Hausdorff dimension of metric spaces and Lipschitz maps onto cubes

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 Urbański 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 ${\dim }_{H}A=\infty$.

Cited by in corpus (1)

• A simple proof of Dvoretzky-type theorem for Hausdorff dimension in doubling spaces