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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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