Hyperbolic entire functions with bounded Fatou components
Dynamical Systems
2016-02-11 v2 Complex Variables
Abstract
We show that an invariant Fatou component of a hyperbolic transcendental entire function is a bounded Jordan domain (in fact, a quasidisc) if and only if it contains only finitely many critical points and no asymptotic curves. We use this theorem to prove criteria for the boundedness of Fatou components and local connectivity of Julia sets for hyperbolic entire functions, and give examples that demonstrate that our results are optimal. A particularly strong dichotomy is obtained in the case of a function with precisely two critical values.
Cite
@article{arxiv.1404.0925,
title = {Hyperbolic entire functions with bounded Fatou components},
author = {Walter Bergweiler and Núria Fagella and Lasse Rempe-Gillen},
journal= {arXiv preprint arXiv:1404.0925},
year = {2016}
}
Comments
27 pages, 5 figures. To appear in Commentarii Mathematici Helvetici. V2: Final accepted manuscript (general revision from V1 throughout)