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

Author: Chengbo Yue
Journal: Trans. Amer. Math. Soc. 348 (1996), 4965-5005
MSC (1991): Primary 58F17; Secondary 58F11, 58F15, 20H10
MathSciNet review: 1348871
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Abstract: This paper studies the ergodic theory at infinity of an arbitrary discrete isometry group $ \Gamma $ 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 $ \Gamma $, dynamics of the geodesic flow and the geometry of $M=H/ \Gamma $. We explore the relationship between the growth exponent of the group, the Hausdorff dimension 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 finite 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 problems. More applications to Mostow rigidity for discrete subgroups of rank 1 noncompact semisimple Lie groups with infinite covolume will be published in subsequent papers by the author.

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Additional Information

Chengbo Yue
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802

Received by editor(s): May 7, 1995
Additional Notes: Research at MSRI supported by NSF Grant #DMS 8505550. Also partially supported by NSF Grant #DMS 9403870 and SFB 170.
Article copyright: © Copyright 1996 American Mathematical Society

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