On multiply connected wandering domains of meromorphic functions

We describe conditions under which a multiply connected wanderingdomain of a transcendental meromorphic function with a finitenumber of poles must be a Baker wandering domain, and we discussthe possible eventual connectivity of Fatou components of transcendentalmeromorphic 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 componentof the boundary of V in . We give examples which show that our results are sharp; for example,we prove that a multiply connected wandering domain can mapto a simply connected wandering domain, and vice versa.