Communications in Mathematical Physics

, Volume 306, Issue 1, pp 35–49 | Cite as

Uniqueness of SRB Measures for Transitive Diffeomorphisms on Surfaces

  • F. Rodriguez Hertz
  • M. A. Rodriguez Hertz
  • A. TahzibiEmail author
  • R. Ures


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.


Lyapunov Exponent Periodic Point Unstable Manifold Absolute Continuity Positive Lebesgue 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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Barreira, L., Pesin, Y.: Lyapunov Exponents and Smooth Ergodic Theory. American Mathematical Society, University Lecture Series, Vol. 23. Providence, RI: Amer. Math. Soc., 2003Google Scholar
  3. 3.
    Benedicks M., Young L.-S.: Sinai-Bowen-Ruelle measure for certain Hénon maps. Invent. Math. 112, 541–576 (1993)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–194 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470, Berlin-Heidelberg-New York: Springer, 1975Google Scholar
  6. 6.
    Cowieson W., Young L.-S.: SRB measures as zero-noise limits. Erg. Th. Dynam. Syst. 25, 1115–1138 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Dolgopyat D.: Limit Theorems for partially hyperbolic systems. Transactions of the American Math. Soc. 356(4), 1637–1689 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Dolgopyat D.: On differentiability of SRB states for partially hyperbolic systems. Invent. Math. 155, 389–449 (2004)MathSciNetADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Furstenberg H.: Strict ergodicity and transformations of the torus. Amer. J. Math. 83, 573–601 (1961)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Hatomoto J.: Diffeomorphisms admitting SRB measures and their regularity. Kodai Math. J. 29, 211–226 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 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)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Katok A.: Lyapunov exponents, entropy and periodic orbits for diffeomorphisms. IHES Publ. Math. 51, 137–173 (1980)MathSciNetzbMATHGoogle Scholar
  14. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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)Google Scholar
  16. 16.
    Pesin Ya., Sinai Ya.G.: Gibbs measures for partially hyperbolic attractors. Erg. Th. Dynam. Syst. 2, 417–438 (1983)MathSciNetGoogle Scholar
  17. 17.
    Pugh C., Shub M.: Ergodic attractors. Trans. Am. Math. Soc. 312, 1–54 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 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)MathSciNetGoogle Scholar
  19. 19.
    Rodriguez Hertz, F., Rodriguez Hertz, M., Tahzibi, A., Ures, R.,: New criteria for ergodicity and non-uniform hyperbolicity, preprint, available at [math.D5], 2009
  20. 20.
    Rohlin, V.A.: On the fundamental ideas of measure theory. AMS Translations 71 (1952)Google Scholar
  21. 21.
    Ruelle D.: A measure associated with Axiom A attractors. Amer. J. Math. 98, 619–654 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Ruelle D.: Thermodynamic Formalism. Addison Wesley, Reading, MA (1978)zbMATHGoogle Scholar
  23. 23.
    Sinai Ya.G.: Gibbs measure in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Tsujii M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194, 37–132 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Whitney H.: Analytic extensions of functions defined in closed sets. Transactions of American Mathematical Society 36(1), 63–89 (1934)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Young L.-S.: What are SRB measures, and which dynamical systems have them?. J. Stat. Phys. 108, 733–754 (2002)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • F. Rodriguez Hertz
    • 1
  • M. A. Rodriguez Hertz
    • 1
  • A. Tahzibi
    • 2
    Email author
  • R. Ures
    • 1
  1. 1.IMERL-Facultad de IngenieríaUniversidad de la RepúblicaMontevideoUruguay
  2. 2.Departamento de MatemáticaICMC-USP São CarlosSão CarlosBrazil

Personalised recommendations