On the iteration of quasimeromorphic mappings

TL;DR

This paper extends the iterative theory of quasimeromorphic mappings to include those with essential singularities and poles, constructing their Julia sets and analyzing their properties in higher dimensions.

Contribution

It introduces a new framework for Julia sets of quasimeromorphic mappings with essential singularities and poles, expanding the scope of complex dynamics in higher dimensions.

Findings

Julia sets share properties with those of transcendental meromorphic functions.

The theory applies to mappings with essential singularities at infinity.

Constructs Julia sets for a broader class of quasimeromorphic mappings.

Abstract

The Fatou-Julia theory for rational functions has been extended both to transcendental meromorphic functions and more recently to several different types of quasiregular mappings in higher dimensions. We extend the iterative theory to quasimeromorphic mappings with an essential singularity at infinity and at least one pole, constructing the Julia set for these maps. We show that this Julia set shares many properties with those for transcendental meromorphic functions and for quasiregular mappings of punctured space.