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On absolutely continuous weakly mixing cocycles over irrational rotations of the circle. (English. Russian original) Zbl 1077.37007

Sb. Math. 194, No. 5, 775-792 (2003); translation from Mat. Sb. 194, No. 5, 139-156 (2003).
Summary: A weakly mixing cocycle over a rotation \(\alpha\) is a measurable function \(\varphi: S^1\to S^1\), where \(S^1= \{z\in\mathbb{C}: |z|=1\}\), such that the equation \[ \varphi^n(z)= c{h(\exp(2\pi i\alpha)z)\over h(z)}\quad\text{for almost all }z\tag{\(*\)} \] has no measurable solutions \(h(\cdot): S^1\to S^1\) for any \(n\in\mathbb{Z}\setminus\{0\}\) and \(c\in\mathbb{C}\), \(|c|= 1\).
If the irrational number \(\alpha\) has bounded convergents in its continued fraction expansion and a function \(M(y)\) increases more slowly than \(y\ln^{1/2}y\), then it is proved that there exists a weakly mixing cocycle of the form \(\varphi(\exp(2\pi ix))= \exp(2\pi i\widetilde\varphi(x))\), where \(\widetilde\varphi: \mathbb{T}\to\mathbb{R}\) belongs to the class \(W^1(M(L)(\mathbb{T}))\). In addition, it is shown that equation \((*)\) (and also the corresponding additive cohomological equation) is solvable for \(\widetilde\varphi\in W^1(L\log^{1/2}_+ L(\mathbb{T}))\).

MSC:

37A25 Ergodicity, mixing, rates of mixing
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
37E10 Dynamical systems involving maps of the circle
11K50 Metric theory of continued fractions
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