Quasiregular semigroups with examples

TL;DR

This paper explores the properties of Julia and Fatou sets in quasiregular semigroups, extending the theory of rational semigroups to higher dimensions and providing illustrative examples of their diverse behaviors.

Contribution

It introduces the study of Julia and Fatou sets in quasiregular semigroups and offers new examples demonstrating various possible dynamics.

Findings

Analysis of Julia and Fatou sets in quasiregular semigroups

Construction of examples illustrating diverse behaviors

Extension of rational semigroup theory to higher dimensions

Abstract

Rational semigroups were introduced by Hinkkanen and Martin as a generalization of the iteration of a single rational map. There has subsequently been much interest in the study of rational semigroups. Quasiregular semigroups were introduced shortly after rational semigroups as analogues in higher real dimensions, but have received far less attention. Each map in a quasiregular semigroup must necessarily be a uniformly quasiregular map. While there is a completely viable theory for the iteration of uniformly quasiregular maps, it is a highly non-trivial matter to construct them. In this paper, we study properties of the Julia and Fatou sets of quasiregular semigroups and, equally as importantly, give several families of examples illustrating some of the behaviours that can arise.