Communications in Mathematical Physics

, Volume 112, Issue 2, pp 317–333 | Cite as

Invariants for smooth conjugacy of hyperbolic dynamical systems, III

  • José Manuel Marco
  • Roberto Moriyón


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.


Neural Network Dynamical System Statistical Physic Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [B]
    Banyaga, A.: Sur la structure du groupe des diffeomorphismes qui preservent une forme symplectique. Commun. Mat. Helv.53, 174–227 (1978)Google Scholar
  2. [C]
    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. 1–26Google Scholar
  3. [CEG]
    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. [CZ]
    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. [F]
    Fenichel, N.: Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.21, 3, 193–226 (1971)Google Scholar
  6. [GK]
    Guillemin, V., Kazhdan, D.: On the cohomology of certain dynamical systems. Topology19, 291–300 (1980)Google Scholar
  7. [H]
    Herman, M.R.: Sur la conjugaison différentiable des diffeomorphismes du cercle à des rotations. Publ. Math.49, 5–234 (1979)Google Scholar
  8. [HP]
    Hirsch, M.W., Pugh, C.C.: Stable manifolds and hyperbolic sets. Proc. Symp. Pure Math.14, AMS, 133–164 (1970)Google Scholar
  9. [HPS]
    Hirsch, M.W., Pugh, C.C., Shub, M.: Invariant manifolds. Lecture Notes in Mathematics, Vol. 583. Berlin, Heidelberg, New York: Springer 1977Google Scholar
  10. [L]
    Livsic, A.: Homology properties ofY-systems. Math. Notes10, 754–757 (1971)Google Scholar
  11. [L1]
    Llave, R. de la: Invariants for smooth conjugacy of hyperbolic dynamical systems, II. Commun. Math. Phys.109, 369–378 (1987)Google Scholar
  12. [LMM]
    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. [M]
    Manning, A.: There are no new Anosov diffeomorphisms on tori. Am. J. Math.96, (3), 422–429 (1974)Google Scholar
  14. [Ma]
    Mather, J.: Appendix to [S], ibidGoogle Scholar
  15. [Mo]
    Moser, J.: On a theorem of Anosov. J. Differ. Equations5, 411–440 (1969)Google Scholar
  16. [MM]
    Marco, J.M., Moriyón, R.: Invariants for smooth conjugacy of hyperbolic dynamical systems, I. Commun. Math. Phys.109, 681–689 (1987)Google Scholar
  17. [P]
    Plante, J.F.: Anosov flows. Am. J. Math.94, 729–754 (1972)Google Scholar
  18. [S]
    Smale, S.: Differentiable dynamical systems. Bull. AMS73, 747–817 (1967)Google Scholar
  19. [Sh]
    Shub, M.: Stabilité globale des systèmes dynamiques. Asterisque56, (1978)Google Scholar
  20. [St]
    Sternberg, S.: On localCr contractions on the real line. Duke Math. J.24, 97–102 (1957)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • José Manuel Marco
    • 1
  • Roberto Moriyón
    • 1
  1. 1.Universidad Autónoma de MadridSpain

Personalised recommendations