The Wiener lemma and cocycles
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- by Marek Ryszard Rychlik PDF
- Proc. Amer. Math. Soc. 104 (1988), 932-933 Request permission
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 \[ \int _{\mathbf {T}} {F(x)F(Tx) \cdots F({T^{n - 1}}x)d\mu (x) \to 1,\quad {\text {as }}\left \| {n\theta } \right \| \to 0,} \] where $\left \| {n\theta } \right \|$ is the distance of $n\theta$ from integers and $\mu$ is the Haar measure.References
- William A. Veech, Strict ergodicity in zero dimensional dynamical systems and the Kronecker-Weyl theorem $\textrm {mod}\ 2$, Trans. Amer. Math. Soc. 140 (1969), 1–33. MR 240056, DOI 10.1090/S0002-9947-1969-0240056-X
- I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ, Ergodic theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 245, Springer-Verlag, New York, 1982. Translated from the Russian by A. B. Sosinskiĭ. MR 832433, DOI 10.1007/978-1-4615-6927-5
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 104 (1988), 932-933
- MSC: Primary 28D05
- DOI: https://doi.org/10.1090/S0002-9939-1988-0964876-6
- MathSciNet review: 964876