## 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

Zbl 0172.072
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