The dynamics of quasiregular maps of punctured space

Nicks, Daniel A. and Sixsmith, David J. (2019) The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal, 68 (1). pp. 323-352. ISSN 0022-2518

[img] PDF - Repository staff only - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (385kB)

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.

Item Type: Article
Schools/Departments: University of Nottingham, UK > Faculty of Science > School of Mathematical Sciences
Identification Number: https://doi.org/10.1512/iumj.2019.68.7556
Depositing User: Eprints, Support
Date Deposited: 26 Apr 2017 11:04
Last Modified: 11 Apr 2019 14:16
URI: http://eprints.nottingham.ac.uk/id/eprint/42305

Actions (Archive Staff Only)

Edit View Edit View