Entropy rigidity of Hilbert and Riemannian metrics

TL;DR

This paper introduces two new characterizations of real hyperbolic n-space using entropy and Poincaré exponent, extending previous results in Hilbert and Riemannian metric spaces.

Contribution

It generalizes existing characterizations of hyperbolic space by employing volume growth entropy and Poincaré exponent in Hilbert and Riemannian metrics.

Findings

Characterization in Hilbert metric space generalizes Crampon's result.

Characterization in Riemannian metrics with Ricci curvature bounds generalizes Ledrappier and Wang.

Provides new geometric criteria for hyperbolic space identification.

Abstract

In this paper we provide two new characterizations of real hyperbolic $n$-space using the Poincar\'e 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.