On the uniform perfectness of the boundary of multiply connected wandering domains

TL;DR

This paper explores conditions under which the boundary of multiply connected wandering domains of entire functions are uniformly perfect, providing criteria and examples that clarify when this property holds or fails.

Contribution

It introduces a general criterion for non-uniform perfectness of boundaries and presents new examples of infinitely connected wandering domains with uniformly perfect boundaries.

Findings

The boundary of certain multiply connected wandering domains is not uniformly perfect.

A criterion is established to determine when the boundary is not uniformly perfect.

Examples of infinitely connected wandering domains with uniformly perfect boundaries are provided.

Abstract

We investigate in which cases the boundary of a multiply connected wandering domain of an entire function is uniformly perfect. We give a general criterion implying that it is not uniformly perfect. This criterion applies in particular to examples of multiply connected wandering domains given by Baker. We also provide examples of infinitely connected wandering domains whose boundary is uniformly perfect.