Genus 2 Cantor sets

TL;DR

This paper constructs a self-similar genus 2 Cantor set in three-dimensional space, demonstrating the first example with uniform local genus 2 and linking it to a new class of quasiregular mappings with such Julia sets.

Contribution

It introduces the first example of a genus 2 Cantor set with uniform local genus and connects it to a novel quasiregular mapping with this Julia set.

Findings

Constructed a self-similar genus 2 Cantor set in .

Established the first example with uniform local genus 2.

Linked the Cantor set to a new quasiregular mapping with this Julia set.

Abstract

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in $\mathbb{R}^3$. This construction is the first for which the local genus is shown to be $2$ at every point of $X$. As an application, we construct, also for the first time, a uniformly quasiregular mapping $f:\mathbb{R}^3 \to \mathbb{R}^3$ for which the Julia set $J(f)$ is a genus $2$ Cantor set.