Entropy of the geodesic flow for metric spaces and Bruhat–Tits buildings

Abstract

Abstract Let (X, dX ) be a geodesically complete Hadamard space endowed with a Borel-measure μ. Assume that there exists a group Γ of isometries of X which acts totally discontinuously and cocompactly on X and preserves μ. We show that the topological entropy of the geodesic flow on the space of (parametrized) geodesics of the compact quotient Γ\X is equal to the volume entropy of μ (if X satisfies a certain local uniformity condition). This extends a result of Manning for riemannian manifolds of nonpositive curvature to the singular case. The result in particular holds for Bruhat–Tits buildings, for which we also compute the entropy explicitly.