Spiders' webs in the Eremenko-Lyubich class

TL;DR

This paper proves that the escaping set of the entire function cosh(z) forms a spider's web, resolving conjectures about the structure of escaping sets and extending results to a broader class of functions.

Contribution

It demonstrates that the escaping set of cosh(z) is a spider's web, disproving a 2020 conjecture and answering a 2012 question about fast escaping sets.

Findings

The escaping set of cosh(z) is a spider's web.

The fast escaping set of cosh(z) is a spider's web.

Results apply to a wider class of functions.

Abstract

Consider the entire function $f(z)=\cosh(z)$. We show that the escaping set of this function - that is, the set of points whose orbits tend to infinity under iteration - has a structure known as a "spider's web". This disproves a conjecture of Sixsmith from 2020. In fact, we show that the "fast escaping set", i.e. the set of points whose orbits tend to infinity at an iterated exponential rate, is a spider's web. This answers a question of Rippon and Stallard from 2012. We also discuss a wider class of functions to which our results apply, and state some open questions.