Commentarii Mathematici Helvetici


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Volume 90, Issue 4, 2015, pp. 799–829
DOI: 10.4171/CMH/371

Published online: 2015-12-03

Hyperbolic entire functions with bounded Fatou components

Walter Bergweiler[1], Núria Fagella[2] and Lasse Rempe-Gillen[3]

(1) Christian-Albrechts-Universität zu Kiel, Germany
(2) Universitat de Barcelona, Spain
(3) University of Liverpool, UK

We show that an invariant Fatou component of a hyperbolic transcendental entire function is a 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.

Keywords: Fatou set, Julia set, transcendental entire function, hyperbolicity, Axiom A, bounded Fatou component, quasidisc, quasicircle, Jordan curve, local connectivity, Laguerre–Pólya class, Eremenko–Lyubich class

Bergweiler Walter, Fagella Núria, Rempe-Gillen Lasse: Hyperbolic entire functions with bounded Fatou components. Comment. Math. Helv. 90 (2015), 799-829. doi: 10.4171/CMH/371