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Rotation number and one-parameter families of circle diffeomorphisms

Published online by Cambridge University Press:  19 September 2008

Masato Tsujii
Masato Tsujii
Affiliation:
Department of Mathematics, Kyoto University, Sakyo-ku, Kyoto, 606, Japan
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Abstract

We consider one-parameter families of circle diffeomorphisms, f1(x) = f(x) + t(t ∈ S1), where f: S1 is a Cr-diffeomorphism (r≥3). We show that, for Lebesgue almost every t ∈ S1 the rotation number of f1, is either a rational number or an irrational number of Roth type. In the former case, f1, has periodic orbits and, in the latter case, f1, is Cr − 2-conjugate to an irrational rigid rotation from well-known theorems of Herman and Yoccoz.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 2 , June 1992 , pp. 359 - 363
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

[A]Arnol'd, V. I.. Small denominators I; On the mappings of a circle into itself. Trans. Amer. Math. Soc. 46 (1961), 213–284.Google Scholar
[H1]Herman, M. R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publ. math. IHES 49 (1979), 5–234.CrossRefGoogle Scholar
[H2]Herman, M. R.. Mesure de Lebesgue et nombre de rotation. Lecture notes in mathematics 597 (Springer: New York, 1977), pp. 271–293.Google Scholar
[T]Tsujii, M.. A measure on the space of mappings and dynamical system theory. J. Math. Soc. Japan, to appear.Google Scholar
[Y]Yoccoz, J. C.. Conjugaison différentiable des difféomorphismes du cercle dont le nombre de rotation vérifie une condition Diophantienne. Ann. Sci. de I' Ec. Norm. Sup. 17 (1984), 333–361.CrossRefGoogle Scholar