完备稠序线性序拓扑空间上不稳定流形的边界点的周期性

A Periodic Properity of Boundary Points of an Unstable Manifold on Completely Densely Ordered Linear Ordered Topological Space

研究完备稠序线性序拓扑空间上连续自映射的不稳定流形的结构.首先证明了连续自映射在不动点处的不稳定流形是连通的.然后指出连续自映射的周期点处的不稳定流形必是有限个区间的并.最后,利用所得结果证明了在具有最大最小元的完备稠序线性序拓扑空间上连续自映射的不稳定流形的边界如果不属于流形本身,则必为该连续自映射的周期点.

In this paper, the structures of the unstable manifold of a continuous self-mapping on a completely densely ordered linear ordered topological space is discussed. First it is proved that the unstable manifold of a fixed point is connected. Next, it is show that the unstable manifold of a periodic point is the union of finite intervals. Finally, using the above conclusions, the following result is obtained： The boundary points of the unstable manifold of any continuous self-mapping, on completely densely ordered linear ordered topological space with the maximal element and the minimal element, are periodic points if they do not belong to the unstable manifold.