Permutable Quasiregular Maps

TL;DR

This paper investigates conditions under which permutable quasiregular maps share the same Julia set, revealing new relationships between their dynamics, especially for transcendental and quasimeromorphic functions.

Contribution

It establishes that commuting quasiregular maps with certain properties have identical Julia sets and explores restrictions on commuting polynomial and transcendental quasiregular maps.

Findings

Permutable quasiregular maps of transcendental type share Julia sets if their fast escaping sets are in their Julia sets.

Commuting quasimeromorphic functions with infinite backward orbit of infinity have the same Julia set.

Polynomial and transcendental quasiregular maps generally do not commute unless their degree is at most their dilatation.

Abstract

Let $f$ and $g$ be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$. We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form $f$ and $g=\phi\circ f$, where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.