Interpolating quasiregular power mappings

TL;DR

This paper constructs a novel quasiregular mapping in three-dimensional space demonstrating unique dynamical properties, including wandering components in the quasi-Fatou set and spherical Julia set components, using a new interpolation technique.

Contribution

It introduces a new quasiregular interpolation method in $\,\mathbb{R}^3$ and applies it to create mappings with diverse growth behaviors and complex dynamical features.

Findings

Quasi-Fatou set contains wandering components.

Julia set includes genuine round spheres.

Mappings can be constructed with arbitrary growth rates.

Abstract

We construct a quasiregular mapping in $\mathbb{R}^3$ that is the first to illustrate several important dynamical properties: the quasi-Fatou set contains wandering components; these quasi-Fatou components are bounded and hollow; and the Julia set has components that are genuine round spheres. The key tool in this construction is a new quasiregular interpolation in round rings in $\mathbb{R}^3$ between power mappings of differing degrees on the boundary components. We also exhibit the flexibility of constructions based on these interpolations by showing that we may obtain quasiregular mappings which grow as quickly, or as slowly, as desired.