%0 Journal Article
%T Hausdorff dimension of metric spaces and Lipschitz maps onto cubes
%A Tam¨¢s Keleti
%A Andr¨¢s M¨¢th¨¦
%A Ond£¿ej Zindulka
%J Mathematics
%D 2012
%I arXiv
%R 10.1093/imrn/rns223
%X 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 $\dim_H A=\infty$.
%U http://arxiv.org/abs/1203.0686v2