Sobolev Embedding of a Sphere Containing An Arbitrary Cantor Set in the image

TL;DR

This paper constructs a broad class of pathological topological spheres in Euclidean space by embedding the sphere in a Sobolev space such that its image contains any given Cantor set, revealing new possibilities for sphere embeddings.

Contribution

It demonstrates that any Cantor set can be contained in the image of a Sobolev $W^{1,n}$ embedding of the sphere, expanding understanding of sphere embeddings with fractal sets.

Findings

Existence of Sobolev embeddings containing arbitrary Cantor sets

Construction of pathological spheres in Euclidean space

Extension of embedding theory to fractal sets

Abstract

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb R}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$ there is a topological embedding $f:{\mathbb S}^n\to{\mathbb R}^{n+1}$ of the Sobolev class $W^{1,n}$ whose image contains the Cantor set $C$.