Skip to main content
SpringerLink
Log in

Linearization of conservative toral homeomorphisms

  • Published:
Inventiones mathematicae Aims and scope

Abstract

We give an equivalent condition for the existence of a semi-conjugacy to an irrational rotation for conservative homeomorphisms of the two-torus. This leads to an analogue of Poincaré’s classification of circle homeomorphisms for conservative toral homeomorphisms with unique rotation vector and a certain bounded mean motion property. For minimal toral homeomorphisms, the result extends to arbitrary dimensions. Further, we provide a basic classification for the dynamics of toral homeomorphisms with all points non-wandering.

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.

Similar content being viewed by others

References

  1. Poincaré, H.: Mémoire sur les courbes définies par une équation différentielle. J. Math. Pures Appl., IV Sér. 1, 167–244 (1885)

    Google Scholar 

  2. Misiurewicz, M., Ziemian, K.: Rotation sets for maps of tori. J. Lond. Math. Soc. 40, 490–506 (1989)

    Article  MathSciNet  Google Scholar 

  3. Fayad, B.: Weak mixing for reparameterized linear flows on the torus. Ergodic Theory Dyn. Syst. 22(1), 187–201 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  4. Jäger, T., Stark, J.: Towards a classification for quasiperiodically forced circle homeomorphisms. J. Lond. Math. Soc. 73(3), 727–744 (2006)

    Article  MATH  Google Scholar 

  5. Jäger, T.: The concept of bounded mean motion for toral homeomorphisms. Preprint, arXiv:0802.4187 (2008)

    Google Scholar 

  6. Kwapisz, J.: A priori degeneracy of one-dimensional rotation sets for periodic point free torus maps. Trans. Am. Math. Soc. 354(7), 2865–2895 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  7. Kwapisz, J.: Combinatorics of torus diffeomorphisms. Ergodic Theory Dyn. Syst. 23, 559–586 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Béguin, F., Crovisier, S., Le Roux, F.: Pseudo-rotations of the closed annulus: variation on a theorem of J. Kwapisz. Nonlinearity 17(4), 1427–1453 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  9. Franks, J., Misiurewicz, M.: Rotation sets of toral flows. Proc. Am. Math. Soc. 109(1), 243–249 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  10. Misiurewicz, M., Ziemian, K.: Rotation sets and ergodic measures for torus homeomorphisms. Fundam. Math. 137(1), 45–52 (1991)

    MATH  MathSciNet  Google Scholar 

  11. Franks, J.: Realizing rotation vectors for torus homeomorphisms. Trans. Am. Math. Soc. 311(1), 107–115 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Bing, R.H.: A homogeneous indecomposable plane continuum. Duke Math. J. 15, 729–742 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  13. Handel, M.: A pathological area preserving C ∞ diffeomorphism of the plane. Proc. Am. Math. Soc. 86(1), 163–168 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  14. Herman, M.: Construction of some curious diffeomorphisms of the Rieman sphere. J. Lond. Math. Soc. 34, 375–384 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. Franks, J.: Generalizations of the Poincaré–Birkhoff theorem. Ann. Math. (2) 128(1), 139–151 (1988)

    Article  MathSciNet  Google Scholar 

  16. Franks, J.: Recurrence and fixed points of surface homeomorphisms. Ergodic Theory Dyn. Syst. 8*, 99–107 (1988)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Collège de France, 3 rue d’Ulm, 75005, Paris, France

    T. Jäger

Authors
  1. T. Jäger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to T. Jäger.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jäger, T. Linearization of conservative toral homeomorphisms. Invent. math. 176, 601–616 (2009). https://doi.org/10.1007/s00222-008-0171-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00222-008-0171-5

Keywords

Search

Navigation