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The maximal spectral type of a rank one transformation. (English) Zbl 0793.28013
The rank one transformations are those invertible measure preserving transformations which may be obtained from the cutting and stacking of an interval using exactly one column. Rank one transformations are always ergodic, have simple spectrum, and may be weakly mixing or even mixing.
The purpose of this paper is to show, using an explicit construction, that the maximal spectral type of a rank one transformation is given by a kind of Riesz product. It is shown how to use this construction to prove that certain examples (such as Chacon’s transformation) have a spectrum which is singular with respect to Lebesgue measure. It is still an open question whether all rank one transformations have singular spectrum, but J. Bourgain [Isr. J. Math. 84, No. 1-2, 53-63 (1993; Zbl 0787.28011)] has recently shown this to be true for Ornstein’s class of rank one mixing transformations.

MSC:
28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
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