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.

目的是我们获得了以下结果。1.1。可解决的谎言组G在封闭的歧管上，它决定了轨道叶该项目从轨道叶的叶状群体学角度来看，保留紧密间隔的差异性是完美的。 c^1的差异1与终点的身份切线。3。plessions of the Condimension Ona闭合3个manifold相交的横向问题并显示出该表，仅仅是一些信息，某些双曲线摩尔甲形成型具有独特的交叉属性，但是双曲线表面的地球流量却没有。

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.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.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: "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。