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On the spectral type of Ornstein’s class one transformations. (English) Zbl 0787.28011
The rank one transformations are, roughly speaking, those measure- preserving transformations which may be obtained from the cutting and stacking of a bounded interval using exactly one column. Many classical examples are well-known, such as those of von Neumann, Kakutani and Chacon. Rank one transformations are always ergodic and it was shown by Baxter that they have simple spectrum. Using probabilistic methods, D. S. Ornstein [Proc. 6 Berkeley Sympos., Math. Stat. Prob., Univ. Calif. Vol. II, 1970, 347-356 (1972; Zbl 0262.28009)] constructed a class of such transformations which are mixing. There has been speculation that the rank one mixing transformations may have non-singular spectrum, or in fact simple Lebesgue spectrum (which would answer a question due to Banach). However, the author shows that any member of Ornstein’s class of rank one mixing transformations has singular spectrum (i.e., its maximal spectral type is singular with respect to Lebesgue measure). The author discusses the question of whether all rank one transformations have singular spectrum, a question which is still open. The paper, which is quite technical, uses harmonic analysis techniques.

28D05 Measure-preserving transformations
47A35 Ergodic theory of linear operators
Full Text: DOI
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