Julia Components of Transcendental Entire Functions with Multiply-Connected Wandering Domains

Abstract

We investigate some topological properties of Julia components, that is, connected components of the Julia set, of a transcendental entire function f with a multiply-connected wandering domain. If C is a Julia component with a bounded orbit, then we show that there exists a polynomial P such that C is homeomorphic to a Julia component of the Julia set of P. Furthermore if C is wandering, then C is a buried singleton component. Also we show that under some dynamical conditions, every such C is full and a buried component. The key for our proof is to show that some iterate of f can be regarded as a polynomial-like map on a suitable arbitrarily large bounded topological disk. As an application of this result, we show that a transcendental entire function having a wandering domain with a bounded orbit cannot have multiply-connected wandering domains.