INVARIANT MEASURES UNDER GEODESIC FLOW HAMID-REZA FANA Ï

Abstract. For a compact Riemannian manifold with negative curvature, the Liouville measure, the Bowen-Margulis measure and the Harmonic measure are three natural invariant measures under the geodesic flow. We show that if any two of the above three measure classes coincide then the space is locally symmetric, provided the function with respect to which the equilibrium state is the Harmonic measure, depends only on the foot points. 1. Preliminaries Let (M, g) be a compact Riemannian manifold with strictly negative curvature. Its geodesic flow is of Anosov type and there are invariant probability measures under the geodesic flow on SM, the unit tangent bundle of M. Among them, three measures are extremely well-known: the Liouville measure, the Bowen-Margulis measure and the Harmonic measure. Let us describe them briefly. The Liouville measure m is the natural Liouville invariant measure corresponding to the contact structure of the geodesic flow, f dm = f(v) dv dx, SM M where dx is the (normalized) Riemannian volume on M and dv is the (normalized) Lebesgue measure on SxM. It is well-known ([7]) that m is the equilibrium state of the Hölder function tr U(v), where U(v) is the second fundamental form of th