Absolute Continuity and Singularity of Spectra for the Flows \(T_t\otimes T_{at}\)

Abstract

Given disjoint countable dense subsets \(C\) and \(D\) of the half-line \((1,+\infty)\), there exists a flow \(T_t\) preserving a sigma-finite measure and such that all automorphisms \(T_1\otimes T_{c}\) with \(c\in C\) have simple singular spectrum and all automorphisms \(T_1\otimes T_{d}\) with \(d\in D\) have Lebesgue spectrum of countable multiplicity.