A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces
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 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.
Keywords
Cite
@article{arxiv.1105.0360,
title = {A generalization of Hausdorff dimension applied to Hilbert cubes and Wasserstein spaces},
author = {Benoit Kloeckner},
journal= {arXiv preprint arXiv:1105.0360},
year = {2012}
}
Comments
v2 Largely expanded version, as reflected by the change of title; all part I on generalized Hausdorff dimension is new, as well as the embedding of Hilbert cubes into Wasserstein spaces. v3 modified according to the referee final remarks ; to appear in Journal of Topology and Analysis