Entropies of compact strictly convex projective manifolds

Mickaël Crampon (1)

1 IRMA - Institut de Recherche Mathématique Avancée

Let M be a compact manifold of dimension n with a strictly convex projective structure. We consider the geodesic flow of the Hilbert metric on it, which is known to be Anosov. We prove that its topological entropy is less than n-1, with equality if and only if the structure is Riemannian, that is hyperbolic. As a corollary, we get that the volume entropy of a divisible strictly convex set is less than n-1, with equality if and only if it is an ellipsoid.

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

Mickaël Crampon. Entropies of compact strictly convex projective manifolds. Journal of modern dynamics, 2009, 3 (4), pp.511-547. ⟨10.3934/jmd.2009.3.511⟩. ⟨hal-00375776⟩