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Orbifold expansion and entire functions with bounded Fatou components

Part of: Complex dynamical systems Entire and meromorphic functions, and related topics Differential topology

Published online by Cambridge University Press:  04 March 2021

LETICIA PARDO-SIMÓN Open the ORCID record for LETICIA PARDO-SIMÓN [Opens in a new window]
LETICIA PARDO-SIMÓN*
Affiliation:
Institute of Mathematics of the Polish Academy of Sciences, ul. Śniadeckich 8, 00-656Warsaw, Poland
*
e-mail: l.pardo-simon@impan.pl
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Abstract

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Many authors have studied the dynamics of hyperbolic transcendental entire functions; these are functions for which the postsingular set is a compact subset of the Fatou set. Equivalently, they are characterized as being expanding. Mihaljević-Brandt studied a more general class of maps for which finitely many of their postsingular points can be in their Julia set, and showed that these maps are also expanding with respect to a certain orbifold metric. In this paper we generalize these ideas further, and consider a class of maps for which the postsingular set is not even bounded. We are able to prove that these maps are also expanding with respect to a suitable orbifold metric, and use this expansion to draw conclusions on the topology and dynamics of the maps. In particular, we generalize existing results for hyperbolic functions, giving criteria for the boundedness of Fatou components and local connectivity of Julia sets. As part of this study, we develop some novel results on hyperbolic orbifold metrics. These are of independent interest, and may have future applications in holomorphic dynamics.

Keywords

hyperbolic orbifoldentire functionFatou setlocal connectivityEremenko– Lyubich class

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30D05: Functional equations in the complex domain, iteration and composition of analytic functions 57R18: Topology and geometry of orbifolds
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 42 , Issue 5 , May 2022 , pp. 1807 - 1846
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction, provided the original article is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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