Hausdorff Dimension of Metric Spaces and Lipschitz Maps onto Cubes

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 always be 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 Urbanski by showing that the transfinite Hausdorff dimension (introduced by him) of an analytic subset A of a complete separable metric space is left perepndiculardim(H)Aright perepndicular if dim(H)A is finite but not an integer, dimHA or dim(H)A-1 if dim(H)A is an integer and at least omega(0) if dim(H)A=infinity.