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Interpolating quasiregular power mappings

Jack Burkart, Alastair N. Fletcher, Daniel A. Nicks

TL;DR

This work constructs a transcendental-type quasiregular map in $\mathbb{R}^3$ whose quasi-Fatou set contains bounded hollow components and whose Julia set features spherical components, achieved through a novel ring-based interpolation between power maps of different degrees. Central to the method is a quasiregular interpolation in round rings that transitions from $p_d$ to $p_{3d}$, combined with Zorich maps and a gluing step to ensure global coherence and growth control. The authors build a global map by layering interpolants and power-type blocks across concentric rings, yielding a dynamical picture with wandering hollow quasi-Fatou components bounded by spherical Julia components, and provide a mechanism to realize prescribed growth rates. Overall, the paper delivers the first higher-dimensional example in which $QF(f)$ contains bounded hollow components and multiple spherical Julia components, advancing understanding of quasi-Fatou/Julia structures in dimension three and offering flexible construction techniques for growth behavior.

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.

Interpolating quasiregular power mappings

TL;DR

This work constructs a transcendental-type quasiregular map in whose quasi-Fatou set contains bounded hollow components and whose Julia set features spherical components, achieved through a novel ring-based interpolation between power maps of different degrees. Central to the method is a quasiregular interpolation in round rings that transitions from to , combined with Zorich maps and a gluing step to ensure global coherence and growth control. The authors build a global map by layering interpolants and power-type blocks across concentric rings, yielding a dynamical picture with wandering hollow quasi-Fatou components bounded by spherical Julia components, and provide a mechanism to realize prescribed growth rates. Overall, the paper delivers the first higher-dimensional example in which contains bounded hollow components and multiple spherical Julia components, advancing understanding of quasi-Fatou/Julia structures in dimension three and offering flexible construction techniques for growth behavior.

Abstract

We construct a quasiregular mapping in 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 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.

Paper Structure

This paper contains 13 sections, 20 theorems, 67 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background
  3. Statement of results
  4. Preliminaries
  5. Quasiregular maps
  6. Zorich maps
  7. Power maps
  8. Quasiregular Interpolation
  9. Spherical Julia components and hollow quasi-Fatou components
  10. Constructing the map
  11. Covering properties
  12. Spherical Julia components
  13. Constructing quasiregular maps meeting growth conditions

Key Result

Theorem 1

There exists a quasiregular map $f:\mathbb{R}^3 \to \mathbb{R}^3$ for which $QF(f)$ contains bounded, hollow components. Moreover, there exist a sequence of quasi-Fatou components $(\Omega_k)_{k=1}^{\infty}$ and a sequence of spheres $(\Gamma_k)_{k=1}^{\infty}$ centered at the origin and with radii

Figures (3)

  • Figure 1: The map $\alpha'$.
  • Figure 2: The initial prism.
  • Figure 3: The image of some of the flaps for the Zorich map $\mathcal{Z}$ based on \ref{['eq:zorich1']}. The red, blue and orange dots indicate the images of the corners of the grid and the green dots indicate the corners of the corresponding flaps.

Theorems & Definitions (42)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • proof : Proof of Theorem \ref{['thm:interp']}
  • Lemma 6
  • ...and 32 more