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Attraction domain for the attractor of a two-color circle rotation

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Abstract

The dynamics of contraction of the unit circle C under iterated two-color circle rotation Sɛ, dependent on a continuous parameter ε ∈ C, is studied. The dynamic distance from the deep hole Dh(ε) of the attraction domain Spir ɛ=C\Att ɛ to the attractor Att ɛ of the rotation Sɛ and the measure of the deep hole |Dh(ε)| are computed. It is proved that as ε ↑ 1, the phenomenon of localization of the deep holes Dh(ε) occurs. It is shown that the process of contraction of the circle,

$$C \supset S_\varepsilon (C) \supset S_\varepsilon ^2 (C) \supset \cdots \supset S_\varepsilon ^k (C) \supset \cdots $$

, goes on in three linear modes if the parameter ε coincides with an eigenvalue εm of a certain B-process, and in four modes in the general case. Bibliography: 7 titles.

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References

  1. M. Boshernitzan and I. Kornfeld, “Interval translation mappings,” Ergod. Theory Dynam. Sys., 15, 821–831 (1995).

    MATH  MathSciNet  Google Scholar 

  2. H. Bruin and S. Troubetzkoy, “The Gauss map on a class of interval translation mappings,” Israel J. Math., 137, 125–148 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Suzuki, S. Ito, and K. Aihara, “Double rotations,” Discr. Contin. Dynam. Sys., 13, 515–532 (2005).

    Article  MATH  MathSciNet  Google Scholar 

  4. V. G. Zhuralev, “One-dimensional Fibonacci tilings,” Izv. Ross. Akad. Nauk, Ser. Mat., 71, No. 2 (2007), 287–321.

    Google Scholar 

  5. V. G. Zhuravlev, “One-dimensional Fibonacci tilings and derivatives of two-colour rotations of a circle,” Preprint Max-Planck-Institut für Mathematik, 59 (2004).

  6. V. G. Zhuravlev, “Two-color rotations of the unit circle,” Izv. Ross. Akad. Nauk, Ser. Mat., in print.

  7. I. P. Kornfel’d, Ya. G. Sinai, and S. V. Fomin, Ergodic Theory [in Russian]. Moscow (1980).

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Authors and Affiliations

  1. Vladimir State Pedagogical Institute, Vladimir, Russia

    V. G. Zhuravlev

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  1. V. G. Zhuravlev
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Correspondence to V. G. Zhuravlev.

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__________

Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 350, 2007, pp. 89–138.

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Zhuravlev, V.G. Attraction domain for the attractor of a two-color circle rotation. J Math Sci 150, 2056–2083 (2008). https://doi.org/10.1007/s10958-008-0122-0

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  • DOI: https://doi.org/10.1007/s10958-008-0122-0

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