Lipschitz images and dimensions

Abstract

We consider the question which compact metric spaces can be obtained as a Lipschitz image of the middle third Cantor set, or more generally, as a Lipschitz image of a subset of a given compact metric space.In the general case we prove that if A and B are compact metric spaces and the Hausdorff dimension of A is bigger than the upper box dimension of B, then there exist a compact set A′⊂A and a Lipschitz onto map f:A′→B.As a corollary we prove that any ‘natural’ dimension in Rn must be between the Hausdorff and upper box dimensions.We show that if A and B are self-similar sets with the strong separation condition with equal Hausdorff dimension and A is homogeneous, then A can be mapped onto B by a Lipschitz map if and only if A and B are bilipschitz equivalent.For given α>0 we also give a characterization of those compact metric spaces that can be obtained as an α-Hölder image of a compact subset of R. The quantity we introduce for this turns out to be closely related to the upper box dimension.