Entropies of strictly convex projective manifolds

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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

Abstract
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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: Hilbert geometry, Anosov geodesic flows, Lyapunov exponents., Entropy.

Mathematics Subject Classification: Primary: 37D40, 53B4.

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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