Spiders' webs in the punctured plane
Vasiliki Evdoridou, David Mart\'i-Pete, David J. Sixsmith

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 . We show that there are many transcendental self-maps of for which the Julia set is such a spider's web, and we construct a transcendental self-map of for which the escaping set 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 for which the fast escaping set is such a spider's web.
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Taxonomy
TopicsMathematical Dynamics and Fractals · Advanced Differential Equations and Dynamical Systems · Meromorphic and Entire Functions
