Foundations for an iteration theory of entire quasiregular maps

TL;DR

This paper extends the iteration theory of transcendental entire functions to quasiregular maps in higher dimensions, defining Julia sets with capacity considerations and establishing their properties for non-polynomial type maps.

Contribution

It introduces an analogous iteration theory for transcendental quasiregular maps, defining Julia sets via capacity and proving their non-emptiness and properties.

Findings

Julia set is non-empty for non-polynomial type maps

Julia set has properties similar to classical transcendental entire functions

Defines Julia set using capacity of complement of forward orbits

Abstract

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.