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

TL;DR

This paper proves that typical smooth, area-preserving flows on surfaces have singular spectrum and are spectrally disjoint, using advanced techniques involving Birkhoff sums, rigidity, and shearing mechanisms.

Contribution

It establishes the singularity of the spectrum and spectral disjointness for a broad class of minimal smooth flows on surfaces, extending previous results.

Findings

Almost every such flow has singular spectrum.

Almost every pair of such flows is spectrally disjoint.

Singularity and disjointness hold for special flows over interval exchange transformations.

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 almost every pair of such flows is spectrally disjoint. More in general, singularity of the spectrum and pairwise disjointness holds for special flows over a full measure set of interval exchange transformations under a roof with symmetric logarithmic singularities. The spectral result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay and the cancellations proved by the last author to prove absence of mixing in this class of flows, by showing that the latter can be combined with rigidity. Disjointness of pairs then follows by producing mixing times (for the second flow), using a new mechanism for shearing based on resonant rigidity times.