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A topological study of generalized dynamical systems

广义动力系统的拓扑研究

基本信息

批准号：
09640090
负责人：
INABA Takashi
金额：
$ 1.79万
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：
1997
资助国家：
日本
起止时间：
1997 至 1998
项目状态：
已结题

项目摘要

By generalized dynamical systems, we mean a wide class of systems including classical dynamical systems, foliations, pseudogroups and relations. In this research we tried to extend various notions and theorems in classical dynamical systems to generalized systems. The results obtained are as follows.1. The Reeb foliation and Anosov foliations have the pseudoleaf tracing property.2. Expansive foliations with the pseudoleaf tracing property are semi-stable (This is a generaliza- tion of Bowen-Walters theorem in classical dynamical systems).3. A group action has the pseudo-orbit tracing property if and only if its suspension foliation has the pseudoleaf tracing property.4. A group action has the semi-stability or the expansivity if and only if its suspension foliation has the same property.5. We recognized that the notion of the center of mass (defined by Cheeger) is useful in the construction of a pseudoleaf from a given countable subset in the ambient manifold.6. It seems that the growth of the number of compact leaves cannot be estimated from above by the geometric entropy. We are now trying to produce a counterexample.7, In the case of foliations it seems that the stability does not imply the pseudoleaf tracing property.These results will be published in Tokyo J.Math. A related result has been published in Erg. Th. Dyn. Sys.We note that the notion of the pseudoleaf (which was introduced in this research) has been applied by Walczak to the new definition of the geometric entropy and the existence of virtual leaves.Hino investigated the stability of processes. Koshikawa studied the cut-and-pasting method of group actions. And Takagi studied the geometry of real hypersurfaces in the complex projective spaces.

广义动力系统是指包括经典动力系统、叶、伪群和关系在内的一类广泛的系统。在本研究中，我们尝试将经典动力系统中的各种概念和定理推广到广义系统。得到的结果如下：1。Reeb叶理和Anosov叶理具有伪叶示踪性。具有伪叶跟踪性质的膨胀叶是半稳定的（这是经典动力系统中Bowen-Walters定理的推广）。当且仅当一个群作用的悬浮叶理具有伪轨道跟踪性质时，该群作用才具有伪轨道跟踪性质。群作用具有半稳定性或扩张性当且仅当它的悬浮叶理具有相同的性质。我们认识到质心的概念（由Cheeger定义）在从环境流形中的给定可数子集构造伪叶时是有用的。似乎密实叶数的增长不能由上面的几何熵来估计。我们现在正试图提出一个反例。在叶面的情况下，稳定性似乎并不意味着假叶的追踪性质。这些结果将发表在东京数学杂志上。相关的研究结果已发表在《Erg》杂志上。Th。直流发电机系统。我们注意到，Walczak将伪叶的概念（在本研究中引入的）应用于几何熵的新定义和虚拟叶的存在。Hino研究了工艺的稳定性。Koshikawa研究了群体行为的剪切粘贴方法。Takagi研究了复射影空间中实超曲面的几何。