Multiply connected wandering domains of meromorphic functions: the pursuit of uniform internal dynamics

TL;DR

This paper investigates the internal dynamics of multiply connected wandering domains of meromorphic functions, demonstrating that uniform behavior can extend across the entire domain and providing the first example of a semi-contracting infinitely connected wandering domain.

Contribution

It proves that uniform internal dynamics in open subsets extend to entire multiply connected wandering domains and constructs the first example of a semi-contracting infinitely connected wandering domain.

Findings

Uniform dynamics in open subsets extend to whole domains.

Existence of semi-contracting infinitely connected wandering domains.

Insight into inhomogeneity of multiply connected wandering domains.

Abstract

Recently, Benini et al. showed that, in simply connected wandering domains of entire functions, all pairs of orbits behave in the same way relative to the hyperbolic metric, thus giving us our first insight into the general internal dynamics of such domains. After proving in a recent manuscript that the same is not true for multiply connected wandering domains, a natural question is: how inhomogeneous can multiply connected wandering domains be? We give an answer to this question, in that we show that uniform dynamics inside an open subset of the domain generalises to the whole wandering domain. As an application of this result, we construct the first example of a meromorphic function with a semi-contracting infinitely connected wandering domain.