Genus $g$ Cantor sets and germane Julia sets

TL;DR

This paper explores the topological properties of Cantor sets in 3, introducing a genus-based obstruction to them being Julia sets of uniformly quasiregular mappings, and constructs new examples with specific genus properties.

Contribution

It introduces a new construction of genus g Cantor sets with uniform local genus and demonstrates their realization as Julia sets, the first such examples for g 3 3.

Findings

Constructed genus g Cantor sets with uniform local genus g

Showed these sets can be realized as Julia sets of quasiregular mappings

Proved that Cantor Julia sets of hyperbolic maps have finite genus g

Abstract

The primary aim of this paper is to give topological obstructions to Cantor sets in $\mathbb{R}^3$ being Julia sets of uniformly quasiregular mappings. Our main tool is the genus of a Cantor set. We give a new construction of a genus $g$ Cantor set, the first for which the local genus is $g$ at every point, and then show that this Cantor set can be realized as the Julia set of a uniformly quasiregular mapping. These are the first such Cantor Julia sets constructed for $g\geq 3$. We then turn to our dynamical applications and show that every Cantor Julia set of a hyperbolic uniformly quasiregular map has a finite genus $g$; that a given local genus in a Cantor Julia set must occur on a dense subset of the Julia set; and that there do exist Cantor Julia sets where the local genus is non-constant.