Israel Journal of Mathematics

, Volume 16, Issue 2, pp 181–197 | Cite as

The central limit theorem for geodesic flows onn-dimensional manifolds of negative curvature

  • M. Ratner


In this paper we prove a central limit theorem for special flows built over shifts which satisfy a uniform mixing of type\(\gamma ^{n^\alpha } \), 0<γ<1, α>0. The function defining the special flow is assumed to be continuous with modulus of continuity of type\(f(z) = \sum\nolimits_{n = 0}^\infty {a_n z^n } \), 0<ρ<1, β>0, andd is the natural metric on sequence space. Geodesic flows on compact manifolds of negative curvature are isomorphic to special flows of this kind.


Central Limit Theorem Compact Manifold Gibbs Measure Negative Curvature Finite Interval 
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  1. 1.
    S. N. Bernstein, Sur l'éxtension du théorème limite du calcul des probabilités aux sommes de quantitiés dépendantes, Math. Ann.97 (1926), 1–59.CrossRefGoogle Scholar
  2. 2.
    R. Bowen,Markov partitions for axiom Adiffeomorphisms, Amer. J. Math.92 (1970), 725–747.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. Bowen,Symbolic dynamics for hyperbolic flows (to appear).Google Scholar
  4. 4.
    W. Feller,An introduction to probability theory and its applications, Vol. 1, New York.Google Scholar
  5. 5.
    B. M. Gurevič,The structure of increasing decompositions for special flows, Theor. Probability Appl.10 (1965), 627–654, MR35 #3034.CrossRefGoogle Scholar
  6. 6.
    I. A. Ibragimov,Some limit theorems for stationary processes, Theor. Probablity Appl.7 (1962), 349–382.CrossRefGoogle Scholar
  7. 7.
    V. P. Leonov,On the dispersion of time-dependent means of a stationary stochastic process, Theor. Probability, Appl.6 (1961), 87–93.CrossRefMathSciNetGoogle Scholar
  8. 8.
    W. Parry,Intrinsic Markov chains, Trans. Amer. Math. Soc.112 (1964), 55–66.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    M. Ratner,Central limit theorem for Anosov flows on three-dimensional manifolds, Soviet Math. Dokl.10 (1969).Google Scholar
  10. 10.
    M. Ratner,Invariant measure with respect to an Anosov flows on a three-dimensional manifold, Soviet Math. Dokl.10 (1969).Google Scholar
  11. 11.
    M. Ratner,Markov partitions for Anosov flows on n-dimensional manifolds (to appear).Google Scholar
  12. 12.
    Y. G. Sinai,The central limit theorem for geodesic flows on manifolds of constant negative curvature, Soviet Math. Dokl.1 (1960), 938–987.Google Scholar
  13. 13.
    Y. G. Sinai,Markov partitions and C-diffeomorphisms, Functional. Anal. Appl.2 (1968), 64–89.CrossRefMathSciNetGoogle Scholar
  14. 14.
    Y. G. Sinai,Gibbs measures in ergodic theory, Uspehi Mat. Nauk (4)27 (1972), 21–63.MathSciNetGoogle Scholar

Copyright information

© The Weizmann Science Press of Israel 1973

Authors and Affiliations

  • M. Ratner
    • 1
  1. 1.Institute of mathematicsThe Hebrew University of JerusalemJerusalemIsrael

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