ admits a finite Galois AT cover of
degree at most $M_n$.

Suppose moreover that $\psi:P\to P$ is a $q$-lift and that, for
each $\gamma\in G$, there is $\delta(\gamma)\in G$ with
$\psi\gamma=\delta(\gamma)\psi$.  Then $\psi$ descends to a
$q$-lift on $P/N$, and the preceding finite cover is equivariant.
\end{proposition}

\begin{proof}
Every morphism from a torus to an abelian variety is constant.
Thus every automorphism of $P$ descends, by faithfully flat descent,
to an automorphism of $A$.  On every torus fiber it is a translation
followed by a monomial automorphism.  The exponent matrix is locally
constant, hence constant on the connected base.  We obtain a
homomorphism
\[
 \lambda:G\longrightarrow
 \operatorname{GL}(H_1(A,\mathbf Z))\times
 \operatorname{GL}(X^*(T))
 \subseteq\operatorname{GL}_{2a+r}(\mathbf Z).
\]
Let $N=\ker\lambda$.  Every element of $N$ induces a translation
on $A$ and is $T$-equivariant on $P$.  Let $H\subset A$ be the
finite translation subgroup obtained in this way, and let $C$ be
the kernel of $N\to H$.  An equivariant automorphism of a
$T$-torsor over the identity of $A$ is a morphism $A\to T$.
Such a morphism is constant, so $C$ is a finite subgroup of $T$.
In particular it is central in $N$.

The quotient $P/C\to A$ is a torsor under the torus $T/C$.
The group $N/C=H$ acts on it equivariantly and covers the free
translation action of $H$ on $A$.  Descent along the finite
\'etale cover $A\to A/H$ shows that
\[
 (P/C)/(N/C)\longrightarrow A/H
\]
is a $T/C$-torsor.  Its total space is $P/N$.  This also explains
why a noncommutative central extension inside $N$ causes no
obstruction: its commutators disappear in the first quotient by $C$.

The finite group $G/N$ embeds in
$\operatorname{GL}_{2a+r}(\mathbf Z)$.  Reduction modulo three is
injective on finite subgroups of this group, since the principal
congruence subgroup of level three is torsion-free.  Padding by
identity blocks gives $|G/N|\le M_n$.

Write $\alpha$ for the isogeny of $A$ covered by $\psi$.
If $\gamma\in N$ covers translation by $h$, then
$\delta(\gamma)$ covers translation by $\alpha(h)$: this follows
from $\alpha\gamma_A=\delta(\gamma)_A\alpha$ and surjectivity of
$\alpha$.  If $M_\gamma$ is the torus exponent matrix, the same
equivariance identity gives
\[
 qM_\gamma=M_{\delta(\gamma)}q.
\]
Thus $\delta(N)\subset N$, and $\psi$ descends to $P/N$.
The descended map acts by $[q]$ on $T/C$ and covers the induced
isogeny of $A/H$.  This isogeny is $q$-polarized, since polarization
descends through an isogeny.  Hence the descended map is a $q$-lift.
\end{proof}


\begin{proposition}[Bounded branch divisor of a dynamical cover]
\label{prop:bounded-branch-dynamical-cover}
Let $X$ be a normal projective variety of dimension $m$, let $H$ be
ample, and let $f:X\to X$ satisfy $f^*H\simeq H^{\otimes q}$,
where $q>1$.  Let $p:W\to X$ be a finite Galois morphism from a
normal variety, and suppose that a finite endomorphism $F:W\to W$
satisfies $pF=fp$.  For a prime divisor $D\subset X$, denote by
$e_D$ its inertia index and set
\[
 \Delta_p=\sum_D(1-e_D^{-1})D.
\]
Then
\[
 f^*\Delta_p-\Delta_p\le R_f,
 \qquad
 0\le\Delta_p\cdot H^{m-1}\le-K_X\cdot H^{m-1}.
\]
In particular the reduced branch divisor satisfies
\[
 \deg_H\operatorname{Br}(p)_{\mathrm{red}}
 \le -2K_X\cdot H^{m-1}.
\]
These bounds are unchanged if $f$ is replaced by any positive
iterate.  If $\deg p\le M$, all coefficients of $\Delta_p$ belong
to the fixed finite set $\{1-1/e:1\le e\le M\}$.
\end{proposition}

\begin{proof}
All pullbacks of Weil divisors in this proof are finite-map
pullbacks, defined by ramification indices of discrete valuations.
They are identities of Weil divisors. Since $p$ is
Galois, those indices are constant above each prime divisor of
$X$, and the codimension-one ramification formula gives
$R_p=p^*\Delta_p$.  Since the characteristic is zero, $R_F$ is
effective.  The ramification
identity for the equal compositions $pF=fp$ gives
\[
 R_F+F^*R_p=R_p+p^*R_f,
 \qquad
 R_F=p^*(R_f+\Delta_p-f^*\Delta_p).
\]
Pullback by a finite surjective morphism detects effectiveness of
Weil divisors.  This proves the first inequality.  Intersections
of a Weil divisor with $H^{m-1}$ are cycle intersections.  The
identities $f_*f^*D=(\deg f)D$ and $f^*H=qH$, followed by the
projection formula, give
$\deg_H(f^*D)=q\deg_H D$ for every Weil divisor $D$.
The codimension-one canonical formula
$K_X\sim f^*K_X+R_f$ consequently gives
\[
 (q-1)\Delta_p\cdot H^{m-1}
 \le R_f\cdot H^{m-1}
 =(1-q)K_X\cdot H^{m-1}.
\]
Every nonzero coefficient of $\Delta_p$ is at least $1/2$, proving
the reduced-branch bound.  The same computation with $q^s$ proves
the assertion about iterates.
\end{proof}


\section{Compatible covers along periodic fiber chains}

\begin{definition}[Compatible covers and periodic returns]
\label{def:compatible-cover-return}
A \emph{finite dynamical cover} of $(X,f)$ is a finite surjective
map $p:Y\to X$ from a connected normal variety, together with
$F:Y\to Y$ satisfying $pF=f^e p$ for some $e\geq1$.
It is \emph{Galois} when $k(Y)/k(X)$ is Galois. It is
\emph{Cartesian} when the normalized full fiber product
$X\times_{f^e,X,p}Y$ is connected and the natural map from $Y$
identifies $Y$ with that normalization. The same terminology is
used for the commuting squares in a finite chain.

For an equivariant fibration, a \emph{marked periodic return}
is the induced map $f^\ell$ on the fibers over a local periodic
section $b(s)$ satisfying $g^\ell b(s)=b(s)$. It is \emph{good}
when these normalized fibers form a flat normal geometrically
integral family on a parameter open, the fiber arrows are finite
and polarized, and the required rational diagrams are defined
generically. In spectral arguments its marked reference point
on the base is a smooth repelling Julia point.

The \emph{birational Albanese} of a normal projective variety is
the Albanese variety of a smooth projective resolution, with its
induced rational map from the original variety. This is independent
of the resolution.
\end{definition}


\begin{lemma}[Stable fields and normalized pullbacks]
\label{lem:stable-field-normalized-pullback}
Let $K\subset L$ be a finite separable extension of degree $D$, and let
$\sigma:L\hookrightarrow L$ be a field embedding preserving $K$.
Suppose
\[
                        K\sigma(L)=L.
\]
Then the natural homomorphism
\[
       K\otimes_{\sigma(K)}\sigma(L)\longrightarrow L
\]
is an isomorphism.  Consequently, for a finite cover of normal
varieties with a compatible dynamical lift, this field condition
means that the normalization of the full dynamical pullback is
connected and is the original cover.

Suppose, moreover, that a return arrow is a composition of finite
dominant arrows around a cycle.  If its normalized pullback of a
degree-$D$ cover is connected, then all intermediate normalized
pullbacks are connected and have degree $D$.
\end{lemma}

\begin{proof}
The tensor product has dimension $D$ over $K$, since
$[\sigma(L):\sigma(K)]=D$.  Its image in $L$ is $K\sigma(L)=L$,
which has the same dimension $D$.  The homomorphism is therefore an
isomorphism.  In characteristic zero the generic algebra of any
intermediate pullback is a finite product of separable field
extensions.  If such a product had a nontrivial idempotent, that
idempotent would remain nontrivial under each subsequent faithfully
flat field base change.  The full pullback could not be a field.
Thus each intermediate algebra is a field, and base change preserves
its total rank $D$.
\end{proof}


\begin{lemma}[A fixed path bounds the location of branching]
\label{lem:finite-path-critical-support}
Let $a_i:X_i\to X_{i+1}$, $0\leq i<N$, be finite maps between
normal projective $m$-folds, with ample Cartier polarizations of
equal volume and $a_i^*H_{i+1}\simeq H_i^q$.
Let $p_i:W_i\to X_i$ be connected finite covers such that every
square is a connected normalized pullback square.  Suppose that
their reduced branch divisors satisfy
\[
                         B_i\cdot H_i^{m-1}\leq C.
\]
If $q^N>C$, then every prime component of $B_N$ is a divisorial
critical value of
\[
                         A=a_{N-1}\circ\cdots\circ a_0.
\]
\end{lemma}

\begin{proof}
Suppose that a prime component $D$ of $B_N$ is not a critical
value of $A$.  The map $A$ has ramification index one at every
prime above $D$.  At those codimension-one discrete valuation
rings it is unramified, since the characteristic is zero.
Unramified base change preserves the nontrivial ramification
index of $p_N$ over $D$.  Since the successive pullbacks are
connected, their composition is the full normalized pullback;
therefore every prime in $A^{-1}(D)$ occurs in $B_0$.
All coefficients of $A^*D$ are one, so
\[
 C\geq B_0\cdot H_0^{m-1}
   \geq A^*D\cdot H_0^{m-1}
   =q^N D\cdot H_N^{m-1}\geq q^N,
\]
a contradiction.  The final inequality uses that the degree of
a nonzero effective integral divisor against an ample Cartier
class is a positive integer.
\end{proof}


\begin{lemma}[Finite parameter space for covers with prescribed branch support]
\label{lem:finite-branch-cover-classes}
Let $\pi:\mathcal X\to B$ be a projective family over a smooth
integral complex variety, with normal geometrically integral generic
fiber, and let $V\subset\mathcal X$ be a closed subset whose generic
fiber is a divisor.  Fix $D\geq1$.  After replacing $B$ by a dense open,
the geometric isomorphism classes of connected normal covers
\[
                     W\longrightarrow X_b
\]
of degree at most $D$, unramified away from
$V_b\cup(X_b)_{\mathrm{sing}}$, are the geometric points of a finite
\'{e}tale algebraic space $H\to B$.  The locus of Galois covers is open
and closed. The covers themselves form a locally constant
Deligne--Mumford stack over $B$ with finite geometric fiber
groupoids and finite \'{e}tale coarse space $H$.
\end{lemma}

\begin{proof}
Resolve the generic pair and spread the resolution.  After shrinking
$B$, this gives a smooth projective family $\widetilde{\mathcal X}\to B$
and a relative simple normal crossings divisor $E$, containing the
inverse image of $V$ and all exceptional divisors, such that
\[
 \widetilde{\mathcal X}\setminus E
       \simeq\mathcal X_{\mathrm{reg}/B}\setminus V.
\]
Set $e=\operatorname{lcm}(1,\ldots,D)$.  Form the $e$th root stack along
every component of $E$.  This is a proper smooth Deligne--Mumford stack
$\mathscr X\to B$.  The root-stack form of Abhyankar's lemma
\cite[Appendix~A]{LO10} identifies degree-at-most-$D$ covers of the displayed open with
finite \'{e}tale covers of $\mathscr X$ of the same degree: all inertia
orders are orders of permutations of at most $D$ letters and therefore
divide $e$.

For each $d\leq D$, the stack of degree-$d$ finite \'{e}tale covers is
the stack $\operatorname{Hom}_B(\mathscr X,\mathrm B\mathfrak S_d)$.
It is algebraic by the Hom-stack theorem \cite{Ols06}.  Finite \'{e}tale covers have
unique infinitesimal liftings, and proper smooth invariance of the
geometric fundamental group \cite[Theorem~A.11]{LO10}
makes this a locally constant stack.
Its geometric fiber is a finite groupoid: the fundamental group of a
proper smooth complex Deligne--Mumford stack is topologically finitely
generated, so it has only finitely many actions on a set of $d$ letters.
Its coarse space is consequently finite \'{e}tale over $B$.
Connectedness and the Galois property are locally constant.  Finally,
normalization in the corresponding function field extends a cover
uniquely from the displayed open to $X_b$.  Purity on $(X_b)_{\rm reg}$
gives the inverse construction.

\end{proof}

\begin{lemma}[A finite class correspondence has a periodic branch]
\label{lem:max-degree-cover-class-selection}
Let $B$ be a normal integral projective variety and let
$g:B\to B$ be finite and dominant.  Let
$\pi:\mathcal X\to B^\circ$ be a projective family with normal
geometrically integral fibers over a dense open $B^\circ\subset B$.
Set $U=B^\circ\cap g^{-1}(B^\circ)$, and suppose there is a finite
morphism over $U$
\[
             a:\mathcal X|_U\longrightarrow (g|_U)^*\mathcal X
\]
whose fiber arrows $a_b:\mathcal X_b\to\mathcal X_{g(b)}$ are
dominant.  Let $H\to B^\circ$ be a finite-type quasi-finite
algebraic space parametrizing geometric isomorphism classes of
connected finite covers of these fibers by normal varieties.
Suppose a constructible relation
\[
        \mathcal R\subset H\mathbin{\times}_{B,g}H
\]
has the following pullback property.  For every geometric pair
$(h,h')\in\mathcal R$ over $b\in U$, the normalization of the full
fiber product
\[
             \mathcal X_b
             \mathbin{\times}_{\mathcal X_{g(b)}}W_{h'}
\]
is connected and represents $h$, where
$W_{h'}\to\mathcal X_{g(b)}$ represents $h'$ and the fiber product
uses $a_b$.  In particular, a target class $h'\in H_{g(b)}$
determines at most one source class $h\in H_b$ with
$(h,h')\in\mathcal R$.
Suppose a set of selected vertices has Zariski-dense image in $B$,
and selected edges in $\mathcal R$ have both endpoints among those
vertices.  Assume that the selected sources are dense in every
dominant component of the closure of the selected vertices.
This holds, in particular, for all the vertices and edges of a
collection of finite cyclic paths with dense image in $B$.

Then there are a finite dominant map $p:B'\to B$ from a normal
integral projective variety, an integer $s>0$, and a finite dominant
map $g':B'\to B'$ such that
\[
                  p g'=g^s p,
\]
and a geometric cover class over the geometric generic fiber after
base change to $B'$ which is invariant under the corresponding
$s$-step normalized pullback.  The base square is itself a connected
normalized pullback square; equivalently, for $K=\mathbf C(B)$ and
$K'=\mathbf C(B')$ one has $K(g')^*K'=K'$.
If $g$ is $q$-polarized, $g'$ is
$q^s$-polarized.
\end{lemma}

\begin{proof}
Let $Z$ be the reduced closure in $H$ of the selected vertices and
discard components whose image is not dense in $B$.  Write its
remaining irreducible components as $Z_1,\ldots,Z_t$, and put
\[
                         d_i=[\mathbf C(Z_i):\mathbf C(B)].
\]
Let $R$ be the closure of the selected edges.  For every $i$, some
irreducible component $R_\alpha$ dominates $Z_i$ by the source
projection.  Its generic point belongs to $\mathcal R$, because
$\mathcal R$ is constructible and the selected edges are dense in
$R_\alpha$.  Its target dominates some $Z_j$.

Forgetting the source class maps $R_\alpha$ generically injectively
to $B\times_{g,B}Z_j$, by the uniqueness assumption.  Thus its
function field is that of one component of this base change, and
\[
 d_i\,[\mathbf C(R_\alpha):\mathbf C(Z_i)]
       =[\mathbf C(R_\alpha):\mathbf C(B)]\leq d_j.
\]
Choose a component of maximal $d_i$.  Every dominating outgoing edge
from it has $d_j=d_i$ and source degree one.  Repeating the choice
within the finite set of maximal-degree components eventually gives
a directed cycle.  Each edge of that cycle therefore defines a
dominant rational map $Z_i\dashrightarrow Z_j$ over $g$; their
composition gives a dominant rational self-map of one $Z_i$ over
$g^s$.  Along each edge, the pullback condition on $\mathcal R$
identifies the source cover with the full connected normalized
pullback of the target cover under the corresponding fiber arrow.
Composing these identifications around the cycle gives invariance
of the selected geometric cover class under the $s$-step normalized
pullback.  Composition is compatible with normalized pullback by
associativity of the corresponding tensor products of function fields.
Equality in the degree estimate
also says that the selected component of $g^*Z_j$ accounts for its
entire generic degree.  Thus each base pullback is connected, and so
is their composition around the cycle.  This is exactly the stated
stable-field equality.

Let $B'$ be the normalization of $B$ in $\mathbf C(Z_i)$.  The
function-field embedding defining this rational self-map preserves
the integral closure of every affine coordinate ring under the
corresponding pullback by $g^s$.  It consequently extends to a
morphism $g':B'\to B'$.  Since $p g'=g^s p$ is finite, $g'$ is
proper and quasi-finite, hence finite.  Finally, if $g^*L\simeq L^q$,
then
\[
                    (g')^*p^*L\simeq(p^*L)^{q^s},
\]
and $p^*L$ is ample.
\end{proof}


\begin{lemma}[Conjugate composita descend a selected class]
\label{lem:conjugate-cover-compositum}
Let $k$ be algebraically closed of characteristic zero, let
$K\subset L$ be a regular extension of finitely generated fields over
$k$, and let $\sigma:L\hookrightarrow L$ preserve $K$.  Fix an
algebraic closure $\Omega$ of $L$ and an extension of $\sigma$ to an
automorphism of $\Omega$, preserving the algebraic closure $\overline K$
of $K$ in $\Omega$.  Suppose a finite extension $K'/K$ in
$\overline K$ satisfies
\[
                 \sigma(K')\subset K',\qquad K\sigma(K')=K'.
\]
Let $E/\overline K L$ be finite Galois and satisfy
$\sigma(E)\subset E$.  Suppose its cover class is invariant under
the group $\operatorname{Gal}(\overline K/K')$.
Then the compositum of all conjugates of $E$ under
$\Gamma=\operatorname{Gal}(\Omega/L)$ is a finite Galois extension
$E^{\mathrm c}/\overline K L$ satisfying
\[
 \gamma(E^{\mathrm c})=E^{\mathrm c}\quad(\gamma\in\Gamma),
 \qquad \sigma(E^{\mathrm c})\subset E^{\mathrm c}.
\]
If a smooth projective model of $E$ over $\overline K$ has positive
irregularity, the same holds for $E^{\mathrm c}$.
\end{lemma}

\begin{proof}
Let $K''$ be the normal closure of $K'/K$ inside $\overline K$.
Then $\sigma(K'')\subset K''$ and $K\sigma(K'')=K''$.  Indeed, if
$a$ is a primitive element of $K'/K$ with minimal polynomial $P$,
then $K(\sigma a)=K'$ makes $\sigma(P)$ irreducible over $K$ of
the same degree.  Its splitting field is both $K''$ and
$K\sigma(K'')$; all its roots belong to $K''$.

Put $\Gamma'=\operatorname{Gal}(\Omega/LK'')$.  This is an open
normal subgroup of $\Gamma$.  With
$\theta(\gamma)=\sigma^{-1}\gamma\sigma$, the stable-field equality
shows that $\theta$ induces an automorphism of the finite quotient
$\Gamma/\Gamma'$.  The class-invariance hypothesis and the Galois
property of $E/\overline K L$ imply that $\Gamma'$ normalizes the
subfield $E$: an isomorphism between two normal extensions inside
$\Omega$ forces equality of the fields.  Thus there are finitely many
$\Gamma$-conjugates of $E$, and their compositum is finite Galois
over $\overline K L$ and is normalized by $\Gamma$.

For any $\gamma\in\Gamma$, choose $\gamma'\in\Gamma$ such that
$\theta(\gamma')\Gamma'=\gamma\Gamma'$.  Since $\Gamma'$ normalizes
$E$, this gives $\gamma E=\theta(\gamma')E$, and hence
\[
            \sigma(\gamma E)=\gamma'\sigma(E)\subset\gamma'E.
\]
Taking the compositum proves $\sigma(E^{\mathrm c})\subset
E^{\mathrm c}$.  Finally, the finite dominant rational map from a
smooth projective model of $E^{\mathrm c}$ to one of $E$ makes
pullback on holomorphic one-forms injective, proving the last assertion.
\end{proof}


\subsection{Descent of an irregular geometric covering class}

We work over an algebraically closed field $k$ of characteristic zero.
For a normal projective variety $V$, write
$\operatorname{Alb}_{\mathrm{bir}}(V)$ for the Albanese variety of a
smooth projective model.  Its torsor and its first rational \(\ell\)-adic
cohomology are birational invariants.
For a polarized abelian variety $(A,\lambda)$, the notation
$\operatorname{Aut}_{\mathrm{gp}}(A,\lambda)$ means the group of
polarization-preserving group automorphisms; every member fixes
the origin.  Affine automorphisms of an Albanese torsor will always
be distinguished from this group.

\begin{lemma}[Constancy of self-isogenous abelian moduli]
\label{lem:cover-class-abelian-moduli}
Let $g\colon B\to B$ be a $q$-polarized endomorphism of a normal
projective variety, with $q>1$, and put $K=k(B)$ and $\sigma=g^*$.
Let
\[
 \nu\colon B\dashrightarrow\mathcal A_{a,\delta}
\]
be a coarse moduli map for polarized abelian varieties of fixed
dimension and polarization type.  Suppose that the geometric abelian
varieties represented by $\nu$ and $\nu\circ g$ are isogenous over
$\overline K$.  Then $\nu$ is constant.
\end{lemma}

\begin{proof}
The assertion is immediate when $\dim B=0$, so assume $\dim B>0$.
The minimal compactification of $\mathcal A_{a,\delta}$ carries an
ample rational Hodge line bundle.  Take an ample integral power
$\mathcal L$.  If $\nu$ is nonconstant, a general section of a high
power of $\mathcal L$ pulls back to a nonzero effective Weil divisor
$D$ on $B$.  Here the pullback is first taken outside the
indeterminacy locus, which has codimension at least two, and then
closed in $B$.

Choose an ample Cartier divisor $H$ with $g^*H\sim qH$, and put
$b=\dim B$.  A general complete-intersection curve in a high multiple
of $H$ avoids the indeterminacy loci of both moduli maps.  After a
finite cover of this curve, the two abelian families exist, their
generic isogeny exists, and both have semistable reduction.  Their
extended Hodge bundles have the same degree.  Indeed, an isogeny and
a quasi-inverse extend to the N\'eron models; the composites on
invariant differentials are multiplication by a nonzero integer,
which is a unit in characteristic zero.  Thus these differential
bundles are isomorphic.  Dividing by the degree of the finite cover
gives
\[
 (g^*D\cdot H^{b-1})=(D\cdot H^{b-1}).
\]
The finite projection formula and $g^*H\sim qH$ give, on the other
hand,
\[
 (g^*D\cdot H^{b-1})=q(D\cdot H^{b-1}).
\]
Hence $D\cdot H^{b-1}=0$, contrary to ampleness and $D\ne0$.
\end{proof}

\begin{theorem}[An irregular Galois covering class gives an independent factor]
\label{thm:irregular-cover-class-descent}
Let $X$ and $B$ be normal projective varieties, let
$f\colon X\to X$ and $g\colon B\to B$ be polarized endomorphisms,
and let $\pi\colon X\dashrightarrow B$ be an equivariant dominant
rational map with geometrically integral generic fiber $Y$.
Put
\[
 K=k(B),\qquad L=k(X)=K(Y),\qquad \sigma=f^*\colon L\hookrightarrow L.
\]
Fix an algebraic closure $\Omega$ of $L$, and let $\overline K$ be
the algebraic closure of $K$ in $\Omega$.  Suppose there is a finite
Galois extension
\[
 E/\overline K L
\]
with the following properties.
\begin{enumerate}
\item The associated connected geometric cover
$W\to Y_{\overline K}$ has an isomorphism class invariant under
$\operatorname{Gal}(\overline K/K)$.
\item There is an automorphism $\widetilde\sigma$ of $\Omega$
extending $\sigma$ such that
\[
 \widetilde\sigma(E)\subset E.
\]
\item $\dim\operatorname{Alb}_{\mathrm{bir}}(W)>0$.
\end{enumerate}
Then there are an abelian variety $A$ over $k$, a finite group $H$
acting affinely on $A$, a polarized endomorphism $h$ of $A/H$, and
an equivariant dominant rational map
\[
 \psi\colon X\dashrightarrow A/H
\]
whose restriction to the geometric generic fiber of $\pi$ is
nonconstant.  In particular, $\psi$ is an exceptional factor
independent of the given phase base.
\end{theorem}

\begin{proof}
Write $f^*H_X\sim qH_X$, with $H_X$ ample and $q>1$.
If $B$ has positive dimension, polarization descent and the degree
formula of Meng--Zhang, Theorem~1.3, give the same polarization
weight $q$ on $B$.
Replacing $X$ by the normalization of the graph of $\pi$ makes
$\pi$ a morphism.  This replacement admits the induced finite
endomorphism and a polarization obtained from polarizations on
$X$ and $B$; the function fields in the statement do not change.
Thus the generic fiber and its normalizations in finite extensions
may be taken projective.

\medskip
\noindent\emph{The arithmetic action.}
Set
\[
 \Gamma=\operatorname{Gal}(\Omega/L),\qquad
 \Gamma_{\mathrm{geo}}=\operatorname{Gal}(\Omega/\overline K L),
 \qquad N=\operatorname{Gal}(\Omega/E).
\]
Geometric integrality gives the exact sequence
\[
 1\longrightarrow\Gamma_{\mathrm{geo}}
 \longrightarrow\Gamma
 \longrightarrow\operatorname{Gal}(\overline K/K)
 \longrightarrow1.
\]
The subgroup $N$ is open and normal in $\Gamma_{\mathrm{geo}}$.
It need not be open in $\Gamma$.

For $\gamma\in\Gamma$, the fields $\gamma(E)$ and $E$ represent
isomorphic covers over $\overline K L$, by the first hypothesis.
Since $E/\overline K L$ is Galois, two such embedded copies are
equal.  Hence $\gamma(E)=E$ and $N$ is normal in $\Gamma$.
Consequently $\Gamma/N$ acts semilinearly on $W$ and, functorially,
on
\[
 J=\operatorname{Alb}_{\mathrm{bir}}(W).
\]
Fix a prime $\ell$ and put
\[
 V=T_\ell(J)\otimes_{\mathbf Z_\ell}\mathbf Q_\ell.
\]
These semilinear automorphisms induce a genuine representation
\[
 r\colon\Gamma\longrightarrow\operatorname{GL}(V),
 \qquad N\subset\ker r.
\]
On $\Gamma_{\mathrm{geo}}/N=\operatorname{Gal}(E/\overline K L)$,
this is exactly the linear action of the deck group on $J$.
We retain the kernel of this deck action in the descent.

The representation is continuous.  Indeed, the cover and its
Albanese descend to a finite extension of $K$.  At every finite
torsion level, all the data and the finitely many torsion points
are defined over a finite extension of $L$.  The subgroup fixing
that extension acts trivially at that level, so the corresponding
kernel is open.

\medskip
\noindent\emph{Constancy of the Albanese and finiteness of the image.}
The polarization of $X$ induces an ample class on $Y$, whose
pullback to $W$ induces a polarization on $J$.  One construction
uses the intersection pairing with the $(\dim W-1)$-st power of
this class on a resolution.  The pulled-back class is nef and big;
the resulting Hermitian form on nonzero holomorphic one-forms is
positive, since it is positive on the dense open set where the
resolution is an isomorphism.  This construction is birationally
invariant and invariant under isomorphisms of the cover.  Thus the
polarized coarse moduli point of $J$ is fixed by
$\operatorname{Gal}(\overline K/K)$ and defines a moduli map over
$K$.

Because $K/g^*K$ is finite,
$\widetilde\sigma(\overline K)=\overline K$.  The second hypothesis
gives a dominant semilinear rational arrow
\[
 W\dashrightarrow\widetilde\sigma^*W.
\]
It induces an isogeny
\[
 u\colon J\longrightarrow\widetilde\sigma^*J.
\]
Indeed, its image contains the target Albanese image, which
generates the target Albanese variety, and the source and target
dimensions are equal.  Lemma~\ref{lem:cover-class-abelian-moduli}
therefore makes the polarized moduli of $J$ constant in the phase
base.

Here is the finite-image argument, including the necessary field
of definition.  Choose a finite extension $K_1/K$ and a projective
model $W_1/K_1$ of the cover, with function field $E_0$ finite over
$L$, such that
\[
 E=\overline K E_0.
\]
After enlarging $K_1$ and $E_0$ by finite extensions, the polarized
Albanese of $W_1$ is isomorphic to $A_0\times_k K_1$ for a fixed
polarized abelian variety $A_0/k$.  The group
\[
 \operatorname{Gal}(\Omega/E_0)
\]
is an open subgroup of $\Gamma$.  Its elements need not fix $E$
pointwise.  They fix the model over $E_0$ and act on its geometric
Albanese through the constant model $A_0$; all torsion points of
$A_0$ lie in the algebraically closed field $k$.  They therefore
act trivially on $V$.  Thus $r$ has finite image.

Equivalently, the residual polarized twisting takes values in
$\operatorname{Aut}_{\mathrm{gp}}(A_0,\lambda_0)$.  This group is
finite, since its integral
homology representation preserves a positive definite Hermitian
form and hence lies in the intersection of a discrete integral
matrix group with a compact group.  The action of this
automorphism group on rational Tate homology is faithful.

\medskip
\noindent\emph{The invariant kernel.}
Define
\[
 \theta(\gamma)=\widetilde\sigma^{-1}\gamma\widetilde\sigma.
\]
For $a\in L$ and $\gamma\in\Gamma$, the element
$\widetilde\sigma(a)$ lies in $L$ and is fixed by $\gamma$.
Hence $\theta(\gamma)(a)=a$, so $\theta$ maps $\Gamma$ into itself.

The isogeny $u$, with the standard semilinear identification of
Tate modules, gives an invertible linear operator $P$ on $V$.
Functoriality and
$\gamma\widetilde\sigma=\widetilde\sigma\theta(\gamma)$ give
\[
 r(\theta(\gamma))P=P\,r(\gamma).
\]
For example, with the convention that the semilinear automorphism
attached to $\gamma$ has comorphism $\gamma^{-1}$, the corresponding
identity of rational maps is
$F\gamma=\theta(\gamma)F$, which gives precisely this formula.
Since $P$ is invertible,
\[
 \theta^{-1}(\ker r)=\ker r.
\]
Thus the forward inclusion $\widetilde\sigma(E)\subset E$
gives the invariant kernel needed for descent.

Put $R=\ker r$ and $L'=\Omega^R$.  Then $L'/L$ is finite Galois.
For $a\in L'$ and $\gamma\in R$,
\[
 \gamma\widetilde\sigma(a)
 =\widetilde\sigma\theta(\gamma)(a)
 =\widetilde\sigma(a),
\]
so $\widetilde\sigma(L')\subset L'$.
Let $p\colon Z\to X$ be the normalization in $L'$.  Normality
extends the field embedding to a regular finite endomorphism
$F_Z\colon Z\to Z$, and
\[
 pF_Z=fp,\qquad F_Z^*p^*H_X\sim q\,p^*H_X.
\]
The induced map on the finite group $\Gamma/R$ is injective,
hence an automorphism.  Accordingly the deck action satisfies
$F_Z\eta=\delta(\eta)F_Z$ for an automorphism $\delta$ of the
deck group.

\medskip
\noindent\emph{The geometric fiber and its constant Albanese.}
Let $K'$ be the relative algebraic closure of $K$ in $L'$, and
normalize $B$ in $K'$.  The geometric generic fiber of $Z$ over
this new phase base has function field $\overline K L'$.
It is birational to
\[
 W/G_0,\qquad
 G_0=(R\cap\Gamma_{\mathrm{geo}})/N.
\]
Every element of $G_0$ acts trivially on $V$.  Faithfulness of the
Tate representation of homomorphisms of abelian varieties implies
that its linear action on $J$ is the identity.  Its action on the
Albanese torsor is consequently a translation.  The translation
images form a finite subgroup $T\subset J$.

The universal property of the Albanese shows that the Albanese
torsor of $W/G_0$ is the quotient of the Albanese torsor of $W$
by $T$, and its abelian variety is $J/T$.  Indeed, maps from
$W/G_0$ to abelian torsors correspond to the maps from $W$ whose
induced homomorphisms kill $T$.  In particular this Albanese has
positive dimension, and its rational Tate module identifies with
$V$.

Let $J'/K'$ be the Albanese of the generic fiber of $Z$ over the
new base.  The exact sequence for $L'/K'$ identifies its arithmetic
Galois action with the action induced by $R$.  This action on
$V$ is trivial.  Moreover $J'$ is isotrivial: geometrically it is
the quotient of the constant $J$ by a finite subgroup, whose
torsion points are constant.  A polarized trivialization of $J'$
over a finite extension has a descent cocycle in
$\operatorname{Aut}_{\mathrm{gp}}(A',\lambda')$ for its constant
polarized model $(A',\lambda')$.  The cocycle acts
trivially on the rational Tate module, so faithfulness makes it
trivial.  Thus
\[
 J'\simeq A'\times_k K'
\]
already over $K'$, for an abelian variety $A'/k$ of positive
dimension.

The Albanese torsor of the generic fiber has finite order in
$H^1(K',A')$.  A closed point splits it over a finite extension,
and restriction--corestriction kills its class by the degree of
that extension.  Pushing out by a suitable multiplication map
therefore gives a rational map
\[
 Z\dashrightarrow A'
\]
which is nonconstant on the geometric generic phase fiber.  This
map induces multiplication by that same integer on the Albanese
$J'=A'_{K'}$ of the generic fiber.  This map need not intertwine
the dynamics.

\medskip
\noindent\emph{Global Albanese and descent to $X$.}
Apply the birational Albanese theorem of Meng--Zhang
(Corollary~1.4 of \emph{Building blocks of polarized endomorphisms
of normal projective varieties}) to $(Z,F_Z)$.  The map
\[
 a\colon Z\dashrightarrow A:=\operatorname{Alb}_{\mathrm{bir}}(Z)
\]
is dominant, and the induced affine endomorphism of $A$ is
polarized.  The preceding map $Z\dashrightarrow A'$ factors
through $a$, so $a$ is nonconstant on the geometric generic
phase fiber.  More precisely, if $j\colon J'\to A_{K'}$ is
the homomorphism induced by the generic-fiber inclusion, the
factorization supplies a homomorphism $b\colon A\to A'$ with
$b_{K'}\circ j=[n]$ for some $n>0$.  Thus $j$ has finite kernel.

The finite deck group of $Z\to X$ acts affinely on $A$, and
$F_Z\eta=\delta(\eta)F_Z$ descends to the corresponding identity
there.  Consequently $a$ induces an equivariant dominant rational
map
\[
 \psi\colon X\dashrightarrow A/H,
\]
where $H$ is the image of the deck group on $A$.  The induced
endomorphism of $A/H$ is polarized by Theorem~1.3 of the same
paper.  The polarized affine endomorphism on $A$ has a fixed
point, since its linear part minus the identity is an isogeny;
translation by that point conjugates it to a polarized isogeny.
Thus the factor is exceptional.  The finite quotient
$A\to A/H$ cannot contract a
positive-dimensional image of a generic phase fiber.  Hence
$\psi$ is nonconstant on that fiber, as required.
\end{proof}


\begin{theorem}[Gluing irregular covers of periodic fibers]
\label{thm:relative-irregular-factor-gluing}
Let $F:X\to X$ and $g:B\to B$ be $q$-polarized endomorphisms of
normal complex projective varieties, with $q>1$, and let
$\pi:X\to B$ be a dominant equivariant morphism with geometrically
integral generic fiber.  Fix a dense open $B_0\subset B$ over which
the fibers are normal and geometrically integral of dimension $m>0$.
Suppose a Zariski-dense set $\mathcal P\subset B_0$ of periodic points
has the following property.  For each $b\in\mathcal P$, some return
$F^{n_b}|_{X_b}$ has a connected finite Galois cover
\[
                      p_b:W_b\longrightarrow X_b
\]
of degree at most one fixed integer $D$, of positive birational
irregularity, whose lifted return is a connected normalized pullback
square.  Equivalently, its function fields satisfy
\[
 \mathbf C(X_b)\,(F_{W_b}^{n_b})^*\mathbf C(W_b)
                           =\mathbf C(W_b).
\]
Then an iterate of $F$ has an exceptional rational factor
\[
                    \rho:X\dashrightarrow Z
\]
which is nonconstant on the geometric generic $\pi$-fiber.
The factor can be enlarged to one for $F$ itself, retaining this
nonconstancy.
\end{theorem}

\begin{proof}
Let $H$ polarize $F$.  Shrink $B_0$ so that the ample volumes of
$H_b=H|_{X_b}$ and the numbers $K_{X_b}\cdot H_b^{m-1}$ are constant.
For the latter, one may use the genus of a general complete-intersection
curve avoiding the singular locus.  Restricting a polarized return to
a fiber, and then lifting it to a finite cover, preserves its
polarization.  Proposition~\ref{prop:bounded-branch-dynamical-cover}
therefore gives a uniform constant $C\geq0$ such that every selected
cover above a point of $B_0$ has reduced branch degree at most $C$.

\emph{Complete chains, including phases outside the good open.}
For a selected $b$, consider the normalizations of the successive
closed images $F^i(X_b)$, with $0\leq i<n_b$.
These are projective $m$-folds, and the induced arrows are finite
and surjective.  They close up because $F^{n_b}(X_b)=X_b$.
Take successive normalized pullbacks of $p_b$ backwards along this
chain.  The stable-field condition and
Lemma~\ref{lem:stable-field-normalized-pullback} imply that all these
pullbacks are connected, Galois, and of the same degree.  They form
a cyclic chain of normalized Cartesian squares.  Each has a
polarized return, and each has positive birational irregularity,
since a partial lifted cycle maps it finitely onto $W_b$.

At a phase $c\in B_0$, the image in question is a closed $m$-dimensional
subvariety of the irreducible $m$-fold $X_c$, and is therefore the whole
fiber.  Thus every state of this cyclic chain whose phase lies in
$B_0$ is an ordinary cover of $X_c$ with the same degree and branch
bounds.  At a phase outside $B_0$ no bound is needed; the intermediate
normalization still supplies the required finite arrow.

\emph{One branch divisor and a finite space of classes.}
Choose $N$ with $q^N>C$.  Form the relative critical-value divisor
$V_N$ of the $N$-step arrow.  After shrinking a dense open
$B_1\subset B_0$, its formation commutes with the fibers, and every
predecessor under $g^N$ of a point of $B_1$ belongs to $B_0$.
For any state over $b\in B_1$, use the predecessor of $b$ in its own
cyclic chain.  The first and last fibers of this length-$N$ path are
good fibers of equal ample volume.  Its full normalized pullback is
connected, and the source branch degree is at most $C$.
Lemma~\ref{lem:finite-path-critical-support} puts the terminal branch
inside $(V_N)_b$.  The proof of that lemma uses only the source bound
and the full length-$N$ arrow, so none of the intervening phases is
required to belong to $B_0$.

By Lemma~\ref{lem:finite-branch-cover-classes}, after another shrink
there is a finite \'{e}tale coarse parameter space $\mathcal H\to B_1$
for the Galois covers of degree at most $D$ with this branch support.
Retain every state of every selected cyclic chain whose phase lies
in $B_1$, and let $Z_i$ be the dominant components of their reduced
closure in $\mathcal H$.  All these components have positive
birational irregularity generically.  To check this, take a finite
atlas of the cover stack, resolve the generic cover, and spread its
resolution to a smooth projective family.  On a dense open its
$h^1(\mathcal O)$ is constant.  The selected positive-irregularity
classes are dense in each retained component.

Take the closure of the one-step edges for which both phase points
are in $B_1$.  Every dominant $Z_i$ has a dominating outgoing edge:
its selected vertices are dense, and discarding those above
$g^{-1}(B\setminus B_1)$ removes only a proper closed subset.
The successor of every remaining selected vertex is retained, since
the complete cyclic chain was retained before restriction.
On $U_1=B_1\cap g^{-1}(B_1)$, the restrictions
\[
                  a_b=F|_{X_b}:X_b\longrightarrow X_{g(b)}
\]
form finite dominant fiber arrows.  Let
$\mathcal R\subset\mathcal H\mathbin{\times}_{B,g}\mathcal H$
consist of those pairs for which the source cover is the full
connected normalized pullback of the target cover along $a_b$.
This relation is constructible, and a target class determines its
source class uniquely.
Consequently the maximum-degree argument of
Lemma~\ref{lem:max-degree-cover-class-selection} applies.  Notice that
this argument needs dense outgoing edges, as just verified; it does
not need a single entire selected cycle to remain in the final open.

We obtain a finite phase extension $K'/K$, where $K=\mathbf C(B)$,
and an iterate with $K\sigma(K')=K'$, together with a dynamically
invariant geometric Galois cover class of positive irregularity.
Here $\sigma$ denotes the induced embedding for that iterate.
Lemma~\ref{lem:conjugate-cover-compositum} replaces it by the compositum
of its finitely many arithmetic conjugates.  This cover class is
invariant over the original field $K$, is preserved by $\sigma$, and
still has positive birational irregularity.

\emph{From the class to an absolute factor.}
Apply Theorem~\ref{thm:irregular-cover-class-descent} to this geometric
cover of the generic $\pi$-fiber.  Its first-cohomology representation
produces an actual finite equivariant Galois cover of $X$; the global
birational Albanese of that cover descends to an exceptional
quotient-of-abelian factor of an iterate of $F$.  The theorem also
gives the required nonconstancy on the geometric generic fiber.
Finally Proposition~\ref{prop:iterate-exceptional-factor-original}
enlarges this to an exceptional factor for $F$ itself.  Its factor
field contains the preceding one, so it remains nonconstant there.
\end{proof}


\section{Periodic observations and the constant quotient}


\subsection{Rational first integrals of the Betti foliation}

Let $S$ be a smooth connected complex algebraic curve and let
$p\colon\mathcal A\to S$ be a polarized abelian scheme.
All statements permit replacing $S$ by a nonempty open subset
and then by a finite cover.  A constant quotient means a
surjective homomorphism of abelian schemes
\[
 c\colon\mathcal A\longrightarrow C\times S,
\]
where $C$ is an abelian variety over $\mathbf C$.  The maximal
such quotient, in the isogeny category, is dual to the maximal
constant abelian subvariety of $\mathcal A^\vee$.

\begin{lemma}[Betti first integrals]
\label{lem:betti-first-integral}
Let $F\in\mathbf C(\mathcal A)$ be a rational function.  Suppose
that, on a simply connected analytic disc $\Delta\subset S$,
$F$ is constant along every Betti leaf on which it is defined.
Then there are a constant quotient $c\colon\mathcal A\to C\times S$
and a rational function $F_0\in\mathbf C(C)$ such that
\[
 F=F_0\circ\operatorname{pr}_C\circ c.
\]
In particular, if the geometric generic fiber of $\mathcal A$
has zero $\mathbf C$-trace, then $F$ is constant.
\end{lemma}

\begin{proof}
We first record the analytic calculation.  Choose normalized
period coordinates on $\Delta$, so that locally on the universal
cover of the fibers
\[
 z=x+\tau(s)y,\qquad x,y\in\mathbf R^a,
\]
where $\tau$ is holomorphic and $\operatorname{Im}\tau>0$.
For a polarization of arbitrary type the first period block is
a fixed diagonal matrix; absorbing this matrix into $x$ gives
the displayed formula.  Write $\Phi(s,z)$ for the meromorphic
lift of $F$.  Constancy along Betti leaves gives, away from the
polar locus,
\begin{equation}
 \partial_s\Phi(s,z)+d_z\Phi(s,z)\,\tau'(s)y=0.
 \label{eq:betti-first-derivative}
\end{equation}
At fixed $s$ one has
\[
 y=(\tau(s)-\overline{\tau(s)})^{-1}(z-\bar z).
\]
Differentiate \eqref{eq:betti-first-derivative} with respect to
$\bar z$.  Since $\Phi$ is meromorphic in $z$, this gives
\[
 d_z\Phi(s,z)\,\tau'(s)
 (\tau(s)-\overline{\tau(s)})^{-1}=0.
\]
Consequently
\begin{equation}
 d_z\Phi(s,z)\,\tau'(s)=0,
 \qquad \partial_s\Phi(s,z)=0.
 \label{eq:betti-period-annihilation}
\end{equation}

Let $K_\eta$ be the identity component of the translation
stabilizer of the rational function $F_\eta$ on the generic
abelian variety.  This is an abelian subvariety: one can take
the stabilizer of the closure of the graph of $F_\eta$ in
$\mathcal A_\eta\times\mathbf P^1$ under translation on the
first factor.  After shrinking $S$, it extends to an abelian
subscheme $\mathcal K\subset\mathcal A$.  Form
\[
 q\colon\mathcal A\longrightarrow\mathcal Q
       :=\mathcal A/\mathcal K.
\]
The function $F$ descends to a rational function $\bar F$ on
$\mathcal Q$, and the translation stabilizer of $\bar F_s$ is
finite for general $s$.  Indeed, a positive-dimensional
connected stabilizer in the quotient would pull back to a
connected translation stabilizer of $F_\eta$ strictly larger
than $K_\eta$.

A homomorphism of abelian schemes respects Betti leaves.
The quotient function $\bar F$ is therefore constant along
Betti leaves of $\mathcal Q$.  Apply
\eqref{eq:betti-period-annihilation} to normalized period
coordinates on $\mathcal Q$.  Every column of its period
derivative is a constant vector field annihilating $\bar F_s$
identically.  Such a vector field lies in the Lie algebra of
the translation stabilizer of $\bar F_s$: its local translation
flow preserves the meromorphic function.  This Lie algebra is
zero.  Hence the period derivative of $\mathcal Q$ vanishes.

It follows that the polarized moduli map of $\mathcal Q$ is
constant on $\Delta$, hence on $S$.  After a finite cover,
$\mathcal Q=C\times S$.  One way to see the finiteness is to
choose a polarized isomorphism with a fixed fiber; the residual
monodromy lies in
$\operatorname{Aut}_{\mathrm{gp}}(C,\lambda)$, the finite group
of polarization-preserving group automorphisms fixing the
origin.  The second equality in
\eqref{eq:betti-period-annihilation}, or directly constancy on
constant sections, now shows that $\bar F$ belongs to
$\mathbf C(C)$.

Finally, a positive-dimensional isotrivial quotient yields a
positive-dimensional isotrivial abelian subvariety after an
isogeny, by Poincar\'e reducibility.  Thus geometric trace zero
forces $C$ to have dimension zero.
\end{proof}

\begin{lemma}[Dense periodic Betti labels]
\label{lem:dense-periodic-betti}
Let $\alpha\colon\mathcal A\to\mathcal A$ be a family of
$q$-polarized isogenies, with $q>1$.  On a simply connected
disc the local sections fixed by some iterate of $\alpha$
are Betti leaves, and their Betti labels are dense in the
real torus of labels.
\end{lemma}

\begin{proof}
The action on integral first homology is a constant integer
matrix $M$.  All its complex eigenvalues have modulus
$\sqrt q>1$: the linear action of a polarized isogeny scales
the positive definite Hermitian form of the polarization by
$q$.  Consequently
\[
 \lVert(M^n-I)^{-1}\rVert\longrightarrow0.
\]
This implication only uses that every eigenvalue has modulus
greater than one; it also follows from the Jordan normal form.
The labels of the fixed points of $\alpha^n$ are
\[
 (M^n-I)^{-1}\mathbf Z^{2a}/\mathbf Z^{2a}.
\]
For any $v\in\mathbf R^{2a}$ choose an integer vector $k_n$
within a fixed bounded distance of $(M^n-I)v$.  Then
$(M^n-I)^{-1}k_n\to v$.  These labels are independent of $s$,
which proves both assertions.
\end{proof}

\begin{corollary}[Periodic observations descend to the constant quotient]
\label{cor:periodic-observations}
Under the hypotheses of Lemma~\ref{lem:dense-periodic-betti},
let $F\colon\mathcal A\dashrightarrow M$ be a rational map to
a fixed complex algebraic variety.  Suppose that for every
local periodic section $u$ on which $F$ is generically defined,
$F\circ u$ is constant.  Then, after a finite cover of $S$,
$F$ factors through a constant abelian quotient of
$\mathcal A$.  If the geometric generic fiber has zero
$\mathbf C$-trace, $F$ is constant.
\end{corollary}

\begin{proof}
First take $M=\mathbf P^1$.  Away from the indeterminacy and
polar loci, the derivative of $F$ in a Betti-leaf direction
depends continuously on the Betti label.  It vanishes for the
dense set of periodic labels, hence for every label.
Lemma~\ref{lem:betti-first-integral} applies.

In general, choose rational coordinates on the closure of the
image of $F$ and apply that lemma to their pullbacks.  There
are finitely many resulting constant quotients.  Their product
is again constant after a finite cover; the image of
$\mathcal A$ in this product is a constant abelian subvariety,
because the corresponding integral homology sublattice is
locally constant.  This image is the required constant quotient.
\end{proof}


\subsection{Finite chains with constant return}

\begin{lemma}[Simultaneous triviality of a finite chain]
\label{lem:constant-return-finite-chain}
Let $S$ be a smooth connected complex algebraic curve.  Let
$\mathcal X_i\to S$, $0\leq i<n$, be families of normal projective
varieties, and let
\[
 A_i:\mathcal X_i\longrightarrow\mathcal X_{i+1},
 \qquad \mathcal X_n=\mathcal X_0,
\]
be finite surjective $S$-morphisms.  If the return map
$R=A_{n-1}\cdots A_0$ is isotrivial, then, after a finite base change
and restriction to a nonempty open subset, the whole diagram is
constant.  In particular every arrow has constant isomorphism class
under independent source and target isomorphisms.
\end{lemma}

\begin{proof}
Make the finite base change trivializing the return, and write
$(\mathcal X_0,R)=(X_0,R_0)\times S$.  Put $K=\mathbf C(S)$,
$E=\mathbf C(X_0)$ and $D=R_0^*E\subset E$.  The extension $E/D$
is finite and separable.  Let $\widetilde E/D$ be its finite Galois
closure, with group $G$, and let $G_0$ fix $E$.

Extension of constants from $\mathbf C$ to $K$ preserves this Galois
closure and its group.  Indeed, function fields of integral complex
varieties are geometrically integral over the algebraically closed
field $\mathbf C$.  Thus $\widetilde E K/DK$ is Galois with group
$G$, and $EK$ is its $G_0$-fixed field.  Every intermediate field
of $EK/DK$ is therefore $(\widetilde E^J)K$ for a subgroup
$G_0\subset J\subset G$.  In particular there are only finitely
many, and each is obtained by extending constants from $\mathbf C$.

Define the prefixes and suffixes
\[
 \pi_i=A_{i-1}\cdots A_0:X_0\times S\longrightarrow\mathcal X_i,
 \qquad
 \rho_i=A_{n-1}\cdots A_i:\mathcal X_i\longrightarrow X_0\times S.
\]
The field $L_i=\pi_i^*K(\mathcal X_i)$ lies between $DK$ and $EK$.
Write $L_i=L_{i,0}K$ with $D\subset L_{i,0}\subset E$, and let
$Y_i$ be the normalization of $X_0$ in the extension $L_{i,0}/D$,
where $D$ is identified with $\mathbf C(X_0)$ using $R_0^*$.
The normal variety $\mathcal X_{i,K}$ is finite over $X_{0,K}$
via $\rho_i$, so uniqueness of normalization gives
\[
 \mathcal X_{i,K}\simeq (Y_i)_K.
\]
The inclusions $L_{i+1}\subset L_i$ are extensions of the inclusions
$L_{i+1,0}\subset L_{i,0}$.  Hence all arrows are the base changes
of the finite morphisms determined by these inclusions.

These finitely many generic-fiber isomorphisms and their inverses
spread over a nonempty open subset of $S$ and satisfy the required
commuting identities there.  Equivalently, after making the total
spaces normal, the finite maps $\rho_i$ identify them directly
with the normalizations of $X_0\times S$ in $L_{i,0}K$, namely
$Y_i\times S$.
\end{proof}

\begin{lemma}[Polarized refinement]
\label{lem:constant-return-polarized-chain}
In Lemma~\ref{lem:constant-return-finite-chain}, suppose ample
relative line bundles $H_i$ satisfy
\[
 A_i^*H_{i+1}\simeq H_i^{\otimes q},\qquad q>1.
\]
Then the simultaneous trivialization can be chosen so that the
polarized arrows are constant, after shrinking $S$.
\end{lemma}

\begin{proof}
First use Lemma~\ref{lem:constant-return-finite-chain} to make the
varieties and arrows constant.  The N\'eron--Severi class of
$H_{0,s}$ is constant.  Since
$R_0^*H_{0,s}=H_{0,s}^{q^n}$, its variation in
$\operatorname{Pic}^0(X_0)$ belongs to
\[
 \ker\bigl(R_0^*-[q^n]\bigr).
\]
This kernel is finite: the pullback of a $Q$-polarized endomorphism
on $\operatorname{Pic}^0$ has analytic eigenvalues of absolute value
$\sqrt Q$, so $Q=q^n>1$ is not an eigenvalue.  Thus $H_{0,s}$ is
constant in the Picard scheme.  A line bundle from $S$ can be removed
after shrinking $S$.

For each $i$, the now constant prefix satisfies
\[
 \pi_i^*H_{i,s}=H_0^{q^i}.
\]
The kernel of $\pi_i^*:\operatorname{Pic}(Y_i)\to
\operatorname{Pic}(X_0)$ is finite.  The norm shows that it is killed
by $\deg\pi_i$; the torsion subgroup of any fixed order in the Picard
group of a normal projective complex variety is finite.  Therefore
$H_{i,s}$ is constant as well.  This proves constancy of the arrows
with their source and target polarizations.
\end{proof}


\subsection{A rational moduli map for polarized fiber arrows}

\begin{proposition}[A separating arrow-moduli map]
\label{prop:polarized-arrow-moduli}
Let $B$ be an irreducible complex algebraic variety, let
$g\colon B\dashrightarrow B$ be dominant, and let
$\pi\colon\mathcal X\to B$ be a projective family with
geometrically integral normal generic fiber.  Let $H$ be a
relatively ample line bundle.  Suppose that, over a nonempty
open subset of $B$, there is a finite surjective fiber arrow
\[
 a_b\colon X_b\longrightarrow X_{g(b)}
\]
varying algebraically with $b$, and satisfying
\[
 a_b^*H_{g(b)}\simeq H_b^{\otimes q},\qquad q>1.
\]
Then there exist an integer $N>0$, a complex algebraic variety
$M$, and a rational map
\[
 \mu\colon B\dashrightarrow M
\]
with the following property on a nonempty open subset $B^\circ$:
\[
 \mu(b)=\mu(c)
\]
if and only if there are polarized isomorphisms
\[
 u\colon(X_b,H_b^N)\xrightarrow{\sim}(X_c,H_c^N),
 \qquad
 v\colon(X_{g(b)},H_{g(b)}^N)
       \xrightarrow{\sim}(X_{g(c)},H_{g(c)}^N)
\]
such that
\[
 v\circ a_b=a_c\circ u.
\]
The two isomorphisms are independent source and target
coordinate changes.
\end{proposition}

\begin{proof}
Restrict to the flat normal-fiber locus and take a sufficiently
high power $L=H^N$ so that $E=\pi_*L$ is locally free of rank $r$
and gives the relative complete embedding. Write
$p(t)=\chi(X_b,L_b^t)$. A graph of $a_b$ has Hilbert polynomial
$p((q+1)t)$ for $\mathcal O(1,1)$, since
$L_b\otimes a_b^*L_{g(b)}\simeq L_b^{q+1}$.
Independent frames of $E$ and $g^*E$ give a map to
\[
 \mathcal H=\operatorname{Hilb}_{p}(\mathbf P^{r-1})^2
 \times\operatorname{Hilb}_{p((q+1)t)}
       (\mathbf P^{r-1}\times\mathbf P^{r-1}).
\]
Let $G=\operatorname{PGL}_r\times\operatorname{PGL}_r$ and take
the closure of the $G$-saturation of the image. Rosenlicht's
theorem \cite[Theorem~2]{Ros56} gives a geometric quotient of a
$G$-invariant dense open of this closure. The resulting rational
map $\mu:B\dashrightarrow M$ is independent of the chosen frames.
Its geometric fibers are exactly the $G$-orbits of graph triples.
Such an orbit records precisely the polarized source and target
isomorphisms commuting with the arrow, because the embeddings
use complete linear systems. This proves the assertion.
\end{proof}


\subsection{Trace-zero abelian factors and their joins}

We work over $k=\overline{\mathbf C(S)}$, retaining the indicated
field of constants $\mathbf C$.  For an abelian variety $A/k$, write
$A_{\mathrm{var}}$ for its quotient by the image of its
$\mathbf C$-trace.  Up to isogeny this is the complementary factor
to the largest constant abelian subvariety.  It has zero
$\mathbf C$-trace.  These constructions are preserved by isogenies
and by all homomorphisms of abelian varieties.

\begin{lemma}[Variation survives the irregular-cover construction]
\label{lem:irregular-extraction-preserves-variation}
In the construction of Theorem~\ref{thm:irregular-cover-class-descent},
let $J/k$ be the abelian variety to which the Albanese variety of
the geometric generic phase-fiber cover becomes isogenous.  Suppose
$J_{\mathrm{var}}\ne0$.  The independent exceptional factor can
then be chosen in the form
\[
                    A_{\mathrm{var}}/\Gamma,
\]
where $A_{\mathrm{var}}$ is a trace-zero abelian variety and
$\Gamma$ is a finite group of affine automorphisms.  The resulting
map is nonconstant on the geometric generic phase fiber.
\end{lemma}

\begin{proof}
Keep the actual normal projective Galois cover $Z$ constructed in
that theorem.  Its construction gives a rational map $Z\dashrightarrow J'$
whose restriction to the geometric generic phase fiber induces
an isogeny on Albanese varieties; here $J'$ is isogenous to $J$.
Indeed, the geometric subgroup killed in the descent acts on the
Albanese torsor by translations.  Quotienting by these translations
is an isogeny, and killing the remaining torsor replaces its
Albanese map by multiplication by a positive integer.  Neither
operation removes a positive-dimensional abelian quotient.

Put $A=\operatorname{Alb}(Z)$.  By the universal property of the
Albanese, the preceding map factors through a homomorphism
$A\to J'$ after a translation.  If $J_Z$ is the Albanese variety
of the geometric generic phase fiber of $Z$, the composite
\[
                     J_Z\longrightarrow A\longrightarrow J'
\]
is an isogeny.  Consequently the first arrow has finite kernel.
If the image of $J_Z$ in $A_{\mathrm{var}}$ were zero, that image
in $A$ would be contained in the constant trace of $A$.  An abelian
subvariety of a constant abelian variety is constant after finite
extension: its integral homology lattice is one of the countably
many sublattices, and an algebraic family of such subvarieties has
constant lattice on a connected parameter open.  Thus $J_Z$, and
hence $J$, would be isotrivial, contrary to $J_{\mathrm{var}}\ne0$.
The map to $A_{\mathrm{var}}$ is therefore nonconstant on the
phase fiber.

Every deck transformation of $Z$ acts affinely on $A$ and preserves
the image of its constant trace.  It consequently acts affinely
on $A_{\mathrm{var}}$.  Taking the quotient by the finite image
$\Gamma$ descends the map to the original variety.  The induced
endomorphism of $A_{\mathrm{var}}$ is polarized: the polarized
Albanese endomorphism descends to this invariant abelian quotient.
A finite quotient of this system is exceptional.  Since the
quotient map is finite, nonconstancy on the phase fiber is preserved.
\end{proof}

\begin{remark}[Detecting variation on selected periodic covers]
\label{rem:periodic-cover-variation}
The same refinement applies to
Theorem~\ref{thm:relative-irregular-factor-gluing} when its selected
periodic cover families have nonisotrivial Albanese varieties.
Here is the specialization point in that assertion.  Spread the
finitely many dominant cover-class components, choose finite
covers carrying actual covering objects, and shrink so that their
relative Albanese schemes and their isogenies are defined.  The
phase-isogeny and Hodge-height argument in
Lemma~\ref{lem:cover-class-abelian-moduli} makes the abelian moduli
of each dynamically selected component independent of phase.
Thus this moduli map depends only on the parameter $S$; restriction
to a good periodic section gives precisely the moduli of its
selected covering family.  If every selected dominant component
had constant abelian moduli over $S$, all such specializations
would be isotrivial.  A Zariski-dense set of selected periodic
families with varying Albanese therefore supplies a component
with $J_{\mathrm{var}}\ne0$.  The finite conjugate compositum and
finite translation quotients used subsequently preserve its
abelian quotient up to isogeny.  The preceding lemma applies.
\end{remark}

\begin{lemma}[Joins of trace-zero abelian-quotient factors]
\label{lem:trace-zero-factor-join}
The join of two polarized rational factors which admit finite
presentations by trace-zero abelian varieties again admits such
a presentation, after an iterate.  Such factors have a maximal
join.  Its function field can be taken relatively algebraically
closed in the function field of the source.
\end{lemma}

\begin{proof}
Lift the normalized joint image to the product of the two finite
abelian presentations.  After an iterate a dominant component is
periodic.  A periodic subvariety of a polarized abelian
endomorphism is an abelian coset, by the abelian case of
Proposition~\ref{prop:periodic-subvariety-at-lift}.  Its abelian
variety has trace zero: a homomorphism from a constant abelian
variety to it, followed by either projection, is zero.  The finite
map from this coset gives the required presentation of the join.

Repeated joins increase dimension until the maximal dimension is
reached; at that dimension they only produce finite extensions.
Take the relative algebraic closure of the resulting factor field
inside the source field, a finite extension.  This extension is
again a trace-zero abelian-quotient factor.  To see this, normalize
its pullback to the abelian presentation and take a periodic
component.  Its map to the abelian variety is finite and
polarized.  Since the abelian endomorphism is \emph{\'etale},
Ji--Song--Xie's Lemma~3.11 makes this cover finite \emph{\'etale};
it is an abelian variety isogenous to the original one, and
therefore still has trace zero.  Maximality now implies that every
other factor of the indicated kind is contained in this one:
otherwise its join would have larger dimension or enlarge a
relatively algebraically closed subfield by a finite extension.
\end{proof}


\subsection{Finite gauge ambiguity and its horizontal derivation}

\begin{proposition}[Horizontal descent through finite ambiguity]
\label{prop:finite-ambiguity-horizontal}
Let $K=\mathbf C(B)$ and let $M=K\mathbf C(S)$, where $S$ is a
smooth complex parameter curve.  Let $\sigma:K\hookrightarrow K$
be a finite field embedding, extended to $M$ by fixing $\mathbf C(S)$.
Suppose that a family of geometrically integral generic phase fibers
has a nonempty finite-type gauge torsor $I$ under $G_M$, where
$G/K$ is an algebraic group.  Suppose first-arrow transport sends
$I$ algebraically to $\sigma^*I$ and sends $T=G^0$ isogenously to
$\sigma^*T$, with $T$ a torus.  Assume these gauges intertwine the
first varying arrow with its fixed reference arrow.

On the rational quotients by the descended $T$-actions, there is a
rational horizontal derivation $W$ which annihilates $K$, projects
to a nonzero parameter derivation, and commutes with the dynamics.
In particular, whenever these quotient fields have normal projective
polarized models, the quotient family is isotrivial over $B$.
\end{proposition}

\begin{proof}
The torus $T$ is normal in $G$.  Changing a gauge by an element of
$G$ therefore leaves the invariant function field unchanged.  The
induced quotient gauges have ambiguity in the finite group scheme
$H=G/T$ over $K$.  They are parameterized by the finite $H_M$-torsor
$I/T$, possibly with an ineffective subgroup in its action on the
quotient field.  The ineffective subgroup does not affect what follows.

Choose a nonzero derivation $\partial$ of $\mathbf C(S)/\mathbf C$
and extend it to $M$ by annihilating $K$.  It extends uniquely to
an algebraic closure $\overline M$.  A quotient gauge $C$ is defined
over a finite extension of $M$.  On the reference quotient field
extended to $\overline M$, define $\partial$ to annihilate the
reference field over $K$, and transport it through $C$ to obtain
$W_C$.  The geometric integrality of the generic phase fiber makes
this extension of scalars a field; equivalently, its function field
is a regular extension of $K$.

Every geometric point of $H$ has coordinates algebraic over $K$.
The derivation $\partial$ annihilates all such coordinates.  Thus
if $C'=C\gamma$ for $\gamma\in H(\overline K)$, then
\[
 (C')_*\partial=C_*\partial.
\]
All Galois conjugates of $C$ give the same derivation.  Consequently
$W_C$ descends to a rational derivation $W$ on the quotient family.
It annihilates $K$ and restricts to $\partial$ on the parameter.

Extend $\sigma$ to an automorphism $\overline\sigma$ of
$\overline M$.  Such an extension exists because $M/\sigma(M)$
is finite.  The equality $\partial\sigma=\sigma\partial$ on $M$
and uniqueness of algebraic extension give
\[
 \partial\overline\sigma=\overline\sigma\partial
 \quad\hbox{on }\overline M.
\]
The transported target gauge and $\overline\sigma(C)$ are both
points of $\sigma^*(I/T)$, hence differ by an element whose
coordinates are algebraic over $K$ and are killed by $\partial$.
The fixed reference arrow commutes with $\partial$.  Differentiating
the gauge identity therefore gives the exact rational identity
\[
 DF\,W=W\circ F.
\]

For completeness, the normal polarized model removes the poles of
$W$.  If $D$ is its polar Weil divisor, the regular differential on
the left cannot increase pole order, while the right multiplies
pole orders by the ramification indices.  Hence $F^*D\leq D$.
Intersecting on the generic parameter fiber with a polarization
forces its horizontal part to vanish.  Delete the finitely many
remaining parameter fibers.  Reflexivity of the tangent sheaf
extends $W$ over codimension two.  Its local flow along the compact
fibers trivializes the family and commutes with $F$.  Since $W$
annihilates $K$, these flows preserve the original map to $B$.
The algebraic conjugacy locus in a relative graph Hilbert scheme
then dominates $S$; a closed point of its generic fiber provides
an algebraic trivialization after a finite parameter extension.
\end{proof}

\subsection{The dynamical splitting field of a torus}

\begin{proposition}[Minimal splitting fields carry the dynamics]
\label{prop:minimal-torus-splitting-dynamics}
Let $M$ be a characteristic-zero field, let $\sigma:M\hookrightarrow M$
be finite, and let $T/M$ be a torus with an isogeny
$T\to\sigma^*T$.  Its minimal Galois splitting field $L/M$ admits
an extension $\widetilde\sigma:L\hookrightarrow L$ of $\sigma$,
and
\[
 L=M\widetilde\sigma(L).
\]
If $M$ is the function field of a normal projective polarized system
whose pullback is $\sigma$, its normalization in $L$ carries a finite
polarized lift of the same weight.

Suppose moreover that $E/M$ is regular and that $E$ is the function
field of a normal projective variety $V$ equipped with a finite
$q$-polarized endomorphism $F$, with $F^*|_M=\sigma$.  Then the
normalization $p:\widetilde V\to V$ in $EL$ carries a finite
$q$-polarized endomorphism $\widetilde F$ satisfying
\[
                 p\widetilde F=Fp.
\]
Its pullback on $EL$ extends both $F^*$ on $E$ and
$\widetilde\sigma$ on $L$.
\end{proposition}

\begin{proof}
Extend $\sigma$ to an automorphism $\overline\sigma$ of an algebraic
closure $\overline M$.  The character lattice of $T$ has finite
Galois image.  Its rational representation determines the kernel
of its integral representation, so isogenous tori have the same
minimal Galois splitting field.  The splitting field of
$\sigma^*T$ is $M\overline\sigma(L)$: its character representation
is obtained by restricting the conjugated representation, whose
kernel fixes exactly that compositum.  Therefore
\[
 L=M\overline\sigma(L),
\]
and $\widetilde\sigma=\overline\sigma|_L$ has image in $L$.

Let $h:Y\to Y$ realize $\sigma$, and let $Y'$ be the normalization
of $Y$ in $L$.  On an affine $U=\operatorname{Spec}A\subset Y$,
write $h^{-1}U=\operatorname{Spec}R$.  The image under
$\widetilde\sigma$ of an element integral over $A$ is integral
over $\sigma(A)\subset R$.  Hence the field embedding gives regular
maps between the two integral closures, and these glue to
$h':Y'\to Y'$.  It is proper and has finite fibers because
$\rho h'=h\rho$; hence it is finite.  The pullback of an ample
polarization of $h$ polarizes $h'$ with the same weight.

Regularity of $E/M$ makes $E$ and $L$ linearly disjoint over $M$,
so $E\otimes_ML=EL$ is a field.  Since $F^*$ and
$\widetilde\sigma$ agree on $M$, the rule
\[
 a\otimes b\longmapsto F^*(a)\widetilde\sigma(b)
\]
defines a nonzero field homomorphism $EL\hookrightarrow EL$.
Let $p:\widetilde V\to V$ be the normalization in this finite
extension of $E$.  The integral-closure argument above, now applied
to the given morphism $F$ on $V$, extends this homomorphism to
$\widetilde F:\widetilde V\to\widetilde V$ with
$p\widetilde F=Fp$.  It is proper, and each of its fibers is
contained in a fiber of the finite map $Fp$.  Thus
$\widetilde F$ is finite; it is surjective because its pullback
on function fields is injective.  If $H$ is an ample line bundle
with $F^*H\simeq H^{\otimes q}$, then $p^*H$ is ample and
\[
 \widetilde F^*p^*H=p^*F^*H\simeq(p^*H)^{\otimes q}.
\]
This gives the asserted $q$-polarized lift on the specified
normalization.
\end{proof}


\subsection{Descent of isotriviality through a finite cover}


\begin{proposition}[A constant finite dynamical cover fixes the quotient]
\label{prop:constant-finite-cover-downstairs}
Let $u:Z\to Z$ be a fixed polarized endomorphism of a normal
projective complex variety.  Let $F:\mathcal X\to\mathcal X$
be an algebraic family of endomorphisms of normal projective
varieties over a smooth curve $S$.  Suppose there is a finite
surjective $S$-morphism
\[
 p:Z\times S\longrightarrow\mathcal X,
 \qquad p(u\times\operatorname{id}_S)=Fp.
\]
Then, after finite parameter base change and restriction, the
family of diagrams $(p_s,F_s)$ is constant up to isomorphism of
the targets.  In particular $(\mathcal X,F)$ is isotrivial.
\end{proposition}

\begin{proof}
Put $n=\dim Z$ and $v=u\times u$ on $Z\times Z$.  The product
endomorphism $v$ is polarized.  For each parameter, let $R_s$
be the closure of the generic reduced equivalence relation of
$p_s$, namely the union of the $n$-dimensional components of
\[
 (Z\times_{X_s}Z)_{\mathrm{red}}\subset Z\times Z.
\]
Each such component dominates $Z$ under either projection.
Equivariance gives $v(R_s)\subset R_s$.  Since $v$ is finite,
the image of any irreducible component has dimension $n$ and
is an irreducible component of $R_s$.

After finite parameter base change, label the finitely many
geometric generic components, spread them out and shrink $S$
so that each gives a flat family of reduced irreducible
subvarieties.  Their finite combinatorial transition under $v$
is constant after shrinking again.  Every component therefore
maps after a fixed number of steps to a component which is
periodic under a fixed iterate of $v$.

Lemma~\ref{lem:fixed-polarized-invariant-image-rigidity} makes
every periodic component family constant.  A preperiodic
component is an irreducible component of the inverse image of
one of these fixed subvarieties under a fixed finite iterate.
There are only finitely many choices, so its irreducible
parameter family is constant as well.  Hence $R_s$ is constant.

Fix $s_0$.  Equality of these generic equivalence relations says
that $p_s$ and $p_{s_0}$ determine the same subfield of
$\mathbf C(Z)$: in a finite separable extension, the functions
constant on its generic equivalence classes are precisely the
target field.  Equivalently, the reduced image $W_s$ of
$(p_{s_0},p_s)$ in $X_{s_0}\times X_s$ is finite and birational
over either target.  Finiteness follows because a fiber of
either projection is the image of a finite fiber of the
corresponding $p_s$.  Normality makes both projections
isomorphisms.  Thus there is a unique isomorphism $c_s$ satisfying
\[
 p_s=c_sp_{s_0}.
\]
The same construction with relative graphs spreads these
isomorphisms algebraically over $S$.  The equality
$p_su=F_sp_s$ and the surjectivity of $p_{s_0}$ then give
$F_sc_s=c_sF_{s_0}$.
\end{proof}


\subsection{Toric extensions of a fixed system}

\begin{lemma}[Weighted vertical divisors]
\label{lem:weighted-vertical-pushforward}
Let $\pi:X\to B$ be a surjective morphism of normal projective
varieties with geometrically integral generic fiber of dimension
$r$.  Let $F$ and $g$ be finite endomorphisms with
$\pi F=g\pi$.  Suppose $L$ is big and nef,
$F^*L\simeq L^q$, and $g$ is $q$-polarized.  Set
$e=(L^r\cdot X_{\eta_B})>0$.  For a vertical Weil divisor $V$
define
\[
 {\mathcal W}_L(V)=\frac1e\sum_{D\subset B}
 \left(\sum_{E\to D}\operatorname{ord}_E(V)
     (L^r\cdot E_{\eta_D})\right)D.
\]
Only divisors $E$ dominating the prime divisor $D$ occur in the
inner sum.  Then, as actual Weil divisors,
\[
 {\mathcal W}_L(\pi^*D)=D,\qquad
 {\mathcal W}_L(F^*V)=g^*{\mathcal W}_L(V).
\]
All coefficients have denominator dividing $e$.
\end{lemma}

\begin{proof}
Localize at a prime divisor of the normal base.  Its local ring
is a discrete valuation ring.  The integral total space is flat
over this ring, since its local rings are torsion-free over it.
Flatness and constancy of the Hilbert polynomial identify the
fiber cycle degree with $e$.  Thus, if
$\pi^*D=\sum m_EE$ at this generic point,
\[
                 \sum m_E(L^r\cdot E_{\eta_D})=e.
\]
This proves the first identity, including all fiber multiplicities.

For the second, fix $D'$ with $g(D')=D$.  Form the base change
$Z=X\times_{B,g}B$ over the corresponding discrete valuation
rings.  It is integral: it is flat and its generic fiber is
geometrically integral.  Write $p:Z\to X$ for the projection and
$u=(F,\pi):X\to Z$.  The map $u$ is finite of degree $q^r$ and
$u^*p^*L=L^q$.  The projection formula therefore gives
\[
 u_*\bigl(F^*V\cdot L^r\bigr)
 =q^{-r}u_*u^*\bigl(p^*V\cdot(p^*L)^r\bigr)
 =p^*\bigl(V\cdot L^r\bigr).
\]
Proper pushforward to the base and flat base change yield
\[
          \pi_*\bigl(F^*V\cdot L^r\bigr)
             =g^*\pi_*\bigl(V\cdot L^r\bigr)
\]
at $D'$.  These are the coefficients in the asserted formula.
For a Weil divisor, perform this calculation on its Cartier locus.
General intersections computing the fiber degree avoid the
non-Cartier locus: over the generic base divisor it has dimension
at most $r-1$, whereas there are $r$ intersections.  Expressing
$L$ as a difference of sufficiently ample line bundles gives the
same calculation for big and nef $L$.  Thus no factoriality
assumption is being used.  The degree numerators are integers,
which proves the denominator assertion.
\end{proof}

\begin{lemma}[A fixed-base divisor equation]
\label{lem:fixed-base-divisor-coboundary}
Let $(B,g)$ be a fixed normal projective $q$-polarized system over
$\mathbf C$.  Suppose $D_s$ is an algebraic curve family of rational
Weil divisors with a common coefficient denominator, and
$c_s\in\mathbf C(B)^*$ varies rationally with $s$, satisfying
\[
                    g^*D_s-qD_s=\operatorname{div}(c_s).
\]
After a finite parameter extension and restriction there exist
$b_s\in\mathbf C(B)^*$ and $a_s\in\mathbf C^*$, rational in $s$,
such that, for one fixed reference $s_0$,
\[
       \frac{c_s}{c_{s_0}}
          =a_s\frac{b_s\circ g}{b_s^q}.
\]
\end{lemma}

\begin{proof}
First reduce $D_s$ modulo integral Weil divisors, writing its
coefficients in a fixed finite subgroup of $\mathbf Q/\mathbf Z$.
Its fractional part $\alpha_s$ satisfies $g^*\alpha_s=q\alpha_s$.
For the primary parts whose primes divide $q$, choose $m$ with
$q^m\alpha_s=0$.  Their supports lie among the branch-value
divisors of $g^m$: an unramified generic inverse image of a
nonzero coefficient would survive pullback.  For the prime-to-$q$
part, a nonzero coefficient at $D$ forces a nonzero coefficient
at $g(D)$; its finite support is therefore forward invariant.
These divisors are preperiodic.

A fixed polarized endomorphism has only countably many
preperiodic subvarieties.  One way to see the countability for
periodic subvarieties is to use density of periodic points on
each such subvariety and the finite-type incidence spaces of
subvarieties of bounded degree: a given subvariety is uniquely
determined among subvarieties of its dimension and bounded
degree by finitely many of its periodic points.  The periodic
points of the ambient map form a countable set.  Taking all
iterated inverse images gives the preperiodic assertion.
An algebraic curve in a Chow variety with countable image is
constant.  Thus the fractional part of $D_s$ is constant after
labeling its components and shrinking the parameter curve.
Choose a fixed representative $D_{\mathrm{frac}}$ and write
$D_s=D_{\mathrm{frac}}+A_s$ with $A_s$ integral.

Choose a projective resolution $\nu:\widetilde B\to B$.
The strict transforms of $A_s-A_{s_0}$ form an algebraic divisor
family and define a connected family
\[
                 M_s\in\operatorname{Pic}^0(\widetilde B).
\]
The divisor equation implies that
$(\widetilde g^*-[q])M_s$ lies in the subgroup generated by the
$\nu$-exceptional divisors.  Here $\widetilde g$ is the rational
lift, whose action on $\operatorname{Pic}^0$ is defined by a graph
resolution.  This action agrees with finite pullback of Weil
classes after pushforward to $B$.

An exceptional divisor representing an algebraically trivial
line bundle is zero: it is numerically trivial, and applying the
negativity lemma to it and its negative proves the assertion.
Hence $(\widetilde g^*-[q])M_s=0$ in
$\operatorname{Pic}^0(\widetilde B)$.  The endomorphism induced
by $g$ on the birational Albanese is $q$-polarized; consequently
the eigenvalues of its dual on $\operatorname{Pic}^0$ have absolute
value $\sqrt q$.  Thus $\widetilde g^*-[q]$ is an isogeny.
Its finite kernel contains the connected family $M_s$, which
contains the identity, so $M_s$ is trivial.

The relative Picard interpretation now makes the line bundle
of the strict-transform difference a pullback from the parameter
curve.  After shrinking, this bundle is trivial.  Its rational
section gives a rational family $b_s$ with
$A_s-A_{s_0}=\operatorname{div}(b_s)$ after pushforward to $B$.
Subtract the two original divisor equations.  The ratio
\[
   (c_s/c_{s_0})\big/\bigl((b_s\circ g)/b_s^q\bigr)
\]
has zero divisor.  A rational function of zero divisor on a
normal projective variety is constant, giving $a_s$.
\end{proof}

\begin{lemma}[Power transport for a polarized torus action]
\label{lem:polarized-torus-power-transport}
Suppose a split torus $T$ acts effectively and rationally on a
normal projective polarized system $(X,F,L)$, and
\[
                    F(t x)=\tau(t)F(x)
\]
for an isogeny $\tau:T\to T$.  The action is regular, and for
some $m>0$ one has $\tau^m=[q^m]$.
\end{lemma}

\begin{proof}
Let $v_1,\dots,v_a$ be the rational fundamental vector fields
of a basis of $\operatorname{Lie}(T)$ and let $P$ be their common
pole divisor, taking the largest pole order at every prime.
Equivariance gives
$DF(v_i)=\sum_j c_{ji}v_j\circ F$, with an invertible constant
matrix $(c_{ji})$.  At a codimension-one point the differential
has regular coefficients.  Invertibility of the constant matrix
therefore gives $F^*P\le P$.  Intersection with $L^{\dim X-1}$
yields $q\deg_LP\le\deg_LP$, whence $P=0$.
Normality extends the derivations across codimension two.
Their complete flows on the projective variety extend the given
rational action; agreement on the initial rational-action domain
makes this an algebraic torus action.

A positive power of $L$ is $T$-linearizable.  Let $\mathcal P$
be its weight polytope.  This polytope has full affine span,
since the action is effective.  The finite map $F$ maps the
fixed locus onto itself: a finite inverse-image set of a fixed
point is stable under the connected torus and consists of fixed
points.  Comparing the linearizations of $F^*L$ and $L^q$ gives
\[
                  \tau^*\mathcal P=q\mathcal P+u
\]
for a character $u$.  The affine automorphism
$x\mapsto q^{-1}(\tau^*x-u)$ permutes the finitely many vertices
of the full-dimensional polytope, so a positive power is the
identity.  Its linear part then gives $(\tau^*)^m=q^m I$,
which is the assertion on $\tau$.
\end{proof}

\begin{proposition}[Toric rigidity over a fixed quotient]
\label{prop:toric-fixed-quotient-rigidity}
Let $\mathcal X\to S$ be an algebraic curve family of normal
projective varieties with a relatively $q$-polarized endomorphism
$F$.  Suppose it admits an equivariant dominant rational map to
one fixed normal projective polarized system $(B,g)$, with
geometrically integral generic fiber.  Assume that a fixed split
torus $T$ acts rationally over $B\times S$, has a dense orbit in
the generic fiber, and satisfies
\[
                    F(t x)=\tau(t)F(x)
\]
for one fixed isogeny $\tau$ of $T$.  Then the family is
isotrivial after finite parameter extension and restriction.
\end{proposition}

\begin{proof}
Let $H$ be the full kernel of the action of $T$ on the geometric
generic fiber, including its finite components. A subgroup of a
split torus is defined by character equations, so $H$ is the base
change of a fixed subgroup of $T$. Equivariance and dominance of
$F$ give $\tau(H)\subseteq H$. Hence $\tau$ induces a surjective
endomorphism, and therefore an isogeny, of the split torus $T/H$.
Replace $T$ and $\tau$ by this quotient and its induced isogeny.
The action is now effective. Since $T$ is commutative, the
stabilizer of a point of its dense geometric orbit fixes that
entire orbit and is trivial. The generic orbit is consequently
a torsor under this effective split torus.

Normalize the equivariant graph of the rational map to $B$.
Its lifted endomorphism is finite, and the sum of the pullbacks
of polarizations on $\mathcal X$ and $B$ is a polarization.
We may therefore use a morphism $\pi$ and a relatively ample
polarization $L$ on this graph.  The rational torus action lifts
to it.  Lemma~\ref{lem:polarized-torus-power-transport}, applied
over the geometric generic parameter, makes its transport $[q]$
after a common iterate; rename the new weight $q$.

A torsor under a split torus over a field is trivial.  Choose
character coordinates $z_1,\dots,z_r$ on its generic orbit.
They identify the full function field with
$\mathbf C(B)(S)(z_1,\dots,z_r)$, and the map is
\[
       (b,z_1,\dots,z_r)\longmapsto
       (g(b),c_1(b,s)z_1^q,\dots,c_r(b,s)z_r^q).
\]
For one character coordinate, separate its divisor into the
horizontal and vertical parts over $B$:
$\operatorname{div}(z)=H_s+V_s$.
The generic normal projective toric fiber gives $F^*H_s=qH_s$.
No vertical component can lie over a horizontal divisor under
a finite equivariant map.  Thus this identity holds as an
identity of horizontal divisors on the whole model, and
\[
                 F^*V_s-qV_s=\pi^*\operatorname{div}(c_s).
\]
Apply Lemma~\ref{lem:weighted-vertical-pushforward}.  The divisors
$D_s={\mathcal W}_{L_s}(V_s)$ satisfy
\[
                    g^*D_s-qD_s=\operatorname{div}(c_s).
\]
They form an algebraic family with one common denominator.
Indeed the divisor of the chosen rational coordinate has
finitely many components on the total family; their images,
multiplicities, and fiber intersection degrees specialize after
shrinking.  The denominator is the fixed integer
$(L_s^r\cdot X_{s,\eta_B})$.

Lemma~\ref{lem:fixed-base-divisor-coboundary}, applied to each
coordinate, gives
\[
 \frac{c_i(b,s)}{c_i(b,s_0)}
    =a_i(s)\frac{b_i(g(b),s)}{b_i(b,s)^q}.
\]
After finite parameter extension choose
$d_i(s)^{1-q}=a_i(s)$.  The rational coordinate change
$z_i\mapsto d_i(s)b_i(b,s)z_i$ conjugates the fixed reference
skew system to the parameter system.  Hence the original
projective family is dynamically birationally isotrivial.
Lemma~\ref{lem:rational-horizontal-normal} extends the transported
parameter derivation over the normal projective models and
makes the family isotrivial.  Finally,
Corollary~\ref{cor:isotrivial-iterate-normal} removes the iterate.
\end{proof}

\section{Flat bundles, foliations, and rational factors}

\subsection{Degrees of connected rational factors}

\begin{lemma}[Degree of a rational factor]
\label{lem:connected-rational-factor-degree}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective variety of dimension $n$. Suppose that
$\pi:X\dashrightarrow Y$ is a dominant rational map with
geometrically integral generic fiber, that $Y$ has dimension $r$,
and that a dominant rational map $g:Y\dashrightarrow Y$ satisfies
$\pi f=g\pi$. Then
\[
                         \deg(g)=q^r.
\]
For general $y$, the finite map between the closures of the fibers
through $y$ and $g(y)$ has degree $q^{n-r}$.
\end{lemma}

\begin{proof}
The Chow reduction of $\pi$ is a normal projective model of
$\mathbf C(Y)$ on which $g$ is a surjective endomorphism, by
\cite[Proposition~7.1 and Theorem~7.2]{MZ22}. Applying the degree
identity in \cite[Theorem~1.3]{MZ18} gives $\deg g=q^r$.
For general fibers the finite restriction of $f$ has degree
$\deg f/\deg g=q^{n-r}$.
\end{proof}


\subsection{Projective models of algebraic foliation quotients}

\begin{proposition}[Polarized regularization of a connected factor]
\label{prop:connected-factor-chow-regularization}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective variety, and let $\pi:X\dashrightarrow Y$ be a
dominant rational map with geometrically integral generic fiber.
Suppose that a dominant rational map $g:Y\dashrightarrow Y$
satisfies $\pi f=g\pi$. Then there is a normal projective model
$Y^{\mathrm c}$ of $\mathbb C(Y)$ on which $g$ is a
$q$-polarized endomorphism. The normalized graph
$X^{\mathrm c}\to X\times Y^{\mathrm c}$ carries a finite
$q$-polarized lift of $f$ and an equivariant surjective morphism
$X^{\mathrm c}\to Y^{\mathrm c}$.
\end{proposition}

\begin{proof}
Take the Chow reduction of $\pi$. Its existence and the regularity
of the descended map are \cite[Proposition~7.1 and Theorem~7.2]{MZ22}.
The resulting normal projective base is birational to $Y$.
The descended endomorphism is $q$-polarized by
\cite[Theorem~1.3]{MZ18}, whose degree identity fixes the weight.
The finite product map $f\times g$ preserves the graph and lifts
to its normalization. The tensor product of the pullbacks of the
two ample polarizations is ample on this normalized graph and is
pulled back to its $q$th power.
\end{proof}

\begin{proposition}[Two algebraic reductions of an invariant foliation]
\label{prop:invariant-foliation-two-algebraic-reductions}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective complex variety of dimension $n$. Let $\mathcal F$
be an invariant rational foliation of rank strictly between
zero and $n$. Each of the following conditions produces an
equivariant connected rational quotient to a positive-dimensional
normal projective variety of dimension less than $n$, equipped
with a $q$-polarized endomorphism:
\begin{enumerate}
\item the Zariski closure of a very general leaf is a proper
subvariety of $X$;
\item the maximal algebraic part $\mathcal F_{\mathrm{alg}}$
has positive rank.
\end{enumerate}
Consequently, the case in which neither construction gives a
proper factor consists of purely transcendental foliations with
Zariski dense very general leaves.
\end{proposition}

\begin{proof}
Bonnet's leaf-closure theorem \cite[Theorem~1.4]{Bon06} gives the
first quotient. The quotient by the maximal algebraic part is
\cite[Section~2.6 and Proposition~2.4]{LBPRT22}. Both constructions
are birational, so they apply on a smooth projective model and
define subfields of $\mathbf C(X)$. Their fibers are irreducible
of dimension strictly between zero and $n$; taking relative
algebraic closure makes the quotient connected. Invariance of
$\mathcal F$ makes its leaf closures and algebraic part invariant,
so $f^*$ preserves the quotient field. Apply
Proposition~\ref{prop:connected-factor-chow-regularization} to
obtain the normal projective polarized quotient.
\end{proof}


\subsection{Invariant flat bundles and algebraic flows}


\begin{lemma}[Entire lines and a codimension-two exceptional set]
\label{lem:lines-codim-two-h-special}
Let $Z$ be a normal quasi-projective complex variety and let
$\theta:\mathbb C^n\to Z$ be a holomorphic map which is locally
finite and has full dimension. Then $Z_{\rm reg}$ is $h$-special.
\end{lemma}
\begin{proof}
The analytic set $A=\theta^{-1}(Z_{\rm sing})$ has complex
codimension at least two. Almost every ordered pair of points of
$\mathbb C^n$ spans an affine complex line avoiding $A$.
Indeed, the incidence space of a point of $A$, a line through that
point, and an ordered pair on the line has complex dimension at most
$(n-2)+(n-1)+2=2n-1$. Covering this incidence space by countably many
relatively compact analytic charts shows that its image in
$\mathbb C^n\times\mathbb C^n$ has Lebesgue measure zero.
Every remaining line gives an entire curve in $Z_{\rm reg}$.
On an open set where $\theta$ is biholomorphic, the resulting pairs
have full measure in a nonempty open subset of
$Z_{\rm reg}\times Z_{\rm reg}$. Thus the entire-curve equivalence
relation is Zariski dense. This is $h$-specialness in the sense of
\cite[Definition~2.2]{CDY25}.
\end{proof}

\begin{proposition}[The algebraic closure of invariant flows]
\label{prop:invariant-flows-semiabelian}
Let $f:X\to X$ be a polarized endomorphism of a normal projective
complex variety. Suppose a nonzero finite-dimensional Lie algebra
of global vector fields is preserved by invertible dynamical
transport. The connected Zariski closure $H$ of its flows in
$\operatorname{Aut}^0(X)$ is a positive-dimensional semiabelian
group. There is an isogeny $\psi:H\to H$ satisfying
$f(hx)=\psi(h)f(x)$.
\end{proposition}
\begin{proof}
Global holomorphic vector fields on the compact complex space $X$
are complete. Dynamical transport gives the stated equivariance
on the analytic subgroup generated by their flows. In the closed
subgroup of $H\times H$ defined by $fh=kf$, projection onto the
first factor is surjective: its image is an algebraic subgroup
containing the Zariski-dense flow subgroup. The second coordinate
is unique since $f$ is surjective. In characteristic zero this
projection is an isomorphism. The second projection is likewise
surjective because transport is invertible. Its kernel is finite,
being contained in the automorphism group of the finite map $f$.
This defines $\psi$.

We first rule out a $\psi$-invariant subgroup
$V\simeq\mathbb G_a$ on which $\psi$ is an automorphism.
A general $V$-orbit is $\mathbb A^1$, and $f$ maps it
isomorphically onto another general orbit. The closures of general
orbits have one fixed positive degree $e$ against an ample
polarization $L$. This follows, for example, by taking their
generic cycle in the Chow variety and restricting to its flat
parameter locus. For $x$ and $f(x)$ in that locus, the induced map
of orbit closures has degree one. The projection formula gives
$qe=e$, a contradiction.

Let $L_H$ be the connected affine part of $H$. Its unipotent
radical is preserved by $\psi$, and the restriction of $\psi$ to
it is an automorphism in characteristic zero. If that radical is
nontrivial, its center has a nonzero vector subgroup invariant
under $\psi$; an eigenline gives the forbidden $V$.

Thus $L_H$ is reductive. If its derived subgroup is nontrivial,
the induced self-isogeny of that semisimple group is an
automorphism. After a power it is inner, say conjugation by $s$.
Choose a Borel containing $s$, and a maximal torus containing its
semisimple Jordan component. In a simple factor, a highest-root
subgroup is centralized by the unipotent radical of the Borel
and normalized by that torus. It is therefore normalized by $s$,
and is a forbidden $\mathbb G_a$ subgroup for this iterate.
Consequently $L_H$ is a torus. A connected algebraic group whose
affine part is a torus is semiabelian, which proves the result.
\end{proof}

\begin{lemma}[First-cohomology weights on a normal invariant open]
\label{lem:normal-open-h1-weights}
Let $f:X\to X$ be $q$-polarized on a normal projective variety,
let $E$ be a reduced divisor satisfying $f^{-1}E=E$, and put
$Y=(X\setminus E)_{\rm reg}$. The rational self-map of $Y$ induced
by $f$ defines an endomorphism of $H^1(Y,\mathbb Q)$. All its
eigenvalues have absolute value $\sqrt q$ or $q$.
\end{lemma}
\begin{proof}
The map is a morphism from $Y\setminus f^{-1}(X_{\rm sing})$
to $Y$. Its omitted subset has codimension at least two, so inclusion
induces an isomorphism on $H^1$, compatibly with mixed Hodge
structures. This defines the asserted endomorphism.

Take a log resolution $\overline Y\to X$ that is an isomorphism
over $Y$. The weight-one part of $H^1(Y)$ is
$H^1(\overline Y)$; its action is induced by the birational
Albanese endomorphism, whose polarization weight is $q$ by
\cite[Corollary~1.4 and Theorem~1.3]{MZ18}. Its eigenvalues have
absolute value $\sqrt q$.
For the weight-two part use the residue description. Projection of
a cohomologically trivial boundary divisor onto the strict
transforms of the components of $E$ is injective: an element of
the kernel is an exceptional numerically trivial divisor, and the
negativity lemma applied to it and its negative makes it zero.
After a fixed iterate, every component of $E$ is fixed and its
pullback multiplicity is $q^k$, by the finite projection formula.
Residue functoriality therefore makes this iterate multiplication
by $q^k$ on the weight-two part. This proves the assertion.
\end{proof}

\begin{proposition}[Unipotent monodromy of an invariant flat bundle]
\label{prop:invariant-flat-unipotent-monodromy}
Let $f:X\to X$ be a polarized endomorphism of a normal projective
complex variety. Let $(\mathcal E,\nabla)$ be a rational integrable
connection with an isomorphism
$f^*(\mathcal E,\nabla)\simeq(\mathcal E,\nabla)$.
Suppose there is a Poincar\'e map
$\theta:\mathbb C^n\to X$, locally finite everywhere, such that
$f^r\theta=\theta P$, where $P$ is an automorphism of
$\mathbb C^n$ whose inverse iterates converge locally uniformly to
$0$. Suppose the connection is holomorphic near $\theta(0)$.
Then, after a finite dynamical cover of $X$ and an iterate, the
monodromy group of the pulled-back connection has connected
unipotent Zariski closure. In particular, the pulled-back rational
flat bundle has a nonzero rational horizontal section.
\end{proposition}
\begin{proof}
Replace $f$ by the indicated return. A horizontal basis near
$\theta(0)$ transforms under the pullback isomorphism by a constant
invertible matrix. Transporting it to the increasing domains
$P^j(B)$ gives a meromorphic horizontal basis for
$\theta^*(\mathcal E,\nabla)$ on all of $\mathbb C^n$.
The bases agree on overlaps by their construction. Thus the
pullback local system has trivial monodromy.

Choose a smooth dense open $U\subset X$ on which the connection
is holomorphic, with monodromy
$\rho:\pi_1(U)\to\operatorname{GL}(V)$ and image $\Gamma$.
The group $\Gamma$ is finitely generated. The dynamical
isomorphism induces conjugation by a matrix $C$ on its image,
giving an injective endomorphism
$\alpha(\gamma)=C\gamma C^{-1}$ of $\Gamma$ with finite-index
image. For the finite-index assertion, remove the critical values
and the images of the singular locus, and use the resulting finite
\'{e}tale covering; removing the additional connection divisors
induces surjections on fundamental groups.

For a prime divisor $D$ in $X\setminus U$, let $T_D$ denote the
image of a small transverse meridian, defined up to conjugacy.
Let $E_\infty$ be the reduced union of those $D$ for which $T_D$
has infinite order. This divisor is totally invariant. Indeed,
at a prime $D'$ mapping to $D$ with ramification index $e$, the
connection isomorphism identifies the local monodromy at $D'$
with a conjugate of $T_D^e$; finite and infinite order are unchanged
under taking positive powers.

The image of $\theta$ avoids $E_\infty$. If it met one component,
local finiteness and openness would give a general smooth point of
that component in the image. A transverse local branch above it
would show that $T_D^e=1$ for some positive integer $e$, because
the pulled-back local system is trivial. This is a contradiction.

By Selberg's lemma choose a torsion-free subgroup of $\Gamma$
of finite index $d$. The intersection of all subgroups of index
at most $d$ is a finite-index normal subgroup $\Gamma_0$ contained
in it and preserved by $\alpha$. Indeed,
$[\Gamma:\alpha^{-1}H]\le[\Gamma:H]\le d$ for each such $H$,
so $\Gamma_0\subset\alpha^{-1}H$ for every $H$ and hence
$\alpha(\Gamma_0)\subset\Gamma_0$. The corresponding finite cover
of $U$ algebraizes and extends by normalization to a finite
dynamical cover $p:X_1\to X$. The lifted endomorphism is polarized
by the pullback of an ample polarization.

The finite cover may moreover be chosen so that an iterate of its
lift commutes with every deck transformation. Replace $\Gamma_0$
by its inverse saturation
\[
 \Gamma_1=\bigcup_{j\ge0}\alpha^{-j}(\Gamma_0).
\]
The subgroups in this union form an increasing chain of finite
index, so the chain stabilizes. Hence
$\Gamma_1=\alpha^{-r}(\Gamma_0)$ for some $r$, and
$\alpha^{-1}(\Gamma_1)=\Gamma_1$. It is normal and torsion free:
a torsion element would inject under $\alpha^r$ into the
torsion-free group $\Gamma_0$. The map induced by $\alpha$ on
the finite group $\Gamma/\Gamma_1$ is therefore an automorphism.
After an iterate this automorphism is the identity. Use this
saturated subgroup, denoted again by $\Gamma_0$, for the cover.

The map $\theta$ lifts holomorphically to
$\theta_1:\mathbb C^n\to X_1$. On $\theta^{-1}(U)$ its lift
exists because the pulled-back monodromy is trivial. Its graph
extends across the omitted analytic set: normalize the finite
fiber product $\mathbb C^n\times_X X_1$ and take the component
selected by this lift. That component is finite and birational
over the normal space $\mathbb C^n$, hence isomorphic to it.
The lift remains locally finite, since its composition with $p$
is locally finite.

Put $Y=(X_1\setminus p^{-1}E_\infty)_{\rm reg}$. The restricted
monodromy factors through $\pi_1(Y)$. Indeed, at each of the
remaining boundary prime divisors its inertia is both finite
and contained in the torsion-free group $\Gamma_0$, and is
therefore trivial. The kernel of the surjection from the
fundamental group of the original smooth open to $\pi_1(Y)$ is
normally generated by these meridians. Its image is still
$\Gamma_0$.

The map $\theta_1$ takes values in
$X_1\setminus p^{-1}E_\infty$. Lemma
\ref{lem:lines-codim-two-h-special} makes $Y$ $h$-special.
Theorem~A of \cite{CDY25} therefore makes $\Gamma_0$ virtually
nilpotent. The finite-index intersection construction, repeated
with a nilpotent subgroup and then with the identity component
of its Zariski closure, gives a compatible cover whose monodromy
closure is a connected nilpotent linear algebraic group
$G=U_G\times T$.
At these later steps perform all finite-index intersections inside
the original monodromy group, and use inverse saturation as well.
Nilpotence is
preserved, since the saturated group injects into the original
nilpotent subgroup under a power of $\alpha$. Connectedness of the
Zariski closure is preserved because this injected image has finite
index, and conjugation by the transport matrix preserves the
identity component of the original monodromy closure. Thus all
covers can be taken Galois with commuting deck action after an
iterate.

Conjugation by $C$ normalizes $G$ and permutes the finitely many
$T$-weights of its faithful matrix representation. A fixed power
fixes every such character. The character restricted to the
monodromy image factors through the abelianization of
$\pi_1(Y)$ and is fixed by the corresponding dynamical action.
Lemma~\ref{lem:normal-open-h1-weights} shows that this action minus
the identity is invertible on rational first homology. Thus every
such fixed character has finite image. Its extension to the
connected group $G$ is trivial by Zariski density. Faithfulness
now gives $T=1$.

A unipotent group acting on the nonzero space $V$ has a nonzero
fixed vector. It gives a single-valued horizontal section. The
connection is regular singular by
Proposition~\ref{prop:invariant-connection-regular-singular}.
Its horizontal sections have moderate growth on a log resolution;
the single-valued section is consequently meromorphic there and
rational on the projective model.
\end{proof}

\begin{corollary}[A finite-type Killing connection supplies global fields]
\label{cor:killing-connection-global-fields}
In the preceding proposition suppose that the horizontal sections
are the local Killing fields of a rational finite-type geometric
structure, through a rational evaluation map
$\mathcal E\dashrightarrow T_X$, and that evaluation is injective
on germs of horizontal sections. Suppose the local Killing algebra
is nonzero. After a finite compatible cover there is a nonzero
finite-dimensional space of rational vector fields $V_0$ preserved
by dynamical transport. Every member of $V_0$ is regular on the
whole normal projective variety.
\end{corollary}
\begin{proof}
Take the full fixed space of the connected unipotent monodromy
closure. It is nonzero and is preserved by the transport matrix,
which normalizes that closure. Its horizontal sections are
rational by the proposition; rational evaluation gives rational
vector fields. Injectivity on horizontal germs makes the resulting
space nonzero. For a basis $W_i$ one has
$Df(W_i)=\sum_j c_{ij}W_j\circ f$ with a constant invertible
matrix $(c_{ij})$. The maximum of their divisorial pole orders
therefore satisfies $f^*P\le P$. Polarization kills $P$, and
normality extends each resulting derivation across codimension
two. Thus the fields are regular everywhere.
\end{proof}


\subsection{Semi-abelian rational factors of polarized maps}

\begin{lemma}[Scalar torus action]
\label{lem:semiabelian-factor-scalar-lift}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal projective
variety. Suppose there is a dominant rational map
$\pi:X\dashrightarrow G$ to a semi-abelian variety and an affine
isogeny $a:G\to G$ such that $\pi f=a\pi$. Write
\[
                1\longrightarrow T\longrightarrow G
                  \stackrel{p}{\longrightarrow} A
                  \longrightarrow0
\]
for its torus and abelian parts. Then an iterate $a^k$ is equivariant
with multiplication by $Q=q^k$ on $T$, and its induced affine
endomorphism $a_A^k$ on $A$ is $Q$-polarized. The extension of $A$
by $T$ is torsion. Moreover, $G$ has a normal projective
compactification on which $a^k$ extends to a $Q$-polarized
endomorphism. A finite group of affine automorphisms of $G$ through
which $a^k$ descends can be included in this compactification.
\end{lemma}

\begin{proof}
The map $a_A$ is a surjective endomorphism of the projective
variety $A$. By \cite[Theorem~1.3]{MZ18}, applied to the rational
map $p\pi$, it is $q$-polarized. Take the normalization
$X_1$ of the closure of the graph of $p\pi$ in $X\times A$.
The product map $f\times a_A$ preserves this graph and induces
a finite endomorphism $f_1:X_1\to X_1$. The sum of the pullbacks
of polarizations on $X$ and $A$ is ample on $X_1$ and has
weight $q$. We may therefore replace $X$ by $X_1$ and assume
that $p\pi:X\to A$ is a morphism.

Let $M$ be the integral matrix of the homomorphism induced by
$a$ on the torus. Trivialize the $T$-torsor $G\to A$ over
the function field of $A$, and choose a basis of its character
lattice. Pullback by $\pi$ gives rational functions
$z_1,\ldots,z_t$ on $X$, with
\[
             z_i\circ f=c_i\prod_j z_j^{M_{ij}},
             \qquad c_i\in\mathbb C(A)^*.
\]
For a horizontal prime divisor $D$ on $X$, put
$v_D=(\operatorname{ord}_D z_i)_i$. Let $\mathcal D$ be the
finite set of horizontal prime divisors for which $v_D\ne0$.
The displayed equations, and the invertibility of $M$ over
$\mathbb Q$, imply
\[
                         f^{-1}\mathcal D=\mathcal D.
\]
Indeed, horizontal divisors remain horizontal under $f$,
vertical divisors remain vertical under inverse image, and the
factors $c_i$ have zero order along every horizontal divisor.

After an iterate, every member of $\mathcal D$ is totally
invariant. If $f^{k*}D=e_DD$, intersection with the $(n-1)$st
power of an ample polarization gives $e_D=q^k$. The valuation
equations therefore read
\[
                             M^k v_D=q^k v_D
                             \qquad(D\in\mathcal D).
\]
These vectors span $\mathbb Q^t$. Otherwise a nonzero integral
character $\chi$ would have zero divisor after pullback to
the projective generic fiber of $X\to A$. It would be algebraic
over $\mathbb C(A)$, whereas dominance of $\pi$ makes every
nonzero torus character transcendental over that field. Thus
$M^k=q^k I$.

Replace $a$ by this iterate and write $Q=q^k$. Translating the
origin in $A$ makes $a_A$ an isogeny: its linear part minus the
identity is an isogeny, since the eigenvalues of a $Q$-polarized
abelian isogeny have modulus $\sqrt Q$. The extension $G$
is given by classes $L_\chi\in\operatorname{Pic}^0(A)$,
one for each torus character. Compatibility with $a$ gives
\[
                            a_A^*L_\chi=L_\chi^{\otimes Q}.
\]
The endomorphism $a_A^*-[Q]$ of $\operatorname{Pic}^0(A)$
is an isogeny, again because its analytic eigenvalues have
modulus $\sqrt Q$ before subtracting $Q$. Hence every $L_\chi$
is torsion. A common multiplication isogeny of $A$ splits the
extension.

Choose a complete projective toric variety $V_\Sigma$ for $T$.
The associated bundle
\[
                     \overline G=G\mathbin{\times^T}V_\Sigma
                     \longrightarrow A
\]
is normal and projective. The scalar map on the torus preserves
every fan, so $a$ extends to a finite endomorphism of
$\overline G$. Pull back by a multiplication
isogeny $[m]:A\to A$ which kills all the extension classes.
The resulting cover is $A\times V_\Sigma$. The affine map
$a_A$ lifts through $[m]$ after choosing a preimage of its
translation part; the lift is $Q$-polarized by pulling back
a polarization on $A$. The induced map on the split torus
bundle has the form $(b,t)\mapsto(a_A^{\prime}(b),c(b)t^Q)$.
The morphism $c:A\to T$ is constant because $A$ is projective.
Consequently this map extends to a $Q$-polarized product map
on $A\times V_\Sigma$. Its finite map to $\overline G$
is equivariant, so $a$ on $\overline G$ is $Q$-polarized by
\cite[Theorem~1.3]{MZ18}.

For the final assertion, the finite group induces a finite group
on the cocharacter lattice. Choose a projective complete fan
invariant under this group, by taking a common refinement of
the translates of a projective complete fan. Translations in
$G$, as well as the affine group automorphisms preserving this
fan, extend to $\overline G$. Thus the finite quotient is
projective, $a$ descends to it, and the descended map is polarized
by \cite[Theorem~1.3]{MZ18}. Its open $G$-quotient has the
finite semi-abelian presentation required for exceptionality.
\end{proof}


\subsection{A unitary flat socle on a normal variety}

\begin{lemma}[An invariant flat subsheaf]
\label{lem:invariant-reflexive-flat-socle}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective complex variety, and let $\mathcal E\ne0$ be reflexive.
If $f^{[*]}\mathcal E\simeq\mathcal E$, there is a nonzero saturated
reflexive subsheaf $\mathcal S\subset\mathcal E$ with the following
properties:
\begin{enumerate}
\item $\mathcal S|_{X_{\rm reg}}$ is a Hermitian flat vector bundle;
\item the given dynamical isomorphism restricts to
$f^{[*]}\mathcal S\simeq\mathcal S$ and preserves its flat connection;
\item the flat connection is rational and has regular singularities.
\end{enumerate}
\end{lemma}

\begin{proof}
Choose an ample Cartier divisor $H$ with $f^*H\sim qH$.
By \cite[Theorem~4.2]{MZ24}, both $\mathcal E$ and
$\mathcal E^*$ are weakly positive. Taking determinants and
intersecting with $H^{n-1}$ gives $\deg_H\mathcal E=0$.
Every torsion-free quotient of a weakly positive sheaf has
nonnegative degree. Thus $\mathcal E$ is $H$-semistable of slope zero.

Let $\mathcal S$ be its socle: the unique maximal saturated
polystable subsheaf of slope zero. The socle is nonzero and
reflexive; write
\[
                  \mathcal S=\bigoplus_{i=1}^{s}\mathcal S_i
\]
with each $\mathcal S_i$ stable of slope zero. The existence and
uniqueness of this saturated socle hold on normal projective
varieties; see \cite[Lemma~1.3.15]{Wei23}. The generically
surjective map $\mathcal E^*\to\mathcal S_i^*$ makes
$\mathcal S_i^*$ weakly positive. Since its degree is zero, its
first Chern class vanishes as a functional on closed
$(n-1,n-1)$-forms. Precisely, apply
\cite[Proposition~4.7]{CDM26} to its weakly positive rank-one
determinant. The zero degree implies that the first Chern class
of the resolved determinant is represented by an exceptional
divisor; its pushforward is therefore the zero functional.
The stable flatness theorem \cite[Theorem~1.2]{CDM26}, whose
ambient hypothesis is normality, now shows that every
$\mathcal S_i^*|_{X_{\rm reg}}$ is Hermitian flat. Consequently
$\mathcal S|_{X_{\rm reg}}$ is Hermitian flat.

Finite separable pullback in characteristic zero preserves
polystability for reflexive sheaves on normal projective varieties,
with the pulled-back polarization; see
\cite[Lemma~2.1(iv)]{Wei24}. Since $f^*H\sim qH$, the sheaf
$f^{[*]}\mathcal S$ is polystable of $H$-slope zero. Its image under
the given dynamical isomorphism is therefore contained in
$\mathcal S$. Both have the same rank and degree. The inclusion
is an isomorphism in codimension one, hence everywhere by
reflexivity. Thus
\[
                         f^{[*]}\mathcal S\simeq\mathcal S.
\]

We next check compatibility with the flat connection. Put
$U=X_{\rm reg}\cap f^{-1}(X_{\rm reg})$. Its complement in
$X_{\rm reg}$ has codimension at least two. The pullback of the
unitary local system of $\mathcal S$ on $U$ extends uniquely to a
unitary local system on $X_{\rm reg}$: removing a complex analytic
subset of codimension at least two from a complex manifold does
not change its fundamental group. Its holomorphic bundle is the
restriction of $f^{[*]}\mathcal S$, by uniqueness of reflexive
extension.

If $A$ is a global holomorphic morphism between two Hermitian
flat bundles on $X_{\rm reg}$, its squared norm is plurisubharmonic.
It extends across $X\setminus X_{\rm reg}$ by normal Hartogs
extension for plurisubharmonic functions
\cite[Theorem~2.1]{CDM26}. Since $X$ is compact, that extension is
constant. In local unitary flat frames the Bochner identity then
gives $\nabla A=0$. Apply this to the dynamical isomorphism to
obtain preservation of the flat connection.

Finally this connection is algebraic. Indeed, subtract a rational
connection on $\mathcal S$; the difference is a holomorphic section
of $\mathcal{H}om(\mathcal S,\mathcal S\otimes\Omega^1)$ away from
the fixed polar divisor of the rational connection. After twisting
by that divisor it extends across codimension two as a section of
a coherent reflexive sheaf on $X$. Projective GAGA makes it
algebraic. Regular singularity follows from the normal
compactification criterion for integrable connections: the
connection has no pole at any codimension-one point of $X$;
see \cite[Theorem~1.1.1]{And07}.
\end{proof}

\begin{theorem}[An invariant foliation gives a factor]
\label{thm:normal-invariant-foliation-factor}
Let $f:X\to X$ be a $q$-polarized endomorphism of a normal
projective complex variety. Suppose that $f$ preserves a
nonzero proper rational foliation $\mathcal F$, and that $f$
admits the locally finite Poincar\'e map used in
Proposition~\ref{prop:invariant-flat-unipotent-monodromy}.
Then either $f$ has a connected rational factor of dimension
strictly between zero and $\dim X$, admitting a normal projective
$q$-polarized model, or $f$ is exceptional.
\end{theorem}

\begin{proof}
The algebraic reductions in
Proposition~\ref{prop:invariant-foliation-two-algebraic-reductions}
already give a factor if the very general leaf has proper
Zariski closure or if $\mathcal F$ has a nonzero algebraic part.
We may therefore assume that $\mathcal F$ is purely
transcendental and that its very general leaf is Zariski dense.

We first show that
\[
                         K_{\mathcal F}\cdot H^{n-1}\ge0.
\]
Let $\rho:Z\to X$ be a projective resolution, let
$\widetilde{\mathcal F}$ be the induced foliation, and put
$\alpha=(\rho^*H)^{n-1}$. If its maximal
Harder--Narasimhan slope were positive, the maximal destabilizing
subsheaf $\mathcal A\subset\widetilde{\mathcal F}$ would be a
foliation. Indeed,
\[
 2\mu_{\alpha,\min}(\mathcal A)
       >\mu_{\alpha,\max}(\widetilde{\mathcal F}/\mathcal A),
\]
and the slope inequality annihilates the bracket morphism
$\wedge^2\mathcal A\to\widetilde{\mathcal F}/\mathcal A$.
The positive-slope algebraicity theorem
\cite[Theorem~1.1]{CP19} makes $\mathcal A$ algebraically integrable
with rationally connected leaves, contradicting pure
transcendence. Thus every Harder--Narasimhan slope is nonpositive,
and the displayed inequality follows from
\[
 K_{\widetilde{\mathcal F}}\cdot\alpha
                         =K_{\mathcal F}\cdot H^{n-1}.
\]

On the other hand, the foliated ramification identity is
\[
 K_{\mathcal F}=f^*K_{\mathcal F}+R_{\mathcal F},
                     \qquad R_{\mathcal F}\ge0.
\]
It gives
$(1-q)K_{\mathcal F}\cdot H^{n-1}
=R_{\mathcal F}\cdot H^{n-1}$, so $R_{\mathcal F}=0$.
The differential therefore induces a reflexive isomorphism
\[
                         f^{[*]}\mathcal F\simeq\mathcal F.
\]
Apply Lemma~\ref{lem:invariant-reflexive-flat-socle} to obtain a
nonzero invariant Hermitian flat subsheaf
$\mathcal S\subset\mathcal F\subset T_X$.

Proposition~\ref{prop:invariant-flat-unipotent-monodromy} makes
its monodromy virtually unipotent. Since that monodromy is
unitary, it is finite. Let
\[
             \rho:\pi_1(X_{\rm reg})\longrightarrow U(s)
\]
be its representation and take the finite Galois cover of
$X_{\rm reg}$ defined by $\ker\rho$. Its normalization over $X$
is a finite Galois quasi-\'etale cover $p:X'\to X$. This cover is
dynamically compatible. Indeed, after choosing a base point and
a path, the flat dynamical isomorphism gives
\[
                  \rho(f_*\gamma)=C\rho(\gamma)C^{-1}.
\]
Here $f_*$ is defined using
$X_{\rm reg}\cap f^{-1}(X_{\rm reg})$, whose inclusion in
$X_{\rm reg}$ induces an isomorphism of fundamental groups.
Thus $f_*^{-1}(\ker\rho)=\ker\rho$, so $f$ lifts to an endomorphism
$f':X'\to X'$. Normality extends the lift from the common smooth
locus, and $p^*H$ is a $q$-polarization of $f'$.

The pulled-back flat bundle is trivial, with a nonzero
finite-dimensional space $V$ of rational horizontal sections.
Their evaluations are regular vector fields on $X'$: on the
\'etale smooth locus this follows from
$\mathcal S\subset T_X$, and normality extends the derivations
across codimension two. Dynamical transport acts on $V$ by an
invertible constant matrix. The deck group $\Gamma$ also preserves
$V$, since it preserves the pulled-back connection. The Lie algebra
generated by $V$ is finite-dimensional as a subspace of
$H^0(X',T_{X'})$, and is preserved both by dynamical transport
and by $\Gamma$.

By Proposition~\ref{prop:invariant-flows-semiabelian}, the connected
algebraic group generated by these flows is a positive-dimensional
semiabelian group $G$, normalized by $f'$ through an isogeny and
by $\Gamma$ through automorphisms. If $G$ has an open orbit, that
orbit is isomorphic to $G$, because the action is effective and
$G$ is commutative. The lifted map is an affine isogeny there.
Lemma~\ref{lem:semiabelian-factor-scalar-lift} and birational
invariance make $f'$ exceptional; exceptionality descends under
the finite equivariant map $p$ by \cite[Theorem~3.9]{JSX26}.

If the general $G$-orbit is proper, put
$K=\mathbb C(X')^G$. The deck group acts on $K$ and
\[
             L=K^\Gamma\subset\mathbb C(X')^\Gamma=\mathbb C(X).
\]
This field has transcendence degree strictly between zero and
$n$. It is relatively algebraically closed in $\mathbb C(X)$:
an element of $\mathbb C(X)$ algebraic over $L$ is algebraic over
$K$, hence belongs to $K$ by connectedness of the Rosenlicht
quotient; being deck-invariant, it belongs to $L$. Conjugation by $C$ induces an automorphism $\beta$ of the finite
deck group, with $f'\delta=\beta(\delta)f'$.
Together with the isogeny normalization of $G$, this gives
$f^*L\subset L$. Thus $L$ defines a connected rational factor of
$f$, and Proposition~\ref{prop:connected-factor-chow-regularization}
provides its normal projective $q$-polarized model.
\end{proof}


\subsection{Finite webs and descent of the transcendental branch}

\begin{lemma}[The splitting cover of an invariant web]
\label{lem:invariant-web-splitting-cover}
Let $f:X\to X$ be a polarized endomorphism of a normal projective
variety. Suppose that, on a dense smooth open, a finite unordered
set of distinct integrable distributions is algebraic and is
preserved by $Df$. There is a finite Galois cover $p:Y\to X$
on which the members become rational foliations. The map $f$
lifts to a polarized endomorphism $F:Y\to Y$. Its action on the
Galois group $\Gamma$ is an automorphism $\beta$, in the sense that
\[
                         F\gamma=\beta(\gamma)F.
\]
After an iterate, each member foliation is preserved.
\end{lemma}
\begin{proof}
On an open where $f$ is \'etale and the members are distinct,
$Df$ acts bijectively on the finite set of members. Take the
finite \'etale cover of orderings and a connected component of
its splitting cover. Its function field is the splitting field
of the algebraic coordinates of the members, and its normalization
over $X$ is finite Galois. The derivative transports an ordering
bijectively to an ordering over $f(x)$; applying its inverse
recovers the original ordering. If the selected connected
component is transported to another component, compose transport
with a constant permutation of the labels taking that component
back to the chosen one. Thus the cover is identified
with its pullback under $f$ on the common generic open. This
identification lifts $f$ and acts bijectively on the finite deck
group. Normalization extends the lift across the boundary.
The pullback of a polarization of $X$ polarizes $F$ with the
same weight. The lift permutes a finite set of foliations,
so an iterate fixes each one.
\end{proof}

\begin{proposition}[The transcendental member of a finite web]
\label{prop:finite-web-transcendental-descent}
In the preceding setting assume that one member foliation on $Y$
is purely transcendental with Zariski dense very general leaves.
Assume the locally finite Poincar\'e hypothesis used in
Theorem~\ref{thm:normal-invariant-foliation-factor} on the relevant
finite dynamical cover. Then the original map $f$ has a proper
connected rational polarized factor, or $f$ is exceptional.
\end{proposition}
\begin{proof}
Take the finite orbit of the chosen member under both the deck
group and dynamical transport. Every member in this orbit is
purely transcendental with dense very general leaves; these
properties are preserved by finite pullback and by deck
transformations. Write the members as $\mathcal F_i$, with $F$
inducing a permutation $\sigma$ of their indices. The slope
argument of Theorem~\ref{thm:normal-invariant-foliation-factor}
gives $\deg_H K_{\mathcal F_i}\ge0$, whereas single-step
ramification gives
\[
 K_{\mathcal F_i}=F^*K_{\mathcal F_{\sigma(i)}}+R_i,
                      \qquad R_i\ge0.
\]
Iterating this identity around each finite cycle shows that all
these degrees and all $R_i$ vanish. Thus the differential
identifies $\mathcal F_i$ with $F^{[*]}\mathcal F_{\sigma(i)}$.

Let $\mathcal S_i$ be the nonzero unitary flat socle of
$\mathcal F_i$. Pullback polystability puts
$F^{[*]}\mathcal S_{\sigma(i)}$ inside $\mathcal S_i$.
The resulting rank inequalities around a cycle are all equalities,
so these inclusions are isomorphisms. Normal Hartogs and the
Bochner argument show that they preserve the flat connections.
The deck group also permutes these connections. Applying the
monodromy argument to an iterate fixing a member shows that
every member socle has finite holonomy.

Take the compositum of their finite holonomy covers over $Y$.
It is finite and Galois over $X$:
the deck group of $Y/X$ permutes the finite set of conjugate
covers, and the holonomy covers themselves are Galois.
Call its normalization $Z$.

This common cover is dynamical. For each socle the holonomy kernel
is exactly inverse-invariant under the induced homomorphism of
fundamental groups. The endomorphism $F$ permutes these kernels
through the deck automorphism $\beta$. Their finite intersection
is therefore exactly inverse-invariant. Thus $F$ lifts to a
polarized endomorphism $\widetilde F:Z\to Z$, and its action on
the full deck group of $Z/X$ is an automorphism.

The map $Z\to Y$ is quasi-\'etale. The horizontal sections of all
conjugate socles therefore give regular vector fields on $Z$,
by extension across codimension two. Their span is nonzero,
finite-dimensional, and preserved by both dynamical transport
and the full deck group. The Lie algebra it generates has the
same invariance properties. Its connected algebraic flow closure
is semiabelian by Proposition~\ref{prop:invariant-flows-semiabelian}.
If it has an open orbit, $\widetilde F$ and hence $f$ are
exceptional. Otherwise its Rosenlicht field, followed by deck
invariants, is a proper relatively algebraically closed subfield
of $\mathbb C(X)$ preserved by $f^*$. The invariant-field argument
at the end of Theorem~\ref{thm:normal-invariant-foliation-factor}
gives the required connected rational factor, and Chow
regularization gives its polarized projective model.
\end{proof}

\begin{lemma}[The algebraic branch of a finite web]
\label{lem:finite-web-algebraic-correspondence}
Let $p:Y\to X$ be the finite Galois dynamical splitting cover of
Lemma~\ref{lem:invariant-web-splitting-cover}. Suppose that a member
foliation supplies a proper connected rational factor of $Y$.
After an iterate let
\[
             \pi_i:Y\dashrightarrow B_i,\qquad 1\le i\le m,
\]
be its distinct translates under the deck group and dynamical
transport, with polarized endomorphisms
$g_i$ on normal projective models of the $B_i$. Put
\[
             K=\mathbb C\bigl(\pi_1^*\mathbb C(B_1),\ldots,
                                  \pi_m^*\mathbb C(B_m)\bigr)
                         \subset\mathbb C(Y).
\]
Then precisely one of the following constructions applies:
\begin{enumerate}
\item If $\operatorname{trdeg}K<\dim X$, its relative algebraic
closure $K^{\rm a}$ in $\mathbb C(Y)$, followed by deck invariants,
gives a proper connected rational polarized factor of $X$.
\item If $\operatorname{trdeg}K=\dim X$, the product map
$(\pi_1,\ldots,\pi_m)$ is generically finite onto its image
$T\subset\prod_i B_i$. The normalization of $T$ is projective
and carries a polarized endomorphism induced by $\prod_i g_i$.
Thus, for that common iterate, $X$ has a finite rational
correspondence with this invariant subvariety of a product of
lower-dimensional polarized systems.
\end{enumerate}
\end{lemma}
\begin{proof}
The finite set of fields is permuted by the deck group and by
single-step dynamical transport; the individual factor maps are
endomorphisms after the stated iterate. Hence $K$ and
its relative algebraic closure $K^{\rm a}$ are invariant under
both. The field $(K^{\rm a})^\Gamma$ is relatively algebraically
closed in $\mathbb C(X)$ by the same field argument as before.
In the first case its transcendence degree is positive and
strictly less than $\dim X$, proving the assertion.

In the second case the image has dimension $\dim X$ and the
product map has finite generic fiber. Take its reduced projective
closure and normalize. The product of the polarized maps is
finite and preserves this closure, and so lifts to its
normalization. The restriction of the sum of the product
polarizations is ample and is multiplied by the common
polarizing weight. The two generically finite rational arrows
from $Y$ to $X$ and to $T$ give the asserted correspondence.
Each arrow can be made finite onto a dense target open after
normalizing the target in the source function field and shrinking.
\end{proof}

\subsection{Spreading finite algebraic presentations}

\begin{lemma}[Countable incidence spreading]
\label{lem:countable-finite-presentation-spread}
Let $S$ be an irreducible algebraic curve over $\mathbb C$, and
let $f:\mathcal X\to\mathcal X$ be an algebraic family of
polarized endomorphisms of normal projective varieties over $S$.
Consider any of the following finite presentations of a fiber:
\begin{enumerate}
\item a finite dynamical cover, optionally with a specified finite
Galois group and deck-equivariance;
\item a connected rational factor with a normal projective
polarized model;
\item a finite equivariant presentation by an AT-system, including
its finite affine group action and chosen projective model;
\item a finite rational correspondence with a projective invariant
subvariety of a product of finitely many lower-dimensional
polarized systems.
\end{enumerate}
If such a presentation exists for uncountably many $s\in S(\mathbb C)$,
then, after a finite dominant base change $S'\to S$ and replacement
of $S'$ by a nonempty open, one such presentation exists
algebraically over $S'$. No uniform bound on the degrees of the
fiberwise covers or on their Hilbert polynomials is required.
The same conclusion holds if a presentation exists at a point
geometrically generic over a countable field of definition of
the original family.
\end{lemma}
\begin{proof}
Choose projective embeddings and graph embeddings for all the
varieties and maps in a presentation. There are countably many
choices of dimensions, Hilbert polynomials, degrees, finite groups,
integers specifying the polarization, and numbers of factors.
For any fixed choice, the graphs belong to finitely many Hilbert
schemes of finite type. Incidence imposes the commuting identities,
group laws, and equivariance identities by algebraic equations.
After the usual flattening stratifications, the required
normality, dominance, finiteness, and geometric integrality of
generic fibers are constructible conditions. Thus their loci are
finite unions of locally closed parameter spaces. For rational
maps, one records their graphs and dense domains of definition;
for finite maps on dense opens, one also records the excluded
closed sets. All these are finite algebraic data.

For an AT-presentation, record the polarized abelian variety with
its group law, the torus rank, the torsor and extension data, and
the affine endomorphisms and finite group action. These data can
likewise be recorded by graphs on a finite affine covering. In
the scalar torus presentation of
Lemma~\ref{lem:semiabelian-factor-scalar-lift}, one may instead
record a finite cover by an abelian variety times a fixed toric
compactification; the toric fans form a countable set. Either
encoding gives countably many finite-type parameter spaces.

Consequently all presentations under consideration are
parameterized by a countable union
\[
                            \coprod_{j\ge1} P_j\longrightarrow S
\]
of finite-type incidence spaces. If no $P_j$ dominates $S$, each
constructible image is a finite set, since $S$ is a curve. Their
union is countable. This contradicts the hypothesis. Hence some
irreducible component $P$ dominates $S$.

The generic fiber $P_\eta$ is a nonempty finite-type scheme over
$\mathbb C(S)$. A closed point of it has residue field a finite
extension $K'/\mathbb C(S)$. Normalize $S$ in $K'$. The chosen
point gives a rational section of the pulled-back incidence space,
which becomes a morphism after shrinking the resulting curve
$S'$. Pulling back its universal data gives the desired algebraic
presentation. Shrinking once more makes all the constructible
conditions valid on the whole remaining base.

For the last assertion, enlarge the chosen countable field to
its algebraic closure $k_0$ and define the family and all incidence
spaces over $k_0$. If $s_0$ is geometrically generic over $k_0$,
a constructible $k_0$-image containing $s_0$ is dense. Choose a
dominant component over $k_0$ and a closed point of its generic
fiber over $k_0(S)$, rather than over $\mathbb C(S)$. Its residue
field is finite over $k_0(S)$. Normalizing the $k_0$-model of $S$
in that field and then extending scalars to $\mathbb C$ gives
the required finite base change defined over $k_0$. Every lift
of $s_0$ is again geometrically generic over $k_0$, so none is
removed by the nonempty $k_0$-opens used to spread the data.
\end{proof}

\begin{corollary}[Spreading with a fixed reference parameter]
\label{cor:pointed-incidence-spread}
Fix a countable algebraically closed field $k_0\subset\mathbb C$
over which the parameter curve, the polarized family, and all
previously constructed algebraic diagrams are defined. Fix
$s_0\in S(\mathbb C)$ geometrically generic over $k_0$. If one of
the finite presentations in
Lemma~\ref{lem:countable-finite-presentation-spread} exists for the
fiber at $s_0$, it can be spread after a finite base change defined
over $k_0$, while retaining a lift of the same reference parameter.
Every further finite dominant base change and nonempty open restriction
defined over $k_0$ again retains every chosen lift of this reference
parameter. The presentation spread need not coincide with the
initially selected presentation of that fiber; its source dynamical
system at the reference parameter remains the prescribed one.
\end{corollary}
\begin{proof}
All the incidence spaces are defined over $k_0$. The point
representing the chosen fiberwise presentation lies in a component
whose constructible image contains $s_0$ and hence dominates $S$.
The closed generic-fiber point and normalization in the preceding
proof are chosen over $k_0$. If $s'_0$ is any point over $s_0$
in a finite base change $S'\to S$, then
\[
           \operatorname{trdeg}_{k_0} k_0(s'_0)
           =\operatorname{trdeg}_{k_0} k_0(s_0)=1.
\]
Thus $s'_0$ lies in every nonempty $k_0$-open of its irreducible
parameter curve. Repeating this argument proves the assertion
for all subsequent steps, without changing the reference fiber.
\end{proof}


\subsection{Algebraic members of a finite invariant web}

\begin{lemma}[Invariant subvarieties of a fixed polarized system are constant]
\label{lem:fixed-polarized-invariant-image-rigidity}
Let $(W,h)$ be a polarized projective system over $\mathbb C$,
and let $K/\mathbb C$ be a field extension. Every geometrically
integral closed subvariety $Z\subset W_K$ with $h_K(Z)=Z$
is defined over $\mathbb C$.
Consequently an algebraic family of reduced invariant subvarieties
of one fixed polarized system is constant after restriction and,
if its components must be marked, finite parameter extension.
\end{lemma}

\begin{proof}
The restriction $h_K|_Z$ is polarized by the restriction of the
ambient ample polarization. Its periodic points over $\overline K$
are Zariski dense by Fakhruddin's periodic-point theorem
\cite[Theorem~5.1]{Fak03}.

For every $m\ge1$, the fixed-point scheme of $h^m$ is
zero-dimensional. Indeed a positive-dimensional fixed locus would
contain a curve on which an ample polarization is both preserved
and multiplied by $q^m>1$, a contradiction. That fixed-point
scheme is defined over the algebraically closed field $\mathbb C$.
Thus all its geometric points over $\overline K$ are constant
$\mathbb C$-points. In particular the dense set of periodic points
of $h_K|_Z$ consists of $\mathbb C$-points of $W$.

The Zariski closure, after any field extension, of a set of
$\mathbb C$-points is the base change of its closure over
$\mathbb C$. For clarity, express the finitely many coefficients
of any polynomial vanishing on the set in a basis linearly
independent over $\mathbb C$; evaluation at a constant point then
shows that every coefficient polynomial over $\mathbb C$ vanishes.
Applying this observation to the dense set above shows that $Z$
is the base change of a subvariety over $\mathbb C$. The family
assertion follows by applying the result to its geometric generic
components and then spreading the equality over a parameter open.
\end{proof}


\section{Stable periodic fibers and varying covers}


\subsection{The relative equilibrium measure}

The relative equilibrium-measure identity supplies the Julia-set
inclusion used for the spectral closure under periodic-fiber returns.
For fibered endomorphisms of projective space, see \cite{DT21}.

\begin{lemma}[Continuous fiber measures]\label{lem:continuous-fiber-measures}
Let $\pi:X\to B$ be a surjective morphism of normal projective
varieties, with $\dim B=r$ and $\dim X=m+r$.  Let $F:X\to X$ and
$g:B\to B$ be $q$-polarized endomorphisms, with $\pi F=g\pi$.
Choose polarizations $L$ and $M$, and denote their canonical currents
by $T$ and $S$.  Let $B^0\subset B_{\rm reg}$ be an open set over
which $\pi$ is flat with normal geometrically integral fibers.
Put $d=(L^m\cdot X_b)$, for $b\in B^0$, and set
\[
                  \nu_b=d^{-1}(T|_{X_b})^m.
\]
Then $b\mapsto\nu_b$, viewed as a family of measures on $X$, is
weakly continuous on $B^0$, and
\begin{equation}\label{eq:normal-fiber-disintegration}
 \mu_F=\int_{B^0}\nu_b\,d\mu_g(b),\qquad
 \mu_F=\frac{T^m\wedge\pi^*S^r}{L^m\cdot\pi^*M^r}.
\end{equation}
For every $b\in B^0\cap\operatorname{supp}\mu_g$,
\begin{equation}\label{eq:every-good-fiber-support}
                  \operatorname{supp}\nu_b
                         \subset\operatorname{supp}\mu_F.
\end{equation}
If $g^n(b)=b$, then $\nu_b$ is the equilibrium measure of
$F^n|_{X_b}$.
\end{lemma}

\begin{proof}
Canonical currents of polarized endomorphisms have continuous
local potentials.  The normalized top wedge of any polarization
of $F$ is its equilibrium measure.  This assertion also holds on
normal varieties: on a projective resolution, normalized pullbacks
of a smooth probability measure converge to the equilibrium
measure, by \cite[Theorem~1.2]{DNT15}; the exceptional and
indeterminacy sets have measure zero.  The uniform convergence of
the canonical potentials identifies this limit with the normalized
top wedge.  In particular this measure is independent of the
chosen polarization.

For every nonnegative rational $t$, the ample rational class
$L+t\pi^*M$ is $q$-polarized and its canonical current is
$T+t\pi^*S$.  Consequently
\[
 (T+t\pi^*S)^{m+r}
             =(L+t\pi^*M)^{m+r}\mu_F.
\]
Comparing the coefficients of $t^r$ proves the second identity in
\eqref{eq:normal-fiber-disintegration}.  Notice that
$L^m\cdot\pi^*M^r=d(M^r)$.

Flatness gives the continuity of the fiber cycles $b\mapsto[X_b]$.
Wedge product with a fixed current having continuous local
potentials preserves this continuity.  Indeed, locally write
$T=dd^cu$.  If $R_j\to R$ are positive closed currents, then
$uR_j\to uR$ because $u$ is continuous and the masses are locally
bounded.  Hence
\[
                   dd^c(uR_j)\longrightarrow dd^c(uR).
\]
Iteration applies to $T^m\wedge[X_b]$; the local mass bounds follow
from positivity and the Chern--Levine--Nirenberg inequality.  This
also proves the assertion on a normal analytic space by embedding
it locally and using a continuous extension of $u$ to define the
products with currents supported on that space.  Thus the
probability measures $\nu_b$ depend weakly continuously on $b$.

On the submersion locus of $\pi$, Fubini's formula for currents
with continuous potentials gives
\[
 T^m\wedge\pi^*S^r
                  =\int_{B^0}(T|_{X_b})^m\,d(S^r)(b).
\]
One may verify this in product coordinates by smooth approximation
of the local potentials.  The relative singular locus has zero
mass for every fiber measure, and also for the measure on the left,
which is a multiple of $\mu_F$.  The inverse image of $B\setminus
B^0$ likewise has zero equilibrium measure.  The displayed formula
therefore gives the first identity in
\eqref{eq:normal-fiber-disintegration} on all of $X$.

Let $x\in\operatorname{supp}\nu_b$, with $b\in B^0\cap
\operatorname{supp}\mu_g$.  In any neighborhood of $x$, choose a
nonnegative continuous function $\chi$ with
$\int\chi\,d\nu_b>0$.  Continuity makes this integral positive
on a neighborhood of $b$, which has positive $\mu_g$-measure.
The disintegration formula gives $\int\chi\,d\mu_F>0$.
This proves \eqref{eq:every-good-fiber-support} at every such $b$.
Finally, on a periodic fiber the restricted current satisfies
\[
          (F^n|_{X_b})^*(T|_{X_b})=q^nT|_{X_b}.
\]
Uniqueness of the continuous canonical metric identifies its
normalized top wedge with the equilibrium measure of the return.
\end{proof}


\subsection{Finite dynamical covers of a fixed system}

\begin{lemma}[A finite dynamical cover of a fixed system]
\label{lem:cover-fixed-target-isotrivial}
Let $(Y,g)$ be a fixed polarized normal projective complex
system.  Let $p:\mathcal Z\to Y\times S$ be a finite surjective
map from a family of normal projective varieties over a smooth
complex curve, with a finite endomorphism $F_Z$ satisfying
\[
                         pF_Z=(g\times\operatorname{id}_S)p.
\]
Then the family $(\mathcal Z,F_Z)$ is isotrivial.  After finite
parameter extension and restriction, the whole diagram is
constant up to isomorphism.
\end{lemma}

\begin{proof}
If $Y$ is a point, a finite parameter extension trivializes the
finite source and the assertion is immediate.  Assume $\dim Y>0$.
Work first over $k=\mathbf C(S)$ and put $K=k(Y)$ and $L=k(Z)$.
Let $\sigma$ be the embedding induced by $F_Z$.  The decreasing
sequence of finite intermediate extensions
\[
                             L_j=K\sigma^j(L)
\]
stabilizes; equality at one step implies equality at all later
steps by applying $\sigma$ and adjoining $K$.  Let $L_0$ denote
its stable value.  Then
\[
            \sigma(L_0)\subset L_0,\qquad K\sigma(L_0)=L_0.
\]
Its normalization $p_0:Z_0\to Y_k$ carries a finite polarized
lift.  The embedding $\sigma^j:L\hookrightarrow L_0$ at a stable
index gives a finite equivariant map $Z_0\to Z$, covering $g^j$
on $Y$.  It is enough to prove that the Cartesian cover $Z_0$
is constant, and then apply
Proposition~\ref{prop:constant-finite-cover-downstairs}.

Let $D$ be the reduced branch divisor of $p_0$ over $\overline k$,
let $H$ polarize $g$, and choose $N$ with
$q^N>(D\cdot H^{\dim Y-1})$. The equality
$K\sigma(L_0)=L_0$ makes every dynamical pullback connected by
Lemma~\ref{lem:stable-field-normalized-pullback}. Applying
Lemma~\ref{lem:finite-path-critical-support} to the length-$N$
constant path shows that $D$ is supported on the fixed divisor
$\operatorname{CritVal}(g^N)$.

Spread this cover over a parameter open. By
Lemma~\ref{lem:finite-branch-cover-classes}, applied to the
constant pair
$(Y,\operatorname{CritVal}(g^N))$, it belongs to a fixed finite
set of geometric cover classes. After finite parameter extension
and restriction the cover is one fixed cover $p_0:Z_0\to Y$:
the class is constant, and its finite isomorphism torsor is
trivialized by a further finite extension. For each $z\in Z_0$,
the image $F_{Z_0,s}(z)$ belongs to the fixed finite fiber over
$g(p_0(z))$, so it is independent of $s$. The lifted system is
therefore fixed.
The finite equivariant map $Z_0\to Z$ now gives the claimed
isotriviality of $Z$.

Once $Z$ has been trivialized, the maps $p_s:Z\to Y$ form a
family of equivariant finite maps.  The graphs are periodic
subvarieties of the fixed polarized product system on $Z\times Y$.
Lemma~\ref{lem:fixed-polarized-invariant-image-rigidity} makes
them constant, proving the assertion about the whole diagram.
\end{proof}

\begin{corollary}[A finite rational factor cannot hide variation]
\label{cor:cover-fixed-rational-target-isotrivial}
A family of polarized normal projective systems admitting a
dominant generically finite rational semiconjugacy to a fixed
polarized projective system is isotrivial.
\end{corollary}

\begin{proof}
Take the equivariant normalized graph and its Stein factorization
over the fixed target.  The finite Stein target is constant by
Lemma~\ref{lem:cover-fixed-target-isotrivial}.  Its birational
conjugacy with the graph and then with the original family
transports a rational parameter-horizontal derivation.
Lemma~\ref{lem:rational-horizontal-normal} extends that derivation
and makes the original projective family isotrivial.
\end{proof}

\subsection{The group of all forward-compatible gauges}

\begin{proposition}[Finite-dimensional forward gauge torsor]
\label{prop:cover-forward-gauge-torsor}
Let $(B,g)$ be a fixed normal projective polarized system.
Suppose an algebraic family over $B\times S$ gives normal
projective generic phase fibers $X_{s,b}$, ample line bundles
$L_{s,b}$, and finite arrows
\[
 a_{s,b}:X_{s,b}\longrightarrow X_{s,g(b)},\qquad
 a_{s,b}^*L_{s,g(b)}\simeq L_{s,b}^{\otimes q}.
\]
Assume that these polarized fibers have bounded projective
embeddings and that a Zariski-dense set of good periodic base
points has isotrivial return families.  Fix a reference parameter
$s_0$ which is good for every selected cycle.
Over $\mathbf C(B)(S)$ there is a nonempty finite-type torsor
$I$ of source gauges which descend through every forward arrow
composition and preserve the chosen line bundles at every target.
Its group $G$ is defined over $\mathbf C(B)$ by the reference
system.  First-arrow transport defines algebraic maps
\[
                  I\longrightarrow g^*I,\qquad
                  G\longrightarrow g^*G.
\]
The latter has finite kernel and is an isogeny on identity
components.  The connected group $G^0$ is a torus.
\end{proposition}

\begin{proof}
For a length-$n$ arrow composition $A_{s,b}^{(n)}$, let
\[
 R_{s,b,n}=
 \bigl(X_{s,b}\times_{X_{s,g^n(b)}}X_{s,b}\bigr)_{\rm red}.
\]
A source isomorphism $C:X_{s_0,b}\to X_{s,b}$ descends to a
unique target isomorphism $C_n$ if and only if it identifies
$R_{s_0,b,n}$ and $R_{s,b,n}$.  Indeed equality of the generic
reduced equivalence relations identifies the two target function
fields inside the source field.  The image of the pair of finite
maps in the product of the targets is finite and birational over
either target; normality makes both projections isomorphisms.

For each $N$, require this condition for $1\leq n\leq N$ and
also require $C_n$ to preserve the specified target line bundle.
These are finite-type algebraic incidence conditions on polarized
isomorphisms and their graphs.  Denote their source-gauge locus
by $I_N$.  Its corresponding reference automorphism group $G_N$
is a closed subgroup of the finite-type linear group
$\operatorname{Aut}_{L_{s_0,b}}(X_{s_0,b})$.
To see closedness, preservation of each reduced relation is a
closed Hilbert-scheme condition.  Descent to its finite normal
quotient gives an algebraic group homomorphism to the target
automorphism group, and preservation of the target Picard class
is closed.  The descent homomorphism is algebraic by the graph
construction; equivalently its graph is a finite-type algebraic
subgroup with trivial first-projection kernel, which in
characteristic zero identifies it with its image.
Whenever $I_N$ is nonempty, two of its points differ by exactly
one element of $G_N$; it is therefore a torsor under $G_N$.

An isotrivial periodic return trivializes its full finite cycle,
including its polarized intermediate arrows, by
Lemmas~\ref{lem:constant-return-finite-chain} and
\ref{lem:constant-return-polarized-chain}.  Thus every selected
good cycle supplies a point of each finite incidence problem.
If an intermediate phase is not in the normal-fiber open, use
the normalization of the corresponding closed image of the
starting fiber.  These normalizations form a complete finite
cycle.  For any fixed $N$, discard the proper closed set where
one of the first $N$ phases is bad; at the retained phases the
normalizations are the whole prescribed fibers.  The image of
the incidence locus in $B\times S$ is constructible and contains
a parameter open above a Zariski-dense set of such base points.
Its closure contains their full cylinders.  Hence its image is
dense and contains an open, proving that $I_N$ is nonempty over
the algebraic closure of the generic field.

The descending chain of closed subgroups $G_N$ stabilizes in
this finite-type group.  After stabilization, the nested nonempty
$I_N$ are torsors under one group and hence are equal.  Their
common value $I$ satisfies every forward condition and is
nonempty over a finite extension of the generic field.
Cancellation in the length-$(n+1)$ identities shows that the
first target gauge belongs to $g^*I$.  It gives the stated
transport maps.  The kernel of group transport lies in the
finite deck group of the first arrow.  The source and target
identity components have the same dimension, because $g$ is
dominant.  The image of a group homomorphism is closed, so
transport is an isogeny on the identity components.

Choose one fixed very ample power of the polarizations, with
projective embedding dimension $M-1$.  If $G^0$ contained an
additive subgroup, choose a nontrivial orbit and its closure $C$.
A unipotent one-parameter subgroup in a projective representation
of dimension $M$ has polynomial orbit coordinates of degree at
most $M-1$.  Consequently
\[
                      0<\deg C\leq M-1.
\]
Every iterated transport carries this additive group isomorphically
to another additive group, since its kernel is finite and
characteristic zero has no nontrivial finite additive subgroup.
The corresponding finite arrow maps $C$ birationally to an
orbit closure $C_n$ with the same degree bound.  Polarization
gives $\deg C_n=q^n\deg C$, a contradiction for large $n$.
Thus $G^0$ has no additive one-parameter subgroup.  It is linear
because it preserves an ample line bundle.  A connected linear
group with no additive subgroup is a torus.
\end{proof}

\subsection{Fixed-base periodic-return rigidity}

\begin{theorem}[Isotrivial periodic returns over a fixed base]
\label{thm:cover-fixed-base-return-rigidity}
Let $F:\mathcal X\to\mathcal X$ be an algebraic one-parameter
family of polarized endomorphisms of normal projective varieties.
Suppose it has an equivariant dominant rational map to one fixed
normal projective polarized system $(B,g)$, with geometrically
integral generic phase fiber.  If the return families on a
Zariski-dense set of good periodic base points are isotrivial,
then $F$ is isotrivial.
\end{theorem}

\begin{proof}
Normalize the equivariant graph of the rational factor.  The sum
of the source and base polarizations gives a finite polarized
lift.  Its fibers over a dense phase open are normal and
geometrically integral, and one fixed sufficiently high power of
the polarization gives bounded embeddings.  Since the periodic
points of a fixed polarized system are countable, choose a
reference parameter outside the exceptional sets of all selected
return trivializations.  Proposition~\ref{prop:cover-forward-gauge-torsor} applies.

Write $T=G^0$.  Transporting the reference action through a gauge
gives a torus action on the varying generic phase fiber.  It is
defined independently of the gauge as a form of $T$: changing the
gauge conjugates the action by an element of $G$, which preserves
its identity component.  Its invariant field is finitely generated
and relatively algebraically closed in the source field.  The
first-arrow isogeny preserves this field.

Take the Chow reduction of the resulting connected rational
quotient.  Meng--Zhang's Chow-reduction theorem
\cite[Proposition~7.1 and Theorem~7.2]{MZ22} makes its induced
rational dynamics a morphism; polarization descent
\cite[Theorem~1.3]{MZ18} gives a polarized normal projective model.  Spread this model over the parameter curve.
On this quotient the gauge ambiguity is the finite group $G/T$.
Proposition~\ref{prop:finite-ambiguity-horizontal} gives a rational
parameter-horizontal derivation, annihilating $\mathbf C(B)$,
which commutes with the quotient dynamics.  Hence this quotient
is isotrivial, with its trivialization over $B$.  Denote its fixed
model by $(Y,h)$.

The generic source fiber over $Y$ is a torus orbit closure.
Indeed a faithful torus action on a projective variety has
trivial generic stabilizer, as is seen from the characters in a
faithful projective embedding; its rational quotient therefore
has a dense torus orbit in the generic fiber.  Pull the torus
to $\mathbf C(Y)(S)$ and take its minimal Galois splitting field.
Proposition~\ref{prop:minimal-torus-splitting-dynamics} gives a
compatible finite polarized cover of $Y\times S$ with Cartesian
normalized pullback.  After finite parameter extension, take an
iterate preserving one geometric component.  Lemma~\ref{lem:cover-fixed-target-isotrivial}
makes this cover, including its lifted dynamics, constant after
a finite parameter extension.  Write it as $(Y',h')$.

Normalize the original source in the compositum with this
splitting field.  Geometric integrality of the generic quotient
fiber makes its function field a regular extension of
$\mathbf C(Y)(S)$, so the compositum is a field.  The original
source already carries a finite polarized endomorphism.  The last
assertion of Proposition~\ref{prop:minimal-torus-splitting-dynamics},
applied to this source model, therefore gives an actual finite
polarized cover $(\mathcal X',F')$ of the source family.
The split torus acts rationally on $\mathcal X'$ over $Y'\times S$
and has a dense generic fiber orbit.  Its first-arrow transport
is one integral exponent matrix, hence a fixed isogeny of the
split torus.  Proposition~\ref{prop:toric-fixed-quotient-rigidity}
now gives isotriviality of $(\mathcal X',F')$.

This last application handles the line-bundle and compactification
data as well.  After the scalar-power iterate, the weighted
vertical divisors satisfy $h'^*D_s-qD_s=\operatorname{div}(c_s)$.
Their fractional parts are fixed, and the integral differences
belong to the finite kernel of $h'^*-[q]$ on the birational
$\operatorname{Pic}^0$.  The resulting rational character
coboundaries conjugate the toric systems; the horizontal-field
lemma extends these conjugacies to the normal projective models.

The constant finite cover descends isotriviality to the source
by Proposition~\ref{prop:constant-finite-cover-downstairs}.
The rational-horizontal-field lemma transfers it through the
initial birational graph, and
Corollary~\ref{cor:isotrivial-iterate-normal} removes any iterate
used to preserve a geometric component or make the torus
transport scalar.
\end{proof}


\subsection{Variation in an exceptional family is abelian}

\begin{lemma}[The birational Albanese of an AT compactification]
\label{lem:cover-at-birational-albanese}
Let $P\to A$ be a torsor under a split torus over an abelian
variety in characteristic zero, and let $Z$ be a normal projective
compactification of $P$.  Then
\[
                 \operatorname{Alb}_{\mathrm{bir}}(Z)\simeq A
\]
after choosing origins.  The assertion is compatible with finite
AT-isogenies and with algebraic families after restriction of the
parameter base.
\end{lemma}

\begin{proof}
A rational map from $Z$ to an abelian variety is regular on a
smooth projective model.  Its restriction to a geometric torus
fiber of $P\to A$ is constant: a morphism from $\mathbf G_m$ to an
abelian variety extends to $\mathbf P^1$ and is constant, and the
coordinate one-parameter subgroups generate the torus.  Hence it
factors rationally through $A$.  A rational map from the smooth
variety $A$ to an abelian variety extends to a morphism, which,
after translation, is a homomorphism.  Conversely the projection
$P\to A$ is a rational map from $Z$ and generates $A$.  These are
precisely the universal properties of the birational Albanese.
The construction is functorial.  In a family, resolve the generic
compactification and spread the resolution and the maps over a
nonempty parameter open; the same universal identities then hold
fiberwise.
\end{proof}

\begin{theorem}[Rigidity of an exceptional family with constant abelian part]
\label{thm:cover-exceptional-constant-abelian}
Let $F:\mathcal X\to\mathcal X$ be an algebraic one-parameter
family of $q$-polarized endomorphisms of normal projective varieties.
Suppose that its geometric generic system has an exceptional
presentation.  If the abelian base of one algebraic family of full
AT presentations is isotrivial, then $F$ is isotrivial.
\end{theorem}

\begin{proof}
All constructions are made after a finite extension of the
parameter field and then spread over a nonempty parameter open.
Replace $F$ by the fixed iterate occurring in the presentation,
and write $Q$ for its polarization weight.  We have
\[
 P\longrightarrow\mathcal X^\circ,\qquad
 \Psi:P\longrightarrow P,\qquad
 P\longrightarrow\mathcal A,
\]
where the first map is finite and surjective, $P$ is a torus torsor
over an abelian scheme $\mathcal A$, and $\Psi$ is a $Q$-lift.
Normalize $\mathcal X$ in the function field of $P$:
\[
                p:\mathcal Z\longrightarrow\mathcal X.
\]
The map $p$ is finite and surjective, its source is normal, and
$P=p^{-1}(\mathcal X^\circ)$.  The lift $\Psi$ extends to a finite
endomorphism $F_{\mathcal Z}$; the pullback of the polarization
of $F$ polarizes $F_{\mathcal Z}$ with weight $Q$.

Trivialize $\mathcal A$ as $A\times S$.  The endomorphism induced
by $\Psi$ has the form $\alpha_s+a_s$, where $\alpha_s$ is a group
endomorphism of the fixed abelian variety.  The rigidity lemma for
homomorphisms of abelian schemes makes $\alpha_s=\alpha$ constant.
The endomorphism $\alpha$ is polarized with weight $Q$, so
$1-\alpha$ is an isogeny.  A finite parameter extension supplies
$c_s$ with $(1-\alpha)c_s=a_s$; translating the base by $c_s$
makes its dynamical system the fixed pair $(A,\alpha)$.

Split the torus after a finite parameter extension.  Its action
on $P$ gives a rational action on $\mathcal Z$ with a dense orbit
in the geometric generic fiber over $A$.  The $Q$-lift identity
is
\[
              F_{\mathcal Z}(t z)=[Q](t)F_{\mathcal Z}(z).
\]
Proposition~\ref{prop:toric-fixed-quotient-rigidity} applies to the
rational quotient $\mathcal Z\dashrightarrow A\times S$ and
shows that $(\mathcal Z,F_{\mathcal Z})$ is isotrivial.
After trivializing this family, Proposition~\ref{prop:constant-finite-cover-downstairs}
applies to its finite equivariant map $p$ and makes
$(\mathcal X,F)$ isotrivial.  Finally
Corollary~\ref{cor:isotrivial-iterate-normal} removes the initial
iterate.
\end{proof}

\begin{corollary}[Bounded varying covers of nonisotrivial exceptional families]
\label{cor:cover-nonconstant-exceptional-varying-cover}
Let $F$ be a nonisotrivial algebraic family of polarized
endomorphisms of normal projective $m$-folds, and suppose its
geometric generic system is exceptional.  After a finite parameter
extension, it has a connected finite Galois cover of degree at most
\[
                    M_m=|\operatorname{GL}_{2m}(\mathbf F_3)|
\]
whose birational Albanese variety is nonisotrivial.  A fixed iterate
of $F$ lifts to this cover and its normalized full pullback is
connected and isomorphic to the original cover.
\end{corollary}

\begin{proof}
Propositions~\ref{prop:galois-at-presentation-audited} and
\ref{prop:bounded-at-galois-degree} give a full Galois AT
presentation of degree at most $M_m$.  Its construction gives
$L\sigma(E)=E$ for the corresponding function fields $E/L$ and
the lifted dynamical embedding $\sigma$; this identity is preserved
when the translation kernel is removed, since that kernel is
preserved by the deck-group automorphism.  Thus its normalization
over the projective source has the asserted connected normalized
pullback property.

If the abelian base of this presentation were isotrivial,
Theorem~\ref{thm:cover-exceptional-constant-abelian} would make
$F$ isotrivial.  Its abelian base therefore varies.  By
Lemma~\ref{lem:cover-at-birational-albanese} that base is the
birational Albanese of the normal projective cover.
\end{proof}

\begin{proposition}[Pulling back a varying exceptional factor]
\label{prop:cover-varying-exceptional-factor}
Let $F$ be an algebraic family of polarized endomorphisms of normal
projective varieties.  Suppose it has a dominant rational
semiconjugacy to a nonisotrivial exceptional family $(B,g)$ of
dimension $b>0$.  After finite parameter extension, a fixed iterate
of $F$ has a connected finite Galois cover of degree at most $M_b$
with connected normalized full pullback and nonisotrivial
birational Albanese.  Geometric connectedness of the rational
factor is not required.
\end{proposition}

\begin{proof}
Use Corollary~\ref{cor:cover-nonconstant-exceptional-varying-cover}
on $(B,g)$.  Write $K$ for the parameter function field, put
$L=K(X)$ and $L_0=K(B)$, and let $E/L_0$ be the resulting Galois
extension.  After the indicated iterate, its full normalized
pullback is itself.  The finite reduced $L$-algebra
\[
                         R=L\otimes_{L_0}E
\]
therefore satisfies
\[
             L\otimes_{L,F^*}R\simeq R.
\]
After finite parameter extension, its connected geometric
components are defined over $K$.  Write $R=\prod_{i=1}^rL_i$.
Each algebra $L\otimes_{L,F^*}L_i$ has at least one field
component.  Since their total number is $r$, each has exactly one
and pullback permutes the $L_i$.  An iterate fixes a chosen
component and its pullback is connected.  Base change of a finite
Galois extension is a product of finite Galois extensions, so
$L_i/L$ is Galois of degree at most $[E:L_0]\leq M_b$.

Normalize $X$ in $L_i$.  The dynamical field embedding extends to
a finite polarized endomorphism on this normal projective cover.
The inclusion $E\hookrightarrow L_i$ gives a dominant rational
map from the cover to the AT compactification associated with
$E$.  Consequently its birational Albanese maps surjectively to
the varying abelian base of that presentation.  A quotient family
of a constant abelian variety is constant after finite parameter
extension: its connected kernel is rigid, and its finite kernel
lies in one fixed torsion group.  The birational Albanese of the
cover therefore cannot be isotrivial.
\end{proof}

\begin{corollary}[A varying exceptional system has a trace-zero abelian factor]
\label{cor:cover-exceptional-trace-zero-factor}
Every nonisotrivial exceptional family has a positive-dimensional
rational factor with a finite presentation by a trace-zero abelian
variety over $\overline{\mathbf C(S)}$.  This factor is itself
nonisotrivial.
\end{corollary}

\begin{proof}
Choose a full Galois AT presentation $P\to X^\circ$ with abelian
base $A$.  Its finite deck group $G$ acts affinely on $A$, as in
Proposition~\ref{prop:bounded-at-galois-degree}.  The abelian
variety $A$ is nonisotrivial by
Theorem~\ref{thm:cover-exceptional-constant-abelian}.  Therefore
its quotient $A_{\rm var}$ by the image of its $\mathbf C$-trace
has positive dimension.  Every group homomorphism of $A$
preserves the trace, so the deck action and the polarized isogeny
descend to affine actions on $A_{\rm var}$.  The map
\[
                   P\longrightarrow A_{\rm var}/G
\]
is deck invariant and descends to a dominant rational factor
of $X$.  The abelian quotient is polarized, and its finite
quotient is exceptional.  If this quotient family were
isotrivial, Lemma~\ref{lem:cover-fixed-target-isotrivial}
would make its finite abelian cover isotrivial.  This contradicts
positive dimension and zero trace.
\end{proof}


\subsection{A closed induction for the varying exceptional factor}

All trace-zero assertions below are geometric: the constant trace
is taken after passage to the algebraic closure of the parameter
function field. In particular a positive-dimensional geometrically
trace-zero abelian variety is not isotrivial.

\begin{lemma}[Descending a varying abelian-quotient factor through a finite cover]
\label{lem:cover-deck-descend-varying-factor}
Let $p:(Y,F)\to(X,f)$ be a finite Galois compatible cover of
polarized one-parameter families. Suppose an iterate of $F$
commutes with the deck group and has a positive-dimensional
rational factor with a finite presentation by a trace-zero
abelian variety. Then an iterate of $f$ has a factor of the same
kind. This descended factor is nonisotrivial.
\end{lemma}

\begin{proof}
Write $\Gamma$ for the deck group and $\rho:Y\dashrightarrow E$
for the given factor. Replace the dynamics by the indicated
iterate. Form all conjugate factors $\rho\gamma$, with
$\gamma\in\Gamma$, using the same projective target $E$.
The normalization $D$ of their joint image in $E^{\Gamma}$
is a polarized factor. The deck group acts on $D$ regularly by
permuting the factors, and its action commutes with the dynamics.
Lemma~\ref{lem:trace-zero-factor-join} gives a finite abelian
presentation $A\to D$, with $A$ of positive dimension and zero
constant trace.

The finite quotient $D\to D/\Gamma$ is equivariant and polarized;
for example, average a polarization over $\Gamma$ and take a
power descending to the quotient. The composite $A\to D/\Gamma$
is a finite abelian presentation of the descended factor, whose
function field is contained in $k(X)$. The factor has the same
positive dimension as $D$. If it were isotrivial,
Lemma~\ref{lem:cover-fixed-target-isotrivial} would make its
finite abelian cover $A$ isotrivial, contradicting its positive
dimension and zero trace.
\end{proof}

\begin{theorem}[The varying exceptional-factor theorem]
\label{thm:reference-spectral-varying-exceptional-factor}
Every nonisotrivial normal projective polarized one-parameter
family in the pointed reference spectral class of
Definition~\ref{def:reference-good-spectra} has a positive-dimensional
rational factor with a finite presentation by a trace-zero abelian
variety. In particular it has a nonisotrivial exceptional rational
factor.
\end{theorem}

\begin{proof}
We prove the statement simultaneously in all dimensions. The
one-dimensional case is
Corollary~\ref{cor:reference-spectral-curves-base}, followed by
Corollary~\ref{cor:cover-exceptional-trace-zero-factor}.
Assume the statement in dimensions less than $N$ and consider a
nonisotrivial family of dimension $N$.

\emph{The finite algebraic alternatives.}
Proposition~\ref{prop:reference-cycle-spectra-local-returns}
applies to the exact pointed spectral hypothesis. The local
alternative and its global data, in
Theorem~\ref{thm:arbitrary-dimensional-local-return-geometry} and
Corollary~\ref{cor:local-alternative-global-data}, therefore apply.
The conjugacy alternative makes the family isotrivial and is
excluded. The remaining alternatives supply one of the following
finite algebraic presentations at the reference parameter:
\begin{enumerate}
\item an exceptional presentation of the source;
\item a proper positive-dimensional connected rational
      polarized factor of the source;
\item a finite Galois dynamical cover $Y\to X$ and a generically
      finite rational map
      \[
             Y\dashrightarrow T\subset\prod_{i=1}^r B_i,
      \]
      where each $B_i$ is a positive-dimensional proper connected
      polarized rational factor of $Y$ and $T$ is invariant.
\end{enumerate}
Here is the reduction of the geometric alternatives to this list.
For a rational Killing connection,
Corollary~\ref{cor:killing-connection-global-fields} and the
deck-stable flow construction give a positive-dimensional
semiabelian action on a finite cover. Its open-orbit case is
exceptional; otherwise its invariant field, followed by deck
invariants, gives the second presentation. For a finite web,
Lemma~\ref{lem:invariant-web-splitting-cover} selects its members
on a finite Galois dynamical cover. The purely transcendental
dense-leaf branch descends by
Proposition~\ref{prop:finite-web-transcendental-descent}.
The other branch supplies proper factors of this cover by
Theorem~\ref{thm:normal-invariant-foliation-factor}.
Lemma~\ref{lem:finite-web-algebraic-correspondence} then gives
presentation (2) or (3), according to the dimension of the join
of their deck conjugates.

Corollary~\ref{cor:pointed-incidence-spread} spreads the selected
presentation over a finite parameter cover defined over the fixed
countable field $k_0$. It retains a lift of the same geometrically
generic reference parameter. Every cover, factor and periodic
return used below remains in the reference spectral class by
Proposition~\ref{prop:reference-spectral-class-closure}.
All dynamics may temporarily be replaced by one common iterate;
the removal of this iterate is described at the end of the proof.

In case (1), the source is nonisotrivial and exceptional, so
Corollary~\ref{cor:cover-exceptional-trace-zero-factor} gives the
assertion.

\emph{A proper factor.}
In case (2), write $\pi:X\dashrightarrow B$ for the connected
factor. If $B$ is nonisotrivial, its dimension is less than $N$,
and the induction hypothesis supplies its trace-zero abelian-
quotient factor. Composing with $\pi$ proves the assertion for
$X$. Suppose instead that $B$ is isotrivial. Trivialize it after
a finite parameter extension, giving one fixed polarized system
$(B,g)$. Use the normalized equivariant graph to make $\pi$ a
morphism. Its generic fiber is geometrically integral, of
dimension
\[
                         m=N-\dim B<N.
\]

At the reference parameter, smooth repelling Julia periodic
points of $g$ in the flat normal-fiber good open form a Zariski-
dense set $\mathcal P$. For each such constant base point, its
return family has the reference spectral property by
Proposition~\ref{prop:reference-spectral-class-closure}.
Let $\mathcal I\subset\mathcal P$ be the points whose return
family is isotrivial. Theorem~\ref{thm:cover-fixed-base-return-rigidity}
shows that $\mathcal I$ cannot be Zariski dense, since the whole
source family is nonisotrivial. Thus
$\mathcal P\setminus\overline{\mathcal I}$ is dense and consists
of nonisotrivial return families of dimension $m$.

The induction hypothesis gives a nonisotrivial trace-zero
abelian-quotient factor of each such return. Proposition
\ref{prop:cover-varying-exceptional-factor} gives a connected
Galois Cartesian cover of its source, of degree at most $M_m$,
with nonisotrivial birational Albanese. The periods and parameter
covers may depend on the selected point; this degree bound does
not. The relative gluing theorem
\ref{thm:relative-irregular-factor-gluing}, together with
Remark~\ref{rem:periodic-cover-variation} and
Lemma~\ref{lem:irregular-extraction-preserves-variation}, gives
a positive-dimensional trace-zero abelian-quotient factor of
$X$, nonconstant on the generic $\pi$-fiber. The gluing theorem
requires only polarization of the fixed base, and does not
require that $(B,g)$ itself be exceptional. This completes
case (2).

\emph{A full-dimensional join upstairs.}
Consider case (3). Every factor family $B_i$ lies in the
reference spectral class and has dimension less than $N$.
If one $B_i$ is nonisotrivial, the induction hypothesis gives a
trace-zero abelian-quotient factor of $B_i$, hence of $Y$.
Lemma~\ref{lem:cover-deck-descend-varying-factor} descends such
a factor to $X$.

It remains to consider the case where all the $B_i$ are
isotrivial. After one common finite parameter extension,
trivialize them as fixed polarized systems. Their product is
fixed. The invariant image family $T$ is then fixed by
Lemma~\ref{lem:fixed-polarized-invariant-image-rigidity}.
Normalize this fixed image; it is again a fixed normal projective
polarized system. The generically finite rational semiconjugacy
from $Y$ to it makes $Y$ isotrivial by
Corollary~\ref{cor:cover-fixed-rational-target-isotrivial}.
Its actual finite equivariant map to $X$ makes $X$ isotrivial by
Proposition~\ref{prop:constant-finite-cover-downstairs}. This
contradicts the standing assumption, so this last subcase does
not occur.

\emph{Returning to the original iterate.}
If a trace-zero abelian-quotient factor $\rho$ was obtained for
$f^r$, join the factor maps
\[
                       \rho,\rho f,\ldots,\rho f^{r-1}.
\]
Their normalized joint image is preserved by the cyclic-shift
map induced by $f$. Proposition
\ref{prop:iterate-exceptional-factor-original} gives its normal
projective $q$-polarized model. Its abelian presentation still
has zero trace by Lemma~\ref{lem:trace-zero-factor-join}, applied
to the common return $f^r$. Thus it is the required factor for
$f$ itself. If any argument above instead made $f^r$ isotrivial,
Corollary~\ref{cor:isotrivial-iterate-normal} makes $f$ isotrivial
as well. This finishes the induction.
\end{proof}

\begin{corollary}[Uniform varying covers without a trichotomy hypothesis]
\label{cor:reference-spectral-uniform-varying-covers}
Every nonisotrivial $m$-dimensional family in the reference
spectral class admits, after finite parameter extension and an
iterate, a connected Galois Cartesian dynamical cover of degree
at most $M_m$ with nonisotrivial birational Albanese. The same
bound applies to every nonisotrivial good periodic-fiber return
of an $N$-dimensional family, with $M_m\leq M_{N-1}$.
\end{corollary}
\begin{proof}
Apply Theorem~\ref{thm:reference-spectral-varying-exceptional-factor}
and Proposition~\ref{prop:cover-varying-exceptional-factor}.
Periodic returns belong to the same class by its closure
proposition, and their dimension is less than $N$.
\end{proof}


\section{Simultaneous coordinates and exceptional extensions}
\label{sec:fixed-fiber-coordinates}

We construct the simultaneous local product coordinates required
in Definition~\ref{def:exceptional-extension}. Applied to a maximal
trace-zero factor, this construction supplies the fixed fiber
system of an exceptional extension.

\begin{lemma}[A stationary representative of constant path classes]
\label{lem:stationary-constant-path}
Consider an algebraic family of finite polarized fiber arrows over
a dominant phase map.  Suppose that every finite path has one
constant geometric polarized isomorphism class defined over
$\mathbf C$.  Then its generic fiber arrows have simultaneous
local product coordinates in the sense of
Definition~\ref{def:simultaneous-fiber-coordinates}.

More precisely, there are a fixed polarized system $(W,L_W,h)$
over $\mathbf C$, an algebraic subgroup
$H\subset\operatorname{Aut}(W,L_W)$, and an integer $n_0$ such that
the length-$n_0$ generic frame scheme is an $H$-torsor representing
all forward frames.  Forward transport is equivariant for an
algebraic endomorphism $\tau:H\to H$ satisfying
\[
                          h\alpha=\tau(\alpha)h.
\]
\end{lemma}

\begin{proof}
Choose compatible constant representatives recursively:
\[
 C_n=(W_0\xrightarrow{h_0}W_1\xrightarrow{h_1}\cdots
                                      \xrightarrow{h_{n-1}}W_n).
\]
Indeed, a representative of length $n+1$ has prefix isomorphic to
the chosen $C_n$, and a constant isomorphism transports this
prefix to $C_n$.  Dominance of the phase map implies that the
shifted length-$n$ chain has the same class as $C_n$.

Let $J_n$ be the space of polarized isomorphisms
$u_0:W_0\to W_1$ which extend to an isomorphism from $C_n$ to its
shift.  Such an extension is unique once $u_0$ is fixed: the
surjectivity of $h_i$ determines $u_{i+1}$ from
\[
                         u_{i+1}h_i=h_{i+1}u_i.
\]
The path-isomorphism scheme is a torsor under the algebraic
path-automorphism group.  Its image at the first vertex is
therefore a coset of an algebraic subgroup in
$\operatorname{Isom}((W_0,L_0),(W_1,L_1))$, and is closed.
The spaces $J_n$ are nonempty and nested.  The ambient isomorphism
scheme is of finite type: polarized isomorphisms are induced by
projective linear transformations after one sufficiently high
power of the polarization.  Noetherianity thus makes $(J_n)$
stationary.  Choose $u_0$ in their intersection and denote its
compatible extensions by $u_i$.

Set $r_0=\operatorname{id}_{W_0}$ and
$r_{i+1}=r_i u_i^{-1}:W_{i+1}\to W_0$.  Then
\[
 r_{i+2}h_{i+1}r_{i+1}^{-1}
             =r_{i+1}h_i r_i^{-1}.
\]
Every transported arrow is consequently the same map
$h=u_0^{-1}h_0$ on $W=W_0$.  The transported polarizations are
$L_W=L_0$, and $h^*L_W\simeq L_W^{\otimes q}$.

Let $H_n\subset\operatorname{Aut}(W,L_W)$ consist of the
first-vertex automorphisms extending through the first $n$ arrows
of the stationary chain.  These are descending closed algebraic
subgroups.  They stabilize at some $H_{n_0}=H$.  Let $I_n$ be the
scheme of polarized isomorphisms from this constant length-$n$
chain to the actual generic path.  It is a geometrically
nonempty $H_n$-torsor. Here $\operatorname{Aut}(W,L_W)$ denotes
the algebraic group of automorphisms preserving the isomorphism
class of $L_W$.  For $n\geq n_0$, prefix restriction
$I_{n+1}\to I_n$ is equivariant for the identity of $H$ and is
therefore an isomorphism.  Thus $I_{n_0}$ represents the compatible
frames of every forward length.

Extend a frame by one arrow and discard its first vertex.  This
gives the rational forward transport of $I_{n_0}$ over the phase map.
Shifting an infinite chain automorphism gives an algebraic
endomorphism $\tau$ of $H$; the first commuting square is precisely
$h\alpha=\tau(\alpha)h$.

The torsor, its transport, and the universal fiber isomorphisms
spread over dense phase opens.  Algebraic groups in characteristic
zero are smooth, so these torsors have sections locally in the
\emph{\'etale} topology.  For a finite path, successive transport
of one such section gives all its vertex frames.  This proves the
coordinate description in
Definition~\ref{def:simultaneous-fiber-coordinates}.
\end{proof}

\begin{proposition}[Constancy of every finite path class]
\label{prop:all-path-classes-constant}
Let $\pi:X\dashrightarrow Z$ be a connected polarized rational
factor of a one-parameter family.  Suppose that the factor has a
finite dynamical presentation by a geometrically trace-zero
abelian variety.  Suppose further that all good marked periodic
fiber returns outside a proper closed phase subset are
isotrivial.  After replacing the phase polarization by one fixed
positive tensor power, every finite polarized path of generic
fiber arrows has one constant geometric isomorphism class defined
over $\mathbf C$.
\end{proposition}

\begin{proof}
Work on an equivariant normal graph model and choose one high
power of its polarization which is relatively very ample over a
dense normal-fiber open.  Fix a length $n$.  Record its $n+1$
embedded fibers and its $n$ arrow graphs in the product of the
corresponding Hilbert schemes.  Applying the Rosenlicht
construction of Proposition~\ref{prop:polarized-arrow-moduli}
with one projective linear group at each vertex gives a rational
map $\mu_n$ separating the geometric polarized path classes.

Pass to the finite abelian presentation $A\to Z$.  Along each
retained marked periodic cycle, the return family is isotrivial.
Lemmas~\ref{lem:constant-return-finite-chain} and
\ref{lem:constant-return-polarized-chain} make its entire
polarized cyclic chain constant over the parameter.  Every
length-$n$ subpath of the repeated cycle consequently has constant
class over $\mathbf C$.  Thus $\mu_n$ is constant along its
periodic Betti section on $A$.

The good periodic Betti labels outside the prescribed proper
closed subset are dense on the locus of definition of $\mu_n$,
by Lemma~\ref{lem:dense-periodic-betti} and the good-cycle
restriction.  Continuity in Betti coordinates makes every local
coordinate of $\mu_n$ constant along Betti leaves.  Apply
Lemma~\ref{lem:betti-first-integral} to its rational affine
coordinates.  The geometric generic abelian variety has zero
$\mathbf C$-trace, so each coordinate is constant.  Hence the
pulled-back path-moduli map has one value over $\mathbf C$.
The same holds downstairs because $A\to Z$ is dominant and
$\mu_n$ separates geometric path classes.  This argument applies
to every $n$ with the same fiber polarization.
\end{proof}

\begin{corollary}[Maximal trace-zero reduction]
\label{cor:reference-spectral-maximal-tracezero-reduction}
Let $f$ be a nonisotrivial family in the pointed reference spectral
class.  Choose a maximal connected rational factor
$\pi:X\dashrightarrow Z$ with a finite presentation by a
geometrically trace-zero abelian variety.  Then $\dim Z>0$, and
the generic fiber arrows of $\pi$ have simultaneous local product
coordinates on dense phase opens, with one fixed polarized fiber
system defined over $\mathbf C$. After a finite parameter
extension, $\pi$ therefore gives an exceptional extension structure
in the sense of Definition~\ref{def:exceptional-extension}.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:reference-spectral-varying-exceptional-factor}
gives a positive-dimensional factor, and
Lemma~\ref{lem:trace-zero-factor-join} gives its maximal connected
join.  If $\dim Z=\dim X$, its geometrically integral generic
fiber is a point, and the coordinate assertion follows with the
point system.

Suppose that $\dim Z<\dim X$.  If the nonisotrivial good periodic
fiber returns were Zariski dense, then
Corollary~\ref{cor:reference-spectral-uniform-varying-covers}
would supply connected Galois Cartesian covers of uniformly
bounded degree with nonisotrivial Albanese varieties.  Relative
gluing in Theorem~\ref{thm:relative-irregular-factor-gluing},
together with Remark~\ref{rem:periodic-cover-variation} and
Lemma~\ref{lem:irregular-extraction-preserves-variation}, would
give a trace-zero abelian-quotient factor nonconstant on the
geometric generic fiber of $\pi$.  Its join with $Z$ has larger
dimension, contrary to maximality.

The closure of the nonisotrivial good returns is therefore a
proper closed phase subset.  Proposition~\ref{prop:all-path-classes-constant}
gives a constant polarized class for every finite generic fiber
path.  Lemma~\ref{lem:stationary-constant-path} supplies the fixed
system $(W,L_W,h)$ and the finite-length frame torsor.  Both are
constructed for the original factor $X\dashrightarrow Z$; their
local sections give the asserted coordinates.
\end{proof}

\section{Exceptional extensions from stable families}
\label{sec:curve-structural-theorem}
\begin{proposition}[Exceptional extensions from stability]\label{prop:curve-structural-theorem}
Let $S$ be a smooth connected complex algebraic curve and let $f$ be a nonisotrivial
$J$-stable polarized family $p:\mathcal X\to S$ with geometrically
integral normal fibers. Put $K=\mathbf C(S)$ and denote its
generic fiber by $X_\eta$.
After a finite extension $K'/K$, the generic system
$(X_{\eta,K'},f_{\eta,K'})$ is an exceptional extension in the
sense of Definition~\ref{def:exceptional-extension}.
Its exceptional base can be chosen maximal among connected
factors with geometrically trace-zero abelian presentations.
\end{proposition}
\begin{proof}
Choose an algebraically closed
countable field $k_0\subset\mathbf C$ over which the family is
defined, and a parameter $s_0$ generic over $k_0$.
Proposition~\ref{prop:stable-spectrum-any-dimension} makes the
marked repelling spectra constant. Thus the family belongs to
the pointed reference spectral class of
Definition~\ref{def:reference-good-spectra}.

Theorem~\ref{thm:reference-spectral-varying-exceptional-factor}
gives a positive-dimensional rational factor with a trace-zero
abelian presentation. By
Lemma~\ref{lem:trace-zero-factor-join}, take its maximal
connected join and write it as
\[
                  \pi:X_{\eta,\overline K}\dashrightarrow Z,
                  \qquad \pi f=g\pi.
\]
Its generic fiber is geometrically integral. The maximal-factor
construction gives $Z$ normal and projective and $g$ regular;
its polarization has weight $q$ by \cite[Theorem~1.3]{MZ18}.
Its abelian presentation consists of a finite surjective map $\eta:A\to Z$
and a polarized affine isogeny $\alpha$ satisfying
$\eta\alpha=g^e\eta$ for some $e\geq1$.

Write $\alpha=t_a\circ\beta$, where $\beta$ is a group
isogeny, and choose an ample line bundle $L_A$ polarizing
$\alpha$. Translations act trivially on the N\'eron--Severi
group, so $\beta^*c_1(L_A)=q^e c_1(L_A)$. If
$(\ker(1-\beta))^0$ had positive dimension, restriction to
this abelian subvariety would contradict ampleness, since
$\beta$ is the identity there. Thus $1-\beta$ is an isogeny.
Choose $b$ satisfying $(1-\beta)b=a$. The conjugate
$t_{-b}\alpha t_b=\beta$ is $q^e$-polarized by $t_b^*L_A$.
Equivalently, take $b$ as the origin of $A$. This proves
that $g$ is exceptional of abelian variety type as defined in
Definition~\ref{def:abelian-type-exceptional}.

Corollary~\ref{cor:reference-spectral-maximal-tracezero-reduction}
applies to this maximal factor. It supplies a single polarized
system $(W,L_W,h)$ over $\mathbf C$ and simultaneous local
product coordinates for all generic fiber paths. Their
finite-length frame torsor and forward transport give the
coordinate data in
Definition~\ref{def:simultaneous-fiber-coordinates}. Since
$N=\dim X_\eta$ and $r=\dim Z$ satisfy $1\leq r\leq N$, the fiber dimension is
$\dim W=N-r\in\{0,\ldots,N-1\}$.

The varieties, morphisms, rational factor, line bundles, abelian
presentation, and finite-length frame torsor are all of finite
presentation. They descend from $\overline K$ to one finite
extension $K'/K$. Their finitely many defining identities spread
over a nonempty open subset of the corresponding finite cover
of $S$. The rational transport then supplies each finite path
on its dense phase domain. Thus $\pi$ is an exceptional extension
structure over $K'$ in the sense of
Definition~\ref{def:exceptional-extension}, proving the proposition.
\end{proof}


\section{Stability of isotrivial families and exceptional extensions}
\label{sec:structural-converse}

This section proves generic $J$-stability for isotrivial families
and for families whose generic systems become exceptional extensions
after a finite parameter extension. For exceptional extensions,
we construct periodic sections whose empirical measures converge
to the equilibrium measure in every fiber, and then apply
Proposition~\ref{normal:generic-periodic-criterion}.

\begin{proposition}[Generic stability of isotrivial families]
\label{prop:isotrivial-implies-stable}
An isotrivial algebraic family of polarized endomorphisms with
geometrically integral normal projective fibers is $J$-stable
on a nonempty Zariski open subset of its parameter space.
\end{proposition}

\begin{proof}
Let $S$ be the parameter space. By isotriviality, a finite
extension of $\mathbf C(S)$ identifies the generic dynamical
system with a fixed complex system $(X_0,h)$. Spread this
isomorphism and its inverse, and then shrink $S$. Since the
extension is separable, this gives a finite surjective \'etale
cover $S'\to S$ and an equivariant isomorphism
\[
              \mathcal X\times_S S'\simeq X_0\times S'.
\]
The map $h$ is polarized: transport the polarization from any
complex fiber of this isomorphism.

A dynamical isomorphism carries equilibrium measures and their
supports to one another. This follows from normalized
inverse-image equidistribution, as used in the proof of
Proposition~\ref{web:dynamics}; see \cite[Theorem~1.2]{DNT15}.
Consequently, the displayed isomorphism identifies the smooth
repelling Julia cycles with the constant cycles of $h$, for
every period simultaneously. Local analytic inverse sections of
$S'\to S$ give the required motions on $S$. This proves
$J$-stability on the chosen parameter open.
\end{proof}


\begin{lemma}[The connected frame group]
\label{lem:converse-frame-torus}
Let $(W,L_W,h)$ be a polarized complex system and let $H$ be its
algebraic group of compatible forward frames.  Write
\[
                     h a=\tau(a)h\qquad(a\in H).
\]
Then $T=H^0$ is a torus, and $\tau|_T$ is an isogeny.  For every
$n\geq1$ and every $a\in H$, the morphism
\[
 T\longrightarrow T,\qquad
 b\longmapsto b^{-1}a\tau^n(b)a^{-1}
\]
is an isogeny.  In particular, the maps $a h^n$, with $a$ in one
connected component of $H$, are conjugate by elements of $T$.
\end{lemma}

\begin{proof}
The kernel of dynamical transport is contained in the finite deck group
of $h$.  Its restriction to $T$ is therefore an isogeny.  The orbit-degree argument in
Proposition~\ref{prop:cover-forward-gauge-torsor} excludes additive
one-parameter subgroups: a nontrivial additive orbit has degree
bounded by a fixed projective embedding dimension, whereas its
images under $h^n$ are birational additive orbits of degree multiplied
by $q^n$. Since $T\subset\operatorname{Aut}(W,L_W)$ is linear,
absence of additive one-parameter subgroups makes $T$ a torus.

The displayed morphism is a homomorphism of tori.  Its kernel consists
of the elements of $T$ commuting with $a h^n$.  This kernel is finite:
the identity component of the automorphism group of a polarized map
fixes every finite set of periodic points, and periodic points are
Zariski dense by \cite{Fak03}.  It therefore acts trivially.  A
homomorphism of tori of equal dimension with finite kernel is an
isogeny.  The final assertion follows from
\[
                  b^{-1}(a h^n)b
                      =\bigl(b^{-1}a\tau^n(b)\bigr)h^n.
\]
\end{proof}

\begin{lemma}[Finite covers of an abelian base]
\label{lem:converse-component-cover}
Let $A$ be a positive-dimensional abelian variety over an
algebraically closed field of characteristic zero.  Suppose $\alpha:A\to A$ is $Q$-polarized and
$p:B\to A$ is finite and surjective, with $B$ normal.  If a surjective
endomorphism $\beta:B\to B$ satisfies $p\beta=\alpha p$, then $p$ is
\'{e}tale.  After choosing an origin, $B$ is an abelian variety and
$\beta$ is an affine isogeny.
\end{lemma}

\begin{proof}
Let $L=p^*L_A$, where $L_A$ polarizes $\alpha$.  Then $L$ is ample and
$\beta^*L\simeq L^{\otimes Q}$.  Pulling back a nonvanishing rational
top form on $A$ identifies the canonical Weil divisor of $B$ with
the effective ramification divisor $R_p$.  The ramification formula
for $\beta$, interpreted at codimension-one points, gives
\[
                         R_p=\beta^*R_p+R_\beta.
\]
If $r=\dim B$, the finite projection formula for Weil divisors and
Cartier intersections gives
\[
       (R_p\cdot L^{r-1})
           =Q(R_p\cdot L^{r-1})+(R_\beta\cdot L^{r-1}).
\]
Ample positivity forces $R_p=R_\beta=0$.  Purity of the branch locus
over the smooth variety $A$ \cite[Tag~0BMB]{Stacks} makes $p$ \'{e}tale.  A connected finite
\'{e}tale cover of an abelian variety is an abelian variety after
choosing a point above the origin; every finite self-map of an
abelian variety is a translation followed by an isogeny.
\end{proof}

\begin{lemma}[Extension of the frame bundle over the abelian base]
\label{lem:converse-global-frame-bundle}
Let $\rho:Y\to A$ be a projective morphism from a normal variety to
an abelian variety.  Let $F:Y\to Y$ and $\alpha:A\to A$ be finite
polarized endomorphisms satisfying $\rho F=\alpha\rho$.  Suppose
that the generic fiber arrows have the simultaneous coordinates
of Definition~\ref{def:simultaneous-fiber-coordinates}, with fixed
system $(W,L_W,h)$.  After a finite \'{e}tale cover of $A$ and an
iterate, the normalization of the pullback of $Y$ is a bundle
associated to a torus torsor and the polarized $T$-variety $W$.
Its forward transport is induced by $h$ and the isogeny $\tau$ of
Lemma~\ref{lem:converse-frame-torus}.
\end{lemma}

\begin{proof}
The stabilization argument in the proof of
Lemma~\ref{lem:stationary-constant-path} supplies the generic
$H$-torsor $I$ of all forward frames and its forward transport.
Put $T=H^0$.  The finite scheme $I/T$ gives a finite cover of $A$.
Normalize $A$ in its component fields.  Forward transport extends
to a finite map of this normalization over $\alpha$: integrality
over the coordinate rings of $A$ gives the extension.  Its image
on the finite set of irreducible components has a periodic
component.  Choose such a component and replace the maps by the
corresponding iterate.  Its self-map is polarized by the pullback
of a polarization on $A$.  Lemma~\ref{lem:converse-component-cover}
makes the resulting connected cover $A'\to A$ \'{e}tale.

Normalize the pullback of $Y$ to $A'$.  The chosen component of
$I/T$ leaves a generic torsor under the split torus $T$, by reduction of
structure group \cite[Section~3.5, Proposition~9]{Ser58}. Hilbert's
Theorem~90 \cite[Tag~03P7]{Stacks} gives it a rational point.  Thus a rational compatible
forward frame exists over $A'$.  Replace notation by $Y,A,F,\alpha$, and replace $h$
by the same iterate throughout the stationary fiber chain.

Call a phase point good if, on an \'{e}tale neighborhood of that
point, the source isomorphism of a frame in the chosen generic
$T$-torsor extends to an isomorphism from the constant polarized
family with fiber $W$.  These neighborhoods form a nonempty open
set $V\subset A$.  We show that $\alpha(V)\subset V$.

Let $U\to V$ be one such neighborhood with source frame
$c_0:U\times W\xrightarrow{\sim}Y_U$.  Since $\alpha$ is \'{e}tale,
the target $Y_{\alpha,U}:=Y\times_{A,\alpha}U$ is normal.  The maps
\[
       F c_0:U\times W\longrightarrow Y_{\alpha,U},
       \qquad \operatorname{id}_U\times h:U\times W\longrightarrow
                                      U\times W
\]
are finite.  Their joint image is finite over both targets.
Over the generic point it is the graph of the next frame, and
both projections are birational.  Normality makes both
projections isomorphisms.  This extends the next frame over $U$.
The extended frame is polarized: its relative Picard class agrees
generically with $L_W$, and the equality persists by separatedness
of the relative Picard functor.  Hence $\alpha(V)\subset V$.

The complement $B=A\setminus V$ is closed and satisfies
$\alpha^{-1}B\subset B$.  It must be empty.  Indeed, take its
components of largest dimension.  Finiteness of their set and
the backward-invariance inclusion imply that their reduced
cycle $C$ satisfies $\alpha^*C=C$.  If its codimension is $c>0$,
intersecting with an ample polarization gives
\[
             (C\cdot L_A^{r-c})
                    =Q^c(C\cdot L_A^{r-c}),
\]
contrary to positivity.  The transition maps between the
extending frames lie generically in the closed subgroup $T$ of
$\operatorname{Aut}(W,L_W)$, and therefore lie in $T$ everywhere.
They define a torus torsor $I\to A$, and the associated-bundle
construction \cite[Section~3.2, Proposition~4]{Ser58} identifies
$Y$ with $I\times^T W$. The forward compatibility identities
extend with the frames.
\end{proof}

\begin{proposition}[Generic stability of exceptional extensions]
\label{prop:abelian-fixed-fiber-converse}
Let $f$ be an algebraic family of polarized endomorphisms with
geometrically integral normal projective fibers. Suppose that,
after a finite parameter extension, its generic system is an
exceptional extension in the sense of
Definition~\ref{def:exceptional-extension}. Then $f$ is $J$-stable
on a nonempty parameter open.
\end{proposition}

\begin{proof}
Work over the finite parameter extension in the hypothesis,
and write $\pi:(X,f)\dashrightarrow(Z,g)$ for the exceptional
extension structure, with fixed fiber system $(W,L_W,h)$.
Choose an equivariant normal projective graph model and pull it
back to the finite abelian presentation of the exceptional base.  Normalize
this pullback and replace the dynamics by the presentation
iterate.  We obtain a polarized map $F$ on a normal projective
variety $Y$, a morphism $\rho:Y\to A$ over a polarized abelian
isogeny, and a proper generically finite dynamical morphism
$p:Y\to X$.  Lemma~\ref{lem:converse-global-frame-bundle} gives,
after another finite parameter extension and an iterate, a global
bundle of copies of $W$ over an abelian variety $A'$.  All these
finite-type data spread over a parameter open.  Shrink that open
so that $A'$ is an abelian scheme, the bundle and its dynamics
are defined everywhere over it, and $p$ remains a proper
generically finite dynamical morphism.  Denote the polarization
weight of the resulting iterate by $Q$ and its fixed fiber map
by $u$.

Work over an analytic coordinate disk $D$ in this parameter
open.  The fixed points of each iterate of the abelian base map
are finite \'{e}tale over $D$.  They therefore give holomorphic
sections on the same disk, for all periods.  Over a section of
period dividing $n$, the torus torsor is holomorphically trivial.
The fiber return in such a trivialization has the form
\[
                              a(s)u^n,
                              \qquad a:D\longrightarrow T.
\]
The torus Lang isogeny in Lemma~\ref{lem:converse-frame-torus}
and simple connectivity of $D$ conjugate this return
holomorphically to $u^n$ on all of $D$.

Let $r=\dim A'$ and $m=\dim W$.  The base has
$Q^{rn}+o(Q^{rn})$ fixed points of its $n$th iterate.  The fixed
system $u$ has $Q^{mn}+o(Q^{mn})$ smooth repelling Julia points
fixed by $u^n$, by periodic equidistribution (and exactly one when $m=0$). Choosing these
points on every base section gives
\[
                         Q^{(r+m)n}+o(Q^{(r+m)n})
\]
distinct holomorphic periodic sections on $D$.  Each section is
smooth and repelling: the horizontal eigenvalues are those of
an abelian isogeny and the vertical ones are those of the chosen
return of $u$.  Each lies in the ambient Julia set by
Lemma~\ref{lem:continuous-fiber-measures}, since the equilibrium
measure of an abelian isogeny has full support.  In every fiber
these sections form an asymptotically full set of repelling
periodic points.  Their normalized counting measures converge
to $\mu_{F_s}$. The collection is permuted by $F$, because it
contains all the smooth repelling Julia points of the returns over
every base point fixed by the $n$th iterate. Its counting measure
is therefore $F$-invariant.

Compose the sections with the spread of $p$. They are holomorphic
periodic sections of the original phase family, and their empirical
measures converge to
\[
                              p_*\mu_{F_s}=\mu_{f_s}.
\]
For this identity, choose a generic point of $X_s$ whose finite
inverse image under $p$ avoids the exceptional sets for backward
equidistribution. Normalized inverse images of this finite fiber,
pushed down by $p$, are the normalized inverse images of that point.
Apply \cite[Theorem~1.2]{DNT15} on resolutions of both varieties;
the exceptional sets have equilibrium measure zero. This gives
$p_*\mu_{F_s}=\mu_{f_s}$.  After restricting the parameter open as in
Proposition~\ref{normal:generic-periodic-criterion}, its relative
periodic-section criterion produces an equilibrium web and hence
$J$-stability. Stability
of an iterate is equivalent to stability of the original map,
because they have the same smooth repelling Julia periodic
points.

Finally, delete finitely many points of $S$ so that the finite
parameter cover is \'etale. Stability descends through its local
analytic inverse sections. Thus the original family is stable
on a nonempty parameter open.
\end{proof}

\section{Proof of the equivalence and the projective-space version}
\label{sec:final-dichotomy}

\begin{proof}[Proof of Theorem~\ref{thm:introduction-normal-reduction}]
If $f$ is generically $J$-stable, restrict to a nonempty open
of $S$ on which it is stable. The function field remains $K$.
If $f$ is nonisotrivial, Proposition~\ref{prop:curve-structural-theorem}
makes its generic system an exceptional extension after a finite
parameter extension. Conversely, isotrivial families are generically
$J$-stable by Proposition~\ref{prop:isotrivial-implies-stable},
and families whose generic systems become exceptional extensions
are generically $J$-stable by
Proposition~\ref{prop:abelian-fixed-fiber-converse}.
\end{proof}

\begin{proof}[Proof of Corollary~\ref{cor:projective-space-stability}]
After restricting to a nonempty parameter open, the family is
polarized by $\mathcal O(1)$ with weight $d$. Every automorphism
of $\mathbf P^N$ belongs to $\operatorname{PGL}_{N+1}$. The equivalence
follows from the main theorem and
Corollary~\ref{cor:stability-zariski-open-extension}, which extends
stability from a nonempty Zariski open to the whole parameter space
for a fixed phase variety. Dominance of $\pi$ gives geometric
unirationality of $Z$. If $\dim Z=1$, its geometric genus is zero,
so $Z_{\overline K}\simeq\mathbf P^1_{\overline K}$. Its abelian
presentation makes an iterate Latt\`es; since the equilibrium
measure is unchanged by iteration, the characterization in
\cite[Theorem~1]{BD05} makes $g$ itself Latt\`es.

Suppose $\dim Z=N$. The connected rational fibration $\pi$ is
then birational. Composing its inverse with an abelian presentation
gives a generically finite rational map
\[
 \varphi:A\dashrightarrow\mathbf P^N,\qquad
 \varphi\alpha=f^e\varphi.
\]
Finite postcomposition does not remove indeterminacy: the
normalized graph has connected fibers over its normal source,
and a positive-dimensional fiber cannot be contracted by a finite
map of the target. Since $\alpha$ is \'etale, it follows that
its indeterminacy locus $I$ satisfies $\alpha^{-1}I=I$.
The intersection argument in
Lemma~\ref{lem:converse-global-frame-bundle} gives $I=\varnothing$.
Thus $\varphi$ is a morphism. The line bundle
$\varphi^*\mathcal O(1)$ is nef and big on $A$, hence ample;
therefore $\varphi$ is finite. Its pushforward of Haar measure is
the equilibrium measure of $f^e$, and is absolutely continuous.
Again $\mu_f=\mu_{f^e}$ and \cite[Theorem~1]{BD05} show that
$f$ is Latt\`es. For a generic system over $\overline K$, apply the analytic
characterization after spreading over a complex parameter open.
Finite abelian presentations are encoded by countably many
finite-type incidence spaces, as in
Lemma~\ref{lem:countable-finite-presentation-spread}. Since their
images contain every complex point of that open, one image is
dominant: a countable union of finite subsets cannot cover a
nonempty complex algebraic curve. A closed point of its generic fiber
gives a presentation after a finite extension of $K$.
The case $\dim Z=1$ descends in the same way.
This proves both assertions over $\overline K$.
\end{proof}

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