On Böttcher coordinates and quasiregular maps

Abstract

Abstract. It is well-known that a polynomial f (z) = a d z d (1 + o(1)) can be conjugated by a holomorphic map φ to w → w d in a neighbourhood of infinity. This map φ 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 [9]. As an application, we prove that if h is affine and c ∈ C, then h(z) 2 + c is not uniformly quasiregular.MSC 2010: 30C65 (Primary), 30D05, 37F10, 37F45 (Secondary).

Superattracting fixed points of quasiregular mappings

“…(ii) In [7], Böttcher coordinates were constructed in a neighbourhood of infinity for degree two quasiregular mappings of the plane with constant complex dilatation. When the dilatation is less than 2 in these examples, infinity is a strongly superattracting fixed point, and convergence to infinity is uniform on a neighbourhood of infinity.…”

Fixed curves near fixed points

“…This is the logarithmic transform of a suitably chosen branch of H −1 • ψ k • f , for some mapping ψ k whose logarithmic transform is φ k . Arguing as in [12], it can be shown that φ k (w) = w + o(1), as Re(w) → −∞,…”

“…This allows the results from the rest of the paper to be applied locally. See [12] for the degree 2 case of such a Böttcher coordinate.…”

“…To do this, we need to make use of a Böttcher coordinate for such a situation. Such coordinates were constructed when the fixed point has local index 2 in [12]. The method employed there works for any local degree.…”

“…The details are rather involved, but follow the same theme (with minor changes) as the proof of the degree 2 case in [12]. For the convenience of the reader, we will sketch a proof.…”