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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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