Rank one transformations with singular spectral type
The class of rank one transformations are, roughly speaking, those that may be obtained from cutting and stacking an interval, using a single column at each stage in the construction. They are known to be ergodic and to have simple spectrum. \textit{J. Bourgain} [Isr. J. Math. 84, No. 1-2, 53-63 (1993; Zbl 0787.28011)] has shown that a subclass of these transformations, namely Ornstein's rank one mixing transformations, have a spectrum singular with respect to Lebesgue measure. It is an open question as to whether all rank one transformations have a singular spectrum. The purpose of this paper is to give very general conditions under which this is true. It is shown that if a rank one transformation \(T\) satisfies \(\sum^\infty_{n= 1}(w_n)^2= \infty\), where \(w_n\) is the number of subcolumns of equal width into which the \(n\)th column is divided to give the \((n+1)\)th column after restacking, then the spectrum of \(T\) is singular. For example, if the cutting is done in a bounded manner then the transformation will have a singular spectrum. It is mentioned that the staircase construction (shown by Adams and Friedman to be mixing) does not satisfy the above criteria, but has been shown by Klemes to have singular spectrum.
scientific article; zbMATH DE number 1033777