The almost sure mutual singularity of the spectrum of Ornstein transformations
type: Item
instance of: scholarly article
title: The almost sure mutual singularity of the spectrum of Ornstein transformations
zbMATH Open document ID: 0967.37004
DOI: 10.1007/BF02773480
published in: Israel Journal of Mathematics
publication date: 2000-08-21T00:00:00Z
review text: This is a joint review for two papers (see also Zbl 0967.37005 below). \textit{D. S. Ornstein} [Proc. 6th Berkeley Sympos. Math. Statist. Probab., Univ. Calif. 1970, Vol. II, 347-356 (1972; Zbl 0262.28009)] constructed a class of rank one transformations using the stacking method and a single column, showing that they are almost surely mixing. Subsequently these maps have been shown to have many important properties, such as minimal self-joinings [\textit{D. J. Rudolph}, J. Anal. Math. 35, 97-122 (1979; Zbl 0446.28018)] and singular spectrum [\textit{J. Bourgain}, Isr. J. Math. 84, 53-63 (1993; Zbl 0787.28011)]. In this paper it is shown that the maximal spectral types of distinct transformations in this class are almost surely (with respect to product measure) mutually singular. It follows that they are pairwise disjoint (in the sense of Furstenburg). In the second of the two papers, the author generalizes the definition of the Ornstein rank one maps to show that if the construction is done so that the cutting parameter is not bounded, then the resulting maps are almost surely totally ergodic, giving a larger collection of mixing Ornstein type rank one maps. In yet another paper [Sankhyā, Ser. A 62, 291-308 (2000)], the author shows that for Ornstein rank one transformations, almost surely the spectral types are singular with respect to their rational translations. Consequently, the powers of these transformations have (almost surely) simple spectrum, giving rise to mixing transformations \(T\) for which \(T^n\) is of rank \(n\) and has simple spectrum, for each \(n\geq 1\).
Mathematics Subject Classification ID: 37A05
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 1358185
zbMATH Keywords: singular spectrum
zbMATH Keywords: Ornstein rank one maps
zbMATH Keywords: simple spectrum
zbMATH Keywords: mixing transformations
author: El Houcein El Abdalaoui
reviewed by: Geoffrey R. Goodson
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02773480
OpenAlex ID: W2365567955
description: scientific article; zbMATH DE number 1358185