Permutable quasiregular maps

Athanasios Tsantaris

doi:10.1017/s0305004121000438

Abstract

AbstractLet f and g be two quasiregular maps in $\mathbb{R}^d$ that are of transcendental type and also satisfy $f\circ g =g \circ f$ . We show that if the fast escaping sets of those functions are contained in their respective Julia sets then those two functions must have the same Julia set. We also obtain the same conclusion about commuting quasimeromorphic functions with infinite backward orbit of infinity. Furthermore we show that permutable quasiregular functions of the form f and $g = \phi \circ f$ , where $\phi$ is a quasiconformal map, have the same Julia sets and that polynomial type quasiregular maps cannot commute with transcendental type ones unless their degree is less than or equal to their dilatation.

Highlights

Introduction and ResultsThe general theory of iteration of holomorphic maps starts from the seminal work of Fatou [16] and Julia [21]
Fatou and Julia initially developed their theory for rational functions and later on Fatou [18] considered iteration of transcendental entire functions
Their proof relies heavily on the properties of the hyperbolic metric under holomorphic maps. Such an approach does not work in higher dimensions. Another interesting question to ask is whether permutable quasiregular maps of polynomial type must have the same Julia set

Summary

Introduction and Results

The general theory of iteration of holomorphic maps starts from the seminal work of Fatou [16] and Julia [21]. Such an approach does not work in higher dimensions Another interesting question to ask is whether permutable quasiregular maps of polynomial type must have the same Julia set. This problem seems harder and is still open. On the other hand if f, g are permutable, uniformly quasiregular maps of polynomial type (see Section 2 for the definition) J ( f ) = J (g) and the proof is almost the same as the one for rational functions in the complex plane which can be found in [3]. In the quasiregular setting we are able to prove the following theorem which can be seen as the analogy to that of Baker and Iyer in higher dimensions.

Background on quasiregular maps and capacity

Examples