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Constructions of cocycles over irrational rotations. (English) Zbl 0891.28011

This paper refines the earlier work of A. Iwanik, M. Lemańczyk and D. Rudolph [Isr. J. Math. 83, No. 1-2, 73-95 (1993; Zbl 0786.28011)] in which (among other things) a continuous coboundary for an irrational circle rotation with non-zero degree and bounded variation was constructed. The example showed that the result of H. Furstenberg [Am. J. Math. 83, 573-601 (1973; Zbl 0178.38404)] that no Lipschitz continuous cocycle with nonzero degree could be a coboundary, for which several authors had weakenend the Lipschitz condition to absolute continuity, could not be weakened any further.
Here the authors construct a coboundary with degree \(1\) and bounded variation which is Hölder continuous for any fixed Hölder exponent less than \(1\), and which is homotopic to the identity. The construction involves making a special Cantor set in the circle associated to the continued fraction expansion of the irrational number defining the rotation.
Reviewer: T.B.Ward (Norwich)

MSC:

28D05 Measure-preserving transformations
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