Hausdorff dimension of metric spaces and Lipschitz maps onto cubes

TL;DR

This paper establishes that compact metric spaces with Hausdorff dimension exceeding k can be Lipschitz-mapped onto a k-dimensional cube, and explores implications for the transfinite Hausdorff dimension of analytic sets.

Contribution

It proves a new dimension-mapping property for compact metric spaces and clarifies the relationship between Hausdorff and transfinite Hausdorff dimensions for analytic sets.

Findings

Compact metric spaces with Hausdorff dimension > k can be Lipschitz-mapped onto a k-cube.

The property does not hold for arbitrary separable metric spaces.

Transfinite Hausdorff dimension of analytic sets relates to their Hausdorff dimension as described.

Abstract

We prove that a compact metric space (or more generally an analytic subset of a complete separable metric space) of Hausdorff dimension bigger than $k$ can be always mapped onto a $k$-dimensional cube by a Lipschitz map. We also show that this does not hold for arbitrary separable metric spaces. As an application we essentially answer a question of Urba\'nski by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset $A$ of a complete separable metric space is the integer part of $\dim_H A$ if $\dim_H A$ is finite but not an integer, $\dim_H A$ or $\dim_H A-1$ if $\dim_H A$ is an integer and at least $\omega_0$ if $\dim_H A=\infty$.