The dynamics of quasiregular maps of punctured space

TL;DR

This paper extends the iteration theory of quasiregular maps to punctured space, defining Julia and Fatou sets with properties similar to classical complex dynamics, and introduces a generalized fast escaping set.

Contribution

It develops a new iteration framework for quasiregular maps of punctured space, defining Julia and Fatou sets with properties akin to classical theory, and generalizes the fast escaping set concept.

Findings

Julia set is non-empty and shares properties with classical Julia sets.

The Julia set equals the boundary of the fast escaping set.

Topological properties of Fatou components are extended.

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.