A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces

Benoit Kloeckner (1)

1 IF - Institut Fourier

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 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 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 push-forward maps. The lower bounds rely on several embedding results; for example we provide bi-Lipschitz embeddings of all powers of any space inside its Wasserstein space, with uniform bound and we prove that the Wasserstein space of a d-manifold has "power-exponential" critical parameter equal to d.

Mots clés: Caratheodory construction; Hausdorff dimension; Minkowski dimension; infinite-dimensional spaces; optimal transport; Wasserstein space; Hilbert cube; bi-Lipschitz embedding

Domaines: Géométrie métrique [math.MG]; Systèmes dynamiques [math.DS]; Géométrie différentielle [math.DG]; Analyse fonctionnelle [math.FA]

Benoit Kloeckner. A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces. Journal of Topology and Analysis, 2012, 04 (2), pp.203. ⟨10.1142/S1793525312500094⟩. ⟨hal-00543790v3⟩