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首页> 外文期刊>Annales scientifiques de l'Ecole normale superieure >COHOMOLOGY CLASSES REPRESENTED BY MEASURED FOLIATIONS, AND MAHLER'S QUESTION FOR INTERVAL EXCHANGES
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COHOMOLOGY CLASSES REPRESENTED BY MEASURED FOLIATIONS, AND MAHLER'S QUESTION FOR INTERVAL EXCHANGES

机译：测得的叶片代表的光学分类，以及间隔交换的马勒问题

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摘要

A translation structure on (S, Σ) gives rise to two transverse measured foliations F, G on S with singularities in Σ, and by integration, to a pair of relative cohomology classes [F], [G] ∈ H~1(S,Σ; R). Given a measured foliation F, we characterize the set of cohomology classes b for which there is a measured foliation G as above with b = [G]. This extends previous results of Thurston [19] and Sullivan [18]. 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 ∈ S_d, the space R_+~d parametrizes the interval exchanges on d intervals with permutation σ. We describe lines l in R_+~d such that almost every point in l is uniquely ergodic. We also show that for σ(i) = d+l-i, for almost every s > 0, the interval exchange transformation corresponding to σ and (s,s~2,..., s~d) is uniquely ergodic. As another application we show that when k = |∑| ≥ 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 R~(k-1) on the set of translation surfaces of type (S, Σ) without horizontal saddle connections. Here B C SL(2, R) is the subgroup of upper triangular matrices.
机译：（S，Σ）上的平移结构在S上产生两个横向测得的叶面F，G，Σ中具有奇点，并通过积分形成一对相对的同调类[F]，[G]∈H〜1（S ，Σ; R）。给定一个测得的叶面F，我们用b = [G]来表征同调类b的集合，其同上具有测得的叶面G。这扩展了Thurston [19]和Sullivan [18]的先前结果。我们将其应用于两个问题：间隔交换和流动在平移曲面的模空间上的独特遍历性。对于固定的置换σ∈S_d，空间R_ +〜d参数化了置换σ的d个间隔上的区间交换。我们在R_ +〜d中描述线l，使得l中的几乎每个点都是唯一遍历的。我们还表明，对于σ（i）= d + 1-i，几乎对于每个s> 0，对应于σ和（s，s〜2，...，s〜d）的区间交换变换是唯一遍历的。作为另一个应用，我们证明当k = | ∑ |时≥2，“水平移动奇异点”的操作在全局上是明确定义的。我们证明，在不具有水平鞍形连接的类型（S，Σ）的平移曲面集上，B <？> R〜（k-1）组有明确定义的动作。在这里，B C SL（2，R）是上三角矩阵的子组。

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