Higher Order Correlations for Group Actions

  • Alexander GorodnikEmail author
Part of the Infosys Science Foundation Series book series (ISFS)


This survey paper discusses behaviour of higher order correlations for one-parameter dynamical systems and more generally for dynamical systems arising from group actions. In particular, we present a self-contained proof of quantitative bounds for higher order correlations of actions of simple Lie groups. We also outline several applications of our analysis of correlations that include asymptotic formulas for counting lattice points, existence of approximate configurations in lattice subgroups, and validity of the Central Limit Theorem for multiparameter group actions.



This survey paper has grown out of the lecture series given by the author at the Tata Institute of Fundamental Research in Spring 2017. I would like to express my deepest gratitude to the Tata Institute for the hospitality and to the organisers of this programme—Shrikrishna Dani and Anish Ghosh—for all their hard work on setting up this event and making it run smoothly. I also would like to thank the referee for carefully reading the draft and for his thoughtful comments.


  1. 1.
    M. Babillot, Points entiers et groupes discrets: de l’analyse aux systèmes dynamiques. Panor. Synthèses, 13, Rigidité, groupe fondamental et dynamique, 1–119, Soc. Math. France, Paris, 2002.Google Scholar
  2. 2.
    U. Bader, A. Furman, A. Gorodnik, B. Weiss, Rigidity of group actions on homogeneous spaces, III. Duke Math. J. 164 (2015), no. 1, 115–155.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    H.-J. Bartels, Nichteuklidische Gitterpunktprobleme und Gleichverteilung in linearen algebraischen Gruppen. Comment. Math. Helv. 57 (1982), no. 1, 158–172.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    B. Bekka, On Uniqueness of Invariant Means. Proc. Amer. Math. Soc. 126 (1998), 507–514.MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    B. Bekka, P. de la Harpe, and A. Valette, Kazhdan’s property (T). New Mathematical Monographs, 11. Cambridge University Press, Cambridge, 2008.Google Scholar
  6. 6.
    B. Bekka and M. Mayer, Ergodic theory and topological dynamics of group actions on homogeneous spaces. London Mathematical Society Lecture Note Series, 269. Cambridge University Press, Cambridge, 2000.Google Scholar
  7. 7.
    M. Björklund, M. Einsiedler, and A. Gorodnik, Quantitative multiple mixing. to appear in J. Eur. Math. Soc.; ArXiv:1701.00945.
  8. 8.
    M. Björklund and A. Gorodnik, Central Limit Theorems for group actions which are exponentially mixing of all orders. accepted to Journal d’Analyse Mathematiques; ArXiv:1706.09167.
  9. 9.
    M. Björklund and A. Gorodnik, Central limit theorems in the geometry of numbers. Electron. Res. Announc. Math. Sci. 24 (2017), 110–122.MathSciNetzbMATHGoogle Scholar
  10. 10.
    M. Björklund and A. Gorodnik, Central limit theorems for Diophantine approximants. ArXiv:1804.06084.
  11. 11.
    A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups. Annals of Mathematics Studies, 94. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980.Google Scholar
  12. 12.
    J. Bourgain, A Szemerédi type theorem for sets of positive density in \(R^k\). Israel J. Math. 54 (1986), 307–316.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    W. Casselman and D. Milicić, Asymptotic behavior of matrix coefficients of admissible representations. Duke Math. J. 49 (1982), no. 4, 869–930.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    G. Cohen and J.-P. Conze, CLT for Random Walks of Commuting Endomorphisms on Compact Abelian Groups. J. Theoret. Probab. 30 (2017), no. 1, 143–195.MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    G. Cohen and J.-P. Conze, Central limit theorem for commutative semigroups of toral endomorphisms. Arxiv:1304.4556.
  16. 16.
    G. Cohen and J.-P. Conze, Almost mixing of all orders and CLT for some \(Z^d\)-actions on subgroups of \(F_p^{Z^d}\). Arxiv:1609.06484.
  17. 17.
    M. Cowling, Sur les coefficients des représentations unitaires des groupes de Lie simples. Analyse harmonique sur les groupes de Lie (Sém., Nancy–Strasbourg 1976–1978), II, Springer, Berlin (1979), 132–178.Google Scholar
  18. 18.
    M. Cowling, U. Haagerup, and R. Howe, Almost \(L^2\) matrix coefficients. J. Reine Angew. Math. 387 (1988), 97–110.MathSciNetzbMATHGoogle Scholar
  19. 19.
    S. G. Dani, Kolmogorov automorphisms on homogeneous spaces. Amer. J. Math. 98 (1976), no. 1, 119–163.MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    S. G. Dani, Spectrum of an affine transformation. Duke Math. J. 44 (1977), no. 1, 129–155.MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    A. del Junco and R. Yassawi, Multiple mixing and rank one group actions. Canad. J. Math. 52 (2000), no. 2, 332–347.MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    J. Delsarte, Sur le gitter fuchsien. C. R. Acad. Sci. Paris 214 (1942), 147–149; Oeuvres de Jean Delsarte, vol. II, Editions du CNRS, Paris, 1971, pp. 829–845.Google Scholar
  23. 23.
    M. Denker, The central limit theorem for dynamical systems. Banach Center Pub. 23 (1989), 33–61.MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Y. Derriennic, Some aspects of recent works on limit theorems in ergodic theory with special emphasis on the “central limit theorem”. Discrete Contin. Dyn. Syst. 15 (2006), no. 1, 143–158.MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    D. Dolgopyat, Limit theorems for partially hyperbolic systems. Trans. Amer. Math. Soc. 356 (2004), no. 4, 1637–1689.MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), no. 1, 143–179.MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    M. Einsiedler, G. Margulis, and A. Venkatesh, Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. Invent. Math. 177 (2009), no. 1, 137–212.MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    A. Eskin and C. McMullen, Mixing, counting, and equidistribution in Lie groups. Duke Math. J. 71 (1993), no. 1, 181–209.MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    D. Fisher, B. Kalinin, and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds. With an appendix by James F. Davis. J. Amer. Math. Soc. 26 (2013), no. 1, 167–198.MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    G. Folland, A course in abstract harmonic analysis. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995.zbMATHGoogle Scholar
  31. 31.
    H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerdi on arithmetic progressions. J. Analyse Math. 31 (1977), 204–256.MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    H. Furstenberg and Y. Katznelson, An ergodic Szemerédi theorem for commuting transformations. J. Analyse Math. 34 (1978), 275–291.MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    H. Furstenberg, Y. Katznelson and B. Weiss, Ergodic theory and configurations in sets of positive density. Mathematics of Ramsey Theory, Algorithms and Combinatorics, Vol. 5, Springer, Berlin, 1990, pp. 184–198.Google Scholar
  34. 34.
    R. Gangolli and V. S. Varadarajan, Harmonic analysis of spherical functions on real reductive groups. Ergebnisse der Mathematik und ihrer Grenzgebiete, 101. Springer-Verlag, Berlin, 1988.zbMATHCrossRefGoogle Scholar
  35. 35.
    A. Gorodnik and A. Nevo, The ergodic theory of lattice subgroups. Annals of Mathematics Studies, 172. Princeton University Press, Princeton, NJ, 2010.Google Scholar
  36. 36.
    A. Gorodnik and A. Nevo, Counting lattice points. J. Reine Angew. Math. 663 (2012), 127–176.MathSciNetzbMATHGoogle Scholar
  37. 37.
    A. Gorodnik and F. Ramirez, Limit theorems for rank-one Lie groups. Proc. Amer. Math. Soc. 142 (2014), no. 4, 1359–1369.MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    A. Gorodnik and R. Spatzier, Exponential mixing of nilmanifold automorphisms, J. Anal. Math. 123 (2014), 355–396.MathSciNetzbMATHCrossRefGoogle Scholar
  39. 39.
    A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms. Acta Math. 215 (2015), no. 1, 127–159.MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    S. Gouëzel, Limit theorems in dynamical systems using the spectral method. Hyperbolic dynamics, fluctuations and large deviations, 161–193, Proc. Sympos. Pure Math., 89, Amer. Math. Soc., Providence, RI, 2015.Google Scholar
  41. 41.
    R. L. Graham, Recent trends in Euclidean Ramsey theory. Trends in discrete mathematics. Discrete Math. 136 (1994), no. 1–3, 119–127.MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    Harish-Chandra, Spherical functions on a semisimple Lie group. I. Amer. J. Math. 80 (1958), 241–310.Google Scholar
  43. 43.
    C. Herz, Sur le phénomène de Kunze-Stein. C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A491–A493.Google Scholar
  44. 44.
    B. Host, Mixing of all orders and pairwise independent joinings of systems with singular spectrum. Israel J. Math. 76 (1991), no. 3, 289–298.MathSciNetzbMATHCrossRefGoogle Scholar
  45. 45.
    B. Host and B. Kra, Nonconventional ergodic averages and nilmanifolds. Ann. Math. 161 (2005), no. 1, 397–488.MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    R. Howe, On a notion of rank for unitary representations of the classical groups. Harmonic analysis and group representations, 223–331, Liguori, Naples, 1982.Google Scholar
  47. 47.
    R. Howe and C. Moore, Asymptotic properties of unitary representations. J. Funct. Anal. 32 (1979), no. 1, 72–96.MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    R. Howe and E.-C. Tan, Nonabelian harmonic analysis. Applications of SL(2,R). Universitext. Springer-Verlag, New York, 1992.Google Scholar
  49. 49.
    H. Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen, I, Math. Ann. 138 (1959), 1–26; II Math. Ann., 142 (1961), 385–398 and 143 (1961), 463–464.Google Scholar
  50. 50.
    S. Kalikow, Twofold mixing implies threefold mixing for rank one transformations. Ergodic Theory Dynam. Systems 4 (1984), no. 2, 237–259.MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    A. Katok and R. Spatzier, First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity. Inst. Hautes Études Sci. Publ. Math. 79 (1994), 131–156.MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    A. Knapp, Representation theory of semisimple groups. An overview based on examples. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 2001.Google Scholar
  53. 53.
    I. Konstantoulas, Effective decay of multiple correlations in semidirect product actions. J. Mod. Dyn. 10 (2016), 81–111.MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    S. Le Borgne, Martingales in hyperbolic geometry. Analytic and probabilistic approaches to dynamics in negative curvature, 1–63, Springer INdAM Ser., 9, Springer, Cham, 2014.Google Scholar
  55. 55.
    F. Ledrappier, Un champ markovien peut être d’entropie nulle et mélangeant. C. R. Acad. Sci. Paris Sér. A-B, 287 (1978), A561–A563.Google Scholar
  56. 56.
    A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergodic Theory Dynam. Systems 25 (2005), no. 1, 201–213.MathSciNetzbMATHCrossRefGoogle Scholar
  57. 57.
    V. P. Leonov, The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes. Dokl. Akad. Nauk SSSR 133 (1960), 523–526.MathSciNetGoogle Scholar
  58. 58.
    V. P. Leonov, On the central limit theorem for ergodic endomorphisms of compact commutative groups. Dokl. Akad. Nauk SSSR 135 (1960), 258–261.MathSciNetGoogle Scholar
  59. 59.
    V. P. Leonov and A.N. Shiryaev, On a method of calculations of semi-invariants. Theory of Probability and its Applications 4 (1959), 319–329.MathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    M. Levin, Central limit theorem for \(Z^{d+}\)-actions by toral endomorphisms. Electron. J. Probab. 18 (2013), no. 35, 42 pp.Google Scholar
  61. 61.
    J.-S. Li, The minimal decay of matrix coefficients for classical groups. Harmonic analysis in China, 146–169, Math. Appl., 327, Kluwer Acad. Publ., Dordrecht, 1995.Google Scholar
  62. 62.
    J.-S. Li and C.-B. Zhu, On the decay of matrix coefficients for exceptional groups. Math. Ann. 305 (1996), no. 2, 249–270.MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms. Ergodic Theory Dynamical Systems 2 (1982), no. 1, 49–68.MathSciNetzbMATHCrossRefGoogle Scholar
  64. 64.
    B. Marcus, The horocycle flow is mixing of all degrees. Invent. Math. 46 (1978), no. 3, 201–209.MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    G. Margulis, Certain applications of ergodic theory to the investigation of manifolds of negative curvature. Funkcional. Anal. i Prilozen. 3 (1969), no. 4, 89–90.MathSciNetGoogle Scholar
  66. 66.
    G. Margulis, On some aspects of the theory of Anosov systems. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2004zbMATHCrossRefGoogle Scholar
  67. 67.
    F. Maucourant, Homogeneous asymptotic limits of Haar measures of semisimple linear groups and their lattices. Duke Math. J. 136 (2007), no. 2, 357–399.MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    F. I. Mautner, Geodesic flows on symmetric Riemann spaces. Ann. Math. 65 (1957), 416–431.MathSciNetzbMATHCrossRefGoogle Scholar
  69. 69.
    C. Moore, Ergodicity of flows on homogeneous spaces. Amer. J. Math. 88 (1966), 154–178.MathSciNetzbMATHCrossRefGoogle Scholar
  70. 70.
    C. Moore, The Mautner phenomenon for general unitary representations. Pacific J. Math. 86 (1980), no. 1, 155–169.MathSciNetzbMATHCrossRefGoogle Scholar
  71. 71.
    C. Moore, Exponential decay of correlation coefficients for geodesic flows. Group representations, ergodic theory, operator algebras, and mathematical physics (Berkeley, Calif., 1984), 163–181, Math. Sci. Res. Inst. Publ., 6, Springer, New York, 1987.Google Scholar
  72. 72.
    S. Mozes, Mixing of all orders of Lie groups actions. Invent. Math. 107 (1992), no. 2, 235–241; erratum: Invent. Math. 119 (1995), no. 2, 399.Google Scholar
  73. 73.
    H. Oh, Tempered subgroups and representations with minimal decay of matrix coefficients. Bull. Soc. Math. France 126 (1998), no. 3, 355–380.MathSciNetzbMATHCrossRefGoogle Scholar
  74. 74.
    H. Oh, Uniform pointwise bounds for matrix coefficients of unitary representations and applications to Kazhdan constants. Duke Math. J. 113 (2002), no. 1, 133–192.MathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    D. Ornstein and B. Weiss, Geodesic flows are Bernoullian, Israel J. Math. 14 (1973), 184–198.MathSciNetzbMATHCrossRefGoogle Scholar
  76. 76.
    S. J. Patterson, A lattice-point problem in hyperbolic space. Mathematika 22 (1975), no. 1, 81–88; erratum: Mathematika 23 (1976), no. 2, 227.MathSciNetzbMATHCrossRefGoogle Scholar
  77. 77.
    F. Pène, Averaging method for differential equations perturbed by dynamical systems. ESAIM Probab. Statist. 6 (2002), 33–88.MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    M. Ratner, The rate of mixing for geodesic and horocycle flows. Ergodic Theory Dynam. Systems 7 (1987), 267–288.MathSciNetzbMATHCrossRefGoogle Scholar
  79. 79.
    V. A. Rokhlin, On endomorphisms of compact commutative groups. Izvestiya Akad. Nauk SSSR. Ser. Mat. 13 (1949), 329–340.MathSciNetGoogle Scholar
  80. 80.
    V. V. Ryzhikov, Joinings and multiple mixing of the actions of finite rank. Funct. Anal. Appl. 27 (1993), no. 2, 128–140.MathSciNetzbMATHCrossRefGoogle Scholar
  81. 81.
    K. Schmidt, Dynamical systems of algebraic origin. Progress in Mathematics, 128. Birkhäuser Verlag, Basel, 1995.Google Scholar
  82. 82.
    K. Schmidt and T. Ward, Mixing automorphisms of compact groups and a theorem of Schlickewei. Invent. Math. 111 (1993), no. 1, 69–76.MathSciNetzbMATHCrossRefGoogle Scholar
  83. 83.
    I. E. Segal and J. von Neumann, A theorem on unitary representations of semisimple Lie groups. Ann. Math. 52 (1950), 509–517.MathSciNetzbMATHCrossRefGoogle Scholar
  84. 84.
    Ya. Sinai, The central limit theorem for geodesic flows on manifolds of constant negative curvature. Soviet Math. Dokl. 1 (1960), 938–987.MathSciNetGoogle Scholar
  85. 85.
    Ya. Sinai, Probabilistic concepts in ergodic theory. 1963 Proc. Internat. Congr. Mathematicians (Stockholm, 1962) pp. 540–559.Google Scholar
  86. 86.
    A. Starkov, Multiple mixing of homogeneous flows. Dokl. Akad. Nauk 333 (1993), no. 4, 442–445; translation in Russian Acad. Sci. Dokl. Math. 48 (1994), no. 3, 573–578.Google Scholar
  87. 87.
    E. Szemerédi, On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975), 199–245.MathSciNetzbMATHCrossRefGoogle Scholar
  88. 88.
    M. Viana, Stochastic dynamics of deterministic systems. Lecture Notes XXI Bras. Math. Colloq. IMPA, Rio de Janeiro, 1997.Google Scholar
  89. 89.
    S. P. Wang, On the Mautner phenomenon and groups with property (T). Amer. J. Math. 104 (1982), no. 6, 1191–1210.MathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    S.P. Wang, The Mautner phenomenon for p-adic Lie groups. Math. Z. 185 (1984), no. 3, 403–412.MathSciNetzbMATHCrossRefGoogle Scholar
  91. 91.
    Z. Wang, Uniform pointwise bounds for matrix coefficients of unitary representations on semidirect products. J. Funct. Anal. 267 (2014), no. 1, 15–79.MathSciNetzbMATHCrossRefGoogle Scholar
  92. 92.
    G. Warner, Harmonic analysis on semi-simple Lie groups. I. Die Grundlehren der mathematischen Wissenschaften, Band 188. Springer-Verlag, New York-Heidelberg, 1972.Google Scholar
  93. 93.
    G. Warner, Harmonic analysis on semi-simple Lie groups. II. Die Grundlehren der mathematischen Wissenschaften, Band 189. Springer-Verlag, New York-Heidelberg, 1972.Google Scholar
  94. 94.
    R. Yassawi, Multiple mixing and local rank group actions. Ergodic Theory Dynam. Systems 23 (2003), no. 4, 1275–1304.MathSciNetzbMATHCrossRefGoogle Scholar
  95. 95.
    T. Ziegler, A non-conventional ergodic theorem for a nilsystem. Ergodic Theory Dynam. Systems 25 (2005), no. 4, 1357–1370.MathSciNetzbMATHCrossRefGoogle Scholar
  96. 96.
    T. Ziegler, Nilfactors of \(R^m\)-actions and configurations in sets of positive upper density in \(R^m\). J. Anal. Math. 99 (2006), 249–266.MathSciNetzbMATHCrossRefGoogle Scholar
  97. 97.
    T. Ziegler, Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (2007), no. 1, 53–97.MathSciNetzbMATHCrossRefGoogle Scholar
  98. 98.
    R. Zimmer, Ergodic theory and semisimple groups. Monographs in Mathematics, 81. Birkhäuser Verlag, Basel, 1984.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.University of ZurichZurichSwitzerland

Personalised recommendations