A GENERALIZATION OF HAUSDORFF DIMENSION APPLIED TO HILBERT CUBES AND WASSERSTEIN SPACES

Abstract

A Wasserstein space 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 made precise.In the first part, we generalize the Hausdorff dimension by defining a family of bi-Lipschitz invariants, called critical parameters, that measure largeness for infinite-dimensional metric spaces. Basic properties of these invariants are given, and they are estimated for a natural set of spaces generalizing the usual Hilbert cube. These invariants are very similar to concepts initiated by Rogers, but our variant is specifically suited to tackle Lipschitz comparison.In the second part, we estimate the value of these new invariants in the case of some Wasserstein spaces, as well as the dynamical complexity of push-forward maps. The lower bounds rely on several embedding results; for example we provide uniform bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d. These arguments are very easily adapted to study the space of closed subsets of a compact metric space, partly generalizing results of Boardman, Goodey and McClure.