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\title{Binder-DeMarco-Dupont Inequality}
\author{}
\date{}
\begin{document}
\maketitle


\begin{abstract}
In the setting of endomorphisms of $\mathbb{P}^k(\mathbb{C})$, a dimension formula for the measure of maximal entropy was conjectured by Binder and DeMarco in 2003 (\cite{BinderDeMarco}), asserting that 
$$\mathrm{dim}_\mathrm{Hausdorff}(\mu)=\log d\sum_{i=1}^k \frac{1}{\chi_i},$$
where $d$ is the degree of the (non-invertible) endomorphism and $\chi_i$'s are the Lyapunov exponents with multiplicity. Indeed, this is well-defined as all Lyapunov exponents are known to be positive due to Briend and Duval.
The most significant progress is due to Dupont, who in complex-dimension 2, proved the dimension lower-bound holds (\cite{Dupont}); And recently by Genes, who proved the equality for two-dimensional polynomial skew-product maps which satisfy a Lyapunov exponent condition (\cite{Clara}).

Recently, He and Bianchi (\cite{HB}) were able to find a family of simple counter-examples. In Appendix A we present a single ``hands on'' description of such a counter example, which is given by a quadratic polynomial skew-product.

In this manuscript, we provide the proof, which was achieved by lightly prompting ChatGPT, of extending Dupont’s inequality to all complex dimensions. We wish to share this manuscript with the community as we believe that this provides a full picture of the recently settled Binder-DeMarco conjecture. We do not claim ownership of the proof.

\end{abstract}

\section{Statement and notation}

Let \(M=\PP^k(\CC)\), equipped with the Fubini--Study K\"ahler form
\(\omega\) and its associated Riemannian metric. Let
\(f:M\to M\) be a holomorphic endomorphism of algebraic degree
\(d\geq2\). Its equilibrium measure is denoted by \(\mu\).
Throughout, logarithms are natural and Hausdorff dimensions are
\emph{real} dimensions. We use the convention
\[
 \dimH\mu=\inf\{\dimH A:A\subset M\text{ is Borel and }\mu(A)=1\}.
\]

\begin{theorem}\label{thm:main}
Let \(\lambda_1\geq\cdots\geq\lambda_k>0\) be the Lyapunov
exponents of \(\mu\), counted with complex multiplicity. Then
\begin{equation}\label{eq:main}
 \boxed{\displaystyle
 \dimH\mu\geq (\log d)\sum_{i=1}^{k}\frac1{\lambda_i}.}
\end{equation}
\end{theorem}

We group equal exponents. Write
\[
 \chi_1>\chi_2>\cdots>\chi_s>0,
 \qquad m_1+\cdots+m_s=k,
 \qquad p_j:=\sum_{a=1}^{j}m_a,
 \qquad L:=\log d,
\]
where \(m_j\) is the complex multiplicity of \(\chi_j\).
Thus \(p_s=k\) and
\[
 \sum_{i=1}^{k}\lambda_i^{-1}
 =\sum_{j=1}^{s}m_j\chi_j^{-1}.
\]
This block notation avoids any simplicity assumption on the spectrum.

The standard properties of the equilibrium measure used below are
ergodicity,
\begin{equation}\label{eq:equilibrium-facts}
 h_\mu(f)=k\log d,\qquad
 \log|\det D_{\RR}f|\in L^1(\mu),\qquad
 \lambda_k>0.
\end{equation}
See, for example, \cite{DS,Dupont}. In particular, the critical set and
its iterated preimages have \(\mu\)-measure zero.

\section{The conditional entropy and dimension theorem used}

Let \((\Mh,\fh,\muh)\) be the natural extension of \((M,f,\mu)\),
and let \(\pi:\Mh\to M\) be the time-zero projection, so that
\(\pi_*\muh=\mu\). Along a regular inverse history there is a strong
unstable flag
\[
 W^1\subset W^2\subset\cdots\subset W^s,
 \qquad \dim_{\CC}W^j=p_j.
\]
These are plaques in the natural extension; their local spatial
projections are embedded holomorphic plaques. Histories are retained
throughout: no assertion that the flag descends to a flag on \(M\)
is needed.

We use the following established part of the endomorphism
Ledrappier--Young theory, due to Qian--Xie; see
\cite{QX}, especially its conditional dimension and local entropy
statements, and Chapter IX of \cite{QXZ}.

\begin{proposition}[Conditional entropy and dimension framework]
\label{prop:LY}
There are partial entropies
\[
 0=H_0\leq H_1\leq\cdots\leq H_s=h_\mu(f)
\]
and exact conditional dimensions \(\delta_j\) on the corresponding
strong unstable plaques. With \(\delta_0=0\), they satisfy
\begin{equation}\label{eq:LY}
 \delta_j-\delta_{j-1}
 =\frac{H_j-H_{j-1}}{\chi_j},\qquad 1\leq j\leq s.
\end{equation}
The dimensions are computed using intrinsic spatial distance on a
plaque, equivalently using a smooth local spatial parametrization.

Moreover, partial entropy has the conditional local-entropy
description. For a typical conditional measure \(\sigma\) on a
\(W^j\)-plaque, write
\[
 B_n^j(y,r)=\{z:\max_{0\leq t<n}
 d_{W_t^j}(f^ty,f^tz)<r\},
\]
intersecting with the conditional-measure domain. Here the forward
leaves and their intrinsic metrics retain the prescribed histories.
Then, for \(\sigma\)-almost every \(y\),
\begin{equation}\label{eq:local-entropy}
 \lim_{r\downarrow0}\liminf_{n\to\infty}
 -\frac1n\log\sigma(B_n^j(y,r))=H_j.
\end{equation}
In particular, after restricting to a set of positive conditional
measure, the upper mass bound associated with
\eqref{eq:local-entropy} can be made uniform in the centre and in all
sufficiently large \(n\).
\end{proposition}

The hypotheses for this framework are supplied by
\eqref{eq:equilibrium-facts}; \(f\) is smooth on the compact manifold
\(M\). There is no requirement of uniform expansion or a global
dominated splitting.

The holomorphicity of the local strong plaques follows from the
holomorphic graph transform. It can also be read from the triangular
normal forms in \cite{BDM}: fixing all slower blocks gives the fast
plaques. After shrinking a local plaque, its parametrization extends
holomorphically to a larger parameter polydisc. Only individual
plaques will be used below, so all constants may depend on that
plaque.

The proof of Theorem~\ref{thm:main} will follow from
\begin{equation}\label{eq:entropy-goal}
 H_j\leq p_jL \qquad (1\leq j<s).
\end{equation}
The next two sections prove this estimate.

\section{Volume and intrinsic packing of holomorphic orbit graphs}

\subsection{The orbit-graph volume estimate}

Write \(\DD^p(a)=\{z\in\CC^p:\max_i|z_i|<a\}\).
Let \(\eta:\DD^p(2)\to M\) be a holomorphic embedding whose image
is contained in a bounded affine coordinate chart. For \(n\geq1\),
define
\[
 G_n(z)=\bigl(\eta(z),f\eta(z),\ldots,f^{n-1}\eta(z)\bigr)
 \in M^n.
\]
The first component makes \(G_n\) an embedding, including at parameter
values whose subsequent images are critical points of \(f\).
Give \(M^n\) the product K\"ahler form
\(\Omega_n=\sum_{t=0}^{n-1}\operatorname{pr}_t^*\omega\), and set
\(\Gamma_n=G_n(\DD^p(1))\), with its induced intrinsic metric.

\begin{lemma}[Orbit-graph volume]\label{lem:graph-volume}
There is a constant \(C_\eta\), independent of \(n\), such that
\begin{equation}\label{eq:graph-volume}
 \vol_{2p}(\Gamma_n)\leq C_\eta d^{pn}.
\end{equation}
\end{lemma}

\begin{proof}
Let \(T=\omega+\ddc g\) be the Green current of \(f\), where \(g\)
is continuous and bounded, and \(f^*T=dT\). It follows that
\[
 (f^t)^*\omega=d^t\omega+\ddc(d^tg-g\circ f^t).
\]
Set \(A_n=\sum_{t=0}^{n-1}d^t\). Choose a smooth local potential
\(\psi\) for \(\omega\) on the chart containing the plaque. On
\(\DD^p(2)\), put
\[
 v_n=\psi\circ\eta+g\circ\eta
       -\frac1{A_n}\sum_{t=0}^{n-1}g\circ f^t\circ\eta.
\]
Then
\begin{equation}\label{eq:potential}
 \ddc v_n=A_n^{-1}G_n^*\Omega_n
 =A_n^{-1}\eta^*\sum_{t=0}^{n-1}(f^t)^*\omega\geq0.
\end{equation}
Thus \(v_n\) is plurisubharmonic. On any fixed intermediate
polydisc, for instance \(\DD^p(3/2)\), it is bounded uniformly in
\(n\), because \(g\) is bounded and \(n/A_n\leq1\).

The Chern--Levine--Nirenberg estimate, applied between
\(\DD^p(1)\) and \(\DD^p(3/2)\), gives
\[
 \int_{\DD^p(1)}(\ddc v_n)^p\leq C.
\]
Equivalently, this follows by \(p\) integrations by parts with nested
cutoff functions, transferring each \(\ddc\) to a cutoff and using
the uniform bound on the potentials. Positivity controls the
intermediate terms. This is the bounded-potential mechanism used
in the growth lemma of \cite[Appendix]{Dupont}; see also \cite{DS}.

Since the Riemannian volume form of a complex \(p\)-dimensional
K\"ahler manifold is its K\"ahler form to the \(p\)-th power divided
by \(p!\), \eqref{eq:potential} implies
\[
 \vol_{2p}(\Gamma_n)
 =\frac{A_n^p}{p!}\int_{\DD^p(1)}(\ddc v_n)^p
 \leq C'A_n^p\leq C_\eta d^{pn}.
\]
\end{proof}

\subsection{A uniform lower volume bound for intrinsic balls}

The next point is necessary because partial entropy uses leafwise
intrinsic distances. An ambient packing estimate alone does not
resolve that issue.

\begin{lemma}[Intrinsic balls in the orbit graph]
\label{lem:intrinsic-volume}
Fix a compact set \(K\Subset\DD^p(1)\). There exist \(r_*>0\)
and \(c>0\), independent of \(n\), such that
\begin{equation}\label{eq:intrinsic-volume}
 \vol_{2p}\bigl(B_{\Gamma_n}(G_n(z),r)\bigr)
 \geq cr^{2p}
 \qquad(z\in K,\ 0<r<r_*).
\end{equation}
\end{lemma}

\begin{proof}
Every \(\Gamma_n\) is a complex submanifold of the K\"ahler manifold
\(M^n\), hence is minimal. The product metrics on \(M^n\) have a
sectional-curvature upper bound independent of \(n\), and their
injectivity radii have a positive lower bound independent of \(n\).

There is also a uniform boundary margin for the centres in question.
Indeed,
\[
 a:=\dist_M\bigl(\eta(K),\eta(\partial\DD^p(1))\bigr)>0.
\]
The length of every graph path is at least the length of its first
coordinate projection. Consequently, the intrinsic graph distance
from \(G_n(K)\) to the boundary of \(\Gamma_n\) is at least \(a\).

Here is the intrinsic monotonicity argument, including its dependence
on constants. Write \(m=2p\), fix a centre \(q=G_n(z)\), and let
\(R\) be ambient product distance from \(q\). For sufficiently small
\(r\), an intrinsic ball of radius \(r\) lies in the ambient ball
of radius \(r\). Hessian comparison for the squared ambient distance,
together with minimality, gives
\[
 \Delta_{\Gamma_n}R^2\geq 2m(1-Cr^2)
 \quad\hbox{on }B_{\Gamma_n}(q,r).
\]
The constant \(C\) depends on the uniform ambient curvature bound
and is independent of \(n\). Also
\(\lvert\nabla_{\Gamma_n}R^2\rvert\leq2r\) there.

Let \(V(r)=\vol_m(B_{\Gamma_n}(q,r))\). Divergence and coarea,
valid for almost every radius (or by approximation across the
intrinsic cut locus), imply
\[
 2m(1-Cr^2)V(r)\leq2rV'(r).
\]
Thus
\[
 \frac{d}{dr}\log\frac{V(r)}{r^m}\geq-mCr.
\]
At each smooth centre,
\(\lim_{r\downarrow0}V(r)/r^m=\omega_m\), the Euclidean unit-ball
volume. Integrating the last inequality gives
\[
 V(r)\geq\omega_m r^m\exp(-mCr^2/2).
\]
Choose \(r_*\) smaller than the uniform ambient comparison radius
and the boundary margin \(a\). This proves the lemma.
\end{proof}

\subsection{Packing with the leafwise dynamical metric}

Suppose now that \(\eta\) parametrizes a local strong unstable
plaque, with its fixed inverse history. Let \(W_t\) denote its
forward leaf at time \(t\), endowed with intrinsic spatial distance.
For \(z,w\in\DD^p(1)\), write
\[
 \rho_n(z,w)
 :=\max_{0\leq t<n}d_{W_t}(f^t\eta(z),f^t\eta(w)).
\]
Distances are taken in the parametrized leaves, retaining histories
when a spatial projection has several sheets.

\begin{lemma}[Leafwise dynamical packing]\label{lem:packing}
For every \(K\Subset\DD^p(1)\) and every sufficiently small fixed
\(r>0\), there exists \(C_{\eta,K,r}<\infty\), independent of
\(n\), such that every \(\rho_n\)-separated set \(E\subset K\)
at separation \(r\) satisfies
\begin{equation}\label{eq:packing}
 \#E\leq C_{\eta,K,r}d^{pn}.
\end{equation}
\end{lemma}

\begin{proof}
For any piecewise smooth parameter path \(\gamma\) joining \(z\)
to \(w\), its image under \(f^t\circ\eta\) is a path in the
corresponding forward leaf. The product metric therefore gives
\[
 \operatorname{length}(G_n\circ\gamma)
 \geq \operatorname{length}(f^t\circ\eta\circ\gamma)
 \geq d_{W_t}(f^t\eta(z),f^t\eta(w))
\]
for every \(t<n\). Taking an infimum over paths gives
\begin{equation}\label{eq:metric-comparison}
 d_{\Gamma_n}(G_n(z),G_n(w))\geq\rho_n(z,w).
\end{equation}
If a forward parametrization has critical points, lengths may be
computed by approximation with paths avoiding the finite union of
proper analytic critical loci. This leaves the inequality unchanged;
the orbit graph itself is always nonsingular because of its first
coordinate.

For \(z\in E\), the intrinsic graph balls of radius \(r/3\)
centred at \(G_n(z)\) are pairwise disjoint by
\eqref{eq:metric-comparison}. Lemmas~\ref{lem:graph-volume} and
\ref{lem:intrinsic-volume} yield
\[
 (\#E)c(r/3)^{2p}
 \leq\vol_{2p}(\Gamma_n)
 \leq C_\eta d^{pn}.
\]
This proves \eqref{eq:packing}.
\end{proof}

\section{Cumulative partial entropies}

\begin{proposition}\label{prop:partial-entropy}
For every level of the strong unstable flag,
\begin{equation}\label{eq:partial-entropy}
 H_j\leq p_j\log d.
\end{equation}
\end{proposition}

\begin{proof}
Fix \(j\) and set \(p=p_j\). Choose a typical conditional measure
\(\sigma\) on a \(W^j\)-plaque. By covering its spatial
parametrization with countably many relatively compact holomorphic
coordinate patches, choose one embedded patch
\(\eta:\DD^p(2)\to M\) and one compact inner subset on which
\(\sigma\) has positive mass. The coordinates may be rescaled so
that the patch satisfies the hypotheses of the preceding section.
The conditional measure is always kept on its prescribed lifted
plaque; notation suppresses that lift.

Let \(\varepsilon>0\). By \eqref{eq:local-entropy}, there exist a
small \(r>0\), an integer \(N\), and a measurable subset \(F\)
of the chosen inner patch with \(\sigma(F)>0\), such that
\begin{equation}\label{eq:uniform-bowen-mass}
 \sigma(B_n^j(y,r))\leq e^{-n(H_j-\varepsilon)}
 \qquad(y\in F,\ n\geq N).
\end{equation}
To obtain this uniformity, first choose one radius from a countable
sequence tending to zero for which the lower entropy rate exceeds
\(H_j-\varepsilon/2\) on a positive-mass set. Then choose a
uniform lower bound on \(n\), losing at most part of that positive
mass. Restricting or normalizing the conditional measure introduces
only a fixed multiplicative constant, which has no entropy cost.

For each \(n\geq N\), select a maximal subset \(E_n\subset F\)
separated at distance \(r/2\) for the intrinsic leafwise dynamical
metric. Lemma~\ref{lem:packing} bounds the cardinality of every
such separated set and hence ensures that the selection terminates
after finitely many points. Maximality gives a cover of \(F\) by
the Bowen balls of radius \(r\) centred at \(E_n\). It follows
from \eqref{eq:uniform-bowen-mass} that
\[
 0<\sigma(F)
 \leq\sum_{y\in E_n}\sigma(B_n^j(y,r))
 \leq C_{\eta,r}d^{pn}e^{-n(H_j-\varepsilon)}.
\]
Taking logarithms, dividing by \(n\), and letting \(n\to\infty\)
gives \(H_j\leq p\log d+\varepsilon\). Let
\(\varepsilon\downarrow0\).
\end{proof}

\begin{remark}
This is a bound for the cumulative entropy \(H_j\), not a bound
\(H_j-H_{j-1}\leq m_j\log d\) on every increment. The latter is
not used. Nor is any equality \(H_j=p_j\log d\) asserted.
\end{remark}

\section{Disintegration and the final dimension estimate}

We first record the measure-theoretic step which avoids any need to
identify a full-unstable conditional with the spatial measure.

\begin{lemma}[Projection of full-measure sets]\label{lem:disintegration}
Let \(\xi\) be a measurable partition of \(\Mh\), with
conditional probabilities \(\muh_{\xi(\xh)}\). Suppose that their
spatial projections
\[
 \nu_{\xh}:=\pi_*\muh_{\xi(\xh)}
\]
have Hausdorff dimension at least \(D\) for \(\muh\)-almost every
\(\xh\). Then \(\dimH\mu\geq D\).
\end{lemma}

\begin{proof}
Let \(A\subset M\) be Borel with \(\mu(A)=1\). Since
\(\muh(\pi^{-1}A)=1\), disintegration implies
\[
 \muh_{\xi(\xh)}(\pi^{-1}A)=1
 \quad\hbox{for almost every }\xh.
\]
Thus \(\nu_{\xh}(A)=1\) for almost every \(\xh\). The definition
of the Hausdorff dimension of a measure yields
\(\dimH A\geq\dimH\nu_{\xh}\geq D\). Taking the infimum over
such \(A\) proves the claim.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:main}]
Take \(\xi\) subordinate to the local full unstable plaques. The
spatial projection is a smooth local parametrization on each plaque;
after restriction to relatively compact coordinate patches it is
bi-Lipschitz. The conditional dimension statement in
Proposition~\ref{prop:LY} therefore gives
\[
 \dimH\nu_{\xh}=\delta_s
 =\sum_{j=1}^{s}\frac{H_j-H_{j-1}}{\chi_j}
 \quad\hbox{for almost every }\xh.
\]
If necessary, exhaust a plaque by countably many such relatively
compact patches; this does not change its almost-everywhere local
dimension. Lemma~\ref{lem:disintegration} implies
\begin{equation}\label{eq:spatial-lower}
 \dimH\mu\geq\delta_s.
\end{equation}

Summation by parts in \eqref{eq:LY} gives
\[
 \delta_s
 =\frac{H_s}{\chi_s}
 -\sum_{j=1}^{s-1}H_j
       \left(\frac1{\chi_{j+1}}-\frac1{\chi_j}\right).
\]
Similarly, since \(p_j=\sum_{a\leq j}m_a\),
\[
 L\sum_{j=1}^{s}\frac{m_j}{\chi_j}
 =\frac{kL}{\chi_s}
 -\sum_{j=1}^{s-1}p_jL
       \left(\frac1{\chi_{j+1}}-\frac1{\chi_j}\right).
\]
Using \(H_s=kL\), subtract the two identities to obtain
\begin{equation}\label{eq:defect-identity}
 \boxed{\displaystyle
 \delta_s-L\sum_{j=1}^{s}\frac{m_j}{\chi_j}
 =\sum_{j=1}^{s-1}(p_jL-H_j)
       \left(\frac1{\chi_{j+1}}-\frac1{\chi_j}\right)\geq0.}
\end{equation}
Every summand is nonnegative by
Proposition~\ref{prop:partial-entropy} and the ordering of the
exponents. Combining \eqref{eq:defect-identity} with
\eqref{eq:spatial-lower} proves \eqref{eq:main}.
\end{proof}

\begin{remark}[Scope of the argument]
The conclusion is the Hausdorff lower bound in
Theorem~\ref{thm:main}. The argument does not assert exact
dimensionality of the spatial measure, an equality in
\eqref{eq:main}, or equality between a plaque conditional and
\(\mu\). It also does not assume that a holomorphic slice is
invariant, or that different histories have the same fast flag.
In complex dimension two there is one nontrivial cumulative
partial entropy; in higher dimensions the same orbit-graph
argument is applied at every level of the flag.
\end{remark}

\clearpage
\appendix
\section{A quadratic example with a strict dimension lower bound}
\label{app:counterexample}
 The proof below is independent
of the general argument in the preceding sections.

Binder--DeMarco's Conjecture~1.3 \cite[p.~615]{BinderDeMarco}
asserts, in degree two and complex dimension two, that
\(\dimH\mu=\log 2/a+\log 2/b\), where \(a\geq b\) are the
two complex Lyapunov exponents, counted once each. We continue to
use natural logarithms and the full-measure convention for
\(\dimH\mu\) stated in Section~1.

\begin{theorem}\label{thm:counterexample}
The holomorphic quadratic endomorphism
\begin{equation}\label{eq:app-map}
 f[Z:W:T]
 =\left[Z^2-10000T^2:
 W^2-100T^2+\tfrac12 ZT:T^2\right]
\end{equation}
has equilibrium measure \(\mu\), complex Lyapunov exponents
\(a>b>0\), and dimension \(D=\dimH\mu\) satisfying
\begin{equation}\label{eq:app-gap}
 \boxed{\displaystyle
 D\geq\frac25>\frac{23}{60}>
 \frac{\log 2}{a}+\frac{\log 2}{b}.}
\end{equation}
In particular, its discrepancy from the conjectured value is
greater than \(1/60\).
\end{theorem}

The proof uses exact inequalities throughout. Its dynamical input
from the literature is the following standard result: a holomorphic
endomorphism of \(\PP^k(\CC)\) of algebraic degree \(d\geq2\)
has topological entropy \(k\log d\), and its equilibrium measure
is its unique measure of maximal entropy
\cite[p.~146, Theorem~2 and the preceding paragraph]{BriendDuval}.
The corrected multiplicity argument is in
\cite{BriendDuvalErratum}. The remaining ingredients are elementary
estimates, the entropy of a Bernoulli shift, and Birkhoff's ergodic
theorem.

\subsection{The invariant set and its equilibrium measure}

The three homogeneous coordinates in \eqref{eq:app-map} have no
common zero outside the origin: the last forces \(T=0\), and the
first two then force \(Z=W=0\). They therefore define a
holomorphic endomorphism of algebraic degree two. In the affine
chart \(T=1\),
\[
 f(x,y)=(x^2-10000,\ y^2-100+x/2).
\]
Regard \(Q=[-101,101]\times[-13,13]\) as a real rectangle in
\(\CC^2\). For \(s,t\in\{-1,1\}\), define an inverse branch on
\(Q\) by
\begin{equation}\label{eq:app-branches}
 g_{s,t}(u,v)=(x,y),\qquad
 x=s\sqrt{10000+u},\qquad
 y=t\sqrt{100+v-x/2}.
\end{equation}
Indeed, \(f\circ g_{s,t}=\id\). Since
\[
 99^2<9899\leq10000+u\leq10101<101^2,
\]
every output satisfies \(99<|x|<101\). If \(s=1\), then
\(73/2<y^2<127/2\), so \(6<|y|<8\). If \(s=-1\), then
\(273/2<y^2<327/2\), so \(11<|y|<13\). Thus the four images
lie in the interior of \(Q\), in the four disjoint
second-coordinate bands
\begin{equation}\label{eq:app-bands}
 (-13,-11),\qquad(-8,-6),\qquad(6,8),\qquad(11,13).
\end{equation}
Points in different images have \(y\)-coordinates differing by
more than \(3\).

Differentiating \eqref{eq:app-branches}, and using \(|2x|>198\)
and \(|2y|>12\), gives the ordinary sup-norm bound
\[
 \|Dg_{s,t}\|_\infty
 \leq\max\left\{\frac1{198},\frac1{12}+\frac1{4752}\right\}
 =\frac{397}{4752}<\frac1{11}.
\]
Convexity of \(Q\) makes this a global Lipschitz bound. For each
infinite word \(\boldsymbol\omega=(\omega_0,\omega_1,\ldots)\)
on the four branch symbols, the nested compact sets
\[
 g_{\omega_0}\circ\cdots\circ g_{\omega_{n-1}}(Q)
\]
have diameters tending to zero, hence have a unique common point
\(\pi(\boldsymbol\omega)\). The coding map is continuous. It
is injective because, after removing the common prefix of two
different words, their points lie in different bands; every inverse
prefix is injective. Consequently \(\pi\) is a homeomorphism
from the compact one-sided four-symbol shift onto
\[
 K=\pi\bigl((\{-1,1\}^2)^{\mathbb N}\bigr).
\]
Moreover, \(f\circ\pi=\pi\circ\sigma\), where \(\sigma\)
deletes the first symbol.

Push the uniform Bernoulli probability forward by \(\pi\),
obtaining \(\nu\). This measure is invariant and ergodic, has
entropy \(\log4\), and assigns mass \(4^{-n}\) to each
length-\(n\) cylinder
\[
 K_{\omega_0\cdots\omega_{n-1}}
 =g_{\omega_0}\circ\cdots\circ g_{\omega_{n-1}}(K).
\]
Since the ambient endomorphism has topological entropy \(\log4\),
the quoted uniqueness theorem identifies \(\nu\) with its
equilibrium measure \(\mu\). In particular, \(\mu(K)=1\).

\subsection{Cylinder separation and the dimension lower bound}

Consider two distinct inputs in \(Q\) with
\(|\Delta u|\leq68|\Delta v|\), and apply the same inverse
branch. Write the outputs as \((x,y)\) and \((x',y')\).
Their corresponding coordinate signs agree, and the exact
difference identities are
\[
 \Delta x=\frac{\Delta u}{x+x'},\qquad
 \Delta y=\frac{\Delta v-\Delta x/2}{y+y'}.
\]
The input cone and distinctness imply \(\Delta v\ne0\). Since
\(|x+x'|>198\) and \(|y+y'|<26\), it follows that
\begin{equation}\label{eq:app-cone-expansion}
 |\Delta y|
 \geq\frac{1-68/396}{26}|\Delta v|
 >\frac1{32}|\Delta v|.
\end{equation}
The last inequality is exact:
\[
 \frac{1-68/396}{26}-\frac1{32}
 =\frac{25}{41184}>0.
\]
Furthermore,
\[
 |\Delta x|\leq\frac{68}{198}|\Delta v|
 <\frac{32\cdot68}{198}|\Delta y|<68|\Delta y|.
\]
Thus a common inverse branch preserves the cone and decreases the
\(y\)-distance by a factor no smaller than \(1/32\).

For points of \(K\) in different first-level cylinders,
\[
 |\Delta x|<202<3\cdot68<68|\Delta y|,
 \qquad |\Delta y|>3.
\]
Now take points in different length-\(n\) cylinders, and let
\(m<n\) be the length of their common prefix. After applying
\(f^m\), the points occupy different first-level cylinders.
Restore the prefix, using cone preservation and
\eqref{eq:app-cone-expansion} at each step. Their original
\(y\)-distance, for every such pair, is greater than
\[
 3\,32^{-m}\geq3\,32^{-(n-1)}.
\]

For any \(0<r\leq1\), choose \(n\geq1\) such that
\(32^{-n}\leq r\leq32^{-(n-1)}\). An ambient Euclidean ball of
radius \(r\) has second-coordinate diameter at most \(2r\),
less than the separation just proved. It therefore meets at most
one length-\(n\) cylinder, and
\begin{equation}\label{eq:app-frostman}
 \mu(B(z,r))\leq4^{-n}
 =(32^{-n})^{2/5}\leq r^{2/5}.
\end{equation}
The centre \(z\) may be arbitrary; open or closed balls give the
same bound.

For completeness, \eqref{eq:app-frostman} bounds the dimension of
the measure itself. If \(E\) is any Borel set of full
\(\mu\)-measure, replace it by \(E\cap K\). A countable cover
of this set by nonempty sets of diameters \(d_i\leq1\) gives, by
enclosing each set in a ball of radius \(d_i\) and using outer
measure,
\[
 1\leq\sum_i d_i^{2/5}.
\]
Zero-diameter sets have zero mass by \eqref{eq:app-frostman}.
Thus \(\mathcal H^{2/5}(E\cap K)>0\), up to the harmless
normalization constant in Hausdorff measure, and
\(\dimH E\geq2/5\). Euclidean and Fubini--Study distances are
bi-Lipschitz equivalent on the compact affine rectangle \(Q\);
hence this conclusion holds for the required metric. Taking the
infimum over full-measure \(E\) gives
\begin{equation}\label{eq:app-dimension}
 D\geq\frac25.
\end{equation}

\subsection{The complex Lyapunov exponents and the strict gap}

Along \(K\) the complex derivative is
\[
 Df(x,y)=\begin{pmatrix}2x&0\\[2pt]1/2&2y\end{pmatrix},
 \qquad198<|2x|<202,\qquad12<|2y|<26.
\]
For \(f^j(p)=(x_j,y_j)\), write
\[
 Df^n(p)=\begin{pmatrix}P_n&0\\E_n&Q_n\end{pmatrix},
 \qquad
 P_n=\prod_{j=0}^{n-1}2x_j,\qquad
 Q_n=\prod_{j=0}^{n-1}2y_j.
\]
With \(r_0=0\) and \(r_n=E_n/P_n\), matrix multiplication
gives
\[
 r_{n+1}=\frac1{4x_n}+\frac{y_n}{x_n}r_n,\qquad
 |r_n|\leq\frac{1/396}{1-26/198}=\frac1{344},\qquad
 \left|\frac{Q_n}{P_n}\right|\leq(26/198)^n.
\]
Let \(\sigma_1(n)\geq\sigma_2(n)>0\) be the two complex
singular values. For \(M=\sqrt{2+344^{-2}}\), the first column
and the Frobenius norm give
\[
 |P_n|\leq\sigma_1(n)\leq M|P_n|.
\]
Since \(\sigma_1(n)\sigma_2(n)=|\det Df^n|=|P_nQ_n|\),
\[
 M^{-1}|Q_n|\leq\sigma_2(n)\leq|Q_n|.
\]
These constants do not depend on \(p\) or \(n\). Birkhoff's
theorem therefore identifies the Lyapunov exponents, for
\(\mu\)-almost every point, as
\begin{equation}\label{eq:app-exponents}
 a=\int_K\log(2|x|)\,d\mu,\qquad
 b=\int_K\log(2|y|)\,d\mu.
\end{equation}
Uniform comparison of tangent norms on \(K\) gives the same
exponents for the Fubini--Study metric. Their order and positivity
follow from
\[
 a>\log198>\log26>b>\log12>0.
\]

The first symbolic \(x\)-sign is positive on two of the four
branches and negative on the other two, each of mass \(1/4\).
Thus each sign has total mass \(1/2\). The sharper band bounds
in \eqref{eq:app-bands} and \eqref{eq:app-exponents} yield
\[
 b>\tfrac12\log12+\tfrac12\log22
   =\tfrac12\log264>4\log2,
 \qquad
 a>\log198>\tfrac{15}{2}\log2.
\]
Here \(264>2^8=256\) and
\(198^2=39204>2^{15}=32768\); no decimal logarithm evaluation
is involved. Consequently
\[
 \frac{\log2}{a}+\frac{\log2}{b}
 <\frac2{15}+\frac14
 =\frac{23}{60}<\frac{24}{60}=\frac25\leq D.
\]
This proves Theorem~\ref{thm:counterexample}, including the strict
discrepancy greater than \(1/60\). \hfill\(\square\)

The strict counterexample is established without computing \(D\)
exactly or assuming any general exact-dimensionality formula.
Every estimate needed for the conclusion is included above.


\clearpage
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\end{document}