Entropy rigidity of Hilbert and Riemannian metrics

Abstract

In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincaré exponent of a discrete group and the volume growth entropy. The first characterization is in the space of Hilbert metrics and generalizes a result of Crampon. The second is in the space of Riemannian metrics with Ricci curvature bounded below and generalizes a result of Ledrappier and Wang.

arXiv comment14 pages, some revisions following the referees remarks. To be published in IMRN