Entropies of strictly convex projective manifolds
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.
Received: April 2009; Revised: December 2009; Available Online: January 2010. |