Genus 2 Cantor sets

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.

20 pages, 6 figures

References in corpus (1)

• Simply Connected 3-Manifolds with a Dense Set of Ends of Specified Genus