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Multiply connected wandering domains of entire functions

Bergweiler, Walter; Rippon, Philip and Stallard, Gwyneth (2013). Multiply connected wandering domains of entire functions. Proceedings of the London Mathematical Society, 107(6) pp. 1261–1301.

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DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1112/plms/pdt010
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Abstract

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 $f$ in any multiply connected wandering domain $U$ of $f$. By introducing a certain positive harmonic function $h$ in $U$, 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 $n$, the image domains $U_n=f^n(U)$ contain large annuli, $C_n$, and that the union of these annuli acts as an absorbing set for the iterates of $f$ in $U$. Moreover, $f$ behaves like a monomial within each of these annuli and the orbits of points in $U$ settle in the long term at particular `levels' within the annuli, determined by the function $h$. We also discuss the proximity of $\partial U_n$ and $\partial C_n$ for large $n$, and the connectivity properties of the components of $U_n \setminus \overline{C_n}$. These properties are deduced from new results about the behaviour of entire functions that omit certain values in an annulus.

Item Type: Journal Article
Copyright Holders: 2013 London Mathematical Society
ISSN: 1460-244X
Project Funding Details:
Funded Project NameProject IDFunding Body
Baker's conjecture and Eremenko's conjecture: a unified approach.EP/H006591/1EPSRC
Not SetNot SetDFG
Not SetNot SetESF
Academic Unit/Department: Mathematics, Computing and Technology > Mathematics and Statistics
Item ID: 35313
Depositing User: Philip Rippon
Date Deposited: 02 Nov 2012 09:06
Last Modified: 13 Feb 2014 16:54
URI: http://oro.open.ac.uk/id/eprint/35313
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