Accessibility of positive-exponent boundary points

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 $U\subset\widehat{\mathbb C}$ be a simply connected domain, $f:U\to U$ a proper holomorphic map of degree $d\geq2$, and suppose its iterates converge to a constant in $U$. Suppose also that $f$ extends holomorphically to a neighbourhood of $\overline U$. A point $x\in\partial U$ is accessible from $U$ if there is a continuous curve $\eta:[0,1]\to\overline U$ with $\eta([0,1))\subset U$ and $\eta(1)=x$. A periodic point satisfies $f^p(x)=x$ for some $p\geq1$. The lower Lyapunov exponent at $x$ is $\underline\chi(x)=\liminf_{n\to\infty}n^{-1}\log\|(f^n)'(x)\|$, with derivative norm in a smooth spherical metric; positivity is independent of a smooth change of metric on the compact invariant boundary.

Statement recorded in the supplied source.

Is every $x\in\partial U$ with $\underline\chi(x)>0$ accessible from $U$? [S2]

Short English statement

Can every boundary point with a positive lower rate of derivative growth be reached from inside the attracting domain?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300049 (AMR-052-0049). 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 1.2, pp. 30; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.