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On an invariant measure for homeomorphisms of a circle with a point of break. (English. Russian original) Zbl 0921.58035
Funct. Anal. Appl. 32, No. 3, 153-161 (1998); translation from Funkts. Anal. Prilozh. 32, No. 3, 11-21 (1998).
The authors study homeomorphisms of the circle \(S^1=[0,1)\) with a simple point of break, that is, with a jump of the first derivative at a point. As is well known in the case of generic rotation numbers, the invariant measure for a \(C^{2+\nu}\)-diffeomorphism is absolutely continuous with respect to the Lebesgue measure. The authors show that the situation becomes diametrically opposite in the presence of a single point of break-type singularity, namely, they prove that an invariant measure is always singular with respect to the Lebesgue measure.

37A99 Ergodic theory
Full Text: DOI
[1] A. Denjoy, ”Sur les courbes definies par les equations différentielles a la surface du tore,” J. Math. Pures Appl.,11, 333–375 (1932). · Zbl 0006.30501
[2] V. I. Arnold, ”Small denominators, I. Mapping the circle onto itself,” Izv. Akad. Nauk SSSR, Ser. Matem.,25, No. 1, 21–86 (1961).
[3] J. Moser, ”A rapidly convergent iteration method, Part II,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3),20, 499–535 (1966). · Zbl 0144.18202
[4] M. Herman, ”Sur la conjugaison différentiable des difféomorphisims du cercle a des rotations,” Publ. Math. IHES,49, 5–233 (1979). · Zbl 0448.58019
[5] M. Herman, ”Résultats récents sur la conjugaison différentiable,” Proc. Intern. Cong. Math., Helsinki, Vol. 2, 1978, pp. 811–820.
[6] J. C. Yoccoz, ”Conjugaison différentiable des difféomorphisims du cercle dont la mombre de rotation vérifie une condition diophantienne,” Ann. Sci. École Norm. Sup. (4),17, 333–359 (1984). · Zbl 0595.57027
[7] K. M. Khanin and Ya. G. Sinai, ”A new proof of M. Herman’s theorem,” Comm. Math. Phys.,112, 89–101 (1987). · Zbl 0628.58021 · doi:10.1007/BF01217681
[8] Ya. G. Sinai and K. M. Khanin, ”Smoothness of conjugacies of diffeomorphisms of the circle with rotations,” Usp. Mat. Nauk,44, No. 1 (265), 57–81 (1989).
[9] Y. Katznelson and D. Ornstein, ”The differentiability of the conjugation of certain diffeomorphisms of the circle,” Ergodic Theory Dynamical Systems,9, No. 4, 643–680 (1989). · Zbl 0819.58033
[10] J. Stark, ”Smooth conjugacy and renormalization for diffeomorphisms of the circle,” Nonlinearity,1, 541–575 (1988). · Zbl 0725.58040 · doi:10.1088/0951-7715/1/4/004
[11] H. Poincaré, Memoire sur les courbes definie par une equation différentielle, I–IV, J. Math. Pures Appl., 1881–1886.
[12] I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian], Nauka, Moscow, 1980.
[13] K. M. Khanin and E. B. Vul, ”Circle homeomorphisms with weak discontinuities,” Adv. Sov. Math., Vol. 3, 57–98 (1991). · Zbl 0733.58026
[14] K. M. Khanin, ”Universal estimates for critical circle mappings,” Chaos,1, 181 (1991). · Zbl 0899.58051 · doi:10.1063/1.165826
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