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arXiv · 1403.7362

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

D. J. Sixsmith
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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.

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D. J. Sixsmith. 2014-03-28. Maximally and non-maximally fast escaping points of transcendental entire functions. https://doi.org/10.1017/s0305004115000018

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