A note on the topology of escaping endpoints

TL;DR

This paper investigates the topological structure of escaping endpoints in the Julia sets of complex exponential functions with parameters less than -1, revealing they are not homeomorphic to the entire set of endpoints and are of first category.

Contribution

It establishes that for transcendental entire functions, the set of escaping endpoints in the Julia set is always of first category, providing new insights into their topological complexity.

Findings

Escaping endpoints are not homeomorphic to the entire set of endpoints.

The set of escaping endpoints is of first category for all transcendental entire functions.

The specific case of exponential functions with parameter a in (-∞, -1) is analyzed.

Abstract

We study topological properties of the escaping endpoints and fast escaping endpoints of the Julia set of complex exponential $\exp(z)+a$ when $a\in (-\infty,-1)$. We show neither space is homeomorphic to the whole set of endpoints. This follows from a general result stating that for every transcendental entire function $f$, the escaping Julia set $I(f)\cap J(f)$ is first category.