On multiply connected wandering domains of meromorphic functions

Rippon, P. J. and Stallard, G. M. (2008). On multiply connected wandering domains of meromorphic functions. Journal of the London Mathematical Society, 77(2) pp. 405–423.

DOI: https://doi.org/10.1112/jlms/jdm118

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) is contained in V, then f maps each component of the boundary of U onto a component of the boundary of V in 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.

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