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Maximally and non-maximally fast escaping points of transcendental entire functions

Published online by Cambridge University Press:  28 January 2015

D. J. SIXSMITH
D. J. SIXSMITH*
Affiliation:
Department of Mathematics and Statistics, The Open University, Walton Hall, Milton Keynes MK7 6AA. e-mail: david.sixsmith@open.ac.uk
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Abstract

We partition the fast escaping set of a transcendental entire function into two subsets, the maximally fast escaping set and the non-maximally fast escaping set. These sets are shown to have strong dynamical properties. We show that the intersection of the Julia set with the non-maximally fast escaping set is never empty. The proof uses a new covering result for annuli, which is of wider interest.

It was shown by Rippon and Stallard that the fast escaping set has no bounded components. In contrast, by studying a function considered by Hardy, we give an example of a transcendental entire function for which the maximally and non-maximally fast escaping sets each have uncountably many singleton components.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 158 , Issue 2 , March 2015 , pp. 365 - 383
Copyright
Copyright © Cambridge Philosophical Society 2015 

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