Smooth mixing flows with purely singular spectra

Abstract

We give a geometric criterion that implies a singular maximal spectral type for a dynamical system on a Riemannian manifold. The criterion, which is based on the existence of fairly rich but localized periodic approximations, is compatible with mixing. Indeed, we check it for an ad hoc class of smooth mixing flows on ${\mathbb{T}}^3$ obtained from linear flows by time change and thus providing natural examples of mixing smooth diffeomorphisms and flows with purely singular spectra