Absolute continuity and singularity of spectra for flows $T_t\otimes T_{at}$

TL;DR

This paper constructs measure-preserving flows with mixed spectral types, demonstrating that sums of certain random variables can have either singular or absolutely continuous distributions depending on parameters, answering a question by V.I. Oseledets.

Contribution

It provides a method to construct flows with prescribed spectral properties, showing the coexistence of singular and Lebesgue spectra within the same flow for different parameters.

Findings

Existence of flows with both singular and Lebesgue spectra for different parameters.

Construction of a random variable whose sum with a scaled version switches between singular and absolutely continuous distributions.

Answer to V.I. Oseledets' question about spectral types of tensor products of flows.

Abstract

Answering the question of V.I. Oseledets, we present a random variable $\xi$ such that the sum $\xi(x)+a\xi(y)$ has a singular distribution for a set of parameters $a$ dense in $(1, +\infty)$, but for another dense set of parameters, this sum has an absolutely continuous distribution. We prove the following assertion: given $C,D$, countable non-intersecting dense subsets of the ray $(1,+\infty)$, there is a measure-preserving flow $T_t$ (acting on the infinite Lebesgue space) such that automorphisms $T_1\otimes T_{c}$ have simple singular spectra for every $c\in C$, and $T_1\otimes T_{d}$ have Lebesgue spectra for all $d\in D$. The spectral measure of this flow plays the role of the distribution of our random variable $\xi$.