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

Sixsmith, Dave (2015). Maximally and non-maximally fast escaping points of transcendental entire functions. Mathematical Proceedings of the Cambridge Philosophical Society, 158(2) pp. 365–383.

DOI: https://doi.org/10.1017/S0305004115000018

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 ntersection 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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