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Introduction to random walks on homogeneous spaces

Abstract

Let a 0 and a 1 be two matrices in SL(2, \({\mathbb{Z}}\)) which span a non-solvable group. Let x 0 be an irrational point on the torus \({\mathbb{T}^2}\). We toss a 0 or a 1, apply it to x 0, get another irrational point x 1, do it again to x 1, get a point x 2, and again. This random trajectory is equidistributed on the torus. This phenomenon is quite general on any finite volume homogeneous space.

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References

  1. 1.

    Auslander L.: Ergodic automorphisms. Amer. Math. Monthly 77, 1–19 (1970)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Y. Benoist and J.-F. Quint, Mesures stationnaires et fermés invariants des espaces homogènes, C. R. Math. Acad. Sci. Paris, 347 (2009), 9–13; Ann. of Math. (2), 174 (2011), 1111–1162.

  3. 3.

    Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (II), preprint, 2011.

  4. 4.

    Y. Benoist and J.-F. Quint, Stationary measures and invariant subsets of homogeneous spaces (III), preprint, 2011.

  5. 5.

    Benoist Y., Quint J.-F.: Random walks on finite volume homogeneous spaces. Invent. Math. 187, 37–59 (2012)

    MathSciNet  MATH  Article  Google Scholar 

  6. 6.

    P. Bougerol and J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators, Progr. Probab. Statist., 8, Birkhäuser Boston, Boston, MA, 1985.

  7. 7.

    Bourgain J., Furman A., Lindenstrauss E., Mozes S.: Invariant measures and stiffness for non-Abelian groups of toral automorphisms. C. R. Math. Acad. Sci. Paris 344, 737–742 (2007)

    MathSciNet  MATH  Article  Google Scholar 

  8. 8.

    Breiman L.: The strong law of large numbers for a class of Markov chains. Ann. Math. Statist. 31, 801–803 (1960)

    MathSciNet  MATH  Article  Google Scholar 

  9. 9.

    A.I. Bufetov, Convergence of spherical averages for actions of free groups, Ann. of Math. (2), 155 (2002), 929–944.

    Google Scholar 

  10. 10.

    Clozel L., Oh H., Ullmo E.: Hecke operators and equidistribution of Hecke points. Invent. Math. 144, 327–351 (2001)

    MathSciNet  MATH  Article  Google Scholar 

  11. 11.

    Dani S.G., Margulis G.: Limit distributions of orbits of unipotent flows and values of quadratic forms. Adv. Soviet Math. 16, 91–137 (1993)

    MathSciNet  Google Scholar 

  12. 12.

    M. Einsiedler, A. Katok and E. Lindenstrauss, Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2), 164 (2006), 513–560.

    Google Scholar 

  13. 13.

    Einsiedler M., Margulis G., Venkatesh A.: Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces. Invent. Math. 177, 137–212 (2009)

    MathSciNet  MATH  Article  Google Scholar 

  14. 14.

    A. Eskin and G. Margulis, Recurrence properties of random walks on finite volume homogeneous manifolds, In: Random Walks and Geometry, Walter de Gruiter, Berlin, 2004, pp. 431–444.

  15. 15.

    A. Eskin, S. Mozes and N.A. Shah, Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2), 143 (1996), 253–299.

    Google Scholar 

  16. 16.

    Eskin A., Oh H.: Ergodic theoretic proof of equidistribution of Hecke points. Ergodic Theory Dynam. Systems 26, 163–167 (2006)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Furstenberg H.: Noncommuting random products. Trans. Amer. Math. Soc. 108, 377–428 (1963)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    H. Furstenberg, Stiffness of group actions, In: Lie Groups and Ergodic Theory, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 105–117.

  19. 19.

    Guivarc’h Y., Raugi A.: Frontière de Furstenberg, propriétés de contraction et théorèmes de convergence. Z. Wahrsch. Verw. Gebiete 69, 187–242 (1985)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Guivarc’h Y., Starkov A.N.: Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms. Ergodic Theory Dynam. Systems 24, 767–802 (2004)

    MathSciNet  MATH  Article  Google Scholar 

  21. 21.

    Ledrappier F.: Mesures stationnaires sur les espaces homogènes. d’après Y. Benoist et J.-F. Quint, Seminaire Bourbaki 1058, 1–17 (2012)

    Google Scholar 

  22. 22.

    É. Le Page, Théorèmes limites pour les produits de matrices aléatoires, In: Probability Measures on Groups, Lecture Notes in Math., 928, Springer-Verlag, 1982, pp. 258–303.

  23. 23.

    G. Margulis, Discrete Subgroups of Semisimple Lie Groups, Ergeb. Math. Grenzgeb. (3), 17, Springer-Verlag, 1991.

  24. 24.

    G. Margulis, Problems and conjectures in rigidity theory, In: Mathematics: Frontiers and Perspectives, Amer. Math. Soc., 2000, pp. 161–174.

  25. 25.

    Margulis G., Tomanov G.M.: Invariant measures for actions of unipotent groups over local fields on homogeneous spaces. Invent. Math. 116, 347–392 (1994)

    MathSciNet  MATH  Article  Google Scholar 

  26. 26.

    Mozes S., Shah N.A.: On the space of ergodic invariant measures of unipotent flows. Ergodic Theory Dynam. Systems 15, 149–159 (1995)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Muchnik R.: Semigroup actions on \({\mathbb{T}^ n}\). Geom. Dedicata 110, 1–47 (2005)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    M.S. Raghunathan, Discrete Subgroups of Lie Groups, Ergeb. Math. Grenzgeb., 68, Springer-Verlag, 1972.

  29. 29.

    M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2), 134 (1991), 545–607.

    Google Scholar 

  30. 30.

    Ratner M.: Raghunathan’s topological conjecture and distributions of unipotent flows. Duke Math. J. 63, 235–280 (1991)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Ratner M.: Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups. Duke Math. J. 77, 275–382 (1995)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    N.A. Shah, Invariant measures and orbit closures on homogeneous spaces for actions of subgroups generated by unipotent elements, In: Lie Groups and Ergodic Theory, Tata Inst. Fund. Res. Stud. Math., 14, Tata Inst. Fund. Res., Bombay, 1998, pp. 229–271.

  33. 33.

    R.J. Zimmer, Ergodic Theory and Semisimple Groups, Monogr. Math., 81, Birkhäuser, Basel, 1984.

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Correspondence to Yves Benoist.

Additional information

This article is based on the 10th Takagi Lectures that the author delivered at Research Institute for Mathematical Sciences, Kyoto University on May 26, 2012.

Communicated by: Toshiyuki Kobayashi

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Benoist, Y., Quint, JF. Introduction to random walks on homogeneous spaces. Jpn. J. Math. 7, 135–166 (2012). https://doi.org/10.1007/s11537-012-1220-9

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Keywords and phrases

  • Lie groups
  • discrete subgroups
  • homogeneous dynamics
  • random walk

Mathematics Subject Classification (2010)

  • 22E40
  • 37C85
  • 60J05