Synthetic studies of foliations and discrete group actions

叶状结构和离散群体行为的综合研究

• 批准号: 13304005
• 负责人: MATSUMOTO Shigenori
• 金额: $ 18.14万
• 依托单位: Nihon University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13304005/
• 关键词: foliations; locally free lie group actions; foliated cohomology; parameter rigidity; horocycle flow; Ruelle invariant; unique ergodicity; minimal set; ルエル不変量; 一竜エルゴート性

The purpose of this project is to study the geometic and dynamical properties of foliations, or more generally, of discrete group actions. Throughout the project, we have obtained the following results.1.When a solvable Lie group G acts on a closed manifold M, it determines the orbit foliation. Assume another G action on M with the same orbit foliation is given. We consider the problem whether or not the two actions are C^∞ conjugate up to an isomorphism of G. We interpreted this problem in terms of the foliated cohomology of the orbit foliations, and gave some examples of actions for which this problem has a positive answer.2.Some groups of the sense preserving diffeomorphisms of the closed interval are shown to be perfect. The examples are, the group of Lipschitz homeomorphism, the group of C^∞ diffeomorphisms which are C^∞ tangent to the identity at the end points, and the group of C^1 diffeomorphisms C^1 tangent to the identity at the end points.3.Suppose two codimension one foliations on a closed 3-manifold intersect transversely. Can it possible to isotope one foliation so that it is also tangent to the other, but in a different way? We considered this problem and have shown that the stable and unstable foliations of the suspension flow of hyperbolic toral automorphisms have unique intersection property, but those of geodesic flows of hyperbolic surfaces have not.

这个项目的目的是研究叶理的几何和动力学性质，或者更一般地，离散群作用的几何和动力学性质。在整个项目中，我们得到了以下结果：1.当一个可解李群G作用在闭流形M上时，它决定了轨道叶层。假设给出了M上具有相同轨道叶层的另一个G作用。我们考虑了这两个作用是否是C^∞共轭直到G的同构的问题，我们用轨道叶的分片上同调来解释这个问题，并给出了这个问题有正答案的一些作用的例子。2.一些闭区间的保义微分同态群被证明是完美的。例如，Lipschitz同胚群，C^∞与端点单位相切的C^∞微分同胚群，以及与端点单位单位相切的C^1微分同胚群。3.设3-闭流形上的两个余维1叶横交。有没有可能对一个叶理进行同位素化，使其也与另一个相切，但以不同的方式？我们考虑了这一问题，并证明了双曲自同构的悬浮流的稳定叶和不稳定叶具有唯一的相交性质，而双曲曲面的测地流的稳定叶和不稳定叶具有唯一的相交性质。

项目成果

S.Matsumoto: "Leafwise cohomology of certain Lie group actions"Ergodic Theory and Dynamical Systems. 23. 1839-1866 (2003)

S.Matsumoto：“某些李群作用的叶向上同调”遍历理论和动力系统。

S.Matsumoto: "On the global rigidity of split Anosov R^n-actions"Journal of the Mathematical Society of Japan. 55. 39-46 (2003)

S.Matsumoto：“论分裂 Anosov R^n-actions 的全局刚性”日本数学会杂志。

S.Matsumoto: "Affine flows on 3-manifolds"Memoirs of American Mathematical Society. (発表予定).

S.Matsumoto：“3-流形上的仿射流”美国数学会回忆录（待提交）。

S.Morita: "Structure of Mapping Class Groups and symplectic representation theory"L'Enseigement Math.Monograph,2001. (2001)

S.Morita：“映射类群的结构和辛表示理论”LEnseigement Math.Monograph，2001。

S.Morita: "Geometry of differential forms"Translation of Mathematical Monographs,201,AMS. 321 (2001)

S.Morita：《微分形式的几何》数学专着译，201，AMS。