Skip to main content
SpringerLink
Log in

Ergodic Theory of Generic Continuous Maps

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

Abstract

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 C 0 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.

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. Abdenur F., Bonatti C., Crovisier S.: Nonuniform hyperbolicity for C 1-generic diffeomorphisms. Israel J. Math. 183, 1–60 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  2. 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  3. 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., 2003

  4. Andersson M.: Robust ergodic properties in partially hyperbolic dynamics. Trans. Amer. Math. Soc. 362(4), 1831–1867 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Prasad, S., Alpern, V.S.: Typical dynamics of volume preserving homeomorphisms. Volume 139 of Cambridge Tracts in Mathematics. Cambridge: Cambridge University Press, 2000

  6. Blank M., Bunimovich L.: Multicomponent dynamical systems: SRB measures and phase transitions. Nonlinearity 16(1), 387–401 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. 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 Chazottes

  8. Bonatti C., Viana M.: SRB measures for partially hyperbolic systems whose central direction is mostly contracting. Israel J. Math. 115, 157–193 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cairns S.S.: On the cellular subdivision of n-dimensional regions. Ann. of Math. (2) 33(4), 671–680 (1932)

    Article  MathSciNet  Google Scholar 

  10. Conley, C.: Isolated invariant sets and the Morse index, Volume 38 of CBMS Regional Conference Series in Mathematics. Providence, RI: Amer. Math. Soc., 1978

  11. 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)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  12. Denker, M., Grillenberger, C., Sigmund, K.: Ergodic theory on compact spaces. Lecture Notes in Mathematics, Vol. 527. Berlin: Springer-Verlag, 1976

  13. Hurley M.: On proofs of the C 0 general density theorem. Proc. Amer. Math. Soc. 124(4), 1305–1309 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hurley M.: Properties of attractors of generic homeomorphisms. Erg. Th. Dyn. Sys. 16(6), 1297–1310 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  15. Järvenpää E., Tolonen T.: Relations between natural and observable measures. Nonlinearity 18(2), 897–912 (2005)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  16. Milnor J.: On manifolds homeomorphic to the 7-sphere. Ann. of Math. (2) 64, 399–405 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  17. Misiurewicz, M.: Ergodic natural measures. In: Algebraic and topological dynamics, Volume 385 of Contemp. Math., Providence, RI: Amer. Math. Soc., 2005, pp. 1–6

  18. Moise, E.E.: Geometric topology in dimensions 2 and 3. Graduate Texts in Mathematics 47, New York: Springer-Verlag, 1977

  19. 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)

    MATH  Google Scholar 

  20. Norton D.E.: The Conley decomposition theorem for maps: a metric approach. Comment. Math. Univ. St. Paul. 44(2), 151–173 (1995)

    MathSciNet  MATH  Google Scholar 

  21. Oxtoby J.C., Ulam S.M.: Measure-preserving homeomorphisms and metrical transitivity. Ann. of Math. (2) 42, 874–920 (1941)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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, 2000

  23. Qiu H.: Existence and uniqueness of SRB measure on C 1 generic hyperbolic attractors. Commun. Math. Phys. 302(2), 345–357 (2011)

    Article  ADS  MATH  Google Scholar 

  24. Ruelle D.: A measure associated with axiom-A attractors. Amer. J. Math. 98(3), 619–654 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ruelle, D.: Historical behaviour in smooth dynamical systems. In: Global analysis of dynamical systems, Inst. Phys., Bristol, 2001, pp. 63–66

  26. Shub M.: Structurally stable diffeomorphisms are dense. Bull. Amer. Math. Soc. 78, 817–818 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  27. Tsujii M.: Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194(1), 37–132 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  28. Yano K.: A remark on the topological entropy of homeomorphisms. Invent. Math. 59(3), 215–220 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departamento de Matemática, PUC-Rio de Janeiro, 22453-900, Rio de Janeiro, RJ, Brazil

    Flavio Abdenur

  2. Universidade Federal Fluminense (GMA), 24020-140, Niterói, RJ, Brazil

    Martin Andersson

Authors
  1. Flavio Abdenur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Martin Andersson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Martin Andersson.

Additional information

Communicated by G. Gallavotti

Partially supported by a CNPq/Brazil productivity-in-research (PQ) grant and by a FAPERJ/“Junior Rio de Janeiro State Scientist” fellowship.

With financial support from Fondation Sciences Mathèmatiques à Paris (France) and FAPERJ (Brazil).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Abdenur, F., Andersson, M. Ergodic Theory of Generic Continuous Maps. Commun. Math. Phys. 318, 831–855 (2013). https://doi.org/10.1007/s00220-012-1622-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-012-1622-9

Keywords

Search

Navigation