Non-escaping endpoints do not explode

TL;DR

This paper investigates the topological structure of certain subsets of Julia sets for exponential maps, revealing that non-escaping endpoints are totally separated and relating this to spider's web structures, with applications to Fatou's function.

Contribution

It demonstrates that non-escaping endpoints, together with infinity, are totally separated, contrasting previous results, and provides a new topological characterization of spider's webs.

Findings

Non-escaping endpoints with infinity are totally separated.

A new topological characterization of spider's webs is provided.

Results apply to Fatou's function, expanding understanding of transcendental dynamics.

Abstract

The family of exponential maps $f_a(z)= e^z+a$ is of fundamental importance in the study of transcendental dynamics. Here we consider the topological structure of certain subsets of the Julia set $J(f_a)$. When $a\in (-\infty,-1)$, and more generally when $a$ belongs to the Fatou set of $f_a$, it is known that $J(f_a)$ can be written as a union of "hairs" and "endpoints" of these hairs. In 1990, Mayer proved for $a\in (-\infty,-1)$ that, while the set of endpoints is totally separated, its union with infinity is a connected set. Recently, Alhabib and the second author extended this result to the case where $a \in F(f_a)$, and showed that it holds even for the smaller set of all escaping endpoints. We show that, in contrast, the set of non-escaping endpoints together with infinity is totally separated. It turns out that this property is closely related to a topological structure known as a `spider's web'; in particular we give a new topological characterisation of spiders' webs that may be of independent interest. We also show how our results can be applied to Fatou's function, $z\mapsto z + 1 + e^{-z}$.