• Elon Lindenstrauss
Conference paper
Part of the NATO Science Series book series (NAII, volume 237)


We give examples of how classifying invariant probability measures for specific algebraic actions can be used to prove density and equidistribution results in number theory.


Probability Measure Invariant Measure Topological Entropy Measure Classification Invariant Probability Measure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. Anantharaman, N. (2004) Entropy and the localization of eigenfunctions, preprint.Google Scholar
  2. Bourgain, J. and Lindenstrauss, E. (2003) Entropy of quantum limits, Comm. Math. Phys. 233, 153–171.MATHCrossRefMathSciNetGoogle Scholar
  3. Burger, M. (1990) Horocycle flow on geometrically finite surfaces, Duke Math. J. 61, 779–803.MATHCrossRefMathSciNetGoogle Scholar
  4. Burger, M. and Sarnak, P. (1991) Ramanujan duals. II, Invent. Math. 106, 1–11.MATHCrossRefMathSciNetGoogle Scholar
  5. Buzzi, J. (1997) Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100, 125–161.MATHMathSciNetGoogle Scholar
  6. Cassels, J. W. S. and Swinnerton-Dyer, H. P. F. (1955) On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. Roy. Soc. London. Ser. A. 248, 73–96.MathSciNetMATHGoogle Scholar
  7. Colin de Verdière, Y. (1985) Ergodicité et fonctions propres du laplacien, Comm. Math. Phys. 102, 497–502.MATHCrossRefMathSciNetGoogle Scholar
  8. Dani, S. G. (1978) Invariant measures of horospherical flows on noncompact homogeneous spaces, Invent. Math. 47, 101–138.MATHCrossRefMathSciNetGoogle Scholar
  9. Dani, S. G. and Margulis, G. A. (1989) Values of quadratic forms at primitive integral points, Invent. Math. 98, 405–424.MATHCrossRefMathSciNetGoogle Scholar
  10. Dani, S. G. and Margulis, G. A. (1990) Values of quadratic forms at integral points: an elementary approach, Enseign. Math. (2) 36, 143–174.MathSciNetMATHGoogle Scholar
  11. Dani, S. G. and Margulis, G. A. (1993) Limit distributions of orbits of unipotent flows and values of quadratic forms, In I. M. Gel’fand Seminar, Vol. 16 of Adv. Soviet Math., Providence, RI, Amer. Math. Soc., pp. 91–137.Google Scholar
  12. De Bièvre, S. (2006) An introduction to quantum equidistribution, in this book.Google Scholar
  13. Donnelly, H. (2003) Quantum unique ergodicity, Proc. Amer. Math. Soc. 131, 2945–2951.MATHCrossRefMathSciNetGoogle Scholar
  14. Einsiedler, M. (2006) Ratner’s theorem on SL(2,R)-invariant measures, arXiv:math.DS/0603483.Google Scholar
  15. Einsiedler, M. and Katok, A. (2003) Invariant measures on G/Γ for split simple Lie groups G, Comm. Pure Appl. Math. 56, 1184–1221, dedicated to the memory of Jürgen K. Moser.MATHCrossRefMathSciNetGoogle Scholar
  16. Einsiedler, M., Katok, A., and Lindenstrauss, E. (2004) Invariant measures and the set of exceptions to Littlewood’s conjecture, Ann. of Math. (2), to appear.Google Scholar
  17. Einsiedler, M. and Lindenstrauss, E. (2006) Diagonal flows on locally homogeneous spaces and number theory, In Proceedings of the International Congress of Mathematicians 2006, to appear.Google Scholar
  18. Einsiedler, M., Lindenstrauss, E., Michel, P., and Venkatesh, A. (2006a) Distribution properties of compact torus orbits. III. Duke’s theorem for cubic fields, in preparation.Google Scholar
  19. Einsiedler, M., Lindenstrauss, E., Michel, P., and Venkatesh, A. (2006b) Distribution properties of compact torus orbits on homogeneous spaces, in preparation.Google Scholar
  20. Elkies, N. D. and McMullen, C. T. (2004) Gaps in √n mod 1 and ergodic theory, Duke Math. J. 123, 95–139.MATHCrossRefMathSciNetGoogle Scholar
  21. Ellenberg, J. and Venkatesh, A. (2006) Local-global principles for representations of quadratic forms, arXiv:math.NT/0604232.Google Scholar
  22. Eskin, A., Margulis, G., and Mozes, S. (1998) Upper bounds and asymptotics in a quantitative version of the Oppenheim conjecture, Ann. of Math. (2) 147, 93–141.MATHCrossRefMathSciNetGoogle Scholar
  23. Eskin, A., Mozes, S., and Shah, N. (1996) Unipotent flows and counting lattice points on homogeneous varieties, Ann. of Math. (2) 143, 253–299.MATHCrossRefMathSciNetGoogle Scholar
  24. Eskin, A. and Oh, H. (2006a) Ergodic theoretic proof of equidistribution of Hecke points, Ergodic Theory Dynam. Systems 26, 163–167.MATHCrossRefMathSciNetGoogle Scholar
  25. Eskin, A. and Oh, H. (2006b) Representations of integers by an invariant polynomial and unipotent flows, preprint.Google Scholar
  26. Furstenberg, H. (1967) Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1, 1–49.MATHCrossRefMathSciNetGoogle Scholar
  27. Furstenberg, H. (1973) The unique ergodicity of the horocycle flow, In Recent advances in topological dynamics, Vol. 318 of Lecture Notes in Math., New Haven, CO, 1972, pp. 95–115, Berlin, Springer.CrossRefGoogle Scholar
  28. Furstenberg, H. (1981) Recurrence in ergodic theory and combinatorial number theory, M. B. Porter Lectures, Princeton, NJ, Princeton Univ. Press.MATHGoogle Scholar
  29. Glasner, E. (2003) Ergodic theory via joinings, Vol. 101 of Math. Surveys Monogr., Providence, RI, Amer. Math. Soc.Google Scholar
  30. Granville, A. and Rudnick, Z. (2006) Uniform distribution, in this book.Google Scholar
  31. Katok, A. and Hasselblatt, B. (1995) Introduction to the modern theory of dynamical systems, Vol. 54 of Encyclopedia Math. Appl., Cambridge, Cambridge Univ. Press, with a supplementary chapter by A. Katok and L. Mendoza.Google Scholar
  32. Katok, A. and Spatzier, R. J. (1996) Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16, 751–778.MATHMathSciNetGoogle Scholar
  33. Kleinbock, D., Shah, N., and Starkov, A. (2002) Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, In Handbook of dynamical systems, Vol. 1A, Amsterdam North-Holland, pp. 813–930.Google Scholar
  34. Kleinbock, D. Y. and Margulis, G. A. (1998) Flows on homogeneous spaces and Diophantine approximation on manifolds, Ann. of Math. (2) 148, 339–360.MATHCrossRefMathSciNetGoogle Scholar
  35. Ledrappier, F. and Sarig, O. (2005) Invariant measures for the horocycle flow on periodic hyperbolic surfaces, Electron. Res. Announc. Amer. Math. Soc. 11, 89–94.CrossRefMathSciNetMATHGoogle Scholar
  36. Lindenstrauss, E. (2005) Rigidity of multiparameter actions, Israel J. Math. 149, 199–226.MATHMathSciNetGoogle Scholar
  37. Lindenstrauss, E. (2006a) Arithmetic quantum unique ergodicity and adelic dynamics, In Proceedings of Current Developments in Mathematics Conference, Harvard, 2004, to appear.Google Scholar
  38. Lindenstrauss, E. (2006b) Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163, 165–219.MATHMathSciNetGoogle Scholar
  39. Lindenstrauss, E. and Schmidt, K. (2005) Symbolic representations of nonexpansive group automorphisms, Israel J. Math. 149, 227–266.MATHMathSciNetGoogle Scholar
  40. Lindenstrauss, E. and Weiss, B. (2000) Mean topological dimension, Israel J. Math. 115, 1–24.MATHMathSciNetGoogle Scholar
  41. Luo,W. and Sarnak, P. (2004) Quantum variance for Hecke eigenforms, Ann. Sci. École Norm. Sup. (4) 37, 769–799.MATHCrossRefMathSciNetGoogle Scholar
  42. Margulis, G. (2000) Problems and conjectures in rigidity theory, In Mathematics: frontiers and perspectives, Providence, RI, Amer. Math. Soc., pp. 161–174.Google Scholar
  43. Margulis, G. A. (1971) The action of unipotent groups in a lattice space, Mat. Sb. (N.S.) 86, 552–556.MathSciNetGoogle Scholar
  44. Margulis, G. A. (1989) Discrete subgroups and ergodic theory, In Number theory, trace formulas and discrete groups, Oslo, 1987, pp. 377–398, Boston, MA, Academic Press.Google Scholar
  45. Margulis, G. A. and Tomanov, G. M. (1994) Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Invent. Math. 116, 347–392.MATHCrossRefMathSciNetGoogle Scholar
  46. Margulis, G. A. and Tomanov, G. M. (1996) Measure rigidity for almost linear groups and its applications, J. Anal. Math. 69, 25–54.MATHMathSciNetGoogle Scholar
  47. Markloff, J. (2006) Distribution modulo one and Ratner’s theorem, in this book.Google Scholar
  48. Morris, D. W. (2005) Ratner’s theorems on unipotent flows, Chicago Lectures in Math., Chicago, IL, Univ. Chicago Press.MATHGoogle Scholar
  49. Mozes, S. and Shah, N. (1995) On the space of ergodic invariant measures of unipotent flows, Ergodic Theory Dynam. Systems 15, 149–159.MATHMathSciNetCrossRefGoogle Scholar
  50. Ornstein, D. S. and Weiss, B. (1987) Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48, 1–141.MATHMathSciNetGoogle Scholar
  51. Raghunathan, M. S. (1972) Discrete subgroups of Lie groups, Vol. 68 of Ergeb. Math. Grenzgeb., New York, Springer.Google Scholar
  52. Ratner, M. (1982) Rigidity of horocycle flows, Ann. of Math. (2) 115, 597–614.CrossRefMathSciNetGoogle Scholar
  53. Ratner, M. (1991a) On Raghunathan’s measure conjecture, Ann. of Math. (2) 134, 545–607.CrossRefMathSciNetGoogle Scholar
  54. Ratner, M. (1991b) Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Math. J. 63, 235–280.MATHCrossRefMathSciNetGoogle Scholar
  55. Ratner, M. (1992) Raghunathan’s conjectures for SL(2,R), Israel J. Math. 80, 1–31.MATHMathSciNetGoogle Scholar
  56. Ratner, M. (1995) Raghunathan’s conjectures for Cartesian products of real and p-adic Lie groups, Duke Math. J. 77, 275–382.MATHCrossRefMathSciNetGoogle Scholar
  57. Rees, M. (1982) Some R2-anosov flows, unpublished.Google Scholar
  58. Roblin, T. (2003) Ergodicité et équidistribution en courbure négative, Vol. 95 of Mém. Soc. Math. Fr. (N.S.), Paris, Soc. Math. France.Google Scholar
  59. Rudnick, Z. (2006) The arithmetic theory of quantum maps, in this book.Google Scholar
  60. Rudnick, Z. and Sarnak, P. (1994) The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161, 195–213.MATHCrossRefMathSciNetGoogle Scholar
  61. Rudolph, D. J. (1990) Fundamentals of measurable dynamics, Oxford Sci. Publ., New York, Oxford Univ. Press.MATHGoogle Scholar
  62. Schmidt, W. M. (1983) Open problems in Diophantine approximation, In Diophantine approximations and transcendental numbers, Vol. 31 of Progr. Math., Luminy, 1982, pp. 271–287, Boston, MA, Birkhäuser Boston.Google Scholar
  63. Shah, N. A. (1996) Limit distributions of expanding translates of certain orbits on homogeneous spaces, Proc. Indian Acad. Sci. Math. Sci. 106, 105–125.MATHMathSciNetGoogle Scholar
  64. Silberman, L. (2005) Arithmetic quantum chaos on locally symmetric spaces, Ph.D. thesis, Princeton University.Google Scholar
  65. Silberman, L. and Venkatesh, A. (2004) On quantum unique ergodicity for locally symmetric spaces. I. A micro local lift, preprint.Google Scholar
  66. Šnirel’man, A. I. (1974) Ergodic properties of eigenfunctions, Uspehi Mat. Nauk 29, 181–182.MATHGoogle Scholar
  67. Vatsal, V. (2002) Uniform distribution of Heegner points, Invent. Math. 148, 1–46.MATHCrossRefMathSciNetGoogle Scholar
  68. Venkatesh, A. (2006) Spectral theory of automorphic forms, a very brief introduction, in this book.Google Scholar
  69. Watson, T. (2001) Rankin triple products and quantum chaos, Ph.D. thesis, Princeton University.Google Scholar
  70. Zelditch, S. (1987) Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Math. J. 55, 919–941.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer 2007

Authors and Affiliations

  • Elon Lindenstrauss
    • 1
  1. 1.Princeton UniversityPrinceton

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