树映射周期点的存在性定理

The existence theorem of the periodic point of the tree map

给定一个闭区间及其上的连续映射,目前的文献已经证明了,如果存在该区间的一个闭子区间满足:它在此区间映射下的象包含其本身,那么这个区间映射一定有不动点.笔者证明了如果一个树映射也满足上述条件,那么这个树映射不一定有不动点,但是它一定有周期点.

Given a closed interval and a continuous map on this interval, the current literature proves that if there exists a closed subinterval of this interval that satisfies that its image contains itself, then the interval map must have a fixed point. Does this conclusion hold for tree maps? In this paper, we prove that if a tree map satisfies the above condition, then this tree map may have no fixed point but must have a periodic point.