Topological invariants for periodic points of torus maps

环面图周期点的拓扑不变量

• 批准号: 13640079
• 负责人: MATSUOKA Takashi
• 金额: $ 0.45万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640079/
• 关键词: Fixed point index; torus map; braid; surface homeomorphism; unstable fixed point; topological entropy; 曲面; 組ひも; トーラス

We considered a homeomorphism on a compact orientable surface which is isotopic to the identity map. Using fixed point index and braid invariant, we investigated the topological property of periodic points of this map, and has obtained the following results on fixed points:1. We introduced a relation on the fixed point set, and showed that this relation is an equivalence relation.2. We proved that the fixed point index of an equivalence class is invariant under the isotopy of the map. This is useful to determine the fixed point indices of equivalence classes of given map3. We showed that every equivalence class has fixed point index at most one4. Applying the above result, we showed that if an equivalence class has at least two fixed points, then one of them must be unstable. This result gives a relationship between the stability of a fixed point and the global topology of the map5. The number of equivalence classes containing an unstable point is greater than half of the number of all equivalence classes minus the genus of the surface6. If these is an equivalence class which has only one point and this point has index one, then the topological entropy of the map is positive. This shows that a topological property of fixed points implies the dynamical property of the map

本文研究了紧致可定向曲面上的同胚，它与恒等映射是同构的。利用不动点指数和辫子不变量，研究了这类映射周期点的拓扑性质，得到了如下关于不动点的结果：1。在不动点集上引入了一个关系，并证明了这个关系是一个等价关系.证明了等价类的不动点指数在映射的合痕下不变。这对于确定给定映射3的等价类的不动点指数是有用的。我们证明了每个等价类的不动点指数至多为1。应用上面的结果，我们证明了如果一个等价类至少有两个不动点，那么其中一个不动点必定是不稳定的。该结果给出了不动点的稳定性与映射的全局拓扑之间的关系5。包含不稳定点的等价类的数量大于所有等价类的数量减去曲面的亏格的一半。如果这是一个只有一个点的等价类，并且这个点的指数为1，则映射的拓扑熵为正。这说明不动点的拓扑性质蕴涵着映射的动力学性质