Singularity of the Spectrum for Smooth Area-Preserving Flows in Genus Two and Translation Surfaces Well Approximated by Cylinders

Jon Chaika,Krzysztof Frączek,Corinna Ulcigrai,Adam Kanigowski

doi:10.1007/s00220-020-03895-x

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 saddles 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.

Highlights

The main result of this paper concerns the nature of the spectrum of locally Hamiltonian flows on genus two surfaces, and, to the best of our knowledge, is the first general spectral result for surfaces of higher genus (g ≥ 2)
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 saddles has singular spectrum
A key ingredient in the proof, which is of independent interest, is a result on translation surfaces well approximated by single cylinders

Summary

Main Results

We state the main result on the spectrum of locally Hamiltonian flows on genus two surfaces (see Sect. 1.1), as well as a result in the language of special flows from which it is deduced, see Sect. 1.2. Fraczek and Lemanczyk in [FL03], considering the same example as Kochergin (special flows with one symmetric logarithmic singularity over rotations), showed that if, in addition to tightness, one can control the tails of the distribution of the centralized Birkhoff sums Sqn ( f )(x) − an, one can prove much stronger results (using joinings and Markov operators) and deduce in particular spectral disjointness from mixing flows, which implies that the spectrum is purely singular. For surfaces of genus two or symmetric permutations, we (implicitly) exploit a very geometric approach to deduce cancellations, based on a simple mechanism which uses in an essential way the hyperelliptic involution: the key idea is that, for any symmetric (of equal backward and forward length) trajectory from a fixed point of the hyperelliptic involution, there are perfect cancellations for Birkhoff sums of the derivative of the roof function

Locally Hamiltonian Flows and Reduction to Special Flows

A Criterion for Singularity in Special Flows

Translation Surfaces Well Approximated by Single Cylinders