Skip to main content
SpringerLink
Log in

Uniqueness of SRB Measures for Transitive Diffeomorphisms on Surfaces

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We give a description of ergodic components of SRB measures in terms of ergodic homoclinic classes associated to hyperbolic periodic points. For transitive surface diffeomorphisms, we prove that there exists at most one SRB measure.

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

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Alves J.F., Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140, 351–398 (2000)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  2. Barreira, L., Pesin, Y.: Lyapunov Exponents and Smooth Ergodic Theory. American Mathematical Society, University Lecture Series, Vol. 23. Providence, RI: Amer. Math. Soc., 2003

  3. Benedicks M., Young L.-S.: Sinai-Bowen-Ruelle measure for certain Hénon maps. Invent. Math. 112, 541–576 (1993)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  4. Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–194 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975

  6. Cowieson W., Young L.-S.: SRB measures as zero-noise limits. Erg. Th. Dynam. Syst. 25, 1115–1138 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dolgopyat D.: Limit Theorems for partially hyperbolic systems. Transactions of the American Math. Soc. 356(4), 1637–1689 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Dolgopyat D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. Furstenberg H.: Strict ergodicity and transformations of the torus. Amer. J. Math. 83, 573–601 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hatomoto J.: Diffeomorphisms admitting SRB measures and their regularity. Kodai Math. J. 29, 211–226 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hu H.Y., Young L.-S.: Nonexistence of SBR measures for some diffeomorphisms that are almost Anosov. Erg. Th. Dynam. Syst. 15(1), 67–76 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kan I.: Open sets of diffeomorphisms having having two attractors, each with an everywhere dense basin. Bull. Amer. Math. Soc. 31, 68–74 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Katok A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. IHES Publ. Math. 51, 137–173 (1980)

    MathSciNet  MATH  Google Scholar 

  14. Ledrappier F., Young L.-S.: The metric entropy of diffeomorphisms Part I: Characterization of measures satisfying Pesin’s entropy formula. Ann. Math. 122, 509–539 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Pesin, Ya.: Characteristic Lyapunov exponents and smooth ergodic theory. Usp. mat. Nauk 32, 55–112, (1977); English transl., Russ. Math. Surv. 32, 55–114, (1977)

  16. Pesin Ya., Sinai Ya.G.: Gibbs measures for partially hyperbolic attractors. Erg. Th. Dynam. Syst. 2, 417–438 (1983)

    MathSciNet  Google Scholar 

  17. Pugh C., Shub M.: Ergodic attractors. Trans. Am. Math. Soc. 312, 1–54 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Rodriguez Hertz F., Rodriguez Hertz M., Tahzibi A., Ures R.: A criterion for ergodicity of non-uniformly hyperbolic diffeomorphisms. Electron. Res. Announc. Math. Sci. 14, 74–81 (2007)

    MathSciNet  Google Scholar 

  19. Rodriguez Hertz, F., Rodriguez Hertz, M., Tahzibi, A., Ures, R.,: New criteria for ergodicity and non-uniform hyperbolicity, preprint, available at http://arxiv.org/abs/0907.4539vL [math.D5], 2009

  20. Rohlin, V.A.: On the fundamental ideas of measure theory. AMS Translations 71 (1952)

  21. Ruelle D.: A measure associated with Axiom A attractors. Amer. J. Math. 98, 619–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ruelle D.: Thermodynamic Formalism. Addison Wesley, Reading, MA (1978)

    MATH  Google Scholar 

  23. Sinai Ya.G.: Gibbs measure in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  24. Tsujii M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194, 37–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. Whitney H.: Analytic extensions of functions defined in closed sets. Transactions of American Mathematical Society 36(1), 63–89 (1934)

    Article  MathSciNet  Google Scholar 

  26. Young L.-S.: What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108, 733–754 (2002)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMERL-Facultad de Ingeniería, Universidad de la República, CC 30, Montevideo, Uruguay

    F. Rodriguez Hertz, M. A. Rodriguez Hertz & R. Ures

  2. Departamento de Matemática, ICMC-USP São Carlos, Caixa Postal 668, 13560-970, São Carlos, SP, Brazil

    A. Tahzibi

Authors
  1. F. Rodriguez Hertz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. A. Rodriguez Hertz
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Tahzibi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. R. Ures
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Tahzibi.

Additional information

Communicated by G. Gallavotti

Rights and permissions

Reprints and permissions

About this article

Cite this article

Rodriguez Hertz, F., Rodriguez Hertz, M.A., Tahzibi, A. et al. Uniqueness of SRB Measures for Transitive Diffeomorphisms on Surfaces. Commun. Math. Phys. 306, 35–49 (2011). https://doi.org/10.1007/s00220-011-1275-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-011-1275-0

Keywords

Search

Navigation