A new class of Ornstein transformations with singular spectrum

It is shown that for any family of probability measures in Ornstein type constructions, the corresponding transformation has almost surely a singular spectrum. This is a new generalization of Bourgain's theorem [J. Bourgain, On the spectral type of Ornstein class one transformations, Israel J. Math. 84 (1993) 53–63], same result is proved for Rudolph's construction [D. Rudolph, An example of a measure-preserving map with minimal self-joining and applications, J. Anal. Math. 35 (1979) 97–122].

RésuméOn montre que pour toute famille de mesures de probabilités dans la construction d'Ornstein, les transformations résultantes ont un spectre presque sûrement singulier. On obtient ainsi une nouvelle généralisation d'un théoréme dû à Bourgain [J. Bourgain, On the spectral type of Ornstein class one transformations, Israel J. Math. 84 (1993) 53–63]. Un résultat similaire est obtenu pour les transformations de Rudolph [D. Rudolph, An example of a measure-preserving map with minimal self-joining and applications, J. Anal. Math. 35 (1979) 97–122].