On B\"{o}ttcher coordinates and quasiregular maps

TL;DR

This paper extends the concept of Böttcher coordinates to compositions of affine maps and polynomials, and demonstrates that certain quadratic polynomials are not uniformly quasiregular, advancing understanding in quasiregular dynamics.

Contribution

It constructs a Böttcher type coordinate for a new class of quasiregular maps formed by affine and polynomial compositions.

Findings

Constructed Böttcher type coordinate for affine-polynomial compositions

Proved quadratic polynomial h(z)^2 + c is not uniformly quasiregular

Extended Böttcher coordinate theory to quasiregular mappings

Abstract

It is well-known that a polynomial f(z)=a_d z^d(1+o(1)) can be conjugated by a holomorphic map phi to w \mapsto w^d in a neighbourhood of infinity. This map phi is called a B\"ottcher coordinate for f near infinity. In this paper we construct a B\"ottcher type coordinate for compositions of affine mappings and polynomials, a class of mappings first studied in "Quasiregular mappings of polynomial type in R^2" by A.Fletcher and D.Goodman. As an application, we prove that if h is affine and c is a complex number, then h(z)^2+c is not uniformly quasiregular.