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) subset of V, then f maps each component of partial derivative U onto a component of the boundary of V in (C) over cap. 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.