Optimal transportation and dynamics of maps acting on measures, whith an emphasis on expanding circle maps

Benoit Kloeckner (1)

1 IF - Institut Fourier

Using optimal transport we study some dynamical properties of maps acting on measures by push-forward. In a first part, the case of expanding circle maps is studied. Ii is shown that they act topologically transitively, that their topological entropy is infinite and, using the definition of the tangent space to the space of measures introduced by Gigli, their derivative at the unique absolutely continuous invariant measure is computed. In particular it is shown that in the simplest cases they are not hyperbolic. In a second part a bi-Lipschitz embedding of any power of any metric space into its Wasserstein space is constructed. A dynamical consequence of this result generalize a result of the first part: a map acting on a compact space with positive topological entropy, acts on its space of measures with infinite topological entropy.

Mots clés: bi-Lipschitz embeddings; Optimal transport; Wasserstein metric; dynamical systems; expanding circle map; topological transitivity; topological entropy; Perron-Frobenius operator; bi-Lipschitz embeddings.

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

Benoit Kloeckner. Optimal transportation and dynamics of maps acting on measures, whith an emphasis on expanding circle maps. 2010. ⟨hal-00494397v1⟩