Rozhdestvenskij, A. V. 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 \(\widetilde\varphi\in W^1(L\log^{1/2}_+ L(\mathbb{T}))\). Cited in 1 Document 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 Keywords:weakly mixing cocycle; rotation; continued fraction expansion PDFBibTeX XMLCite \textit{A. V. Rozhdestvenskij}, Sb. Math. 194, No. 5, 775--792 (2003; Zbl 1077.37007); translation from Mat. Sb. 194, No. 5, 139--156 (2003) Full Text: DOI