Journal of Modern Dynamics (JMD)

Entropies of strictly convex projective manifolds

Pages: 511 - 547, Issue 4, October 2009      doi:10.3934/jmd.2009.3.511

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Mickaël Crampon - IRMA, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France (email)

Abstract: 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.

Keywords:  Entropy, Hilbert geometry, Anosov geodesic flows, Lyapunov exponents.
Mathematics Subject Classification:  Primary: 37D40, 53B40.

Received: April 2009;      Revised: December 2009;      Available Online: January 2010.