A study of minimal sets in differentiable flows and foliations

可微流和叶状结构中最小集的研究

• 批准号: 11640062
• 负责人: INABA Takashi
• 金额: $ 2.11万
• 依托单位: CHIBA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640062/
• 关键词: flow; invariant fiber measure; Ruelle invariant; amenable group; 極小集合

In this research we made measure theoretic and topological investigation of differentiable flows on 3-dimensional manifolds. Through an effort to understand more concretely Zimmer's ergodic theory on group actions, we established a basis of finding new aspects of dynamics of flows. Our method is as follows : Given a flow we lift it to the total space of its projectivized normal bundle. Then, by disintegrating an invariant measure on the total space, we obtain a measure on each fiber. We Call the family of measures thus obtained an invariant fiber measure. An invariant fiber measure provides us with a setup for measuring the twisting phenomenon of the flow.In this research we established some fundamental theorems on invariant fiber measures, giving careful and accessible proofs for all of them. We also obtained a new representation formula of the Ruelle invariant by making use of an invariant fiber measure, Moreover, as an application of this formula, we showed the following result : Consider the group of measure preserving diffeomorphisms of the 2-dimensional disk and the Ruelle map defined on this group. Then the map becomes a homomorphism whenever we restrict it to an amenable subgroup.We gave a one-hour talk on this result at an international symposium on foliations held at Warsaw in 2000.

在这项研究中，我们对 3 维流形上的可微流进行了测度理论和拓扑研究。通过努力更具体地理解齐默关于群体行为的遍历理论，我们为寻找流动动力学的新方面奠定了基础。我们的方法如下：给定一个流，我们将其提升到其投影法线丛的总空间。然后，通过分解总空间上的不变测度，我们获得每个纤维上的测度。我们将由此获得的测度系列称为不变纤维测度。不变纤维测量为我们提供了测量流动扭曲现象的装置。在这项研究中，我们建立了一些关于不变纤维测量的基本定理，为所有这些定理提供了仔细且易于理解的证明。我们还利用不变纤维测度得到了 Ruelle 不变量的新表示公式，此外，作为该公式的应用，我们得到了以下结果：考虑保留二维圆盘微分同胚的测度组和在该组上定义的 Ruelle 图。然后，每当我们将其限制为一个合适的子群时，该映射就成为同态。2000 年在华沙举行的国际叶状结构研讨会上，我们就这个结果进行了一个小时的演讲。

项目成果

Takashi Inaba: "An example of a flow on a non-compact surface without minimal set"Ergodic Theory and Dynamical Systems. 19・1. 31-33 (1999)

Takashi Inaba：“无最小集的非紧表面上的流的示例”遍历理论和动力系统 19・1 (1999)。

U-H.Ki and R Takagi: "Real hypersurfaces satisfying ∇_ζS=0 in a complex space form II"Korean Ann.Math.. 16. 207-221 (1999)

U-H.Ki 和 R Takagi：“在复空间形式 II 中满足 ∇_ζS=0 的实超曲面”韩国 Ann.Math.. 16. 207-221 (1999)

U-H.Ki, H.Song and R Takagi: "Submanifolds of codimension 3 admitting almost contact metric structure in a complex projective space"Nihonkai Math.J.. 11-1. 57-86 (2000)

U-H.Ki、H.Song 和 R Takagi：“在复射影空间中承认几乎接触度量结构的余维 3 的子流形”Nihonkai Math.J.. 11-1。

Yoshiyuki Hino and Satoru Murakami: "Skew product flows of quesi-processes and stabilities"Fields Institute Communications. 21. 269-278 (1999)

Yoshiyuki Hino 和 Satoru Murakami：“扭曲问题流程和稳定性的产品流”菲尔兹通讯研究所。

Y.Hino and S.Murakami: "Quasi-processes and stabilities in functional differential equations"Nonlinear Analysis. (toappear).

Y.Hino 和 S.Murakami：“泛函微分方程中的拟过程和稳定性”非线性分析。