Singularity of the spectrum of typical minimal smooth area-preserving flows in any genus

Abstract

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces, and show that almost every such locally Hamiltonian flow with only simple saddles has singular spectrum. Furthermore, we prove that for almost every pair of such flows, the elements of the pair are spectrally disjoint. More generally, the results from which these statements are deduced are singularity of the spectrum and pairwise spectral disjointness for special flows over full measure sets of interval exchange transformations under a roof with symmetric logarithmic singularities. Spectral singularity is proved using a criterion based on tightness of Birkhoff sums with exponential tail decay. The assumptions of the criterion are verified exploiting the cancellations proved by the last author to prove the absence of mixing in this class of flows, by showing that the latter can be combined with rigidity by exploiting the local product structure of Rauzy-Veech induction. Pairwise spectral disjointness then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on what we call resonant rigidity times.