On B\"{o}ttcher coordinates and quasiregular maps
Complex Variables
2012-05-10 v1 Dynamical Systems
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.
Cite
@article{arxiv.1205.1978,
title = {On B\"{o}ttcher coordinates and quasiregular maps},
author = {Alastair Fletcher and Robert Fryer},
journal= {arXiv preprint arXiv:1205.1978},
year = {2012}
}
Comments
to appear in {\it Contemp. Math.}, volume title: "Quasiconformal Mappings, Riemann Surfaces, and Teichm\"{u}ller Spaces" (in honor of Clifford Earle's 75th birthday)