Basing upon the correspondence between the invariant measures of the geodesic flow on a negatively curved manifold and the measures on the sphere at infinity, we give new constructions of the maximal entropy and the harmonic invariant measures of the geodesic flow En partant de la correspondance entre les mesures invariantes pour le flot geodesique sur une variete a courbure negative et les mesures sur la sphere a l'infini, nous donnons des constructions nouvelles des mesures invariantes (la mesure d'entropie maximale et la mesure harmonique) pour le flot geodesique

Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds

Consider closed Riemannian manifolds with negative sectional curvature. There are three natural dynamics associated with the Riemannian structure: the geodesic flow on the unit tangent bundle, the dynamics of the invariant foliations of the geodesic flow, and the Brownian motion on the universal cover of the manifold. These dynamics define global asymptotic objects such as growth rates or measures at infinity. For locally symmetric negatively curved spaces, these objects are easy to compute and to describe. In this paper, we survey some of their properties and relations in the general case.

Applications of Dynamics to Compact Manifolds of Negative Curvature

Let M be an n-dimensional compact Riemannian manifold of negative sectional curvature. The geodesic flow (pt is a smooth dynamical system on the unit tangent bundle T1M of M, generated by the geodesic spray X. Recall tha t T1M admits four natural foliations W s~, W ~, W s, W s~ which are invariant under the geodesic flow. The leaf W~(v) containing vETIM of the strong stable foliation W ~ consists of all points wETIM with the property tha t the distance between Otw and Otv converges to zero as t--*c~ (where we may use the distance on TIM induced by the Sasaki metric). The leaf W~(v) through v of the stable foliation W ~ is W~(v)=UteR~tw~(v ) , and the strong unstable foliation W ~ (or the unstable foliation W ~) is the image of W ~ (or W ~) under the flip .~: w--*-w. The leaf W~(v) of W i (i=ss, su, s, u) is a smoothly immersed submanifold of TIM depending continuously on v in the C~topo logy (see [Sh]). Moreover the tangent bundle T W i of W ~ is a HSlder-continuous subbundle of TTIM. The purpose of this paper is to investigate ergodic and analytic properties of secondorder differential operators L on TIM with HSlder-continuous coefficients and without zero-order terms which are subordinate to the stable foliation in the following sense:

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Harmonic measures for compact negatively curved manifolds

We describe some properties of the harmonic measures associated with the stable and the strong stable foliations of a geodesic flow on a negatively-curved manifold.

Ergodic properties of the stable foliations

We study cohomology classes of Holder continuous closed leafwise 1-forms on the stable foliation of an Anosov geodesic flow. Each class contains a harmonic 1-form and is determined by its periods. Asymptotic quantities are computed in terms of the Pressure function defined by the geodesic flow.

Ursula Hamenstädt

Papers

An explicite description of harmonic measure