Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-18T06:22:54.046Z Has data issue: false hasContentIssue false
  • English
  • Français

The differentiability of the conjugation of certain diffeomorphisms of the circle

Published online by Cambridge University Press:  19 September 2008

Y. Katznelson  and
D. Ornstein
Y. Katznelson
Affiliation:
Mathematics Department, Stanford University, Stanford CA 94305, USA
D. Ornstein
Affiliation:
Mathematics Department, Stanford University, Stanford CA 94305, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Our purpose in this paper is to present a more or less complete solution to the problem of the smoothness of the conjugation of aperiodic diffeomorphisms of the circle. We show that the rotation number and the smoothness of the diffeomorphism guarantee a certain smoothness for the homeomorphism which conjugates it with a rigid rotation, and obtain the best smoothness that can be guaranteed.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 4 , December 1989 , pp. 643 - 680
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[A]Arnold, V.I.. Small denominators I, on the mapping of a circle into itself. Ivestijia Akad. Nauk. serie Math. 25(1) (1961), 21–86.Google Scholar
Translation Amer. Math. Soc. 2nd series, 46, 213–284.Google Scholar
[C]Carleson, L.. A remark on Denjoy's inequality and Herman's theorem. Publ. Math. I.H.E.S. 49 (1979), 235–241.CrossRefGoogle Scholar
[H]Herman, M.R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. Math. I.H.E.S. 49 (1979), 5–234.CrossRefGoogle Scholar
[H2]Herman, M.. Sur les courbes invariantes par les difféomorphismes de l'anneau. Astérisque 104 (1983), chap. VIII.Google Scholar
[HS]Hawkins, J. & Schmidt, K.. On C2-diffeomorphisms which are of type III1. Invent. Math. 66 (1982), 511–518.CrossRefGoogle Scholar
[K]Katznelson, Y.. The action of diffeomorphisms of the circle on the Lebesgue measure. J. D Anal. Math. 36 (1979), 156–166.CrossRefGoogle Scholar
[KO]Katznelson, Y. & Ornstein, D.. The absolute continuity of the conjugation of certain diffeomorphisms of the circle. Ergod. Th. & Dynam. Sys.Google Scholar
[KS]Khanin, K.M. & Sinai, Ya. G.. A new proof of M. Herman's theorem. Comm. Math. Phys. 112 (1987), 89–101.CrossRefGoogle Scholar
[LA]Lazutkin, V.F.. Examples of diffeomorphisms of a circle. Vestnik Leningrad Univ. Math. 10 (1982), 55–62.Google Scholar
[LI]Livcic, A.N.. Some homology properties of U-systems. Mat. Zametki 10 (1971), 555–564;Google Scholar
Math. Notes 10 (1971), 758–763.Google Scholar
[M]Moser, J.. A rapidly convergent iteration method, part II. Ann. Scuola Norm. Sup. di Pisa 20 (1966), 499–535;Google Scholar
Moser, J.. A new technique for the construction of solutions of non-linear differential equations. Proc. Nat. Acad. Sci. 41 (1963), 1824–1831;Google Scholar
Moser, J.. Perturbation theory for almost periodic solutions for undamped non linear differential equations. Inter. Symp. Nonlinear Differential Equations and Nonlinear Mechanics, pp. 71–79. New York, 1963.CrossRefGoogle Scholar
[Y]Yoccoz, J.-C.. Conjugaison differentiable des diffeomorphismes du cercle dont le nombre de rotation verifie une condition diophantienne. Ann. Sci. Ecole Norm. Sup. 4 17 (1984), 333–359.CrossRefGoogle Scholar