On Böttcher 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 Böttcher coordinate for f near infinity. In this paper we construct a Böttcher 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.

to appear in {\it Contemp. Math.}, volume title: "Quasiconformal Mappings, Riemann Surfaces, and Teichmüller Spaces" (in honor of Clifford Earle's 75th birthday)