Boundaries of escaping Fatou components

TL;DR

This paper investigates the boundary behavior of escaping Fatou components for transcendental entire functions, showing most boundary points are escaping and linking boundary escaping points to the nature of the component.

Contribution

It establishes a connection between boundary points and escaping behavior in Fatou components, providing new insights into their structure and properties.

Findings

Most boundary points of escaping wandering domains are escaping.

If enough boundary points are escaping, the domain is escaping.

The union of the escaping set and infinity is connected.

Abstract

Let $f$ be a transcendental entire function and $U$ be a Fatou component of $f$. We show that if $U$ is an escaping wandering domain of $f$, then most boundary points of $U$ (in the sense of harmonic measure) are also escaping. In the other direction we show that if enough boundary points of $U$ are escaping, then $U$ is an escaping Fatou component. Some applications of these results are given; for example, if $I(f)$ is the escaping set of $f$, then $I(f)\cup\{\infty\}$ is connected.