Optimal transport and dynamics of expanding circle maps acting on measures

Benoit Kloeckner (1, 2)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 IF - Institut Fourier

Using optimal transport we study some dynamical properties of expanding circle maps acting on measures by push-forward. 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 1 is an eigenvalue of infinite multiplicity, so that the invariant measure admits many deformations into nearly invariant ones. As a consequence, we obtain counter-examples to an infinitesimal version of Furstenberg's conjecture. We also prove that this action has positive metric mean dimension with respect to the Wasserstein metric.

Mots clés: Furstenberg conjecture; Perron-Frobenius operator; Optimal transport; Wasserstein metric; dynamical systems; expanding circle map; metric mean dimension; Perron-Frobenius operator.

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

Benoit Kloeckner. Optimal transport and dynamics of expanding circle maps acting on measures. Ergodic Theory and Dynamical Systems, 2013, 33 (02), pp 529-548. ⟨10.1017/S014338571100109X⟩. ⟨hal-00494397v4⟩