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Cyclic approximation of ergodic step cocycles over irrational rotations. (English) Zbl 0817.28008

Let \(T\) be an automorphism of the Lebesgue probability space \((X,{\mathcal F},\mu)\). A. B. Katok and A. M. Stepin [Usp. Mat. Nauk 22, No. 5(137), 81-106 (1967; Zbl 0172.072)] defined the speed of cyclic approximation which \(T\) admits, and introduced the invariant \[ d(T)= \sup\{r> 0: T\text{ admits a cyclic approximation with speed }\textstyle{{1\over n^ r}}\}. \] This notion is applied to ergodic maps of the 2-torus \({\mathbf T}^ 2\) (where \({\mathbf T}\) is identified with \([0, 1)\), using Borel measurable sets and Lebesgue measure). Let \(\alpha\in {\mathbf T}\) and \(\phi: {\mathbf T}\to {\mathbf T}\) be a measurable function, then the corresponding Anzai skew product \(T_ \phi\) with cocycle \(\phi\) is defined by \[ T_ \phi(x, y)= (x+ \alpha, y+ \phi(x)). \] \(T_ \phi\) is said to be a weakly mixing extension (of the rotation \(x\to x+ \alpha\)) if it is ergodic, and its only eigenvalues are \(\exp(2\pi in\alpha)\), \(n\in \mathbb{Z}\). The aim of the paper is to show that certain types of non-constructive results may be obtained in a restrictive class of cocycles such as step functions. In particular, the author constructs a step cocycle \(\phi(x)= \gamma 1_{[0,\beta)}(x)\) such that \(T_ \phi\) admits a cyclic approximation with speed controlled by \(\alpha\), and is a weakly mixing extension. In addition, for any value \(d(T)\geq 3/2\), \(T_ \phi\) is constructed such that \(d(T_ \phi)\) differs from \(d(T)\) by at most 1/2. It is also shown that for almost every \(\alpha\), \(T_ \phi\) is rank one and rigid.

MSC:

28D05 Measure-preserving transformations

Citations:

Zbl 0172.072
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