A Wasserstein spaces is a metric space of sufficiently concentratedprobability measures over a general metric space. The main goal of this paperis 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 ofbi-Lipschitz invariants, called critical parameters, that measure largeness forinfinite-dimensional metric spaces. Basic properties of these invariants aregiven, and they are estimated for a naturel set of spaces generalizing theusual Hilbert cube. In a second part, we estimate the value of these newinvariants in the case of some Wasserstein spaces, as well as the dynamicalcomplexity of push-forward maps. The lower bounds rely on several embeddingresults; for example we provide bi-Lipschitz embeddings of all powers of anyspace inside its Wasserstein space, with uniform bound and we prove that theWasserstein space of a d-manifold has "power-exponential" critical parameterequal to d.
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