Topological invariants for periodic points of torus maps

我们考虑了一个紧致可定向表面上的同胚，它是恒等映射的同位素。利用不动点索引和辫状不变量，研究了该映射周期点的拓扑性质，得到了不动点上的如下结果：我们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索引在映射的同位素下是不变的。这对于确定给定map3的等价类的不动点指数是有用的。我们证明了每一个等价类都有不动点指标，且不动点指标最多为1。应用上述结果，我们证明了如果一个等价类至少有两个不动点，那么其中一个一定是不稳定的。该结果给出了不动点的稳定性与map5的全局拓扑之间的关系。含有不稳定点的等价类的数目大于所有等价类的数目减去曲面的属数的一半。如果这是一个只有一个点的等价类，并且这个点的索引为1，则映射的拓扑熵为正。这表明不动点的拓扑性质蕴涵着映射的动力学性质

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