A Unified Approach to Quasiregular Linearization in the Plane

TL;DR

This paper extends classical complex dynamics theorems to quasiregular mappings in the plane using a unified approach that does not depend on local injectivity, employing logarithmic transforms and specific restrictions.

Contribution

It generalizes K"onig's and B"ottcher's theorems to quasiregular maps without relying on local injectivity, introducing a unified method via logarithmic transforms.

Findings

Unified approach to quasiregular linearization

Extension of classical theorems to broader class of maps

Conditions for linearizability in quasiregular dynamics

Abstract

We generalize the classical K\"onig's and B\"ottcher's Theorems in complex dynamics to certain quasiregular mappings in the plane. Our approach to these results is unified in the sense that it does not depend on the local injectivity, or not, of the map at the fixed point. By passing to the logarithmic transform we obtain a quasiconformal mapping in either case. Certain restrictions on the quasiregular mappings are needed in order for there to be a candidate to linearize to. These are provided by requiring a simple infinitesimal space of the mapping at the fixed point and restricting to the BIP mappings introduced by the authors in prior work.