Braid invariant for periodic points of surface maps and its applications

曲面图周期点的辫状不变量及其应用

基本信息

批准号：
09640115
负责人：
MATSUOKA Takashi
金额：
$ 1.54万
依托单位：
Naruto University of Education
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 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640115/
关键词：
periodic point 2-dimensional embedding braid fixed point index pseudo-Anosov map unstable fixed point 不動点埋め込み写像組ひも 2次元力学系位相的エントロピー絡み数

项目摘要

We studied the topological structure of the set of periodic points for embeddings of a 2-dimensional disk to itself, by exploiting the braid invariant, which is one of the topological invariants defined for periodic points. Our results are the following :1. Thurston's theory has shown that if the embedding has a periodic point which is topologically complicated (i.e., its braid invariant has a pseudo-Anosov component in its decomposition), then there exist infinitely many periodic points. We proved that if P(n), the set of periodic points with period less than or equal to an integer n, is topologically complicated, then P(n) has at least 2n+3 points.2. When P(n) has at most 2n+2 points, the above result shows that this set is topologically simple. In this case, we determined all the possible forms of the braid invariant of P(n).3. We studied fixed points from a viewpoint different from the above, and obtained the following :(1) Using the notion of a braid, we introduced an equivalence relation on the fixed point set, and proved that the braid invariant of each equivalence class is of a simple type. This implies that the study of the structure of the fixed point set is divided into two parts: the study of the property of each equivalence class and that of how the equivalence classes are combined together.(2) We proved that the fixed point index of each equivalence class is less than two. As an application of this result, we studied the relationship between the topological property and stability of fixed points, and showed that every equivalence class with at least two points must contain an unstable fixed point. Moreover, we showed that the number of equivalence classes having an unstable fixed point is greater than that of the equivalence classes containing no unstable fixed points.

我们通过利用辫子不变式来研究了二维磁盘本身嵌入的周期点的拓扑结构，这是定义为周期性点定义的拓扑不变的之一。我们的结果如下：1。 Thurston的理论表明，如果嵌入具有拓扑复杂的周期点（即，其辫子不变性在其分解中具有伪 - anosov组件），那么存在无限的周期性点。我们证明，如果p（n）（p（n）的周期点小于或等于整数n）在拓扑上是复杂的，那么p（n）至少具有2n+3点。2。当p（n）最多具有2N+2点时，以上结果表明该集合在拓扑上很简单。在这种情况下，我们确定了p（n）.3的辫子不变的所有可能形式。我们从与上述不同的视点研究了固定点，并获得了以下内容：（1）使用辫子的概念，我们在固定点集上引入了等价关系，并证明了每个等价类的辫子不变性是一个简单的类型。这意味着对固定点集的结构的研究分为两个部分：每个等价类别的特性的研究以及如何将等价类组合在一起。（2）我们证明了每个等效类的固定点索引小于两个。作为此结果的应用，我们研究了固定点的拓扑特性和稳定性之间的关系，并表明每个具有至少两个点的等效类都必须包含一个不稳定的固定点。此外，我们表明，具有不稳定固定点的等效类的数量大于没有不稳定固定点的等效类的数量。