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

Yue, Chengbo

doi:10.1090/S0002-9947-96-01614-5

The ergodic theory of discrete isometry groups on manifolds of variable negative curvature
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Trans. Amer. Math. Soc. 348 (1996), 4965-5005 Request permission

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

Lars V. Ahlfors, Complex analysis: An introduction of the theory of analytic functions of one complex variable, 2nd ed., McGraw-Hill Book Co., New York-Toronto-London, 1966. MR 0188405
Alano Ancona, Sur les fonctions propres positives des variétés de Cartan-Hadamard, Comment. Math. Helv. 64 (1989), no. 1, 62–83 (French). MR 982562, DOI 10.1007/BF02564664
A. Ancona, Théorie du potentiel sur les graphes et les variétés, École d’été de Probabilités de Saint-Flour XVIII—1988, Lecture Notes in Math., vol. 1427, Springer, Berlin, 1990, pp. 1–112 (French). MR 1100282, DOI 10.1007/BFb0103041
Kiyoshi Hayashi, A topological invariant related to the number of orthogonal geodesic chords, Osaka J. Math. 24 (1987), no. 2, 263–270. MR 909017
Michael T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701–721 (1984). MR 730923
Michael T. Anderson and Richard Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429–461. MR 794369, DOI 10.2307/1971181
Werner Ballmann, Mikhael Gromov, and Viktor Schroeder, Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61, Birkhäuser Boston, Inc., Boston, MA, 1985. MR 823981, DOI 10.1007/978-1-4684-9159-3
B. H. Bowditch, Geometrical finiteness with variable negative curvature, I.H.E.S. Preprint.
M. Coornaert, Sur les groups proprement discontinus d’isometriés des espaces hyperboliques au sens de Gromov, Thèse (1990).
Kevin Corlette, Hausdorff dimensions of limit sets. I, Invent. Math. 102 (1990), no. 3, 521–541. MR 1074486, DOI 10.1007/BF01233439
Patrick Eberlein, Geodesic flows on negatively curved manifolds. I, Ann. of Math. (2) 95 (1972), 492–510. MR 310926, DOI 10.2307/1970869
Patrick Eberlein, Geodesic flows on negatively curved manifolds. II, Trans. Amer. Math. Soc. 178 (1973), 57–82. MR 314084, DOI 10.1090/S0002-9947-1973-0314084-0
V. R. Portnov, Conditions for normal solvability in the sense of Hausdorff of a certain nonlinear problem, Dokl. Akad. Nauk SSSR 209 (1973), 555–557 (Russian). MR 0336467
W. M. Goldman and J. J. Millson, Local rigidity of discrete groups acting on complex hyperbolic space, Invent. Math. 88 (1987), no. 3, 495–520. MR 884798, DOI 10.1007/BF01391829
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Ursula Hamenstädt, Time-preserving conjugacies of geodesic flows, Ergodic Theory Dynam. Systems 12 (1992), no. 1, 67–74. MR 1162399, DOI 10.1017/S0143385700006581
Vadim A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), no. 4, 361–393 (English, with French summary). Hyperbolic behaviour of dynamical systems (Paris, 1990). MR 1096098
Yuri Kifer and François Ledrappier, Hausdorff dimension of harmonic measures on negatively curved manifolds, Trans. Amer. Math. Soc. 318 (1990), no. 2, 685–704. MR 951889, DOI 10.1090/S0002-9947-1990-0951889-7
F. Ledrappier, Ergodic properties of Brownian motion on covers of compact negatively-curve manifolds, Bol. Soc. Brasil. Mat. 19 (1988), no. 1, 115–140. MR 1018929, DOI 10.1007/BF02584822
F. Ledrappier, Harmonic measures and Bowen-Margulis measures, Israel J. Math. 71 (1990), no. 3, 275–287. MR 1088820, DOI 10.1007/BF02773746
F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I. Characterization of measures satisfying Pesin’s entropy formula, Ann. of Math. (2) 122 (1985), no. 3, 509–539. MR 819556, DOI 10.2307/1971328
G. A. Margulis, Thesis, In Russian, Moscow University, 1970.
Brian Marcus, Unique ergodicity of some flows related to Axiom A diffeomorphisms, Israel J. Math. 21 (1975), no. 2-3, 111–132. MR 413183, DOI 10.1007/BF02760790
Peter J. Nicholls, The ergodic theory of discrete groups, London Mathematical Society Lecture Note Series, vol. 143, Cambridge University Press, Cambridge, 1989. MR 1041575, DOI 10.1017/CBO9780511600678
P. Pansu, Quasiisométriés des variétés à courbure négative, Thèse Université de Paris 7 (1987).
S. J. Patterson, Lectures on measures on limit sets of Kleinian groups, Analytical and geometric aspects of hyperbolic space (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 111, Cambridge Univ. Press, Cambridge, 1987, pp. 281–323. MR 903855
Marina Ratner, Raghunathan’s conjectures for $\textrm {SL}(2,\mathbf R)$, Israel J. Math. 80 (1992), no. 1-2, 1–31. MR 1248925, DOI 10.1007/BF02808152
Y. G. Sinai, Classical systems with Lebesgue spectrum II, AMS. Transl. 2 68 (1968), 34–88.
Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
Dennis Sullivan, The density at infinity of a discrete group of hyperbolic motions, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 171–202. MR 556586, DOI 10.1007/BF02684773
Dennis Sullivan, Discrete conformal groups and measurable dynamics, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 57–73. MR 634434, DOI 10.1090/S0273-0979-1982-14966-7
Dennis Sullivan, Related aspects of positivity in Riemannian geometry, J. Differential Geom. 25 (1987), no. 3, 327–351. MR 882827
Dennis Sullivan, Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics, Acta Math. 149 (1982), no. 3-4, 215–237. MR 688349, DOI 10.1007/BF02392354
Lai Sang Young, Dimension, entropy and Lyapunov exponents, Ergodic Theory Dynam. Systems 2 (1982), no. 1, 109–124. MR 684248, DOI 10.1017/s0143385700009615
Cheng Bo Yue, Brownian motion on Anosov foliations and manifolds of negative curvature, J. Differential Geom. 41 (1995), no. 1, 159–183. MR 1316554
C. B. Yue, On Sullivan’s conjecture, Random and Comp. Dyn. 1 (1992), 131-145.
C. B. Yue, Dimension and rigidity of quasi-Fuchsian representations, Annals of Mathematics 143 (1996), 331–355.
C. B. Yue, Mostow rigidity for rank 1 discrete subgroups with ergodic Bowen-Margulis measure, To appear in Inv. Math.