Skip to main content
SpringerLink
Log in

Invariants for smooth conjugacy of hyperbolic dynamical systems, III

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

Abstract

We give a characterization of Anosov diffeomorphisms smoothly conjugated to a toral automorphism in dimension two in term of the Lyapunov exponents of periodic points. We also give necessary and sufficient conditions for the regularity of solutions of the vector cohomology equations associated to an Anosov flow in three dimensions. This allows us to prove a corresponding conjugation theorem.

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. Banyaga, A.: Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique. Commun. Mat. Helv.53, 174–227 (1978)

    Google Scholar 

  2. Calabi, E.: On the group of automorphisms of a symplectic manifold. In: Problems in Analysis (symposium in honour of S. Bochner). Princeton, NJ: Princeton University Press 1970, pp. 1–26

    Google Scholar 

  3. Collet, P., Epstein, H., Gallavotti, G.: Perturbations of geodesic flows on surfaces of constant negative curvature and their mixing properties. Commun. Math. Phys.95, 61–112 (1984)

    Google Scholar 

  4. Conley, C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V.I. Arnold. Invent. Math.73, 33–49 (1983)

    Google Scholar 

  5. Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.21, 3, 193–226 (1971)

    Google Scholar 

  6. Guillemin, V., Kazhdan, D.: On the cohomology of certain dynamical systems. Topology19, 291–300 (1980)

    Google Scholar 

  7. Herman, M.R.: Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations. Publ. Math.49, 5–234 (1979)

    Google Scholar 

  8. Hirsch, M.W., Pugh, C.C.: Stable manifolds and hyperbolic sets. Proc. Symp. Pure Math.14, AMS, 133–164 (1970)

    Google Scholar 

  9. Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977

    Google Scholar 

  10. Livsic, A.: Homology properties ofY-systems. Math. Notes10, 754–757 (1971)

    Google Scholar 

  11. Llave, R. de la: Invariants for smooth conjugacy of hyperbolic dynamical systems, II. Commun. Math. Phys.109, 369–378 (1987)

    Google Scholar 

  12. Llave, R. de la, Marco, J. M., Moriyón, R.: Canonical perturbation theory of Anosov systems and regularity results for the Livsic cohomology equation. Ann. Math.123, 537–611 (1986)

    Google Scholar 

  13. Manning, A.: There are no new Anosov diffeomorphisms on tori. Am. J. Math.96, (3), 422–429 (1974)

    Google Scholar 

  14. Mather, J.: Appendix to [S], ibid

    Google Scholar 

  15. Moser, J.: On a theorem of Anosov. J. Differ. Equations5, 411–440 (1969)

    Google Scholar 

  16. Marco, J.M., Moriyón, R.: Invariants for smooth conjugacy of hyperbolic dynamical systems, I. Commun. Math. Phys.109, 681–689 (1987)

    Google Scholar 

  17. Plante, J.F.: Anosov flows. Am. J. Math.94, 729–754 (1972)

    Google Scholar 

  18. Smale, S.: Differentiable dynamical systems. Bull. AMS73, 747–817 (1967)

    Google Scholar 

  19. Shub, M.: Stabilité globale des systèmes dynamiques. Asterisque56, (1978)

  20. Sternberg, S.: On localCr contractions on the real line. Duke Math. J.24, 97–102 (1957)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universidad Autónoma de Madrid, Spain

    José Manuel Marco & Roberto Moriyón

Authors
  1. José Manuel Marco
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Moriyón
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by J.N. Mather

Supported by CAICyT grant #397/84

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marco, J.M., Moriyón, R. Invariants for smooth conjugacy of hyperbolic dynamical systems, III. Commun.Math. Phys. 112, 317–333 (1987). https://doi.org/10.1007/BF01217815

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01217815

Keywords

Search

Navigation