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This paper studies the ergodic theory at innity of an arbitrary discrete isometry group acting on any Hadamard manifold H of pinched variable negative curvature. Most of the results obtained by Sullivan in the constant curvature case are generalized to the case of variable curvature. We describe connections between measures supported on the limit set of , dy- namics of the geodesic flow and the geometry of M = H=. We explore the relationship between the growth exponent of the group, the Hausdor dimen- sion of the limit set and the topological entropy of the geodesic flow. The equivalence of various descriptions of an analogue of the Hopf dichotomy is proved. As applications, we settle a question of J. Feldman and M. Ratner about the horocycle flow on a nite volume surface of negative curvature and obtain an asymptotic formula for the counting function of lattice points. At the end of this paper, we apply our results to the study of some rigidity prob- lems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with innite covolume will be published in subsequent papers by the author.

The ergodic theory of discrete isometry groups on manifolds of variable negative curvature