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

We construct a large class of pathological $n$-dimensional topological spheres in ${\mathbb{R}}^{n+1}$ by showing that for any Cantor set $C\subset {\mathbb R}^{n+1}$thereisatopologicalembedding$f:{\mathbb S}^n\to{\mathbb R}^{n+1}$oftheSobolevclass$W^{1,n}$whoseimagecontainstheCantorset$C$.