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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Wiener lemma and cocycles

Author: Marek Ryszard Rychlik
Journal: Proc. Amer. Math. Soc. 104 (1988), 932-933
MSC: Primary 28D05
MathSciNet review: 964876
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Abstract: We give a sufficient and necessary condition for a function with its values in the unit circle to be a multiplicative coboundary. This theorem generalizes the following result of Veech [1]. Let $ T:{\mathbf{T}} \to {\mathbf{T}}$ be a rotation of the unit circle $ {\mathbf{T}}$ by an irrational angle $ \theta $. Let $ F:{\mathbf{T}} \to {\mathbf{T}}$ be a measurable function. Then $ F$ is a multiplicative coboundary iff

$\displaystyle \int_{\mathbf{T}} {F(x)F(Tx) \cdots F({T^{n - 1}}x)d\mu (x) \to 1,\quad {\text{as }}\left\Vert {n\theta } \right\Vert \to 0,} $

where $ \left\Vert {n\theta } \right\Vert$ is the distance of $ n\theta $ from integers and $ \mu $ is the Haar measure.

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Keywords: Wiener lemma, multiplicative cocycles
Article copyright: © Copyright 1988 American Mathematical Society