Spiders' webs in the Eremenko-Lyubich class
Lasse Rempe
TL;DR
The paper addresses whether escaping structures in the Eremenko--Lyubich class can form a spider's web, focusing on the function $f(z)=\cosh(z)$; it constructs a topological model linking $h(z)=\cosh(z)$ to the disjoint-type map $g(z)=\cosh(z)/2$ and transfers disconnectedness from $\hat{J}(g)\setminus A(g)$ to $\hat{\mathbb{C}}\setminus A(h)$ via a continuous surjection. The key result shows that $A(\cosh)$ is a spider's web, hence $I(\cosh)$ is a spider's web, thereby disproving Sixsmith's conjecture for $f\in\mathcal{B}$ and answering Rippon--Stallard's question in the affirmative for this setting. The work further extends the spider's web phenomenon to a wider class of finite-order entire functions with bounded critical values, indicating a broader landscape where $A(f)$ (and often $I(f)$) are spider's webs. The findings contribute new examples and a topological framework for understanding the escaping sets of transcendental entire functions in the Eremenko--Lyubich class.
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.