Riesz products and spectral decompositions for rank 1 measure preserving transformations

Abstract

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

arXiv comment24 pages, extended version