The size and topology of quasi-Fatou components of quasiregular maps
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- by Daniel A. Nicks and David J. Sixsmith PDF
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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
- MR Author ID: 862157
- Email: Dan.Nicks@nottingham.ac.uk
- David J. Sixsmith
- Affiliation: School of Mathematical Sciences, University of Nottingham, NG7 2RD, United Kingdom
- MR Author ID: 952973
- Email: David.Sixsmith@open.ac.uk
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 749-763
- MSC (2010): Primary 37F10; Secondary 30C65, 30D05
- DOI: https://doi.org/10.1090/proc/13253
- MathSciNet review: 3577875