Fast escaping points of entire functions

Rippon, Philip and Stallard, Gwyneth (2012). Fast escaping points of entire functions. Proceedings of the London Mathematical Society (In press).

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DOI (Digital Object Identifier) Link:	http://dx.doi.org/doi:10.1112/plms/pds001
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Abstract

Let $f$ be a transcendental entire function and let $A(f)$ denote the set of points that escape to infinity `as fast as possible' under iteration. By writing $A(f)$ as a countable union of closed sets, called `levels' of $A(f)$ , we obtain a new understanding of the structure of this set. For example, we show that if $U$ is a Fatou component in $A(f)$ , then $\partial U\subset A(f)$ and this leads to significant new results and considerable improvements to existing results about $A(f)$ . In particular, we study functions for which $A(f)$ , and each of its levels, has the structure of an `infinite spider's web'. We show that there are many such functions and that they have a number of strong dynamical properties. This new structure provides an unexpected connection between a conjecture of Baker concerning the components of the Fatou set and a conjecture of Eremenko concerning the components of the escaping set.

Item Type:	Journal Article
Copyright Holders:	2012 London Mathematical Society
ISSN:	1460-244X
Funders:	EPSRC
Academic Unit/Department:	Mathematics, Computing and Technology > Mathematics and Statistics
Related URLs:	http://arxiv.org/abs/1009.5081(Other) http://plms.oxfordjournals.org/(Other)
Item ID:	31138
Depositing User:	Philip Rippon
Date Deposited:	17 Jan 2012 12:01
Last Modified:	24 Jan 2013 12:27
URI:	http://oro.open.ac.uk/id/eprint/31138

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