Applying methods of harmonic analysis we give a simple proof of the multidimensional version of the Rokhlin-Sinaǐ theorem which states that a Kolmogorov -action on a Lebesgue space has a countable Lebesgue spectrum. At the same time we extend this theorem to -actions. Next, using its relative version, we extend to -actions some other general results connecting spectrum and entropy.
An example of type III cocycle without unbounded gaps of an ergodic probability measure preserving transformation will be shown.
Utilizing the cut-and-stack techniques we construct explicitly a weakly mixing rigid rank-one transformation T which is conjugate to T². Moreover, it is proved that for each odd q, there is such a T commuting with a transformation of order q. For any n, we show the existence of a weakly mixing T conjugate to T² and whose rank is finite and greater than n.
Suppose is a finite positive rotation invariant Borel measure on the open unit disc , and that the unit circle lies in the closed support of . For the Bergman space is the collection of functions in holomorphic on . We show that whenever a Gaussian power series almost surely lies in but not in , then almost surely: a) the zero set of is not contained in any zero set (, and b) is not contained in any zero set.