Spectral Type of Ornstein's Class One Transformations

摘要

The Note is concerned with the spectral analysis of a class of transformationsintroduced in a paper by D. Ornstein. These transformations called class one or rank one have spectral multiplicity one. It is apparently an unsolved problem whether the spectral type may be absolutely continuous with respect to Lebesgue measure (such an example would be very meaningful to Banach's well known problem whether a dynamical system (Omega,mu,T) may have simple Lebesgue spectrum). In the paper by Ornstein, an example of a mixing rank one transformation is obtained. The construction makes essential use of probabilistic techniques. It is the authors' purpose here to show that for such a random construction the maximal spectral type is always singular. The investigation will be mainly oriented towards harmonic analysis.