Periodic domains of quasiregular maps

TL;DR

This paper studies the dynamics of transcendental quasiregular maps in higher dimensions, establishing bounds on escape rates in periodic domains, and constructing novel examples of such maps with unique iterative behaviors.

Contribution

It provides the first example of a transcendental quasiregular map in three dimensions with a periodic domain where all iterates tend to infinity, and improves bounds on escape rates in periodic components.

Findings

Bound on escape rates in periodic components of quasiregular maps.

Construction of a 3D quasiregular map with all iterates tending to infinity in a periodic domain.

Existence of a quasiregular map equal to the identity in a half-space.

Abstract

We consider the iteration of quasiregular maps of transcendental type from $\mathbb{R}^d$ to $\mathbb{R}^d$. We give a bound on the rate at which the iterates of such a map can escape to infinity in a periodic component of the quasi-Fatou set. We give examples which show that this result is best possible. Under an additional hypothesis, which is satisfied by all uniformly quasiregular maps, this bound can be improved to be the same as those in a Baker domain of a transcendental entire function. We construct a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ with a periodic domain in which all iterates tend locally uniformly to infinity. This is the first example of such behaviour in a dimension greater than two. Our construction uses a general result regarding the extension of biLipschitz maps. In addition, we show that there is a quasiregular map of transcendental type from $\mathbb{R}^3$ to $\mathbb{R}^3$ which is equal to the identity map in a half-space.