On the spectral type of Ornstein's class one transformations
type: Item
instance of: scholarly article
title: On the spectral type of Ornstein's class one transformations
zbMATH Open document ID: 0787.28011
DOI: 10.1007/BF02761690
published in: Israel Journal of Mathematics
publication date: 1994-01-02T00:00:00Z
review text: 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, \textit{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.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 474983
zbMATH Keywords: spectral multiplicity
zbMATH Keywords: rank one transformations
zbMATH Keywords: measure-preserving transformations
zbMATH Keywords: cutting
zbMATH Keywords: stacking
zbMATH Keywords: rank one mixing transformations
zbMATH Keywords: Ornstein's class
zbMATH Keywords: singular spectrum
zbMATH Keywords: maximal spectral type
author: Jean Bourgain
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
cites work: Mixing of all orders and pairwise independent joinings of systems with singular spectrum
cites work: Twofold mixing implies threefold mixing for rank one transformations
cites work: Hardy's inequality and the \(L^ 1\) norm of exponential sums
cites work: Q5677743
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description: scientific article; zbMATH DE number 474983