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 Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 1, Pages 79–120 (Mi izv601)

Two-colour rotations of the unit circle

V. G. Zhuravlev

Abstract: We consider two-colour, or double, rotations $S_{(\alpha,\beta,\varepsilon)}(x)$ of the unit circle $C$ the colouring of which depends on a continuous parameter $\varepsilon\in C$ and each area of which is given its own rotation angle, $\alpha$ or $\beta$. We choose as a model the one-parameter family of two-colour rotations $S_\varepsilon(x)=S_{(2\tau,\tau,\varepsilon)}(x)$, where $\tau=(1+\sqrt{5} )/2$ is the golden ratio, which have rotation rank $d=2$. It is proved that the first-return map $S_\varepsilon|\mathrm{Att}_\varepsilon$ (the restriction of the rotation $S_\varepsilon(x)$ to its attractor $\mathrm{Att}_\varepsilon$) is isomorphic to the integral map $T_\varepsilon=T(S^{\pm1},d_\varepsilon)$ constructed from the simple rotation $S$ of the circle through the angle $\pm \tau$ and some piecewise-constant function $d_\varepsilon$. An exact formula is obtained for the function $\nu(\varepsilon)$ of frequency distribution of points of the orbits under the action of $S_\varepsilon$.

Keywords: two-colour (double) rotations, ITM-maps (interval translation maps), distribution of fractional parts, Fibonacci tilings.

DOI: https://doi.org/10.4213/im601

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English version:
Izvestiya: Mathematics, 2009, 73:1, 79–120

Bibliographic databases:

UDC: 514
MSC: 37E10, 37B10, 37E45, 11B85
Revised: 23.10.2007

Citation: V. G. Zhuravlev, “Two-colour rotations of the unit circle”, Izv. RAN. Ser. Mat., 73:1 (2009), 79–120; Izv. Math., 73:1 (2009), 79–120

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv601
• https://doi.org/10.4213/im601
• http://mi.mathnet.ru/eng/izv/v73/i1/p79

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This publication is cited in the following articles:
1. V. G. Zhuravlev, “The attraction domain for the attractor of a two-color circle rotation”, J. Math. Sci. (N. Y.), 150:3 (2008), 2056–2083
2. V. G. Zhuravlev, “Geometrizatsiya teoremy Gekke”, Chebyshevskii sb., 11:1 (2010), 126–144
3. A. V. Shutov, “Drobi Fareya i perestanovki, porozhdennye drobnymi dolyami $\{i\alpha\}$”, Chebyshevskii sb., 15:1 (2014), 195–203
4. V. G. Zhuravlev, “Bounded remainder sets”, J. Math. Sci. (N. Y.), 222:5 (2017), 585–640
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