Dynamics of mappings with constant dilatation

TL;DR

This paper investigates the dynamics of quadratic quasiregular maps with constant dilatation, revealing the number of fixed external rays, their dynamic behavior, and the non-conjugacy of most such mappings near infinity.

Contribution

It extends complex dynamics techniques to quadratic quasiregular maps with constant dilatation, classifying fixed external rays and analyzing conjugacy properties.

Findings

Mappings have 1, 2, or 3 fixed external rays.

All cases of fixed external rays occur.

Mappings are nowhere uniformly quasiregular near infinity.

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.