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

TL;DR

This paper introduces a new family of invariants called critical parameters to measure the 'largeness' of infinite-dimensional metric spaces, applies them to Wasserstein spaces and Hilbert cubes, and estimates their values and properties.

Contribution

It generalizes Hausdorff dimension through critical parameters, providing tools to analyze the size and complexity of Wasserstein spaces and related metric spaces.

Findings

Defined bi-Lipschitz invariants called critical parameters.

Estimated invariants for spaces generalizing Hilbert cubes.

Established embeddings and calculated invariants for Wasserstein spaces of manifolds.

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.