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Cyclic approximation of analytic cocycles over irrational rotations. (English) Zbl 0845.28009
It has been established that for “most” $$f\in C^r(\mathbb{T})$$, where $$\mathbb{T}$$ is the circle, the Anzai skew product $$T_f(x, y)= (x+ \alpha, y+ f(x))$$ $$\text{mod }1$$ defined on the 2-torus $$\mathbb{T}^2$$ admits a good cyclic approximation by periodic transformations but the cocycle is weakly mixing, provided the irrational $$\alpha$$ admits a sufficiently good approximation by rationals.
The present author extends this result as follows: Fix $$\alpha$$ and a sequence of rationals $$p_n/q_n\to \alpha$$ with $$q_n> 0$$ and $$p_n$$ and $$q_n$$ relatively prime. Let $$E\subset C'(\mathbb{T})$$ be an additive subgroup with topology stronger than $$C'$$-convergence and such that (1) $$E$$ is a complete metric group, (2) $$E$$ contains the constants, (3) $$E$$ has a dense subset of trigonometric polynomials. Suppose $$|\alpha- p_n/q_n|= o(\varepsilon(q_n t_n)/q^2_n)$$, where $$\varepsilon(n)$$ is a non-increasing sequence of positive numbers and $$t_n\to \infty$$. Then the set of cocycles $$f\in E$$ such that $$T_f$$ admits cyclic approximation with speed $$o(\varepsilon(n)/n)$$ is residual in $$E$$.
Special cases of $$E$$ allow one to prove, for example, that the weakly mixing cocycles form a dense $$G_\delta$$ set in $$E$$.

MSC:
 28D05 Measure-preserving transformations
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