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\title{Dimension of the equilibrium measure on $\mathbb{P}^n$}
\author{}
\date{}
\begin{document}
\maketitle

\begin{abstract}
Consider the dynamics of endomorphisms $f: \PP^n\to \PP^n$ of degree $d\geq 2$. 
This note presents an explicit and elementary counterexample to the dimension formula for the equilibrium measure $\mu$ conjectured by Binder--DeMarco \cite{BinderDeMarco}, coming from the same family of quadratic skew products that appeared recently in Bianchi--He \cite{HB}.  In addition, the note shows that the one-sided inequality
\[
  \dim_{\mathrm H}\mu
  \geq (\log d)\sum_{i=1}^n \lambda_i^{-1}
\]
holds for the equilibrium measure of every degree-$d$ holomorphic endomorphism of $\mathbb P^n$. The proof adapts Dupont's volume-growth method \cite{Dupont} to the unstable flag: combined with Gromov's orbit-graph argument \cite{Gromov}, it bounds the cumulative partial
entropies. Qian--Xie's extension \cite{QX} of Ledrappier--Young theory \cite{LY} then yields the inequality.
\end{abstract}

\noindent\emph{Provenance.} This revision was produced by ChatGPT to clarify the sources and condense the exposition. 

\section{Introduction}
Let $f:\mathbb P^n(\mathbb C)\to\mathbb P^n(\mathbb C)$ be a holomorphic
endomorphism of degree $d\geq2$, with equilibrium measure $\mu$ and
Lyapunov exponents $\lambda_1\geq\cdots\geq\lambda_n>0$, counted with
complex multiplicity. Binder--DeMarco \cite[Conjecture 1.3]{BinderDeMarco}
conjectured that
\begin{equation}\label{eq:conjecture}
 \dimH\mu=(\log d)\sum_{i=1}^n\frac1{\lambda_i},
\end{equation}
for the Hausdorff dimension of $\mu$, defined by 
\[
 \dimH\mu=\inf\{\dimH A:A\text{ is Borel},\ \mu(A)=1\}.
\]
Bianchi--He \cite{HB} recently disproved this conjecture using quadratic
skew products. The Appendix proves directly that the example (from the same family) 
\[
 f(x,y)=(x^2-10000,\ y^2-100+x/2)
\]
has $\dimH\mu\geq2/5$, whereas the right-hand side of \eqref{eq:conjecture}
is less than $23/60$.

The purpose of this note is to observe that expression in 
\eqref{eq:conjecture} provides a lower bound on the dimension in general.
\begin{theorem}\label{thm:main}
For every holomorphic endomorphism $f$ as above,
\begin{equation}\label{eq:main}
 \dimH\mu\geq(\log d)\sum_{i=1}^n\frac1{\lambda_i}.
\end{equation}
\end{theorem}
The proof adapts the method of Dupont \cite{Dupont}, who established
this inequality for $n=2$ and the bound
$(n-1)\log d/\lambda_1+\log d/\lambda_n$ in arbitrary dimension.
His argument counts inverse branches meeting a small ball by means of
holomorphic slices, volume growth, and the normal forms of
Berteloot--Dupont--Molino \cite{BDM}. The same volume-growth
principle applies at every level of the fast unstable flag.

The partial-entropy formulation is already suggested in Dupont's
\S1.1, which compares \eqref{eq:conjecture} with the Ledrappier--Young
formula and singles out the values $H_i=i\log d$.
The cumulative bounds $H_j\leq p_j\log d$ suffice; here $p_j$
is the dimension of the corresponding fast plaque. These follow by
combining Dupont's local growth estimate \cite[Appendix]{Dupont} with
Gromov's orbit-graph packing argument \cite[\S\S1--3]{Gromov} and the
conditional version of the Brin--Katok local-entropy method \cite{BK,QX}.
The dimension estimate then follows from Qian--Xie's endomorphism
extension \cite{QX,QXZ} of Ledrappier--Young theory \cite{LY}.
Bianchi--He \cite[Proposition 2.3]{HB} use the analogous entropy-defect
identity in dimension two and, under their hypotheses, deduce
$H_1\leq\log d$ from Dupont's inequality. Equality in
\eqref{eq:conjecture} does hold for skew products whose fiber exponent
dominates the base exponent, by Genes \cite{Clara}.

\paragraph{Notation.}
Put $M=\mathbb P^n(\mathbb C)$ and equip it with the Fubini--Study form
$\omega$ and its associated metric. Group equal exponents as
\[
 \chi_1>\cdots>\chi_s>0,\qquad
 \sum_{j=1}^s m_j=n,\qquad p_j=\sum_{a=1}^jm_a,\qquad L=\log d,
\]
where $m_j$ is the complex multiplicity of $\chi_j$. Thus
$\sum_i\lambda_i^{-1}=\sum_jm_j/\chi_j$. The measure $\mu$ is ergodic, with
\begin{equation}\label{eq:equilibrium-facts}
 h_\mu(f)=nL,\qquad \log|\det D_{\mathbb R}f|\in L^1(\mu),
 \qquad\lambda_n>0.
\end{equation}
See \cite{DS,BriendDuval,BriendDuvalErratum}, and Briend--Duval
\cite{BDexponents} for positivity of the exponents.
In particular the critical set and its iterated preimages have zero
$\mu$-measure.

\section{Conditional entropy and dimension}
Let $(\Mh,\fh,\muh)$ be the natural extension, with time-zero projection
$\pi:\Mh\to M$. A regular inverse history determines a strong unstable
flag $W^1\subset\cdots\subset W^s$, where $\dim_{\mathbb C}W^j=p_j$.
The local spatial projections of these plaques are holomorphic, by
holomorphic graph transform; compare the normal forms in \cite{BDM}.
Their parametrizations extend to slightly larger polydiscs after
shrinking. The flag is kept on the natural extension and may depend on
the inverse history.

Qian--Xie's endomorphism version of Ledrappier--Young theory gives
exact conditional dimensions and their entropy relations
\cite[Proposition 2.4 and equation (19)]{QX}, and conditional local
entropy \cite[Proposition 4.4]{QX}; see also \cite[Chapter IX]{QXZ}.

\begin{proposition}[Qian--Xie framework]\label{prop:LY}
There are cumulative partial entropies
$0=H_0\leq H_1\leq\cdots\leq H_s=h_\mu(f)$ and exact conditional
dimensions $\delta_j$ on the strong unstable plaques such that
\begin{equation}\label{eq:LY}
 \delta_j-\delta_{j-1}=\frac{H_j-H_{j-1}}{\chi_j},
 \qquad 1\leq j\leq s,\qquad\delta_0=0.
\end{equation}
Dimensions are measured in the intrinsic spatial metric on a plaque.
For a typical plaque conditional $\sigma$, define
\[
 B_N^j(y,r)=\{z:\max_{0\leq t<N}d_{W_t^j}(f^ty,f^tz)<r\}
\]
inside its conditional-measure domain. The forward leaves and their
intrinsic metrics retain the prescribed histories. Then
\begin{equation}\label{eq:local-entropy}
 \lim_{r\downarrow0}\liminf_{N\to\infty}
 -\frac1N\log\sigma(B_N^j(y,r))=H_j
 \quad\text{for }\sigma\text{-almost every }y.
\end{equation}
\end{proposition}
The hypotheses are supplied by \eqref{eq:equilibrium-facts}. It remains
to prove
\begin{equation}\label{eq:entropy-goal}
 H_j\leq p_jL\qquad(1\leq j<s).
\end{equation}

\section{Volume growth and orbit graphs}
Gromov's orbit graphs and minimal-volume argument
\cite[\S\S1--3]{Gromov} are localized by Dupont's bounded-potential method
\cite[Appendix]{Dupont}. Intrinsic graph distances are needed to
control the intrinsic leafwise dynamical metric.

\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.
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 \cite{CLN}, applied between
\(\DD^p(1)\) and \(\DD^p(3/2)\), gives
\[
 \int_{\DD^p(1)}(\ddc v_N)^p\leq C.
\]
By \eqref{eq:potential},
\[
 \vol_{2p}(\Gamma_N)
 =\frac{A_N^p}{p!}\int_{\DD^p(1)}(\ddc v_N)^p
 \leq C'A_N^p\leq C_\eta d^{pN}.
\]
\end{proof}

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

The following intrinsic version of the minimal-volume estimate is proved
explicitly to keep the constants independent of the orbit length.
\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 \(a\).
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
\[
 (\#E)c(r/3)^{2p}
 \leq\vol_{2p}(\Gamma_N)
 \leq C_\eta d^{pN}.
\]
\end{proof}

\section{The cumulative entropy bound}
\begin{proposition}\label{prop:partial-entropy}
For each level of the strong unstable flag, $H_j\leq p_j\log d$.
\end{proposition}
\begin{proof}
Fix $j$, set $p=p_j$, and choose a typical conditional $\sigma$ on a
lifted $W^j$-plaque. A countable cover by holomorphic coordinate patches
provides an embedding $\eta:\DD^p(2)\to M$ as above and an inner compact
set of positive $\sigma$-measure. Keep this conditional on its fixed
lifted plaque.

For $\varepsilon>0$, \eqref{eq:local-entropy} gives a small $r>0$, an
integer $N_0$, and a positive-measure subset $F$ of the inner compact set
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_0).
\end{equation}
Indeed, first choose a radius from a countable sequence tending to zero
on which the lower entropy rate exceeds $H_j-\varepsilon/2$ on a set of
positive measure, then choose a uniform $N_0$. Restriction or
normalization contributes only a fixed multiplicative constant.

A maximal $(r/2)$-separated set $E_N\subset F$ for the leafwise dynamical
metric has at most $C_{\eta,r}d^{pN}$ points by Lemma~\ref{lem:packing}.
Its radius-$r$ Bowen balls cover $F$, so
\[
 0<\sigma(F)\leq\sum_{y\in E_N}\sigma(B_N^j(y,r))
 \leq C_{\eta,r}d^{pN}e^{-N(H_j-\varepsilon)}.
\]
Letting $N\to\infty$ and then $\varepsilon\downarrow0$ proves the claim.
This is the conditional version of the local-entropy covering argument
of Brin--Katok \cite{BK}, with the leafwise input of \cite{QX}.
\end{proof}
\begin{remark}
The estimate is cumulative. Bounds
$H_j-H_{j-1}\leq m_j\log d$ for individual increments do not follow
and are not needed.
\end{remark}

\section{Proof of the dimension inequality}
\begin{lemma}\label{lem:disintegration}
Let $\xi$ be a measurable partition of $\Mh$. If the projected
conditionals $\nu_{\xh}=\pi_*\muh_{\xi(\xh)}$ have Hausdorff dimension at
least $D$ for almost every $\xh$, then $\dimH\mu\geq D$.
\end{lemma}
\begin{proof}
For any Borel set $A$ with $\mu(A)=1$, disintegration of
$\muh(\pi^{-1}A)=1$ gives $\nu_{\xh}(A)=1$ almost everywhere. Hence
$\dimH A\geq D$; take the infimum over $A$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:main}]
Take $\xi$ subordinate to full unstable plaques. Projection to $M$ is
bi-Lipschitz on relatively compact spatial coordinate patches, so
Proposition~\ref{prop:LY} and Lemma~\ref{lem:disintegration} give
\begin{equation}\label{eq:spatial-lower}
 \dimH\mu\geq\delta_s
 =\sum_{j=1}^s\frac{H_j-H_{j-1}}{\chi_j}.
\end{equation}
Use a countable exhaustion by patches if necessary. Summation by parts,
using $H_s=nL$ and $p_j=\sum_{a\leq j}m_a$, yields
\begin{equation}\label{eq:defect-identity}
 \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}
Each summand is nonnegative by Proposition~\ref{prop:partial-entropy}.
Together with \eqref{eq:spatial-lower}, this proves the theorem.
\end{proof}
The identity \eqref{eq:defect-identity} extends the two-exponent entropy
defect in \cite[Proposition 2.3]{HB}. Equality in \eqref{eq:main} would
require $H_j=p_jL$ for every $j<s$, together with equality in
\eqref{eq:spatial-lower}.

\appendix
\section{An explicit counterexample}
\label{app:counterexample}
The following counterexample to \cite[Conjecture 1.3]{BinderDeMarco}
belongs to the quadratic skew-product family of Bianchi--He \cite{HB}.
The proof is independent of the conditional entropy argument above.
\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}
\end{theorem}
The proof uses uniqueness of the measure of maximal entropy
\cite[Theorem 2]{BriendDuval}, with its erratum \cite{BriendDuvalErratum}.
\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}
These satisfy $f\circ g_{s,t}=\id$ and $99<|x|<101$.
For $s=1$, $6<|y|<8$; for $s=-1$, $11<|y|<13$.
Thus the images are compactly contained in $Q$ and occupy the disjoint 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
\[
 \|Dg_{s,t}\|_\infty
 \leq\max\left\{\frac1{198},\frac1{12}+\frac1{4752}\right\}
 =\frac{397}{4752}<\frac1{11}.
\]
Since $Q$ is convex, these are contractions. Their disjoint images give a
homeomorphic coding of the invariant compact set $K$ by the one-sided
shift on four symbols. Pushing forward the uniform Bernoulli measure
gives an invariant ergodic probability of entropy $\log4$, hence the
equilibrium measure $\mu$ by uniqueness. Every cylinder
$K_{\omega_0\cdots\omega_{N-1}}=g_{\omega_0}\circ\cdots\circ
 g_{\omega_{N-1}}(K)$ has mass $4^{-N}$.
\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}
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}
For any full-measure Borel set $E\subset K$, a cover by sets of diameters
$d_i\leq1$ therefore satisfies $1\leq\sum_i d_i^{2/5}$. This proves
$\dimH E\geq2/5$. Euclidean and Fubini--Study distances are bi-Lipschitz
equivalent on $Q$, so
\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 \(\sigma_1(N)\sigma_2(N)=|\det Df^N|=|P_NQ_N|\),
\[
 M^{-1}|Q_N|\leq\sigma_2(N)\leq|Q_N|.
\]
Birkhoff's theorem therefore gives
\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
\[
 a>\log198>\log26>b>\log12>0.
\]
Each $x$-sign has mass $1/2$. The band bounds give
\[
 b>\tfrac12\log12+\tfrac12\log22
   =\tfrac12\log264>4\log2,
 \qquad
 a>\log198>\tfrac{15}{2}\log2.
\]
Since $264>2^8$ and $198^2>2^{15}$,
\[
 \frac{\log2}{a}+\frac{\log2}{b}
 <\frac2{15}+\frac14
 =\frac{23}{60}<\frac{24}{60}=\frac25\leq D.
\]
This proves Theorem~\ref{thm:counterexample}. \hfill\(\square\)

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\end{document}