Escaping sets are not sigma-compact

TL;DR

This paper proves that for transcendental entire functions, the escaping set is not sigma-compact, answering a question posed in 2009 and advancing understanding of the structure of these dynamical sets.

Contribution

The paper establishes that the escaping set of transcendental entire functions is not sigma-compact, resolving a long-standing open problem in complex dynamics.

Findings

Escaping set $I(f)$ is not sigma-compact for transcendental entire functions.

Addresses a question posed by Rippon in 2009.

Provides new insights into the topological structure of escaping sets.

Abstract

Let $f$ be a transcendental entire function. The escaping set $I(f)$ consists of those points that tend to infinity under iteration of $f$. We show that $I(f)$ is not $\sigma$-compact, resolving a question of Rippon from 2009.