Geodesic
Constructions of smooth and analytic cocycles over irrational circle rotations
Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 4, pp. 745-764.

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We define a class of step cocycles (which are coboundaries) for irrational rotations of the unit circle and give conditions for their approximation by smooth and real analytic coboundaries. The transfer functions of the approximating (smooth and real analytic) coboundaries are close (in the supremum norm) to the transfer functions of the original ones. This result makes it possible to construct smooth and real analytic cocycles which are ergodic, ergodic and squashable (see [Aaronson, Lemańczyk, Volný]), of type $III_0$, or which are coboundaries with nonintegrable transfer functions. The cocycles are constructed as sums of coboundaries.
Classification : 11K50, 28D05, 60F05
Keywords: smooth cocycle; real analytic cocycle; transfer function; type $III_0$; ergodic and squashable; distributions of a cocycle 
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     author = {Voln\'y, Dalibor},
     title = {Constructions of smooth and analytic cocycles over irrational circle rotations},
     journal = {Commentationes Mathematicae Universitatis Carolinae},
     pages = {745--764},
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Volný, Dalibor. Constructions of smooth and analytic cocycles over irrational circle rotations. Commentationes Mathematicae Universitatis Carolinae, Tome 36 (1995) no. 4, pp. 745-764. http://geodesic.mathdoc.fr/item/CMUC_1995__36_4_a12/