Genus 2 Cantor sets

Alastair Fletcher and Daniel Stoertz

Vol. 321 (2022), No. 2, 283–307

DOI: 10.2140/pjm.2022.321.283

Abstract

We construct a geometrically self-similar Cantor set $X$ of genus $2$ in ${ℝ}^3$. 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 $f:{ℝ}^3\to {ℝ}^3$ for which the Julia set $J(f)$ is a genus $2$ Cantor set.

Keywords

Cantor sets, Julia sets, uniformly quasiregular mappings

Mathematical Subject Classification

Primary: 54C50

Secondary: 30C65, 37F10, 37F50

Milestones

Received: 25 August 2021

Revised: 31 October 2022

Accepted: 5 November 2022

Published: 21 March 2023