On the geometry of simply connected wandering domains

TL;DR

This paper explores the geometric properties of wandering domains in entire functions, demonstrating that any suitable bounded domain can serve as such a domain, including those with Jordan curve boundaries.

Contribution

It proves that all bounded, connected, regular open sets with connected complements can be realized as wandering domains of entire functions, including escaping and oscillating types.

Findings

Any bounded connected regular open set with connected complement can be a wandering domain.

Every Jordan curve can be the boundary of a wandering Fatou component.

Constructs entire functions with prescribed wandering domains.

Abstract

We study the geometry of simply connected wandering domains for entire functions and we prove that every bounded connected regular open set, whose closure has a connected complement, is a wandering domain of some entire function. In particular such domain can be realized as an escaping or an oscillating wandering domain. As a consequence we obtain that every Jordan curve is the boundary of a wandering Fatou component of some entire function.