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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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