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 Izv. RAN. Ser. Mat., 2010, Volume 74, Issue 2, Pages 65–108 (Mi izv621)

One-dimensional Fibonacci tilings and induced two-colour rotations of the circle

V. G. Zhuravlev

Abstract: We study two-colour rotations $S_\varepsilon(a,b)$ of the unit circle that take $x\in[0,1)$ to the point $\langle x+a\tau\rangle$ if $x\in[0,\varepsilon)$ and to $\langle x+b\tau\rangle$ if $x\in[\varepsilon,1)$. The rotations $S_\varepsilon(a,b)$ depend on discrete parameters $a,b\in\mathbb Z$ and a continuous parameter $\varepsilon\in[0,1)$ and we choose $\tau$ to be the golden ratio $\frac{1+\sqrt5}2$. We shall show that the $S_\varepsilon(a,b)$ have an invariance property: the induced maps or first-return maps for $S_\varepsilon(a,b)$ are again two-colour rotations $S_{\varepsilon'}(a',b')$ with renormalized parameters $\varepsilon'\in[0,1)$, $a',b'\in\mathbb Z$. Moreover, we find conditions under which the induced maps $S_{\varepsilon'}(a',b')$ have the form $S_{\varepsilon'}(a,b)$, that is, the $S_\varepsilon(a,b)$ are isomorphic to their induced maps and thus have another property, namely, that of self-similarity. We describe the structure of the attractor $\operatorname{Att}(S_\varepsilon(a,b))$ of a rotation $S_\varepsilon(a,b)$ and prove that the restriction of a rotation to its attractor is isomorphic to a certain family of integral isomorphisms $T_\varepsilon$ obtained by lifting the simple rotation of the circle $S(x)=\langle x+\tau\rangle$. A corollary is the uniform distribution of the $S_\varepsilon(a,b)$-orbits on the attractor $\operatorname{Att}(S_\varepsilon(a,b))$. We find a connection between the measure of the attractor $\operatorname{Att}(S_\varepsilon(a,b))$ and the frequency distribution function $\nu_\varepsilon(\theta_1,\theta_2)$ of points in $S_\varepsilon(a,b)$-orbits over closed intervals $[\theta_1,\theta_2]\subset[0,1)$. Explicit formulae for the frequency $\nu_\varepsilon(\theta_1,\theta_2)$ are obtained in certain cases.

Keywords: Fibonacci tilings, double rotations of the circle, induced and integral maps, frequency distribution.

DOI: https://doi.org/10.4213/im621

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English version:
Izvestiya: Mathematics, 2010, 74:2, 281–323

Bibliographic databases:

UDC: 511.218
MSC: 37E10, 37E05, 37B10, 11B85
Revised: 03.06.2008

Citation: V. G. Zhuravlev, “One-dimensional Fibonacci tilings and induced two-colour rotations of the circle”, Izv. RAN. Ser. Mat., 74:2 (2010), 65–108; Izv. Math., 74:2 (2010), 281–323

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv621
• https://doi.org/10.4213/im621
• http://mi.mathnet.ru/eng/izv/v74/i2/p65

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This publication is cited in the following articles:
1. V. G. Zhuravlev, “Bounded remainder sets on the double covering of the Klein bottle”, J. Math. Sci. (N. Y.), 207:6 (2015), 857–873
2. V. G. Zhuravlev, “Bounded remainder sets”, J. Math. Sci. (N. Y.), 222:5 (2017), 585–640
3. Gorodetski A., Kleptsyn V., “Synchronization Properties of Random Piecewise Isometries”, Commun. Math. Phys., 345:3 (2016), 781–796
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