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A class of real cocycles having an analytic coboundary modification. (English) Zbl 0817.28009
Let \((X,{\mathcal B},\mu)\) be the unit interval with Lebesgue measurable sets and Lebesgue measure and \(T: X\to X\) an irrational rotation, \(Tx= x+ \alpha\text{ mod }1\). Denote by \(G\) a locally compact Abelian metric group with Haar measure \(m\) (usually \(G= \mathbb{R}\) or \(S^ 1\)), and \(\phi: X\to G\) a measurable function. Then \(T_ \phi: X\times G\to X\times G\) defined by \[ T_ \phi(x, g)= (Tx, g\cdot \phi(x)) \] is called a \(G\)- extension of \(T\) with cocycle \(\phi\). If \(G= S^ 1\) then \(T_ \phi\) is called an Anzai skew product. Two cocycles \(\phi\) and \(\psi\) are cohomologous if there exists a measurable function \(f: X\to G\) satisfying \(\phi/\psi(x)= f(Tx)/f(x)\), \(x\in X\).
In Israel J. Math. 80, No. 1-2, 33-64 (1992; Zbl 0774.28008), the same authors constructed two Anzai skew products that are weakly isomorphic but not isomorphic (in fact a class of such examples over a dense \(G_ \delta\) of admissible base rotations). It was shown that the cocycles in those examples may be smoothed to give \(C^ \infty\) cocycles with the same properties. In this paper the authors abstract these properties using the concept of an “almost analytic cocycle construction procedure” (a.a.c.c.p.). In particular, it is shown that each real valued cocycle may be smoothed (i.e., may be replaced by a cohomologous real analytic cocycle), and in fact the construction in the earlier paper, on a perhaps smaller residual set of \(\alpha\)’s, is an a.a.c.c.p.
In addition, by modifying the methods of A. Iwanik and J. Serafin [Colloq. Math. 66, No. 1, 63-76 (1993)], the construction may be done to give examples of Anzai skew products which are rank one and have partially continuous spectrum.

MSC:
28D05 Measure-preserving transformations
37A99 Ergodic theory
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