Singular continuous spectrum and quantitative rates of weak mixing

Abstract

We prove that for a dense Gδ of shift-invariant measures on AZd, all d shifts have purely singular continuous spectrum and give a new proof that in the weak topology of measure preserving Zd transformations, a dense Gδ is generated by transformations with purely singular continuous spectrum. We also give new examples of smooth unitary cocycles over an irrational rotation which have purely singular continuous spectrum. Quantitative weak mixing properties are related by results of Strichartz and Last to spectral properties of the unitary Koopman operators.