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

Nicks, Daniel A.; Sixsmith, David J.

Authors

David J. Sixsmith



Abstract

We consider the iteration of quasiregular maps of transcendental type from Rd to Rd. 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 com- plementary 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.

Journal Article Type Article
Publication Date Jan 1, 2017
Journal Proceedings of the American Mathematical Society
Print ISSN 0002-9939
Electronic ISSN 1088-6826
Publisher American Mathematical Society
Peer Reviewed Peer Reviewed
Volume 145
Issue 2
APA6 Citation Nicks, D. A., & Sixsmith, D. J. (2017). The size and topology of quasi-Fatou components of quasiregular maps. Proceedings of the American Mathematical Society, 145(2), https://doi.org/10.1090/proc/13253
DOI https://doi.org/10.1090/proc/13253
Publisher URL http://www.ams.org/journals/proc/2017-145-02/S0002-9939-2016-13253-X/
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf
Additional Information First published in Proceedings of the American Mathematical Society in volume 145, no. 2, published by the American Mathematical Society.

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Copyright Statement
Copyright information regarding this work can be found at the following address: http://eprints.nottingham.ac.uk/end_user_agreement.pdf





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