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BROWNIAN MOTION ON ANOSOV FOLIATIONS AND MANIFOLDS OF NEGATIVE CURVATURE

We study ergodic properties of Anosov foliations. Some rigidity results are obtained, including applications to manifolds of negative curvature, and an integral formula for topological entropy. We also show that the in function c{x) in Margulis's asymptotic formula c(x) = lim^^^ e S{x,R) is almost always not constant. In dimension 2, c(x) is a con- stant function if and only if the manifold has constant negative curvature. Generally, if the Ledrappier-Patterson-Sullivan measure is a flip invari- ant, then c{x) is constant. Our entropy formula yields an upper bound of Gromov's simplicial volume in terms of scalar curvature.