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Two-colour rotations of the unit circle. (English) Zbl 1187.37058
Izv. Math. 73, No. 1, 79-120 (2009); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 73, No. 1, 79-120 (2009).
Let $$I=[0,1]$$ and identify on it the endpoints and let $$\alpha,\beta,\varepsilon\in I$$. A two-colour or double shift is defined by: $$S_{(\alpha,\beta,\varepsilon)}=\langle x+\alpha\rangle$$ if $$0\leq x<\varepsilon$$ and $$S_{(\alpha,\beta,\varepsilon)}=\langle x+\beta\rangle$$ when $$\varepsilon\leq x<1$$ ($$\langle z\rangle$$ denotes de fractional part of $$z\in I$$).
The author focuses on the one parameter family of two-colours $$S_\varepsilon=S_{(2\phi,\phi,\varepsilon)}$$ for any $$\varepsilon\in I$$ and $$\phi=\frac{1+\sqrt{5}}{2}$$ (the golden ratio). The following results are proved:
(A)
For any $$\varepsilon\in I$$ the sequence $$I\supset S_\varepsilon (I) \supset S^2_\varepsilon (I)\supset\dots \supset S^k_\varepsilon (I)\supset \cdots$$ is finite ($$S^k_\varepsilon$$ is the $$k$$-fold composite of $$S_\varepsilon$$)
(B)
$$I$$ is the disjoint union of $$\mathrm{Att}_{\varepsilon}$$ and $$\mathrm{Spir}_\varepsilon$$, respectively the attractor and the basin of attraction of $$S_\varepsilon$$. Any point $$x\in \mathrm{Spir}_\varepsilon$$ enters $$\mathrm{Att}_\varepsilon$$ after a finite number of iterates.
(C)
$$S_\varepsilon|_{\mathrm{Att}_\epsilon}$$ defines a one-to-one mapping of $$\mathrm{Att}_\varepsilon$$ onto itself. Moreover an isomorphism of this map with one integral map is given.
(D)
For any $$\varepsilon$$ there exists an interval $$J_\varepsilon\subset\mathrm{Att}_\varepsilon$$ such that $$S_\varepsilon|_{J_\varepsilon}$$ is isomorphic either to the shift $$S^{+1}(x)=\langle x+\tau\rangle$$ or $$S^{-1}(x)=\langle x-\tau\rangle$$ (both defined on $$I$$).
(E)
The points of the orbit of any $$x\in\mathrm{Att}_\varepsilon$$ are uniformly distributed in $$\mathrm{Att}_\varepsilon$$ with respect to the Haar measure on $$I$$ restricted to $$\mathrm{Att}_\varepsilon$$.

##### MSC:
 37E10 Dynamical systems involving maps of the circle 37B10 Symbolic dynamics 37E45 Rotation numbers and vectors 11B85 Automata sequences
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