Derivatives of circle rotations, and similarity of orbits. (Russian, English) Zbl 1082.37040

Shutov, A. V.

Zap. Nauchn. Semin. POMI 314, 272-284, 291 (2004); translation in J. Math. Sci., New York 133, No. 6, 1765-1771 (2006).

In the paper, arithmetic and geometric properties of an irrational circle rotation \[ R_{\alpha}:x\to x-\alpha \pmod 1 \] are studied. For a set \(Y\), the derivative of \(R_{\alpha}\) on \(Y\) is defined as the first return map \[ d_YR_{\alpha}(x)=R_{\alpha}^{n_Y(x)}(x)\quad\text{with}\quad n_Y(x)=\min\{n\in\mathbb N:R_{\alpha}^n(x)\in Y\}. \] Intervals of the unit circle are taken as sets \(Y\). The main result is the following theorem: Let \(R_{\alpha}:x\to x-\alpha \pmod 1\) be a circle rotation. If \(I=I_m\) is a proper interval of differentiation then the derivative \(d_IR_{\alpha}(x)\) is also a circle rotation. If \(I\neq I_m\) is an improper interval then \(d_IR_{\alpha}(x)\) is a noncyclic 3-interval exchange.
As a geometric consequence of the theorem, the similarity of orbits for rotations with different \(\alpha\) is established.

Reviewer: Georgy Osipenko (St. Peterburg)

MSC:

37E10	Dynamical systems involving maps of the circle
37A45	Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
37B10	Symbolic dynamics

Keywords:

derivatives of circle rotations; symbolic dynamics; generalized Fibonacci tiling; continued fractions