Periodic Points and Braid Theory
Periodic Points and Braid Theory
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DOI：
10.1007/1-4020-3222-6_5
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发表时间：
2005
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影响因子：
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通讯作者：
T. Matsuoka
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文献类型：
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作者：
T. Matsuoka
本文综述了辫子理论在曲面上迭代同胚的周期轨道结构研究中的应用。在动力系统中，经常使用拓扑不变量来研究系统的定性和定量性质。辫子是这样的不变量之一，表征拓扑行为的周期轨道的情况下，表面同胚。设M是紧曲面，f：M→ M是同胚.设S是f的一个有限不变集，即f的1/4个周期轨道的并集。假设f是恒等映射id的同位素，选择并固定一个将id变形为f的同位素{ft f} 0≤ t≤ 1。然后，当t从0变到1时，S在ft f下的像在M上移动并返回到初始位置。这个运动定义了M中的辫子。这个辫子的共轭类称为S关于同胚f的辫子类型。假设f是单射的，并且与id是同位素的，这是定义辫型所必需的，对于动力学来说是相当自然的，例如，周期性强迫微分方程的解的时间映射满足这个假设。表面动力学中辫型的研究始于20世纪80年代初，并已发展到低维动力系统理论的一个广泛领域。本研究主要有两个方面。一个是辫子群的矩阵表示的应用，它与Fadell，Husseini和Fried在80年代初建立的广义Lefschetz数理论密切相关。事实上，S的辫子的矩阵的迹与f到M− S的限制的广义莱夫谢茨数的阿贝尔化一致。广义Lefschetz数描述了不属于S的周期轨道与S的联系。因此，通过计算矩阵，可以获得周期轨道的存在性和链接行为的信息。这是广义Lefschetz数理论的一个重要应用。
This article surveys applications of the braid theory to the study of the periodic orbit structure of iterated homeomorphisms on surfaces. In dynamical systems, it is often the case that topological invariants are used to study qualitative and quantitative properties of the system. The braid is one of such invariants characterizing topological behavior of periodic orbits in the case of surface homeomorphisms. Let M be a compact surface, and f: M→ M a homeomorphism. Let S be a finite invariant set of f, ie a union of finitely many periodic orbits of f. Assume that f is isotopic to the identity map id, and choose and fix an isotopy {ft f} 0≤ t≤ 1 deforming id to f. Then, the image of S under ft f move on M and return to the initial position while t varies from 0 to 1. This motion defines a braid in M. The conjugacy class of this braid is called the braid type of S with respect to the homeomorphism f. The assumption that f is injective and isotopic to id, which is necessary to define the braid type, is rather natural for dynamics as, for example, the time one map of the solution to a periodically forced differential equation satisfies this assumption.The study of braid types in surface dynamics was started in the early 1980’s by several people, and has been developed to an extensive area in the theory of low dimensional dynamical systems. There are two main streams in this study. One is the application of matrix representations of braid groups which are closely related to the theory of generalized Lefschetz number established by Fadell, Husseini, and Fried in the early 1980’s. In fact, the trace of the matrix of the braid of S coincides with the abelianization of the generalized Lefschetz number of the restriction of f to M− S. The generalized Lefschetz number describes how periodic orbits not belonging to S link with S. Hence, by computing the matrix one can obtain information of the existence and linking behavior of periodic orbits. This provides a remarkable application of the theory of generalized Lefschetz number.