On multiply connected wandering domains of meromorphic functions

TL;DR

This paper investigates the structure and connectivity properties of wandering domains in transcendental meromorphic functions, establishing conditions under which these domains are Baker wandering domains and exploring their boundary behaviors.

Contribution

It introduces new criteria for multiply connected wandering domains to be Baker wandering domains and analyzes their connectivity and boundary mapping properties.

Findings

Multiply connected wandering domains can be Baker wandering domains.

Boundary components of Fatou components map onto each other.

Examples demonstrate sharpness of the theoretical results.

Abstract

We describe conditions under which a multiply connected wandering domain of a transcendental meromorphic function with a finite number of poles must be a Baker wandering domain, and we discuss the possible eventual connectivity of Fatou components of transcendental meromorphic functions. We also show that if $f$ is meromorphic, $U$ is a bounded component of $F(f)$ and $V$ is the component of $F(f)$ such that $f(U)\subset V$, then $f$ maps each component of $\partial U$ onto a component of the boundary of $V$ in $\hat{\C}$. We give examples which show that our results are sharp; for example, we prove that a multiply connected wandering domain can map to a simply connected wandering domain, and vice versa.