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

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.