Singularity of the spectrum for smooth area-preserving flows in genus two and translation surfaces well approximated by cylinders

TL;DR

This paper proves that smooth area-preserving flows on genus two surfaces generally have singular spectrum, using properties of translation surfaces and interval exchange transformations.

Contribution

It establishes the generic singularity of spectrum for genus two flows and introduces new results on translation surfaces approximated by single cylinders.

Findings

Almost every genus two flow with non-degenerate saddle has singular spectrum.

Singular spectrum holds for a full measure set of interval exchange transformations with symmetric logarithmic singularities.

Translation surfaces can be well approximated by single cylinders in almost every direction.

Abstract

We consider smooth flows preserving a smooth invariant measure, or, equivalently, locally Hamiltonian flows on compact orientable surfaces and show that, when the genus of the surface is two, almost every such locally Hamiltonian flow with two non degenerate isomorphic saddle has singular spectrum. More in general, singularity of the spectrum holds for special flows over a full measure set of interval exchange transformations with a hyperelliptic permutation (of any number of exchanged intervals), under a roof with symmetric logarithmic singularities. The result is proved using a criterion for singularity based on tightness of Birkhoff sums with exponential tails decay. A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders. We show that for almost every translation surface in any connected component of any stratum there exists a full measure set of directions which can be well approximated by a single cylinder of area arbitrarily close to one. The result, in the special case of the stratum $\mathcal{H}(1,1)$, yields rigidity sets needed for the singularity result.