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 Mat. Sb., 2003, Volume 194, Number 5, Pages 139–156 (Mi msb738)

On absolutely continuous weakly mixing cocycles over irrational rotations

A. V. Rozhdestvenskii

M. V. Lomonosov Moscow State University

Abstract: A weakly mixing cocycle over a rotation $\alpha$ is a measurable function $\varphi\colon S^1\to S^1$, where $S^1=ż\in\mathbb C:|z|=1\}$, such that the equation
\varphi^n(z)=c\frac{h(\exp(2\pi i\alpha)z)}{h(z)} \quadfor almost all $z$ \tag{1}
has no measurable solutions $h( \cdot )\colon S^1\to S^1$ 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 $\widetilde\varphi\colon\mathbb T\to\mathbb R$ belongs to the class $W^1(M(L)(\mathbb T))$. In addition, it is shown that equation (1) (and also the corresponding additive cohomological equation) is soluble for $\widetilde\varphi\in W^1(L\log_+^{1/2}L(\mathbb T))$.

DOI: https://doi.org/10.4213/sm738

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English version:
Sbornik: Mathematics, 2003, 194:5, 775–792

Bibliographic databases:

UDC: 517.987.5
MSC: Primary 28D04; Secondary 42Axx

Citation: A. V. Rozhdestvenskii, “On absolutely continuous weakly mixing cocycles over irrational rotations”, Mat. Sb., 194:5 (2003), 139–156; Sb. Math., 194:5 (2003), 775–792

Citation in format AMSBIB
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• https://doi.org/10.4213/sm738
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This publication is cited in the following articles:
1. A. V. Rozhdestvenskii, “On the additive cohomological equation and time change for a linear flow on the torus with a Diophantine frequency vector”, Sb. Math., 195:5 (2004), 723–764
2. A. V. Rozhdestvenskii, “On non-trivial additive cocycles on the torus”, Sb. Math., 199:2 (2008), 229–251
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