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 $A^{\prime }\subset A$ and a Lipschitz onto map $f:A^{\prime }\to B$. As a corollary we prove that any `natural' dimension in ${\mathbb{R}}^n$ 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 $\alpha$-Hölder image of a compact subset of $\mathbb{R}$. The quantity we introduce for this turns out to be closely related to the upper box dimension.

19 pages. Minor modifications; Lemma 2.2 is added and the proof of Theorem 7.7 is augmented