Exponential Decay of Correlation Coefficients for Geodesic Flows

  • Calvin C. Moore
Part of the Mathematical Sciences Research Institute Publications book series (MSRI, volume 6)


We obtain exponential decay bounds for correlation coefficients of geodesic flows on surfaces of constant negative curvature (and for all Riemannian symmetric spaces of rank one), answering a question posed by Marina Ratner. The square integrable functions on the unit sphere bundle of M are allowed to satisfy weak differentiability conditions. The methods are those of unitary representation theory and invoke the notion of Sobelev vectors of representations. In the course of the discussion we obtain a new characterization of tempered irreducible representation of semi-simple groups.


Irreducible Representation Unitary Representation Borel Subgroup Maximal Compact Subgroup Geodesic Flow 
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  1. [1]
    V. Bargmann, Irreducible unitary representations of the Lorentz group, Ann. of Math. 48, (1947), 568–640.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    A. Borei and H. Garland, Laplacian and the discrete spectrum of an arithmetic group, Amer. J. of Math. v.105, (1983), 309–335.CrossRefGoogle Scholar
  3. [3]
    A. Borei and N. Wallach, Continuous Cohomology, Discrete Subgroups, and Representations of Reduct ive Groups, Ann. of Math. Studies 94, (1980), Princeton University Press.Google Scholar
  4. [4]
    J. Brezin and C.C. Moore, Flows on homogeneous spaces: a new look, Amer. J. of Math, v.103, (1981), 571–613.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    M. Cowling, Surles coefficients des representations unitaires des groupes de Lie simple, Lecture Notes in Mathematics 739, (1979), 132–178, Springer-Verlag.Google Scholar
  6. [6]
    J. Dixmier, Les C *-Algebres et leur Représentations, Gautheir-villars, Paris, 1969.Google Scholar
  7. [7]
    P. Eymard, Moyennes Invariantes et Représentations Unitaires, Lecture Notes in Math. 300, (1972), Springer-Verlag.Google Scholar
  8. [8]
    J.M.G. Fell, Weak containment and induced representation of groups, Canadian Math. J. 14, (1962), 237–268.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    I.M. Gelfand and S. Fomin, Geodesic flows on manifolds of constant negative curvature, Uspekhi Matem. Nauk 7, (1952), 118–137.MathSciNetGoogle Scholar
  10. [10]
    Harish Chandra, Spherical functions on a semi-simple group, I, Amer. J. of Math. 80, (1958), 241–310.MATHCrossRefGoogle Scholar
  11. [11]
    T. Hirai, On irreducible representations of the Lorentz group of nth order, Proc. Japan Acad. 38, (1962), 83–87.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    H. Kraljevic, Representations of the universal covering group of the group SU(n,l), Glasnik Mat., Ser. 3, 8, (28), (1973), 23–72.MathSciNetGoogle Scholar
  13. [13]
    H. Kralijevic, On representations of the group SU(n,l), Trans. Amer. Math. Soc. 221, (1976), 433–448.MathSciNetGoogle Scholar
  14. [14]
    A.W. Knapp and E.M. Stein, Intertwining operators for semi-simple groups, Ann. of Math. 93, (1971), 489–578.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    A.W. Knapp and G. Zuckerman, Classification of irreducible tempered representations of semi-simple Lie groups, Proc. Nat. Acad. Sci., USA 73, (1976), 2178–2180.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    A.W. Knapp and G. Zuckerman, Classification theorems for representations of semi-simple Lie groups, Lecture Notes in Math. 587, (1977), 138–159, Springer-Verlag.Google Scholar
  17. [17]
    R. Langlands, On the classification of irreducible representations of real algebraic groups, notes, Institute for Advanced Study, Princeton, New Jersey, 1973.Google Scholar
  18. [18]
    C.C. Moore, Amenable subgroups of semi-simple groups and proximal flows, Israel J. Math. 34, (1979), 121–138.MathSciNetMATHCrossRefGoogle Scholar
  19. [19]
    M. Ratner, Rigidity of time changes for horocycle flows, to appear, Acta Math.Google Scholar
  20. [20]
    E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, New Jersey, 1970.MATHGoogle Scholar
  21. [21]
    P. Trombi, The tempered spectrum of a real semi-simple Lie group, Amer. J. of Math. 99, (1977), 57–75.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    P. Trombi and V. Varadarajan, Spherical transforms on semi-simple Lie groups, Ann. of Math. 94, (1971), 246–303.MathSciNetMATHCrossRefGoogle Scholar
  23. [23]
    P. Trombi, and V. Varadarajan, Asymptotic behavior of eigenfunctions on a semi-simple Lie group: the discrete spectrum. Acta Math. 129, (1972), 237–280.MathSciNetMATHCrossRefGoogle Scholar
  24. [24]
    G. Warner, Harmonic Analysis on Semi-simple Lie groups, II, Grund. Math. Wiss. 189, Springer, 1972.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1987

Authors and Affiliations

  • Calvin C. Moore
    • 1
  1. 1.Mathematical Sciences Research Institute and Department of MathematicsUC BerkeleyUSA

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