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The size and topology of quasi-Fatou components of quasiregular maps


Authors: Daniel A. Nicks and David J. Sixsmith
Journal: Proc. Amer. Math. Soc. 145 (2017), 749-763
MSC (2010): Primary 37F10; Secondary 30C65, 30D05
DOI: https://doi.org/10.1090/proc/13253
Published electronically: August 22, 2016
MathSciNet review: 3577875
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Abstract: We consider the iteration of quasiregular maps of transcendental type from $ \mathbb{R}^d$ to $ \mathbb{R}^d$. In particular we study quasi-Fatou components, which are defined as the connected components of the complement of the Julia set.

Many authors have studied the components of the Fatou set of a transcendental entire function, and our goal in this paper is to generalise some of these results to quasi-Fatou components. First, we study the number of complementary components of quasi-Fatou components, generalising, and slightly strengthening, a result of Kisaka and Shishikura. Second, we study the size of quasi-Fatou components that are bounded and have a bounded complementary component. We obtain results analogous to those of Zheng, and of Bergweiler, Rippon and Stallard. These are obtained using techniques which may be of interest even in the case of transcendental entire functions.


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Additional Information

Daniel A. Nicks
Affiliation: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom
Email: Dan.Nicks@nottingham.ac.uk

David J. Sixsmith
Affiliation: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom
Email: David.Sixsmith@open.ac.uk

DOI: https://doi.org/10.1090/proc/13253
Received by editor(s): January 13, 2016
Received by editor(s) in revised form: March 17, 2016, and April 18, 2016
Published electronically: August 22, 2016
Additional Notes: Both authors were supported by Engineering and Physical Sciences Research Council grant EP/L019841/1.
Communicated by: Jeremy Tyson
Article copyright: © Copyright 2016 American Mathematical Society