Robert Fryer | Searcharxiv

arXiv2012

Dynamics of mappings with constant dilatation

Let h:C \to C be an R-linear map. In this article, we explore the dynamics of the quasiregular mapping H(z)=h(z)^2. Via the Böttcher type coordinate constructed in "On Böttcher coordinates and quasiregular maps" by Fletcher and Fryer, we are able to obtain results for any degree two mapping of the plane with constant complex dilatation. We show that any such mapping has either one, two or three fixed external rays, that all cases can occur, and exhibit how the dynamics changes in each case. We use results from complex dynamics to prove that these mappings are nowhere uniformly quasiregular in a neighbourhood of infinity. We also show that in most cases, two such mappings are not quasiconformally conjugate on a neighbourhood of infinity.

math.CV↗

arXiv2012

On Böttcher coordinates and quasiregular maps

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.

math.CV↗