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.

通过通用的动力学系统，我们的意思是一类广泛的系统，包括经典的动态系统，叶子，假群和关系。在这项研究中，我们试图将经典动力学系统中的各种概念和定理扩展到广义系统。获得的结果如下1。 Reeb Foliation和Anosov叶子具有伪eaf的追踪特性。2。使用假叶跟踪特性的膨胀叶子是半稳定的（这是经典动力学系统中Bowen-Walters定理的概括）.3。当且仅当其悬浮叶叶具有伪eaf的追踪属性时，小组行动具有伪轨追踪属性。4。当且仅当其悬浮叶叶具有相同的特性时，小组行动具有半稳定性或膨胀性。5。我们认识到，质量中心（由Cheeger定义）的概念可用于构造从环境歧管中给定的可计数子集的伪叶。6。似乎无法通过几何熵从上面估计的紧凑叶子数量的生长。现在，我们正在尝试产生反例。7在叶子的情况下，稳定性似乎并不意味着伪eaf的追踪特性。这些结果将在东京J.Math发表。相关结果已在ERG中发布。 Th。 dyn。我们注意，沃尔卡克（Walczak）应用了伪eak的概念（这是在这项研究中引入的），用于几何熵的新定义和虚拟叶子的存在。Hino研究了过程的稳定性。 Koshikawa研究了小组动作的切割方法。高吉研究了复杂的投影空间中真实超曲面的几何形状。