# American Institute of Mathematical Sciences

October  2009, 3(4): 511-547. doi: 10.3934/jmd.2009.3.511

## Entropies of strictly convex projective manifolds

 1 IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Received  April 2009 Revised  December 2009 Published  January 2010

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 hyperbolic. As a corollary, the volume entropy of a divisible strictly convex set is less than $n-1$, with equality if and only if it is an ellipsoid.
Citation: Mickaël Crampon. Entropies of strictly convex projective manifolds. Journal of Modern Dynamics, 2009, 3 (4) : 511-547. doi: 10.3934/jmd.2009.3.511
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