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.

中文翻译：

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属 2 中平滑保面积流和圆柱体很好逼近的平移面的谱的奇异性

我们考虑保持平滑不变测度的平滑流，或者等效地，紧凑可定向表面上的局部哈密顿流，并表明，当表面的属为 2 时，几乎每个具有两个非退化同构鞍座的局部哈密顿流都具有奇异谱。更一般地说，在具有对称对数奇点的屋顶下，具有超椭圆置换（任意数量的交换区间）的区间交换变换的完整测量集上的特殊流的频谱奇点成立。使用基于具有指数尾衰减的 Birkhoff 和的紧密性的奇异性标准证明了结果。证明中具有独立意义的一个关键要素是在由单个圆柱体很好地近似的平移表面上的结果。我们表明，对于任何层的任何连接组件中的几乎每个平移表面，都存在一个完整的方向测量集，可以通过任意接近一个区域的单个圆柱体很好地近似。结果，在层 $\mathcal{H}(1,1)$ 的特殊情况下，产生奇异结果所需的刚性集。