
Mathematics 2011
A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spacesDOI: 10.1142/S1793525312500094 Abstract: A Wasserstein spaces is a metric space of sufficiently concentrated probability measures over a general metric space. The main goal of this paper is to estimate the largeness of Wasserstein spaces, in a sense to be precised. In a first part, we generalize the Hausdorff dimension by defining a family of biLipschitz invariants, called critical parameters, that measure largeness for infinitedimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a naturel set of spaces generalizing the usual Hilbert cube. In a second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of pushforward maps. The lower bounds rely on several embedding results; for example we provide biLipschitz embeddings of all powers of any space inside its Wasserstein space, with uniform bound and we prove that the Wasserstein space of a dmanifold has "powerexponential" critical parameter equal to d.
