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

Nicks, Daniel A.; Sixsmith, David J.

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.

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), 749-763. https://doi.org/10.1090/proc/13253

• Journal Article Type: Article
• Acceptance Date: Apr 20, 2016
• Online Publication Date: Aug 21, 2016
• Publication Date: Jan 1, 2017
• Deposit Date: Jul 19, 2016
• Publicly Available Date: Aug 21, 2016
• 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
• Pages: 749-763
• DOI: https://doi.org/10.1090/proc/13253
• Public URL: https://nottingham-repository.worktribe.com/output/830878
• Publisher URL: http://www.ams.org/journals/proc/2017-145-02/S0002-9939-2016-13253-X/
• Additional Information: First published in Proceedings of the American Mathematical Society in volume 145, no. 2, published by the American Mathematical Society.
• Contract Date: Jul 19, 2016