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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量（为周期点定义的拓扑不变量之一），研究了将二维圆盘嵌入到其自身的周期点集的拓扑结构。我们的结果如下：1．瑟斯顿的理论表明，如果嵌入有一个拓扑复杂的周期点（即，其辫子不变量在其分解中具有伪阿诺索夫分量），则存在无限多个周期点。我们证明，如果周期小于或等于整数n的周期点的集合P(n)是拓扑复杂的，则P(n)至少有2n+3个点。 2．当P(n)最多有2n+2个点时，上面的结果表明这个集合是拓扑简单的。在这种情况下，我们确定了 P(n).3 辫子不变量的所有可能形式。我们从与上述不同的角度研究了不动点，得到了以下结论：(1)利用辫子的概念，在不动点集上引入了等价关系，并证明了每个等价类的辫子不变量都是简单类型的。这意味着对不动点集结构的研究分为两部分：研究各等价类的性质和研究等价类如何组合在一起。(2)证明了各等价类的不动点指数小于2。作为这一结果的应用，我们研究了不动点的拓扑性质和稳定性之间的关系，并表明每个至少有两个点的等价类必须包含一个不稳定的不动点。此外，我们还表明，具有不稳定不动点的等价类的数量大于不包含不稳定不动点的等价类的数量。