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

2015/07/19 by Hajłasz, Piotr, Zhou, Xiaodan
#46E35 #57A50 #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1507.05319

Abstract

We construct a large class of pathological n-dimensional topological spheres in \mathbb Rⁿ⁺¹ by showing that for any Cantor set C⊂ \mathbb Rⁿ⁺¹ there is a topological embedding f:\mathbb Sⁿ→\mathbb Rⁿ⁺¹ of the Sobolev class W^1,n whose image contains the Cantor set C.

Related

• A Cantor set in the unit sphere in ℂ² with large polynomial hull
• On mappings generating embedding operators in Sobolev classes on metric measure spaces
• Sobolev extensions over Cantor-cuspidal graphs
• Embedding of self-similar ultrametric Cantor sets
• On embeddings in the sphere