A topological study of generalized dynamical systems

我们所说的广义动力系统是指包括经典动力系统、叶系、伪群和关系在内的一大类系统。在这项研究中，我们试图将经典动力系统中的各种概念和定理推广到广义系统。取得的结果如下：1.研究结果。Reeb面理和Anosov面理具有伪叶面层示踪性质。具有伪叶跟踪性质的扩张叶系是半稳定的(这是经典动力系统中Bowen-Walters定理的推广)。群作用具有伪轨跟踪性质的充要条件是它的悬浮叶理具有伪叶跟踪性质。群作用具有半稳定性或扩张性的充要条件是它的悬浮叶具有相同的性质。我们认识到，质心的概念(由Cheeger定义)在由环境流形中给定的可数子集构造伪叶时是有用的。似乎不能从上面用几何熵来估计紧凑型叶片数的增长。我们现在正试图产生一个反例。7，在叶序的情况下，这种稳定性似乎并不意味着伪叶跟踪性质。这些结果将发表在东京数学杂志上。一项相关结果已发表在ERG上。TH.戴恩。我们注意到Walczak引入的伪叶的概念已经被应用于几何熵的新定义和虚拟叶的存在性。Koshikawa研究了群体动作的剪切粘贴方法。和Takagi研究了复射影空间中实超曲面的几何。

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.