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Iterates of meromorphic functions on escaping Fatou components

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

Published online by Cambridge University Press:  22 November 2022

Jian-Hua Zheng  and
Cheng-Fa Wu Open the ORCID record for Cheng-Fa Wu [Opens in a new window]
Jian-Hua Zheng
Affiliation:
Department of Mathematical Sciences, Tsinghua University, Beijing 100084, P. R. China (zheng-jh@mail.tsinghua.edu.cn)
Cheng-Fa Wu*
Affiliation:
Institute for Advanced Study, Shenzhen University, Shenzhen 518060, P. R. China College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, P. R. China (cfwu@szu.edu.cn)
*
*Corresponding author.
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Abstract

In this paper, we prove that the ratio of the modulus of the iterates of two points in an escaping Fatou component could be bounded even if the orbit of the component contains a sequence of annuli whose moduli tend to infinity, and this cannot happen when the maximal modulus of the meromorphic function is uniformly large enough. In this way we extend certain related results for entire functions to meromorphic functions with infinitely many poles.

Keywords

Escaping setsmeromorphic functionsFatou setJulia set

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
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 153 , Issue 6 , December 2023 , pp. 1906 - 1928
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Copyright
Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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