Skip to main content
SpringerLink
Log in

More Denjoy minimal sets for area preserving diffeomorphisms

  • Published:
Commentarii Mathematici Helvetici

Abstract

For an area preserving, monotone twist diffeomorphism and an irrational number ω, we prove that if there is no invariant circle of angular rotation number ω, then there are uncountably many Denjoy minimal sets of angular rotation number ω. For each pair of positive integersn andR we prove that the space (with the vague topology) of Denjoy minimal sets of angular rotation number ω and intrinsic rotation number (ω+R)/n (mod. 1) contains a disk of dimensionn−1.

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. R. Abraham andJ. Marsden,Foundations of Mechanics, 2nd ed. Benjamin Cummings, Reading, Mass., 1978.

    MATH  Google Scholar 

  2. V. I. Arnold andA. Avez,Ergodic Problems of Classical Mechanics. Benjamin, NY, 1968.

    Google Scholar 

  3. S. Aubry,The Twist Map, the Extended Frenkel-Kontorova Model, and the Devil's Staircase. Physica7D (1983) 240–258.

    MathSciNet  Google Scholar 

  4. S. Aubry andP. Y. Le Daeron,The Discrete Frenkel-Kontorova Model and its Extensions. Physica8D (1983) 381–422.

    Google Scholar 

  5. S. Aubry, P. Y. Le Daeron, andG. André,Classical Ground-States of a One-Dimensional Model for Incommensurate, Structures. Preprint (1982).

  6. G. D. Birkhoff,Proof of Poincaré's Geometric Theorem. Trans. AMS14, 14–22. Reprinted inCollected Works, vol. I, AMS, New York, (1950) 673–681.

  7. G. D. Birkhoff,Sur quelques courbes fermees remarquables. Bull. Soc. Math. France60, 1–26. Reprinted in Collected Works, vol. II, AMS, (1950) 418–443.

    MathSciNet  Google Scholar 

  8. N. Bourbaki,Elements de mathematique, 6,Intégration, 2. éd. Paris, Hermann (1965).

    Google Scholar 

  9. A. Chenciner,La dynamique au voisinage d'un point fixe elliptique, conservatif: de Poincaré et Birkhoff à Aubry et Mather. Séminaire N. Bourbaki,1983–1984, exposé 622 (1984).

  10. R. Douady,Applications du théorème des tores invariants. Thèse 3ème cycle, Université Paris VII (1982).

  11. G. A. Hedlund,Geodesics on a two-dimensional Riemannian manifold with periodic coefficients. Annals of Math.33 (1932) 719–739.

    Article  MathSciNet  Google Scholar 

  12. M. R. Herman,Sur la conjugaison differentiable des difféomorphismes du cercle à des rotations. Publications Maths. de l'I.H.E.S.49 (1979) 5–234.

    MATH  MathSciNet  Google Scholar 

  13. M. R. Herman,Sur les courbes invariantes par les difféomorphismes de l'anneau, vol. 1. astérisque,103–104 (1983).

    Google Scholar 

  14. A. Katok,Periodic and Quasi-periodic Orbits for Twist Maps. Dynamical Systems and Chaos, Proceedings, Sitges, 1982, L. Garrido, ed. Springer-Verlag, Berlin (1983).

    Google Scholar 

  15. J. L. Kelley,General Topology, New York, Van Nostrand (1955).

    MATH  Google Scholar 

  16. J. N. Mather,Existence of Quasi-Periodic Orbits for Twist Homeomorphisms of the Annulus. Topology21 (1982) 457–467.

    Article  MATH  MathSciNet  Google Scholar 

  17. J. N. Mather,Concavity of the Lagrangian for Quasi-periodic Orbits. Comment. Math. Helvetici57 (1982) 356–376.

    Article  MATH  MathSciNet  Google Scholar 

  18. J. N. Mather A Criterion for the Non-existence of Invariant Circles. Preprint (1982).

  19. J. N. Mather,Non-uniqueness of Solutions of Percival's Euler-Lagrange Equation. Commun. Math. Phys.86 (1982) 465–476.

    Article  MATH  MathSciNet  Google Scholar 

  20. J. N. Mather,Amount of Rotation about a Point and the Morse Index. Commun. Math. Physics94 (1984), 141–153.

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Morse,A fundamental class of geodesics on any closed surface of genus greater than one. Trans. AMS,26, (1924) 25–60.

    Article  MathSciNet  Google Scholar 

  22. I. C. Percival,Variational principles for invariant tori and contori. Symp. on Nonlinear Dynamics and Beam-Beam Interactions. (M. Month and J. C. Herrara, eds.) No. 57, American Institute of Physics Conf. Proc (1980) 310–320.

  23. I. C. Percival,A rational principle for invariant tori of fixed frequence J. Phys. A. Math. and Gen.12 (1979) L.57.

    MathSciNet  Google Scholar 

  24. M. Reed andB. Simon,Methods of modern mathematical physics, vol. 1. New York, Academic Press (1980).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept of Mathematics, Princeton University, Fine Hall Washington Road, 08544, Princeton, N.J., USA

    John N. Mather

Authors
  1. John N. Mather
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Partially supported by NSF contract #MCS82-01604.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mather, J.N. More Denjoy minimal sets for area preserving diffeomorphisms. Commentarii Mathematici Helvetici 60, 508–557 (1985). https://doi.org/10.1007/BF02567431

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02567431

Keywords

Search

Navigation