用测量叶理表示的上同调类和区间交换的Mahler问题
Cohomology classes represented by measured foliations, and Mahler's question for interval exchanges
摘要：
（S，S）上的平移曲面产生S上的两个横向测量叶理\FF，\GG，其奇点在S上，通过积分，产生一对上同调类[ \FF ]，\，[ \GG ] \in H ^1（S，S; \R）。给定一个测量的叶理\FF，我们用\B = [ \GG ]来表征上同调类\B的集合，对于该集合，存在如上所述的测量的叶理\GG。这推广了Thurston和Sullivan的结果。我们将其应用到两个问题：唯一遍历的间隔交换和流动的模空间的翻译表面。对于一个固定的置换σ \in \mathcal{S}_d，空间\R^d_+用置换σ参数化d个区间上的区间交换。我们在\R^d_+中描述线\ell，使得\ell中几乎每个点都是唯一遍历的。我们还证明了对σ（i）= d+1-i，几乎对任何s>0，对应于σ和（s，s^2，\ldots，s^d）的区间交换变换是唯一遍历的.作为另一个应用，我们表明，当k=| Σ| \geq 2，“水平移动奇点”的操作是全局良好定义的。本文证明了B群\ltimes \R^{k-1}在不带水平鞍联络的（S，S）型平移曲面集上的作用.这里B \subset \SL（2，\R）是上三角矩阵的子群。
英文摘要：
A translation surface on (S, Σ) gives rise to two transverse measured foliations \FF, \GG on S with singularities in Σ, and by integration, to a pair of cohomology classes [\FF], \, [\GG] \in H^1(S, Σ; \R). Given a measured foliation \FF, we characterize the set of cohomology classes \B for which there is a measured foliation \GG as above with \B = [\GG]. This extends previous results of Thurston and Sullivan. We apply this to two problems: unique ergodicity of interval exchanges and flows on the moduli space of translation surfaces. For a fixed permutation σ\in \mathcal{S}_d, the space \R^d_+ parametrizes the interval exchanges on d intervals with permutation σ. We describe lines \ell in \R^d_+ such that almost every point in \ell is uniquely ergodic. We also show that for σ(i) = d+1-i, for almost every s>0, the interval exchange transformation corresponding to σand (s, s^2, \ldots, s^d) is uniquely ergodic. As another application we show that when k=|Σ| \geq 2, the operation of `moving the singularities horizontally' is globally well-defined. We prove that there is a well-defined action of the group B \ltimes \R^{k-1} on the set of translation surfaces of type (S, Σ) without horizontal saddle connections. Here B \subset \SL(2,\R) is the subgroup of upper triangular matrices.
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