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The absolute continuity 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
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Let f be an orientation preserving ℋ-diffeomorphism of the circle. If the rotation number α = ρ(f) is irrational and log Df is of bounded variation then, by a wellknown theorem of Denjoy, f is conjugate to the rigid rotation Rα. The conjugation means that there exists an essentially unique homeomorphism h of the circle such that f = h−lRαh. The general problem of relating the smoothness of h to that of f under suitable diophantine conditions on α has been studied extensively (cf. [H1], [KO], [Y] and the references given there). At the bottom of the scale of smoothness for f there is a theorem of M. Herman [H2] which states that if Df is absolutely continuous and D log Df ∈ Lp, p > 1, α = ρ (f) is of ‘constant type’ which means ‘the coefficients in the continued fraction expansion of α are bounded’, and if f is a perturbation of Rα, then h is absolutely continuous. Our purpose in this paper is to give a different proof and an improved version of Herman's theorem. The main difference in the result is that we do not need to assume that f is close to Rα; the proof is very different from Herman's and is very much in the spirit of [KO].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 9 , Issue 4 , December 1989 , pp. 681 - 690
Copyright
Copyright © Cambridge University Press 1989

References

REFERENCES

[C]Carleson, L.. On mappings, conformal at the boundary. J. d'Analyse Math. 19 (1967), 1–13.CrossRefGoogle Scholar
[H]Herman, M.R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Eludes Sci. Publ. Math. 49 (1979), 5–234.CrossRefGoogle Scholar
[H2]Herman, M.R.. Sur les difféomorphismes du cercle de nombre de rotation de type constant. Conference on Harmonic Analysis in Honor of A. Zygmund, Vol. II, (1981), 708–725.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 diffeomorphism of the circle on the Lebesgue measure. J. d'Analyse Math. 36 (1979), 156–166.CrossRefGoogle Scholar
[KO]Katznelson, Y. & Ornstein, D.. The differentiability of the conjugation of certain diffeomorphisms of the circle. Ergod. Th. & Dynam. Sys. this issue, preceding paper.Google Scholar
[L]Lazutkin, V.F.. Examples of diffeomorphisms of a circle. Vestnik Leningrad Univ. Math. 10 (1982), 55–62.Google 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