In this article we consider Wasserstein spaces (with quadratic transportation cost) as intrinsic metric spaces. We are interested in usual geometric properties: curvature, rank and isometry group, mostly in the case of Euclidean spaces. Our most striking result is that the Wasserstein space of the line admits "exotic" isometries, which do not preserve the shape of measures.

Given a finite-to-one map T acting on a compact metric space Ω and an appropriate Banach space of functions X (Ω), one classically constructs for each potential A ∈ X a transfer operator L A acting on X (Ω). Under suitable hypotheses, it is well-known that L A has a maximal eigenvalue λ A , has a spectral gap and defines a unique Gibbs measure µ A . Moreover there is a unique normalized potential of the form B := A + f − f • T + c acting as a representative of the class of all potentials defining the same Gibbs measure.The goal of the present article is to study the geometry of the set of normalized potentials N , of the normalization map A → B, and of the Gibbs map A → µ A . We give an easy proof of the fact that N is an analytic submanifold of X and that the normalization map is analytic; we compute the derivative of the Gibbs map; last we endow N with a natural weak Riemannian metric (derived from the asymptotic variance) with respect to which we compute the gradient flow induced by the pressure with respect to a given potential, e.g. the metric entropy functional. We also apply these ideas to recover in a wide setting existence and uniqueness of equilibrium states, possibly under constraints.

Abstract. -We study the geometry of the space of measures of a compact ultrametric space X, endowed with the L p Wasserstein distance from optimal transportation. We show that the power p of this distance makes this Wasserstein space affinely isometric to a convex subset of ℓ 1 . As a consequence, it is connected by 1 p -Hölder arcs, but any α-Hölder arc with α > 1 p must be constant. This result is obtained via a reformulation of the distance between two measures which is very specific to the case when X is ultrametric; however thanks to the Mendel-Naor Ultrametric Skeleton it has consequences even when X is a general compact metric space. More precisely, we use it to estimate the size of Wasserstein spaces, measured by an analogue of Hausdorff dimension that is adapted to (some) infinite-dimensional spaces. The result we get generalizes greatly our previous estimate that needed a strong rectifiability assumption.The proof of this estimate involves a structural theorem of independent interest: every ultrametric space contains large co-Lipschitz images of regular ultrametric spaces, i.e. spaces of the form {1, . . . , k} N with a natural ultrametric.We are also lead to an example of independent interest: a space of positive lower Minkowski dimension, all of whose proper closed subsets have vanishing lower Minkowski dimension.