On Multiply Connected Wandering Domains of Meromorphic Functions

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) ⊂ V , then f maps each component of ∂U onto a component of the boundary of V in Ĉ. 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.