Dr Daniel Nicks Dan.Nicks@nottingham.ac.uk
LECTURER
The dynamics of quasiregular maps of punctured space
Nicks, Daniel; Sixsmith, David J.
Authors
David J. Sixsmith
Abstract
The Fatou-Julia iteration theory of rational and transcendental entire functions has recently been extended to quasiregular maps in more than two real dimensions. Our goal in this paper is similar; we extend the iteration theory of analytic self-maps of the punctured plane to quasiregular self-maps of punctured space. We define the Julia set as the set of points for which the complement of the forward orbit of any neighbourhood of the point is a finite set. We show that the Julia set is non-empty, and shares many properties with the classical Julia set of an analytic function. These properties are stronger than those known to hold for the Julia set of a general quasiregular map of space. We define the quasi-Fatou set as the complement of the Julia set, and generalise a result of Baker concerning the topological properties of the components of this set. A key tool in the proof of these results is a version of the fast escaping set. We generalise various results of Marti-Pete concerning this set, for example showing that the Julia set is equal to the boundary of the fast escaping set.
Citation
Nicks, D., & Sixsmith, D. J. (2019). The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal, 68(1), 323-352. https://doi.org/10.1512/iumj.2019.68.7556
Journal Article Type | Article |
---|---|
Acceptance Date | Mar 9, 2017 |
Publication Date | 2019 |
Deposit Date | Apr 26, 2017 |
Journal | Indiana University Mathematics Journal |
Print ISSN | 0022-2518 |
Electronic ISSN | 1943-5258 |
Publisher | Indiana University Mathematics Journal |
Peer Reviewed | Peer Reviewed |
Volume | 68 |
Issue | 1 |
Pages | 323-352 |
DOI | https://doi.org/10.1512/iumj.2019.68.7556 |
Keywords | General Mathematics |
Public URL | https://nottingham-repository.worktribe.com/output/849441 |
Contract Date | Apr 26, 2017 |
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