The dynamics of quasiregular maps of punctured space

Nicks, Daniel; Sixsmith, David J.

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: 0022-2518
• 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