Skip to main content
SpringerLink
Log in

Invariant Radon measures for horocycle flows on Abelian covers

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We classify the ergodic invariant Radon measures for horocycle flows on ℤd–covers of compact Riemannian surfaces of negative curvature, thus proving a conjecture of M. Babillot and F. Ledrappier. An important tool is a result in the ergodic theory of equivalence relations concerning the reduction of the range of a cocycle by the addition of a coboundary.

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. Aaronson, J., Nakada, H., Sarig, O., Solomyak, R.: Invariant measures and asymptotics for some skew products. Isr. J. Math. 128, 93–134 (2002). Corrections: Isr. J. Math. 138, 377–379 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Aaronson, J., Sarig, O., Solomyak, R.: Tail-invariant measures for some suspension semiflows. Discrete Contin. Dyn. Syst. 8, 725–735 (2002)

    MathSciNet  MATH  Google Scholar 

  3. Anosov, D.V.: Geodesic flows on closed Riemann manifolds with negative curvature. Proc. Steklov Inst. Math. 90 (1967). Translated from the Russian by S. Feder. Providence, R.I.: AMS 1969

  4. Arveson, W.: An invitation to C*–Algebra. Grad. Texts Math. 39. Springer 1976

  5. Babillot, M.: On the classification of invariant measures for horospherical foliations on nilpotent covers of negatively curved manifolds. Preprint

  6. Babillot, M., Ledrappier, F.: Geodesic paths and horocycle flows on Abelian covers. Lie groups and ergodic theory (Mumbai, 1996), pp. 1–32, Tata Inst. Fund. Res. Stud. Math. 14, Tata Inst. Fund. Res., Bombay 1998

  7. Bowen, R., Marcus, B.: Unique ergodicity for horocycle foliations. Isr. J. Math. 26, 43–67 (1977)

    MATH  Google Scholar 

  8. Bowen, R.: Symbolic dynamics for hyperbolic flows. Am. J. Math 95, 429–460 (1973)

    MATH  Google Scholar 

  9. Bowen, R.: Anosov foliations are hyperfinite. Ann. Math. 106, 549–565 (1977)

    MATH  Google Scholar 

  10. Bowen, R., Ruelle, D.: The ergodic theory of Axiom A flows. Invent. Math. 29, 181–202 (1975)

    MATH  Google Scholar 

  11. Burger, M.: Horocycle flow on geometrically finite surfaces. Duke Math. J. 61, 779–803 (1990)

    MathSciNet  MATH  Google Scholar 

  12. Coudene, Y.: Cocycles and stable foliations of Axiom A flows. Ergodic Theory Dyn. Syst. 21, 767–775 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dani, S.G.: Invariant measures of horospherical flows on noncompact homogeneous spaces. Invent. Math. 47, 101–138 (1978)

    MATH  Google Scholar 

  14. Feldman, J., Moore, C.: Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Am. Math. Soc. 234, 289–324 (1977)

    MATH  Google Scholar 

  15. Furstenberg, H.: The unique ergodicity of the horocycle flow. Springer Lect. Notes 318, 95–115 (1972)

    MATH  Google Scholar 

  16. Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, Vol. I: Structure of topological groups, integration theory, group representations, 2nd edn. Grundlehren Math. Wiss. 115. Berlin, New York: Springer 1979

  17. Kaimanovich, V.: Ergodic properties of the horocycle flow and classification of Fuchsian groups. J. Dyn. Control Syst. 6, 21–56 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kaimanovich, V., Schmidt, K.: Ergodicity of cocycles. 1: General Theory. Preprint

  19. Kuratowski, C.: Topologie, Vol. 1, 4th edn., Monogr. Mat., Warszawa 20. Warszawa: PWN 1958

  20. Marcus, B.: Unique ergodicity of the horocycle flow: variable negative curvature case. Isr. J. Math. 21, 133–144 (1975)

    MATH  Google Scholar 

  21. Marcus, B.: Ergodic properties of horocycle flows. Ann. Math. 105, 81–105 (1977)

    MATH  Google Scholar 

  22. Margulis, G.A.: Certain measures associated with U–flows on compact manifolds. Funct. Anal. Appl. 4, 133–144 (1970)

    Google Scholar 

  23. Parry, W., Pollicott, M.: Zeta functions and the periodic orbit structure of hyperbolic dynamics. Astérisque 187–188 (1990)

  24. Parry, W., Schmidt, K.: Natural coefficients and invariants for Markov-shifts. Invent. Math. 76, 15–32 (1984)

    MathSciNet  MATH  Google Scholar 

  25. Plante, J.: Anosov flows. Am. J. Math. 94, 729–754 (1972)

    MATH  Google Scholar 

  26. Pollicott, M.: ℤd–covers of horosphere foliations. Discrete Contin. Dyn. Syst. 6, 147–154 (2000)

    MathSciNet  MATH  Google Scholar 

  27. Pollicott, M.: Ergodicity of stable manifolds for nilpotent extensions of Anosov flows. Discrete Contin. Dyn. Syst. 8, 599–604 (2002)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  29. Schmidt, K.: Cocycles on ergodic transformation groups. Macmillan Lectures in Mathematics, Vol. 1, pp. 202. Delhi: Macmillan Company of India, Ltd. 1977

  30. Series, C.: The Poincaré flow of a foliation. Am. J. Math. 102, 93–128 (1980)

    MathSciNet  MATH  Google Scholar 

  31. Sharp, R.: Closed orbits in homology classes for Anosov flows. Ergodic Theory Dyn. Syst. 13, 387–408 (1993)

    MathSciNet  MATH  Google Scholar 

  32. Solomyak, R.: A short proof of ergodicity of Babillot-Ledrappier measures. Proc. Am. Math. Soc. 129, 3589–3591 (2001)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Department, Penn State University, University Park, PA, 16802, USA

    Omri Sarig

Authors
  1. Omri Sarig
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Omri Sarig.

Additional information

Dedicated to the memory of M. Babillot

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sarig, O. Invariant Radon measures for horocycle flows on Abelian covers. Invent. math. 157, 519–551 (2004). https://doi.org/10.1007/s00222-004-0357-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-004-0357-4

Keywords

Search

Navigation