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

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 $C$.