Distinguishing endpoint sets from Erd\H{o}s space

TL;DR

This paper investigates the topological structure of endpoint sets in Julia sets for exponential functions, revealing non-homeomorphism to Erdős space and path-connectedness of certain orbit sets.

Contribution

It proves that the endpoint set of the Julia set for $f(z)= ext{exp}(z)-1$ is not homeomorphic to Erdős space and extends these results to other exponential functions.

Findings

Endpoint set of Julia set not homeomorphic to Erdős space

Set of points with escaping or attracting orbits is path-connected

Results extend to many exponential family functions

Abstract

We prove that the set of all endpoints of the Julia set of $f(z)=\exp(z)-1$ which escape to infinity under iteration of $f$ is not homeomorphic to the rational Hilbert space $\mathfrak E$. As a corollary, we show that the set of all points $z\in \mathbb C$ whose orbits either escape to $\infty$ or attract to $0$ is path-connected. We extend these results to many other functions in the exponential family.