Lipschitz images and dimensions

TL;DR

This paper investigates which compact metric spaces can be obtained as Lipschitz images of other spaces, establishing relationships between dimensions and mappings, and characterizing spaces via H"older images with connections to fractal dimensions.

Contribution

It provides new conditions relating Hausdorff and box dimensions for Lipschitz images, and characterizes spaces obtainable through H"older maps, advancing understanding of metric space mappings.

Findings

Lipschitz images exist when Hausdorff dimension exceeds box dimension.

Any natural dimension in  is between Hausdorff and box dimensions.

Characterization of spaces as ta-Hf6lder images related to box dimension.

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 $A'\subset A$ and a Lipschitz onto map $f\colon A'\to B$. 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 $\alpha$-H\"older 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.