Boundaries of univalent Baker domains

TL;DR

This paper investigates the boundary behavior of univalent Baker domains for transcendental entire functions, showing that non-escaping boundary points are negligible in harmonic measure and deriving new conditions related to the escaping set and Eremenko's conjecture.

Contribution

It introduces a novel boundary behavior result for conformal maps of Baker domains and connects this to the structure of the escaping set and Eremenko's conjecture.

Findings

Non-escaping boundary points have harmonic measure zero.

Provides a new sufficient condition for the connectedness of the escaping set.

Establishes a general result supporting Eremenko's conjecture.

Abstract

Let $f$ be a transcendental entire function and let $U$ be a univalent Baker domain of $f$. We prove a new result about the boundary behaviour of conformal maps and use this to show that the non-escaping boundary points of $U$ form a set of harmonic measure zero with respect to $U$. This leads to a new sufficient condition for the escaping set of $f$ to be connected, and also a new general result on Eremenko's conjecture.