Skip to main content
SpringerLink
Log in

Rigidity of multiparameter actions

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We survey some of the recent progress in understanding diagonalizable algebraic actions of multidimensional abelian groups, a subject pioneered by Hillel Furstenberg.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Aaronson,An Introduction to Infinite Ergodic Theory, Volume 50 of Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 1997.

    Google Scholar 

  2. R. L. Adler and B. Weiss,Entropy, a complete metric invariant for automorphisms of the torus, Proceedings of the National Academy of Sciences of the United States of America57 (1967), 1573–1576.

    Article  MATH  MathSciNet  Google Scholar 

  3. D. Berend,Multi-invariant sets on tori, Transactions of the American Mathematical Society280 (1983), 509–532.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Berend,Minimal sets on tori, Ergodic Theory and Dynamical Systems4 (1984), 499–507.

    MATH  MathSciNet  Google Scholar 

  5. D. Berend,Multi-invariant sets on compact abelian groups, Transactions of the American Mathematical Society286 (1984), 505–535.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Bourgain and E. Lindenstrauss,Entropy of quantum limits, Communications in Mathematical Physics233 (2003), 153–171.

    Article  MATH  MathSciNet  Google Scholar 

  7. Y. Colin de Verdière,Ergodicité et fonctions propres du laplacien, Communications in Mathematical Physics102 (1985), 497–502.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. W. S. Cassels and H. P. F. Swinnerton-Dyer,On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philosophical Transactions of the Royal Society of London, Series A248 (1955), 73–96.

    Article  MathSciNet  MATH  Google Scholar 

  9. S. G. Dani and G. A. Margulis,Values of quadratic forms at integral points: an elementary approach, L’Enseignement Mathématique (2)36 (1990), 143–174.

    MathSciNet  MATH  Google Scholar 

  10. M. Einsiedler and A. Katok,Invariant measures on G/Γfor split simple Lie groups G, Communications on Pure and Applied Mathematics56 (2003), 1184–1221. Dedicated to the memory of Jürgen K. Moser.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Einsiedler, A. Katok and E. Lindenstrauss,Invariant measures and the set of exceptions to Littlewood’s conjecture, Annals of Mathematics, to appear.

  12. M. Einsiedler and E. Lindenstrauss,Rigidity properties of Z d-actions on tori and solenoids, Electronic Research Announcements of the American Mathematical Society9 (2003), 99–110.

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Feldman,A generalization of a result of R. Lyons about measures on [0, 1), Israel Journal of Mathematics81 (1993), 281–287.

    Article  MATH  MathSciNet  Google Scholar 

  14. H. Furstenberg,Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Mathematical Systems Theory1 (1967), 1–49.

    Article  MATH  MathSciNet  Google Scholar 

  15. H. Furstenberg,The unique ergodicity of the horocycle flow, inRecent Advances in Topological Dynamics (Proc. Conf., Yale Univ., New Haven, Conn., 1972; in honor of Gustav Arnold Hedlund), Lecture Notes in Mathematics318, Springer, Berlin, 1973, pp. 95–115.

    Google Scholar 

  16. G. Hedlund,Dynamics of geodesic flows, Bulletin of the American Mathematical Society45 (1939), 241–260.

    MATH  MathSciNet  Google Scholar 

  17. B. Host,Nombres normaux, entropie, translations, Israel Journal of Mathematics91 (1995), 419–428.

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Host,Some results of uniform distribution in the multidimensional torus, Ergodic Theory and Dynamical Systems20 (2000), 439–452.

    Article  MATH  MathSciNet  Google Scholar 

  19. W. Hurewicz,Ergodic theorem without invariant measure, Annals of Mathematics (2)45 (1944), 192–206.

    Article  MathSciNet  Google Scholar 

  20. D. Jakobson,Equidistribution of cusp forms on PSL2(ℤ)\PSL2(ℝ), Annales de l’Institut Fourier (Grenoble)47 (1997), 967–984.

    MATH  MathSciNet  Google Scholar 

  21. A. S. A. Johnson,Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers, Israel Journal of Mathematics77 (1992), 211–240.

    MATH  MathSciNet  Google Scholar 

  22. A. Johnson and D. J. Rudolph,Convergence under × q of × p invariant measures on the circle, Advances in Mathematics115 (1995), 117–140.

    Article  MATH  MathSciNet  Google Scholar 

  23. B. Kalinin and A. Katok,Invariant measures for actions of higher rank abelian groups, inSmooth Ergodic Theory and its Applications (Seattle, WA, 1999), Volume 69 of Proceedings of Symposia in Pure Mathematics, American Mathematical Society, Providence, RI, 2001, pp. 593–637.

    Google Scholar 

  24. B. Kalinin and A. Katok,Measurable rigidity and disjointness for ℤk actions by toral automorphisms, Ergodic Theory and Dynamical Systems22 (2002), 507–523.

    Article  MATH  MathSciNet  Google Scholar 

  25. A. Katok, S. Katok and K. Schmidt,Rigidity of measurable structure for ℤd actions by automorphisms of a torus, Commentarii Mathematici Helvetici77 (2002), 718–745.

    Article  MATH  MathSciNet  Google Scholar 

  26. B. Kalinin and R. Spatzier,Rigidity of the measurable structure for algebraic actions of higher rank abelian groups, arXiv:math.DS/0207137; Ergodic Theory and Dynamical Systems25 (2005), 175–200.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Katok and R. J. Spatzier,Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory and Dynamical Systems16 (1996), 751–778.

    Article  MATH  MathSciNet  Google Scholar 

  28. A. Katok and R. J. Spatzier,Corrections to: “Invariant measures for higher-rank hyperbolic abelian actions” [Ergodic Theory and Dynamical Systems 16 (1996), 751–778; MR 97d:58116], Ergodic Theory and Dynamical Systems18 (1998), 503–507.

    Article  MathSciNet  Google Scholar 

  29. D. Kleinbock, N. Shah and A. Starkov,Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory, inHandbook of Dynamical Systems, Vol. 1A, North-Holland, Amsterdam, 2002, pp. 813–930.

    Google Scholar 

  30. R. Kenyon and A. Vershik,Arithmetic construction of sofic partitions of hyperbolic toral automorphisms, Ergodic Theory and Dynamical Systems18 (1998), 357–372.

    Article  MATH  MathSciNet  Google Scholar 

  31. E. Lindenstrauss,On quantum unique ergodicity for Γ\ℍ × ℍ, International Mathematics Research Notices17 (2001), 913–933.

    Article  MathSciNet  Google Scholar 

  32. E. Lindenstrauss,p-adic foliation and equidistribution, Israel Journal of Mathematics122 (2001), 29–42.

    MATH  MathSciNet  Google Scholar 

  33. E. Lindenstrauss,Invariant measures and arithmetic quantum unique ergodicity, Annals of Mathematics, to appear.

  34. E. Lindenstrauss,Recurrent measures and measure rigidity, inProceeding of the II Workshop on Dynamics and Randomness, Santiago de Chile, Dec. 9–13, 2002 (A. Maass, S. Martinez and J. San Martin, eds.), to appear.

  35. F. Ledrappier and E. Lindenstrauss,On the projections of measures invariant under the geodesic flow, International Mathematics Research Notices9 (2003), 511–526.

    Article  MathSciNet  Google Scholar 

  36. E. Lindenstrauss, D. Meiri and Y. Peres,Entropy of convolutions on the circle, Annals of Mathematics (2)149 (1999),871–904.

    Article  MATH  MathSciNet  Google Scholar 

  37. W. Z. Luo and P. Sarnak,Quantum ergodicity of eigenfunctions on PSL2(ℤ)\ℍ, Publications Mathématiques de l’Institut des Hautes Études Scientifiques81 (1995), 207–237.

    Article  MATH  MathSciNet  Google Scholar 

  38. E. Lindenstrauss and K. Schmidt,Invariant sets and measures of nonexpansive group automorphisms, Israel Journal of Mathematics144 (2004), 29–60.

    MATH  MathSciNet  Google Scholar 

  39. E. Lindenstrauss and B. Weiss,On sets invariant under the action of the diagonal group, Ergodic Theory and Dynamical Systems21 (2001), 1481–1500.

    Article  MATH  MathSciNet  Google Scholar 

  40. R. Lyons,On measures simultaneously 2-and 3-invariant, Israel Journal of Mathematics61 (1988), 219–224.

    Article  MATH  MathSciNet  Google Scholar 

  41. G. A. Margulis,Discrete subgroups and ergodic theory, inNumber Theory, Trace Formulas and Discrete Groups (Oslo, 1987), Academic Press, Boston, MA, 1989, pp. 377–398.

    Google Scholar 

  42. G. A. Margulis,Oppenheim conjecture, inFields Medallists’ Lectures, Volume 5 of World Scientific Series in 20th Century Mathematics, World Scientific Publishing, River Edge, NJ, 1997, pp. 272–327.

    Google Scholar 

  43. G. Margulis,Problems and conjectures in rigidity theory, inMathematics: Frontiers and Perspectives American Mathematical Society, Providence, RI, 2000, pp. 161–174.

    Google Scholar 

  44. D. Meiri,Entropy and uniform distribution of orbits in T d, Israel Journal of Mathematics105 (1998), 155–183.

    MATH  MathSciNet  Google Scholar 

  45. D. W. Morris,Ratner’s Theorem on Unipotent Flows, Chicago Lectures in Mathematics, University of Chicago Press, to appear; arXiv:math.DS/0310402.

  46. S. Mozes,On closures of orbits and arithmetic of quaternions, Israel Journal of Mathematics86 (1994), 195–209.

    MATH  MathSciNet  Google Scholar 

  47. S. Mozes,Actions of Cartan subgroups, Israel Journal of Mathematics90 (1995), 253–294.

    MATH  MathSciNet  Google Scholar 

  48. D. Meiri and Y. Peres,Bi-invariant sets and measures have integer Hausdorff dimension, Ergodic Theory and Dynamical Systems19 (1999), 523–534.

    Article  MATH  MathSciNet  Google Scholar 

  49. G. A. Margulis and G. M. Tomanov,Invariant measures for actions of unipotent groups over local fields on homogeneous spaces, Inventiones Mathematicae116 (1994), 347–392.

    Article  MATH  MathSciNet  Google Scholar 

  50. R. Muchnik,Orbits of zariski dense semigroup of SL(n,ℤ), Ergodic Theory and Dynamical Systems, to appear.

  51. R. Muchnik,Semigroup actions on \(\mathbb{T}^n \), Geometriae Dedicata, to appear.

  52. D. S. Ornstein and B. Weiss,Entropy and isomorphism theorems for actions of amenable groups, Journal d’Analyse Mathématique48 (1987), 1–141.

    MATH  MathSciNet  Google Scholar 

  53. M. Ratner,Factors of horocycle flows, Ergodic Theory and Dynamical Systems2 (1982), 465–489.

    MATH  MathSciNet  Google Scholar 

  54. M. Ratner,Rigidity of horocycle flows, Annals of Mathematics (2)115 (1982), 597–614.

    Article  MathSciNet  Google Scholar 

  55. M. Ratner,Horocycle flows, joinings and rigidity of products, Annals of Mathematics (2),118 (1983), 277–313.

    Article  MathSciNet  Google Scholar 

  56. M. Ratner,On Raghunathan’s measure conjecture, Annals of Mathematics (2)134 (1991), 545–607.

    Article  MathSciNet  Google Scholar 

  57. M. Ratner,Raghunathan’s topological conjecture and distributions of unipotent flows, Duke Mathematical Journal63 (1991), 235–280.

    Article  MATH  MathSciNet  Google Scholar 

  58. M. Ratner,Raghunathan’s conjectures for SL(2, ℝ), Israel Journal of Mathematics80 (1992), 1–31.

    MATH  MathSciNet  Google Scholar 

  59. M. Rees,Some R 2-anosov flows, unpublished, 1982.

  60. Z. Rudnick and P. Sarnak,The behaviour of eigenstates of arithmetic hyperbolic manifolds, Communications in Mathematical Physics161 (1994), 195–213.

    Article  MATH  MathSciNet  Google Scholar 

  61. D. J. Rudolph,Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold, Ergodic Theory and Dynamical Systems2 (1982), 491–512.

    MATH  MathSciNet  Google Scholar 

  62. D. J. Rudolph, ×2and ×3invariant measures and entropy, Ergodic Theory and Dynamical Systems10 (1990), 395–406.

    MATH  MathSciNet  Google Scholar 

  63. P. Sarnak,Estimates for Rankin-Selberg L-functions and quantum unique ergodicity, Journal of Functional Analysis184 (2001), 419–453.

    Article  MATH  MathSciNet  Google Scholar 

  64. K. Schmidt,Dynamical Systems of Algebraic Origin, Volume 128 of Progress in Mathematics, Birkhäuser Verlag, Basel, 1995.

    Google Scholar 

  65. K. Schmidt,Algebraic coding of expansive group automorphisms and two-sided beta-shifts, Monatshefte für Mathematik129 (2000), 37–61.

    Article  MATH  Google Scholar 

  66. K. Schmidt,Algebraic ℤd-actions, preprint, available for download at http://www.pims.math.ca/publications/distchair/schmidt notes.pdf.

  67. A. Selberg,Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, Journal of the Indian Mathematical Society (N.S.)20 (1956), 47–87.

    MathSciNet  MATH  Google Scholar 

  68. A. I. Šnirel’man,Ergodic properties of eigenfunctions, Uspekhi Matematicheskikh Nauk29(6(180)) (1974), 181–182.

    MathSciNet  MATH  Google Scholar 

  69. A. Starkov,Orbit closures of toral automorphism group, preprint, 1999.

  70. A. Starkov,Orbit closures of toral automorphism group II, preprint, 2000.

  71. N. A. Sidorov and A. M. Vershik,Bijective coding of automorphisms of a torus, and binary quadratic forms, Uspekhi Matematicheskikh Nauk53(5(323)) (1998), 231–232.

    MathSciNet  Google Scholar 

  72. L. Silberman and A. Venkatesh,On quantum unique ergodicity for locally symmetric spaces I: a micro local life, preprint, 2004.

  73. G. Tomanov and B. Weiss,closed orbits for action of maximal tori on homogeneous spaces, Duke Mathematical Journal119 (2003), 367–392.

    Article  MATH  MathSciNet  Google Scholar 

  74. T. Watson,Rankin triple products and quantum chaos, Ph.D. thesis, Princeton University, 2001.

  75. S. A. Wolpert,The modulus of continuity for Γ0(m)\ℍsemi-classical limits, Communications in Mathematical Physics216 (2001), 313–323.

    Article  MATH  MathSciNet  Google Scholar 

  76. S. Zelditch,Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Mathematical Journal55 (1987), 919–941.

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Princeton University, 08540, Princeton, NJ, USA

    Elon Lindenstrauss

  2. Courant Institute of Mathematical Sciences, New York University, 10012, New York, NY, USA

    Elon Lindenstrauss

Authors
  1. Elon Lindenstrauss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elon Lindenstrauss.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lindenstrauss, E. Rigidity of multiparameter actions. Isr. J. Math. 149, 199–226 (2005). https://doi.org/10.1007/BF02772541

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02772541

Keywords

Search

Navigation