Abstract: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, Lema\'nczyk, Voln\'y]), of type $III_0$, or which are coboundaries with nonintegrable transfer functions. The cocycles are constructed as sums of coboundaries.
Keywords: smooth cocycle, real analytic cocycle, transfer function, type $III_0$, ergodic and squashable, distributions of a cocycle
AMS Subject Classification: 28D05, 11K50, 60F05