A sharp growth condition for a fast escaping spider's web

TL;DR

This paper establishes a growth condition under which the fast escaping set of a transcendental entire function forms a spider's web, providing new insights into longstanding conjectures in complex dynamics.

Contribution

It identifies a sharp growth criterion ensuring the fast escaping set is a spider's web and provides counterexamples showing the optimality of this condition.

Findings

Fast escaping set forms a spider's web under certain growth conditions.

Counterexamples show the growth condition is optimal.

Results shed light on Baker's and Eremenko's conjectures.

Abstract

We show that the fast escaping set $A(f)$ of a transcendental entire function $f$ has a structure known as a spider's web whenever the maximum modulus of $f$ grows below a certain rate. We give examples of entire functions for which the fast escaping set is not a spider's web which show that this growth rate is best possible. By our earlier results, these are the first examples for which the escaping set has a spider's web structure but the fast escaping set does not. These results give new insight into a conjecture of Baker and a conjecture of Eremenko.