Spiders’ Webs in the Eremenko–Lyubich Class

Consider the entire function f(z) = cosh(z). We show that the escaping set I(f) —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(f) , consisting of 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.