On quasiregular linearizers

TL;DR

This paper explores the concept of linearization in quasiregular dynamics, showing how different linearizers related to the same mapping and fixed point are interconnected, especially those solving Schr"oder equations exhibiting automorphic properties.

Contribution

It introduces the relationship between linearizers of uniformly quasiregular mappings and their automorphic nature, extending classical linearization concepts into the quasiregular setting.

Findings

Linearizers from the same uqr mapping are related.

Linearizers solving Schr"oder equations are automorphic.

The study extends classical complex dynamics to quasiregular mappings.

Abstract

Linearization is a well-known concept in complex dynamics. If $p$ is a polynomial and $z_0$ is a repelling fixed point, then there is an entire function $L$ which conjugates $p$ to the linear map $z\mapsto p'(z_0)z$. This notion of linearization carries over into the quasiregular setting, in the context of repelling fixed points of uniformly quasiregular mappings. In this article, we investigate how linearizers arising from the same uqr mapping and the same repelling fixed point are related. In particular, any linearizer arising from a uqr solution to a Schr\"oder equation is shown to be automorphic with respect to some quasiconformal group.