×

Entropies of strictly convex projective manifolds. (English) Zbl 1189.37034

Summary: 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.

MSC:

37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)