Braid invariant for periodic points of surface maps and its applications

利用周期点拓扑不变量中的辫状不变量，研究了二维圆盘嵌入到其自身的周期点集的拓扑结构.我们的研究结果如下：1. Thurston的理论已经表明，如果嵌入具有拓扑复杂的周期点（即，其辫子不变量在分解中具有伪Anosov分量），则存在无穷多个周期点。证明了如果周期小于等于整数n的周期点集P（n）是拓扑复杂的，则P（n）至少有2n+3个点.当P（n）至多有2n+2个点时，上述结果表明该集合是拓扑单的.在这种情况下，我们确定了P（n）的辫子不变量的所有可能形式。3.本文从与上述不同的角度研究了不动点，得到如下结果：（1）利用辫的概念，在不动点集上引入了一个等价关系，证明了每个等价类的辫不变量都是单型的。这意味着对不动点集结构的研究分为两部分：研究每个等价类的性质和研究等价类如何组合在一起。(2)我们证明了每个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与稳定性之间的关系，证明了每个至少有两个点的等价类必包含一个不稳定不动点.此外，我们还证明了具有不稳定不动点的等价类的数目大于不含不稳定不动点的等价类的数目。

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.