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Cyclic approximation of irrational rotations. (English) Zbl 0807.28008
It was shown by A. B. Katok and A. M. Stepin [Uspehi Mat. Nauk 22, No. 5(137), 81-106 (1967; Zbl 0172.072)] that almost every irrational rotation $T: [0,1]\to [0,1],\quad Tx= x+ \alpha\text{ mod }1,$ admits a cyclic approximation with speed $$o\left({1\over n}\right)$$ (called a good cyclic approximation). It was observed by del Junco that in fact every irrational rotation admits a good cyclic approximation, i.e., there exists a sequence of partitions $$\zeta_ n\to \varepsilon$$ as $$n\to\infty$$, $$\zeta_ n= \{C_ 0,C_ 1,\dots,C_{h_ n-1}\}$$, and a transformation $$T_ n$$ which cyclically permutes the elements of $$\zeta_ n$$, i.e., $$C_ j= T^ j_ n C_ 0$$, $$j= 0,1,\dots, h_ n- 1$$, satisfying $\sum^{h_ n- 1}_{j=0} \mu(TC_ j \Delta T_ n C_ j)= o\left({1\over n}\right).$ In this paper, the author shows that the speed of the cyclic approximation of $$T$$ is essentially as good as the speed of rational approximation of $$\alpha$$, i.e., $$T$$ admits cyclic approximation with speed $$o(f(n))$$ iff there exists a sequence of rational numbers $$p/q\to \alpha$$ such that $$\left|\alpha-{p\over q}\right|= o(f(q))$$. Applications to Anzai skew products are given.

##### MSC:
 28D05 Measure-preserving transformations 37A99 Ergodic theory
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##### References:
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