Spiders' webs in the punctured plane

TL;DR

This paper extends the concept of spider's webs to the punctured plane, characterizes their properties, and explores their occurrence in transcendental self-maps of * with implications for Julia and escaping sets.

Contribution

It introduces a new adaptation of spider's webs to the punctured plane and analyzes their connection with transcendental dynamics in *.

Findings

Existence of transcendental self-maps of * with Julia sets as spider's webs

Construction of a self-map with an escaping set that is a spider's web

Conjecture that fast escaping sets do not form spider's webs in *

Abstract

Many authors have studied sets, associated with the dynamics of a transcendental entire function, which have the topological property of being a spider's web. In this paper we adapt the definition of a spider's web to the punctured plane. We give several characterisations of this topological structure, and study the connection with the usual spider's web in $\mathbb{C}$. We show that there are many transcendental self-maps of $\mathbb{C}^*$ for which the Julia set is such a spider's web, and we construct a transcendental self-map of $\mathbb{C}^*$ for which the escaping set $I(f)$ has this structure and hence is connected. By way of contrast with transcendental entire functions, we conjecture that there is no transcendental self-map of $\mathbb{C}^*$ for which the fast escaping set $A(f)$ is such a spider's web.