On multiply connected wandering domains of meromorphic functions

Details On multiply connected wandering domains of meromorphic functions On multiply connected wandering domains of meromorphic functions

2007-11-16

arxiv.org/abs/0711.2582v1

Mathematics

Complex Variables

18 pages. To be published in the Journal of the London Mathematical Society

Scientific paper

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.

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