Boundary theorems for geometric coding trees

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. A Riemann map is a biholomorphic map $R:\mathbb D\to V$ onto a simply connected proper plane domain, where $\mathbb D=\{w:|w|<1\}$. Its radial boundary values are limits of $R(r\zeta)$ as $r\uparrow1$ for $|\zeta|=1$. In the coding-tree dictionary, a prime end corresponds to a geometric branch, its impression to the intersection of the closures of images of longer and longer symbolic cylinders, and its principal points to limit points of the branch vertices.

Statement recorded in the supplied source.

Which theorems about boundary behaviour of Riemann maps have analogues for geometric coding trees and their coding maps $z_\infty$? [S2]

Short English statement

Determine which results about approaching the boundary through a Riemann map also hold when the approach is through branches of a geometric coding tree.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 5300057 (AMR-052-0057). 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 2.6, pp. 33; arXiv:math/9205209v1. https://arxiv.org/pdf/math/9205209.