Dynamics of mappings with constant dilatation

Abstract

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 B\"{o}ttcher type coordinate constructed in "On B\"{o}ttcher 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.