Lipschitz images and dimensions

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 ' subset of A and a Lipschitz onto map f : A ' -> B. As a corollary we prove that any 'natural' dimension in R-n 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 alpha > 0 we also give a characterization of those compact metric spaces that can be obtained as an alpha-H & ouml;lder image of a compact subset of R. The quantity we introduce for this turns out to be closely related to the upper box dimension. (c) 2024 The Author(s). Published by Elsevier Inc. This is an