Journal d'Analyse Mathématique

, Volume 135, Issue 1, pp 123–183 | Cite as

Multi-invariant measures and subsets on nilmanifolds

  • Zhiren Wang


Given a Zr-action α on a nilmanifold X by automorphisms and an ergodic α-invariant probability measure μ, we show that μ is the uniform measure on X unless, modulo finite index modification, one of the following obstructions occurs for an algebraic factor action
  1. (1)

    the factor measure has zero entropy under every element of the action

  2. (2)

    the factor action is virtually cyclic.

We also deduce a rigidity property for invariant closed subsets.


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  1. [AR62]
    L. M. Abraomov and V. A. Rohlin, Rohlin, Entropy of a skew product of mappings with invariant measure (Russian) Vestnik Leningrad. Univ. 17 (1962), 5–13.Google Scholar
  2. [BQ11]
    Y. Benoist and J.-F. Quint, Mesures stationnaires et fermés invariants des espaces homogènes, Ann. of Math. (2) 174 (2011), 1111–1162.MathSciNetCrossRefMATHGoogle Scholar
  3. [BQ13]
    Y. Benoist and J. F. Quint, Stationary measures and invariant subsets of homogeneous spaces (II), J. Amer.Math. Soc. 26 (2013), 659–734.MathSciNetCrossRefMATHGoogle Scholar
  4. [Ber83]
    D. Berend, Multi-invariant sets on tori, Trans. Amer. Math. Soc. 280 (1983), 509–532.MathSciNetCrossRefMATHGoogle Scholar
  5. [Ber84]
    D. Berend, Multi-invariant sets on compact abelian groups, Trans. Amer. Math. Soc. 286 (1984), 505–535.MathSciNetCrossRefMATHGoogle Scholar
  6. [CG90]
    L. J. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications. Part I, Cambridge Univ. Press, Cambridge, 1990.MATHGoogle Scholar
  7. [Ein06]
    M. Einsiedler, Ratner’s theorem on SL(2,R)-invariant measures, Jahresber. Deutsch. Math.-Verein. 108 (2006), 143–164.MathSciNetMATHGoogle Scholar
  8. [EK03]
    M. Einsiedler and A. Katok, Invariant measures on G/Γ for split simple Lie groups G, Comm. Pure Appl. Math. 56 (2003), 1184–1221.MathSciNetCrossRefMATHGoogle Scholar
  9. [EKL06]
    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.MathSciNetCrossRefMATHGoogle Scholar
  10. [EL03]
    M. Einsiedler and E. Lindenstrauss, Rigidity properties of Zd -actions on tori and solenoids, Electron. Res. Announc. Amer. Math. Soc. 9 (2003), 99–110 (electonic).MathSciNetCrossRefMATHGoogle Scholar
  11. [EL10]
    M. Einsiedler and E. Lindenstrauss, Diagonal actions on locally homogeneous spaces, Homogeneous Flows, Moduli Spaces and Arithmetic, Amer. Math. Soc., Providence, RI, 2010, pp. 155–241.Google Scholar
  12. [ELW]
    M. Einsiedler, E. Lindenstrauss, and Z. Wang, Rigidity properties of abelian actions on tori and solenoids, in preparation.Google Scholar
  13. [EW11]
    M. Einseidler and T. Ward, Ergodic Theory with a View Towards Number Theory, Springer-Verlag, London, 2011.Google Scholar
  14. [Fel93]
    J. Feldman, A generalization of a result of R. Lyons about measures on [0, 1), Israel J. Math. 81 (1993), 281–287.MathSciNetCrossRefMATHGoogle Scholar
  15. [FKS11]
    D. Fisher, B. Kalinin, and R. Spatzier, Totally nonsymplectic Anosov actions on tori and nilmanifolds, Geom. Topol. 15 (2011), 191–216.MathSciNetCrossRefMATHGoogle Scholar
  16. [FKS13]
    D. Fisher, B. Kalinin, and R. Spatzier, Global rigidity of higher rank Anosov actions on tori and nilmanifolds, J. Amer.Math. Soc. 26 (2013), 167–198.MathSciNetCrossRefMATHGoogle Scholar
  17. [Fur67]
    H. Furstenberg, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Math. Systems Theory 1 (1967), 1–49.MATHGoogle Scholar
  18. [Gre61]
    L. W. Green, Spectra of nilflows, Bull. Amer. Math. Soc. 67 (1961), 414–415.MathSciNetCrossRefMATHGoogle Scholar
  19. [Hoc12]
    M. Hochman, Geometric rigidity of ×m invariant measures, J. Eur. Math. Soc. (JEMS) 14 (2012), 1539–1563.MathSciNetCrossRefMATHGoogle Scholar
  20. [HS12]
    M. Hochman and P. Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2) 175 (2012), 1001–1059.MathSciNetCrossRefMATHGoogle Scholar
  21. [Hos95]
    B. Host, Nombres normaux, entropie, translations, Israel J. Math. 91 (1975), 419–428.MathSciNetCrossRefMATHGoogle Scholar
  22. [Hum75]
    J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1995.MATHGoogle Scholar
  23. [Joh92]
    A. S. A. Johnson, Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers, Israel J. Math. 77 (1992), 211–240.MathSciNetCrossRefMATHGoogle Scholar
  24. [JR95]
    A. Johnson and D. J. Rudolph, Convergence under ×q of ×p invariant measures on the circle, Adv. Math. 115 (1995), 117–140.MathSciNetCrossRefMATHGoogle Scholar
  25. [KK01]
    B. Kalinin and A. Katok, Invariant measures for actions of higher rank abelian groups, Smooth Ergodic Theory and its Applications, Amer. Math. Soc. Providence RI, 2001, pp. 593–637.CrossRefGoogle Scholar
  26. [KS05]
    B. Kalinin and R. Spatzier, Rigidity of the measurable structure for algebraic actions of higher-rank abelian groups, Ergodic Theory Dynam. Systems 25 (2005), 175–200.MathSciNetCrossRefMATHGoogle Scholar
  27. [KN11]
    A. Katok and V. Niţică, Rigidity in Higher Rank Abelian Group Actions. Volume I: Introduction and Cocycle Problem, Cambridge Univ. Press, Cambridge, 2011.CrossRefMATHGoogle Scholar
  28. [KS96]
    A. Katok and R. J. Spatzier, Invariant measures for higher-rank hyperbolic abelian actions, Ergodic Theory Dynam. Systems 16 (1996), 751–778.MathSciNetCrossRefMATHGoogle Scholar
  29. [LS82]
    F. Ledrappier and J.-M. Strelcyn, A proof of the estimation from below in Pesin’s entropy formula, Ergodic Theory Dynam. Systems 2 (1982), 203–219.MathSciNetCrossRefMATHGoogle Scholar
  30. [LY88]
    F. Ledrappier and L.-S. Young, The metric entropy of diffeomorphisms. I., II., Ann. of Math. (2) 122 (1985), 509–539, 540–574.MathSciNetCrossRefMATHGoogle Scholar
  31. [Lin06]
    E. Lindenstrauss, Invariant measures and arithmetic quantum unique ergodicity, Ann of Math. (2) 163 (2006), 165–219.MathSciNetCrossRefMATHGoogle Scholar
  32. [LW12]
    E. Lindenstrauss and Z. Wang, Topological self-joining of Cartan actions by toral automorphisms, Duke Math. J. 161 (2012), 1305–1350.MathSciNetCrossRefMATHGoogle Scholar
  33. [Lyo88]
    R. Lyons, On measures simultaneously 2- and 3-invariant, Israel J. Math. 61 (1988), 219–224.MathSciNetCrossRefMATHGoogle Scholar
  34. [Mau10]
    F. Maucourant, A nonhomogeneous orbit closure of a diagonal subgroup, Ann. of Math. (2) 171 (2010), 557–570.MathSciNetCrossRefMATHGoogle Scholar
  35. [Par69]
    W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.MathSciNetCrossRefMATHGoogle Scholar
  36. [Par70]
    W. Parry, Dynamical systems on nilmanifolds, Bull. London Math. Soc. 2 (1970), 37–40.MathSciNetCrossRefMATHGoogle Scholar
  37. [Par96]
    W. Parry, Squaring and cubing the circle—Rudolph’s theorem, Ergodic Theory of Zd actions, Cambridge Univ. Press, Cambridge, 1996, pp. 177–183.Google Scholar
  38. [Rag72]
    M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag, New York, 1972.CrossRefMATHGoogle Scholar
  39. [Rat91]
    M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. (2) 134 (1991), 545–607.MathSciNetCrossRefMATHGoogle Scholar
  40. [R-HW14]
    F. Rodriguez Hertz and Z. Wang, Global rigidity of higher rank Anosov algebraic actions, Invent. Math. 198 (2014), 65–209.MathSciNetCrossRefMATHGoogle Scholar
  41. [Ros61]
    M. Rosenlicht, On quotient varieties and the affine embedding of certain homogeneous spaces, Trans. Amer.Math. Soc. 101 (1961), 211–223.MathSciNetCrossRefMATHGoogle Scholar
  42. [Rud90]
    D. J. Rudolph, ×2 and ×3 invariant measures and entropy, Ergodic Theory Dynam. Systems 10 (1990), 395–406.MathSciNetCrossRefMATHGoogle Scholar
  43. [Sch95]
    K. Schmidt, Dynamical Systems of Algebraic Origin, Birkhäuser Verlag, Basel, 1995.CrossRefMATHGoogle Scholar
  44. [Sta99]
    A. N. Starkov, The first cohomology group, mixing, and minimal sets of the commutative group of algebraic actions on a torus, J. Math. Sci. 7 (1999), 2567–2582.MathSciNetGoogle Scholar
  45. [WZ92]
    T. Ward and Q. Zhang, The Abramov-Rokhlin entropy addition formula for amenable group actions, Monatsh. Math. 114 (1992), 317–329.MathSciNetCrossRefMATHGoogle Scholar

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© Hebrew University Magnes Press 2018

Authors and Affiliations

  1. 1.Pennsylvania State UniversityUniversity ParkUSA

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