Bergweiler, Walter; Rippon, Philip and Stallard, Gwyneth
Multiply connected wandering domains of entire functions.
Proceedings of the London Mathematical Society
The dynamical behaviour of a transcendental entire function in any periodic component of the Fatou set is well understood. Here we study the dynamical behaviour of a transcendental entire function in any multiply connected wandering domain of . By introducing a certain positive harmonic function in , related to harmonic measure, we are able to give the first detailed description of this dynamical behaviour. Using this new technique, we show that, for sufficiently large , the image domains contain large annuli, , and that the union of these annuli acts as an absorbing set for the iterates of in . Moreover, behaves like a monomial within each of these annuli and the orbits of points in settle in the long term at particular `levels' within the annuli, determined by the function . We also discuss the proximity of and for large , and the connectivity properties of the components of . These properties are deduced from new results about the behaviour of entire functions that omit certain values in an annulus.
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