Maximally and non-maximally fast escaping points of transcendental entire functions
Dynamical Systems
2019-02-20 v1 Complex Variables
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.
Keywords
Cite
@article{arxiv.1403.7362,
title = {Maximally and non-maximally fast escaping points of transcendental entire functions},
author = {D. J. Sixsmith},
journal= {arXiv preprint arXiv:1403.7362},
year = {2019}
}