Communications in Mathematical Physics

, Volume 112, Issue 1, pp 89–101 | Cite as

A new proof of M. Herman's theorem

  • K. M. Khanin
  • Ya. G. Sinai


A new proof of the M. Herman theorem on the smooth conjugacy of a circle map is presented here. It is based on the thermodynamic representation of dynamical systems and the study of the ergodic properties for the corresponding radom variables.


Neural Network Dynamical System Statistical Physic Complex System Nonlinear Dynamics 
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  1. 1.
    Vul, E.B., Sinai, Ya. G., Khanin, K.M.: Feigenbaum universality and the thermodynamic formalism. Usp. Mat. Nauk.39:3, 3–37 (1984) [English transl. Russ. Math. Surv.39, 1–40 (1984)]Google Scholar
  2. 2.
    Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.I.: Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A33, 2, 1141–1151 (1986)Google Scholar
  3. 3.
    Collet, P., Lebowitz, J., Porzio, A.: Dimension spectrum for some dynamical systems. Rutgers Univ. PreprintGoogle Scholar
  4. 4.
    Khanin, K.M., Sinai, Ya. G.: Renormalization group method and the K.A.M. theory. In: Nonlinear phenomena in plasma physics and hydrodynamics. Sagdeev, R.Z. (ed.). Moscow: Mir 1986Google Scholar
  5. 5.
    Sinai, Ya.G., Khanin, K.M.: Renormalization group method in the theory of dynamical systems. In: Proceedings of the conference “Renormalization Group 86“, Dubna 1986 (in press)Google Scholar
  6. 6.
    Herman, M.: Sur la conjugaison différentiable des difféomorphism du cercle à des rotations. Pub. Mat. I.H.E.S.49, 5–233 (1979)Google Scholar
  7. 7.
    Herman, M.: Simple proofs of local conjugacy theorems for diffeomorphisms of the circle with almost every rotation number. Bol. Soc. Bras. Mat.16, 45–83 (1985);Google Scholar
  8. 7a.
    Sue les difféomorphisms du cercle de nombre de rotation de type constant. In: Papers in Honor of A. Zygmund. Becker, A. et al. (eds.). Belmont: Wadsworth 1983Google Scholar
  9. 8.
    Yoccoz, J.C.: Centralisateurs et conjugaison différentiable des diffeomorphisms du cercle. Thesis Univ. Paris Sud (1985) unpublished;Google Scholar
  10. 8a.
    C 1 conjugaison des difféomorphisms du cercle. In: Geometry and dynamic. Palis, J. (ed.). Lecture Notes in Mathematics, Vol. 1007. Berlin, Heidelberg, New York: Springer 1983;Google Scholar
  11. 8b.
    Conjugaison différentiable des difféomorphisms du cercle dont le nombre de rotation vérifie une condition diophantienne. Ann. Sci. Éc. Norm. Super.17, 333–359 (1984)Google Scholar
  12. 9.
    Cornfeld, I.P., Sinai, Ya.G., Fomin, S.V.: Ergodic theory. Moscow: Nauka 1980 (English transl.: Berlin, Heidelberg, New York: Springer 1982)Google Scholar
  13. 10.
    Ruelle, D.: Thermodynamic Formalism. Reading: Addison-Wesley 1978Google Scholar
  14. 11.
    Arnol'd, V.I.: Small denominators. I. Mappings of the circumference onto itself. Izv. Mat. Nauk.25:1, 25–96 (1961) [English Transl. Am. Math. Soc.49, 213–284]Google Scholar
  15. 12.
    Hawkins, J., Schmidt, K.: OnC 2 diffeomorphisms of the circle which are of type III1. Invent. math.66, 511–518 (1966)Google Scholar

Copyright information

© Springer-Verlag 1987

Authors and Affiliations

  • K. M. Khanin
    • 1
  • Ya. G. Sinai
    • 1
  1. 1.L. D. Landau Institute for Theoretical PhysicsMoscow, V-334USSR

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