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On slow escaping and non-escaping points of quasimeromorphic mappings

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

Published online by Cambridge University Press:  22 January 2020

LUKE WARREN Open the ORCID record for LUKE WARREN [Opens in a new window]
LUKE WARREN*
Affiliation:
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK email [email protected]
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Abstract

We show that for any quasimeromorphic mapping with an essential singularity at infinity, there exist points whose iterates tend to infinity arbitrarily slowly. This extends a result by Nicks for quasiregular mappings, and Rippon and Stallard for transcendental meromorphic functions on the complex plane. We further establish a new result for the growth rate of quasiregular mappings near an essential singularity, and briefly extend some results regarding the bounded orbit set and the bungee set to the quasimeromorphic setting.

Keywords

quasiregular mappingsquasimeromorphic mappingsescaping setbungee setslow escape

MSC classification

Primary: 37F10: Polynomials; rational maps; entire and meromorphic functions
Secondary: 30C65: Quasiconformal mappings in $^n$, other generalizations 30D05: Functional equations in the complex domain, iteration and composition of analytic functions
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 4 , April 2021 , pp. 1190 - 1216
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Copyright
© Cambridge University Press, 2020

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