Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [Ah] L. V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian Groups, Proc. Nat. Acad. Sci. U.S.A. 55 (1966), 251-254. MR 32:5844
  • [A1] A. Ancona, Sur les fonctions propres positives des variétés de Cartan-Hadamard, Comm. Math. Helv. 64 (1989), 62-83. MR 90k:53069
  • [A2] A. Ancona, Théorie du potentiel sur les graphes et les variétés, Lect. Notes in Math. 1427, Springer, Berlin, 1990. MR 92g:31012
  • [A3] A. Ancona, Negatively curved Manifolds, Elliptic operators, and the Martin boundary, Ann. Math. 121 (1987), 495-536. MR 88k:53069
  • [And] M. Anderson, The Dirichlet problem at infinity, J. Diff. Geom. 18 (1983), 701-721. MR 85m:58178
  • [AS] M. Anderson and R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. Math. 121 (1985), 429-461. MR 87a:58151
  • [BGS] W. Ballmann, M. Gromov and V. Schroeder, Manifolds of non-positive curvature, Progress in Mathematics 61, Birkhäuser (1985). MR 87h:53050
  • [B] B. H. Bowditch, Geometrical finiteness with variable negative curvature, I.H.E.S. Preprint.
  • [C] M. Coornaert, Sur les groups proprement discontinus d'isometriés des espaces hyperboliques au sens de Gromov, Thèse (1990).
  • [Co] K. Corlette, Hausdorff dimension of limit sets I, Invent. Math. 102 (1990), 521-542. MR 91k:58067
  • [E1] P. Eberlein, Geodesic flows on negatively curved manifolds; I, Ann. of Math. 95 (1972), 151-170. MR 46:10024
  • [E2] P. Eberlein, Geodesic flows on negatively curved manifolds; II, Trans. of AMS 178 (1973), 57 - 82. MR 47:2636
  • [E-O'N] P. Eberlein and B. O'Neil, Visibility manifolds, Pacific J. Math 46 (1973), 45-109. MR 49:1241
  • [GM] W.M. Goldman and J.J. Millson, Local rigidity of discrete groups acting on complex hyperbolic spaces, Invent. Math. 88 (1987), 495-520. MR 88f:22027
  • [G] M. Gromov, Hyperbolic groups, Essays in group theory, Springer, 1987. MR 89e:20070
  • [H] U. Hamenstädt, Time preserving conjugacies of geodesic flows, Erg. Th. and Dyn. Sys. 12 (1992), 67-74. MR 93g:58115
  • [K] V. A. Kaimanovich, Invariant measures for the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. Henri Poincaré, Physique Théorique 53, no. 4 (1990), 361-393. MR 92b:58176
  • [KL] Y. Kifer and F. Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, Trans. of AMS 318, No. 2 (1990), 685 - 704. MR 91a:58205
  • [L1] F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively curved manifolds, Bol. Soc. Bras. Mat. 19 (1988), 115-140. MR 91e:58210
  • [L2] F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), 275-287. MR 92a:58107
  • [LY] F. Ledrappier and L. S. Young, The metric entropy of diffeomorphisms I, Ann. of Math. 122 (1985), 509-539. MR 87i:58101a
  • [M] G. A. Margulis, Thesis, In Russian, Moscow University, 1970.
  • [M1] B. Marcus, Unique ergodicity of the horocycle flow: variable negative curvature case, Israel J. Math. 21 (1975), 133-144. MR 54:1302
  • [N] P. J. Nicholls, The ergodic theory of discrete groups, Cambridge University Press, 1989. MR 91i:58104
  • [Pan] P. Pansu, Quasiisométriés des variétés à courbure négative, Thèse Université de Paris 7 (1987).
  • [P] S. J. Patterson, Measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space, Cambridge University Press, 1987, pp. 291-323. MR 89b:58122
  • [R] M. Ratner, Raghunathan's conjectures for $SL(2,R)$, Israel J. Math. 80 (1992), 1-37.MR 94k:22024
  • [Si1] Y. G. Sinai, Classical systems with Lebesgue spectrum II, AMS. Transl. 2 68 (1968), 34-88.
  • [Si2] Y. G. Sinai, Gibbs measure in ergodic theory, Russ. Math. Surveys 27, no. 4 (1972), 21-69. MR 53:3265
  • [S1] D. Sullivan, The density at infinity of a discrete group of hyperbolic motions, Publ. Math. I.H.E.S. 50 (1979), 171-202. MR 81b:58031
  • [S2] D. Sullivan, Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc. 6 (1982), 57-73. MR 83c:58066
  • [S3] D. Sullivan, Related aspects of positivity in Riemannian geometry, J. Diff. Geom. 25 (1987), 327-351. MR 88d:58132
  • [S4] D. Sullivan, Disjoint spheres, diophantine approximation, and the logarithm law for geodesics, Acta Math. 149 (1982), 215-237. MR 84j:58097
  • [Y] L. S. Young, Dimension, entropy and Lyapunov exponents, Erg. Th. Dynam. Sys. 2 (1982), 109-129. MR 84h:58087
  • [Yu1] C. B. Yue, Brownian motion on Ansov foliations and manifolds of negative curvature, J. of Diff. Geom. 41 (1995), 159-183. MR 95k:58123
  • [Yu2] C. B. Yue, On Sullivan's conjecture, Random and Comp. Dyn. 1 (1992), 131-145.
  • [Yu3] C. B. Yue, Dimension and rigidity of quasi-Fuchsian representations, Annals of Mathematics 143 (1996), 331-355. CMP 96:10
  • [Yu4] C. B. Yue, Mostow rigidity for rank 1 discrete subgroups with ergodic Bowen-Margulis measure, To appear in Inv. Math.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 58F17, 58F11, 58F15, 20H10

Retrieve articles in all journals with MSC (1991): 58F17, 58F11, 58F15, 20H10

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