Hyperbolic entire functions with bounded Fatou components

Bergweiler, Walter, Fagella, Núria and Rempe-Gillen, Lasse ORCID: 0000-0001-8032-8580 (2015) Hyperbolic entire functions with bounded Fatou components. Commentarii Mathematici Helvetici, 90 (4). pp. 799-823.

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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.

Item Type:	Article
Additional Information:	27 pages, 5 figures. To appear in Commentarii Mathematici Helvetici. V2: Final accepted manuscript (general revision from V1 throughout)
Uncontrolled Keywords:	Faou set, Julia set, transcendental entire function, hyperbolicity, Axiom A, bounded Fatou component, quasidisc, quasicircle, Jordan curve, local connectivity, Laguerre–Pólya class, remenko–Lyubich class
Subjects:	?? QA ??
Depositing User:	Symplectic Admin
Date Deposited:	06 Jul 2015 08:05
Last Modified:	26 Feb 2024 11:44
DOI:	10.4171/CMH/371
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2015681