Information geometry of barycenter map

Abstract

Using barycenter of the Busemann function we define a map, called the barycenter map from a space P + of probability measures on the ideal boundary ∂X to an Hadamard manifold X. We show that the space P + carries a fibre space structure over X from a viewpoint of information geometry. Following the idea of [7, 9] and [8] we present moreover a theorem which states that under certain hypotheses of information geometry a homeomorphism Φ of ∂X induces, via the push-forward for probability measures, an isometry of X whose ∂X-extension coincides with Φ.