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Fatou components and singularities of meromorphic functions

Published online by Cambridge University Press:  23 January 2019

Krzysztof Barański
Affiliation:
Institute of Mathematics, University of Warsaw, ul. Banacha 2, 02-097Warszawa, Poland (baranski@mimuw.edu.pl)
Núria Fagella
Affiliation:
Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB) and Barcelona Graduate School of Mathematics (BGSMath). Gran Via 585, 08007Barcelona, Catalonia, Spain (nfagella@ub.edu; xavier.jarque@ub.edu)
Xavier Jarque
Affiliation:
Departament de Matemàtiques i Informàtica, Institut de Matemàtiques de la Universitat de Barcelona (IMUB) and Barcelona Graduate School of Mathematics (BGSMath). Gran Via 585, 08007Barcelona, Catalonia, Spain (nfagella@ub.edu; xavier.jarque@ub.edu)
Bogusława Karpińska
Affiliation:
Faculty of Mathematics and Information Science, Warsaw University of Technology, ul. Koszykowa 75, 00-662Warszawa, Poland (bkarpin@mini.pw.edu.pl)

Abstract

We prove several results concerning the relative position of points in the postsingular set P(f) of a meromorphic map f and the boundary of a Baker domain or the successive iterates of a wandering component. For Baker domains we answer a question of Mihaljević-Brandt and Rempe-Gillen. For wandering domains we show that if the iterates Un of such a domain have uniformly bounded diameter, then there exists a sequence of postsingular values pn such that ${\rm dist} (p_n, U_n)\to 0$ as $n\to \infty $ . We also prove that if $U_n \cap P(f)=\emptyset $ and the postsingular set of f lies at a positive distance from the Julia set (in ℂ), then the sequence of iterates of any wandering domain must contain arbitrarily large disks. This allows to exclude the existence of wandering domains for some meromorphic maps with infinitely many poles and unbounded set of singular values.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2019

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