Research Repository

See what's under the surface

The dynamics of quasiregular maps of punctured space

Nicks, Daniel A.; Sixsmith, David J.

Authors

Daniel A. Nicks

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.

Journal Article Type Article
Publication Date 2019
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
Institution Citation Nicks, D. A., & Sixsmith, D. J. (2019). The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal, 68(1), 323–352. doi:10.1512/iumj.2019.68.7556
DOI https://doi.org/10.1512/iumj.2019.68.7556
Copyright Statement Copyright information regarding this work can be found at the following address: http://eprints.nottingh.../end_user_agreement.pdf

Downloadable Citations