SPIDERS’ WEBS IN THE EREMENKO–LYUBICH CLASS

Consider the entire function ƒ(z) = cosh(z). We show that the escaping set I(ƒ) – that is, the set of points whose orbits tend to infinity under iteration of f – 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 A(ƒ), i.e. the subset of I(ƒ) consisting 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.