Approximating quasi-repeller dimension by measures

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $U\subset\widehat{\mathbb C}=\mathbb C\cup\{\infty\}$ be open and $f:U\to f(U)$ holomorphic and proper (inverse images of compact sets are compact), with $f(U)\supset U$. Choose a root $z\in f(U)$, $d\geq2$ distinct preimages $z^1,\ldots,z^d\in U$, and curves $\gamma^i\subset f(U)$ from $z$ to $z^i$. Here $\mathbb N_0=\{0,1,2,\ldots\}$ and $\Sigma_d=\{1,\ldots,d\}^{\mathbb N_0}$, with left shift $\sigma$. A geometric coding tree consists of choices of successive lifted curves $\gamma_n(\alpha)$ satisfying $\gamma_0(\alpha)=\gamma^{\alpha_0}$, $f\circ\gamma_{n+1}(\alpha)=\gamma_n(\sigma\alpha)$ and $\gamma_{n+1}(\alpha)(0)=\gamma_n(\alpha)(1)$. Write $z_n(\alpha)=\gamma_n(\alpha)(1)$, $D(z_\infty)=\{\alpha:z_n(\alpha)\text{ converges in }\overline U\}$ and $z_\infty(\alpha)=\lim_n z_n(\alpha)$. The construction does not impose absence of critical values when suitable lifts can be chosen. Put $\Lambda=z_\infty(D(z_\infty))$. If $f$ extends holomorphically to a neighbourhood of $\overline\Lambda$, the source calls $\Lambda$ a holomorphic quasi-repeller. Let $\mathcal M(\Lambda)$ be the probability measures on $\overline\Lambda$ invariant under $f$ and ergodic (every invariant measurable set has measure zero or one), and let $\mathcal M^+(\Lambda)=\{m\in\mathcal M(\Lambda):h_m(f)>0\}$, where $h_m(f)$ is measure-theoretic entropy. For a probability measure, $\dim_Hm=\inf\{\dim_HE:m(E)=1\}$. The Hausdorff dimension $\dim_H E$ is the threshold exponent for Hausdorff measures $\mathcal H^s(E)$, defined using covers by sets of small diameter and sums of their $s$th powers. Planar Lebesgue measure is denoted by $m_2$.

Statement recorded in the supplied source.

Is $\sup_{m\in\mathcal M^+(\Lambda)}\dim_Hm=\dim_H\overline\Lambda$? Does replacing $\mathcal M^+(\Lambda)$ by all of $\mathcal M(\Lambda)$ help, that is, does the corresponding supremum over all invariant ergodic probability measures equal $\dim_H\overline\Lambda$? [S2]

Short English statement

Can positive-entropy invariant ergodic measures approximate the full dimension of the quasi-repeller closure? Does allowing zero-entropy ergodic measures improve the answer?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300058 (AMR-052-0058). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Feliks Przytycki, "On Invariant Measures for Iterations of Holomorphic Maps", in Ben Bielefeld and Mikhail Lyubich (eds.), Problems in Holomorphic Dynamics (1992), Problem 2.7, pp. 33; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.