Communications in Mathematical Physics

, Volume 318, Issue 3, pp 831–855 | Cite as

Ergodic Theory of Generic Continuous Maps



We study the ergodic properties of generic continuous dynamical systems on compact manifolds. As a main result we prove that generic homeomorphisms have convergent Birkhoff averages under continuous observables at Lebesgue almost every point. In spite of this, when the underlying manifold has dimension greater than one, generic homeomorphisms have no physical measures—a somewhat strange result which stands in sharp contrast to current trends in generic differentiable dynamics. Similar results hold for generic continuous maps.

To further explore the mysterious behaviour of C0 generic dynamics, we also study the ergodic properties of continuous maps which are conjugated to expanding circle maps. In this context, generic maps have divergent Birkhoff averages along orbits starting from Lebesgue almost every point.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. ABC.
    Abdenur F., Bonatti C., Crovisier S.: Nonuniform hyperbolicity for C 1-generic diffeomorphisms. Israel J. Math. 183, 1–60 (2011)MathSciNetMATHCrossRefGoogle Scholar
  2. ABV.
    Alves J.F., Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly expanding. Invent. Math. 140(2), 351–398 (2000)MathSciNetADSMATHCrossRefGoogle Scholar
  3. AHK.
    Akin, E., Hurley, M., Kennedy, J.A.: Dynamics of topologically generic homeomorphisms. Mem. Amer. Math. Soc. 164(783), viii+130, Providence RI: Amer. Math. Soc., 2003Google Scholar
  4. And.
    Andersson M.: Robust ergodic properties in partially hyperbolic dynamics. Trans. Amer. Math. Soc. 362(4), 1831–1867 (2010)MathSciNetMATHCrossRefGoogle Scholar
  5. AP.
    Prasad, S., Alpern, V.S.: Typical dynamics of volume preserving homeomorphisms. Volume 139 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2000Google Scholar
  6. BB.
    Blank M., Bunimovich L.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity 16(1), 387–401 (2003)MathSciNetADSMATHCrossRefGoogle Scholar
  7. Bow.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Volume 470 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, revised edition, 2008, with a preface by David Ruelle, Edited by Jean-René ChazottesGoogle Scholar
  8. BV.
    Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–193 (2000)MathSciNetMATHCrossRefGoogle Scholar
  9. Cai.
    Cairns S.S.: On the cellular subdivision of n-dimensional regions. Ann. of Math. (2) 33(4), 671–680 (1932)MathSciNetCrossRefGoogle Scholar
  10. Con.
    Conley, C.: Isolated invariant sets and the Morse index, Volume 38 of CBMS Regional Conference Series in Mathematics. Providence, RI: Amer. Math. Soc., 1978Google Scholar
  11. CQ.
    Campbell J.T., Quas A.N.: A generic C 1 expanding map has a singular S-R-B measure. Commun. Math. Phys. 221(2), 335–349 (2001)MathSciNetADSMATHCrossRefGoogle Scholar
  12. DGS.
    Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, Vol. 527. Berlin: Springer-Verlag, 1976Google Scholar
  13. Hur1.
    Hurley M.: On proofs of the C 0 general density theorem. Proc. Amer. Math. Soc. 124(4), 1305–1309 (1996)MathSciNetMATHCrossRefGoogle Scholar
  14. Hur2.
    Hurley M.: Properties of attractors of generic homeomorphisms. Erg. Th. Dyn. Sys. 16(6), 1297–1310 (1996)MathSciNetMATHCrossRefGoogle Scholar
  15. JT.
    Järvenpää E., Tolonen T.: Relations between natural and observable measures. Nonlinearity 18(2), 897–912 (2005)MathSciNetADSMATHCrossRefGoogle Scholar
  16. Mil.
    Milnor J.: On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64, 399–405 (1956)MathSciNetMATHCrossRefGoogle Scholar
  17. Mis.
    Misiurewicz, M.: Ergodic natural measures. In: Algebraic and topological dynamics, Volume 385 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2005, pp. 1–6Google Scholar
  18. Moi.
    Moise, E.E.: Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics 47, New York: Springer-Verlag, 1977Google Scholar
  19. MYNPV.
    Muñoz-Young E., Navas A., Pujals E., Vásquez C.H.: A continuous Bowen-Mañé type phenomenon. Disc. Cont. Dyn. Sys. 20(3), 713–724 (2008)MATHGoogle Scholar
  20. Nor.
    Norton D.E.: The Conley decomposition theorem for maps: a metric approach. Comment. Math. Univ. St. Paul. 44(2), 151–173 (1995)MathSciNetMATHGoogle Scholar
  21. OU.
    Oxtoby J.C., Ulam S.M.: Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42, 874–920 (1941)MathSciNetMATHCrossRefGoogle Scholar
  22. Pal.
    Palis, J.: A global view of dynamics and a conjecture on the denseness of finitude of attractors, Géométrie complexe et systèmes dynamiques (Orsay, 1995), Astérisque, (261):xiii–xiv, 335–347, 2000Google Scholar
  23. Qiu.
    Qiu H.: Existence and uniqueness of SRB measure on C 1 generic hyperbolic attractors. Commun. Math. Phys. 302(2), 345–357 (2011)ADSMATHCrossRefGoogle Scholar
  24. Rue1.
    Ruelle D.: A measure associated with axiom-A attractors. Amer. J. Math. 98(3), 619–654 (1976)MathSciNetMATHCrossRefGoogle Scholar
  25. Rue2.
    Ruelle, D.: Historical behaviour in smooth dynamical systems. In: Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 63–66Google Scholar
  26. Shu.
    Shub M.: Structurally stable diffeomorphisms are dense. Bull. Amer. Math. Soc. 78, 817–818 (1972)MathSciNetMATHCrossRefGoogle Scholar
  27. Tsu.
    Tsujii M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1), 37–132 (2005)MathSciNetMATHCrossRefGoogle Scholar
  28. Yan.
    Yano K.: A remark on the topological entropy of homeomorphisms. Invent. Math. 59(3), 215–220 (1980)MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticaPUC-Rio de JaneiroRio de JaneiroBrazil
  2. 2.Universidade Federal Fluminense (GMA)NiteróiBrazil

Personalised recommendations