Hyperbolic entire functions with bounded Fatou components

  • Walter Bergweiler

    Christian-Albrechts-Universität zu Kiel, Germany
  • Núria Fagella

    Universitat de Barcelona, Spain
  • Lasse Rempe-Gillen

    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.

Cite this article

Walter Bergweiler, Núria Fagella, Lasse Rempe-Gillen, Hyperbolic entire functions with bounded Fatou components. Comment. Math. Helv. 90 (2015), no. 4, pp. 799–829

DOI 10.4171/CMH/371