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.

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