Riesz products and spectral decompositions for rank 1 measure preserving transformations

TL;DR

This paper explicitly constructs spectral decompositions for rank one measure preserving transformations, revealing their spectral type as purely singular Riesz products, and enhances understanding of their spectral properties.

Contribution

It provides explicit spectral decompositions for rank one transformations, clarifying their spectral type as purely singular Riesz products, which was previously understood only in a generic sense.

Findings

Spectral type of $U(g)$ is purely singular

Spectral decompositions are explicitly constructed

Rank one transformations have Riesz product spectral measures

Abstract

We consider rank one measure preserving transformations $g$ and the corresponding unitary operators $U(g)$. It is known that a generic (in the sense of Baire category) measure preserving transformation has rank one, spectral type of $U(g)$ is purely singular and is given by a Riesz product. We write explicitly spectral decompositions of $U(g)$.