An analogue of Green's Functions for Quasiregular Maps

TL;DR

This paper constructs an analogue of Green's functions for certain quasiregular maps, enabling analysis of their escaping sets and revealing unique dynamical behaviors not seen in polynomial dynamics.

Contribution

It introduces a new type of Green's function for degree-two planar quasiregular maps with constant dilatation, linking them to the real squaring map.

Findings

Green's functions semi-conjugate quasiregular maps to the real squaring map

Boundary properties of escaping sets are analyzed using these functions

Examples show behaviors absent in quadratic polynomial dynamics

Abstract

Green's functions are highly useful in analyzing the dynamical behavior of polynomials in their escaping set. The aim of this paper is to construct an analogue of Green's functions for planar quasiregular mappings of degree two and constant complex dilatation. These Green's functions are dynamically natural, in that they semi-conjugate our quasiregular mappings to the real squaring map. However, they do not share the same regularity properties as Green's functions of polynomials. We use these Green's functions to investigate properties of the boundary of the escaping set and give several examples to illustrate behavior that does not occur for the dynamics of quadratic polynomials.