# Strongly automorphic mappings and Julia sets of uniformly quasiregular mappings

## TL;DR

This paper extends classical complex dynamics results to higher dimensions by studying strongly automorphic and uniformly quasiregular mappings, characterizing their Julia sets, automorphy groups, and establishing a Denjoy-Wolff type theorem in three dimensions.

## Contribution

It proves an analogue of Ritt's theorem for quasiregular mappings, characterizes automorphy groups via orbifolds, and generalizes the Denjoy-Wolff theorem to three-dimensional quasiregular mappings.

## Key findings

- Characterization of automorphy groups via crystallographic orbifolds
- Classification of Julia set behaviors for quasiregular mappings
- First generalization of the Denjoy-Wolff theorem in $\

## Abstract

A theorem of Ritt states the a linearizer of a holomorphic function at a repelling fixed point is periodic only if the holomorphic map is conjugate to a power of $z$, a Chebyshev polynomial or a Latt\`es map. The converse, except for some exceptions, is also true. In this paper, we prove the analogous statement in the setting of strongly automorphic quasiregular mappings and uniformly quasiregular mappings in $\mathbb{R}^n$. Along the way, we characterize the possible automorphy groups that can arise via crystallographic orbifolds and a use of the Poincar\'e conjecture. We further give a classification of the behaviour of uniformly quasiregular mappings on their Julia set when the Julia set is a quasisphere, quasidisk or all of $\mathbb{R}^n$ and the Julia set coincides with the set of conical points. Finally, we prove an analogue of the Denjoy-Wolff Theorem for uniformly quasiregular mappings in $\mathbb{B}^3$, the first such generalization of the Denjoy-Wolff Theorem where there is no guarantee of non-expansiveness with respect to a metric.