On Julia limiting directions in higher dimensions

TL;DR

This paper explores Julia limiting directions for transcendental quasiregular mappings in higher dimensions, providing conditions for all directions to be limiting and addressing the inverse problem in three dimensions.

Contribution

It introduces the first study of Julia limiting directions in higher dimensions and advances understanding of sectorial domains and their polynomial growth bounds.

Findings

Every direction can be a Julia limiting direction under certain conditions.

Sectorial domains in quasi-Fatou components imply polynomial growth bounds.

Characterization of compact subsets of S^2 related to Julia limiting directions.

Abstract

In this paper we study, for the first time, Julia limiting directions of quasiregular mappings in $\mathbb{R}^n$ of transcendental-type. First, we give conditions under which every direction is a Julia limiting direction. Along the way, our methods show that if a quasi-Fatou component contains a sectorial domain, then there is a polynomial bound on the growth in the sector. Second, we give a contribution to the inverse problem in $\mathbb{R}^3$ of determining which compact subsets of $S^2$ can give rise to Julia limiting directions. The methods here will require showing that certain sectorial domains in $\mathbb{R}^3$ are ambient quasiballs, which is a contribution to the notoriously hard problem of determining which domains are the image of the unit ball $\mathbb{B}^3$ under an ambient quasiconformal map of $\mathbb{R}^3$ to itself.