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Ergodicity and irrational rotations. (English) Zbl 0807.28009

The author is interested in the form of the cocyle given by the function \(\exp(\pi i\chi_{[0,t)})\) where \(0< \theta< 1\) is an irrational number, \(r\in \mathbb{Z}\), and \(t= r\theta\) modulo 1.
Let \(\Gamma= \{n\theta\}_{n\in \mathbb{Z}}\) be the subgroup of \(\mathbb{T}= \mathbb{R}/\mathbb{Z}\) (which is identified with the half-open interval \([0,1)\)). Let \(G\) be a compact Abelian group. A \(G\)-cocycle \(A\) on \(\Gamma\) is a function \(A: \Gamma\times\mathbb{T}\to G\) which satisfies \(A(\gamma_ 1+ \gamma_ 2,x)= A(\gamma_ 1,x)A(\gamma_ 2,x- \gamma_ 1)\) for all \(\gamma_ 1,\gamma_ 2\in \Gamma\) and a.e. \(x\in \mathbb{T}\). \(A\) is a \(G\)- coboundary if \(A(\theta,x)= q(x- \theta)^{-1} q(x)\) for some function \(q: \mathbb{T}\to G\). \(A\) is \(G\)-trivial if \(A\) is a constant multiple of a \(G\)-boundary. Let \(\mathbb{Z}_ n\) be the subgroup of the unit circle consisting of the \(n\)th roots of unity.
The author investigates various aspects of this situation, the corresponding skew products arising from the rotation by \(\theta\), and implications for uniform distribution modulo 1. The main result is that if \(r\) is a rational number, then the \(\mathbb{Z}_ 2\)-cocycle given by \(A(x,\theta)= \exp(\pi i\chi_{[0,t)})\), \(t= \{r\theta\}\), is \(\mathbb{Z}_ 2\)-trivial if and only if \(r\) is an even integer.

MSC:

28D05 Measure-preserving transformations
11K06 General theory of distribution modulo \(1\)
37A99 Ergodic theory
54H20 Topological dynamics (MSC2010)
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