The dynamics of quasiregular maps of punctured space

Nicks, Daniel A. and Sixsmith, David J. (2017) The dynamics of quasiregular maps of punctured space. Indiana University Mathematics Journal . ISSN 0022-2518 (In Press)

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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
Depositing User: Eprints, Support
Date Deposited: 26 Apr 2017 11:04
Last Modified: 28 Apr 2017 12:39
URI: http://eprints.nottingham.ac.uk/id/eprint/42305

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