The almost sure mutual singularity of the spectrum of Ornstein transformations
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 \(n\geq 1\).
scientific article; zbMATH DE number 1358185