On multiply connected wandering domains of meromorphic functions

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.