Sankhya: The Indian Journal of Statistics

2000, Volume 62, Series A, Pt. 3, pp.291--306

ON THE SPECTRUM OF THE POWERS OF ORNSTEIN TRANSFORMATIONS

By

EL HOUCEIN EL ABDALAOUI, *Université de Rouen-Mathématiques, Colbert*

*SUMMARY. *Generalizing Bourgain's methods we
prove that the singularity problem of the spectrum of rank one
transformations is related to the singularity problem of the
maximal spectral type of rank one transformations with respect to
its translations. For the Ornstein transformations it is proved
that almost surely the spectral types of these transformations are
singular with respect to their rational translations, so the
powers of these transformations have almost surely simple
spectrum. We deduce that for any positive integer *n*, there exist
mixing transformations of rank *n* with simple spectrum.

*AMS (1991) subject classification.* Primary 28D05; secondary
47A35.

*Key words and phrases. * Dynamical system, simple
spectrum, mixing maps, rank one transformation, Lebesgue spectrum.