关于压缩族和Stein的一个不动点问题
On contractive families and a fixed-point question of Stein
摘要：
在本文中，我们证明了压缩映射定理的下列错误推广（由于J.D. Stein Jr.）：设X是完全度量空间，F是X的有限自映射族.假设存在一个严格小于小于1的正常数，使得对于X的任意两个点x和y，F中的某个成员将这两个点收缩至多为该常数的因子。则F的成员的某个合成有一个不动点。我们还证明了，上述情况对只包含两个映射的（连续）交换F也成立。我们猜想它对任何有限大小的交换F都成立。
英文摘要：
In this paper we disprove the following conjectured generalization of the Contraction Mapping Theorem (due to J.D. Stein Jr.): Let X be a complete metric space and let F be a finite family of self-maps of X. Suppose there is a postive constant strictly less than 1 such that, for any two points x and y of X, some member of F contracts those points by a factor of at most that constant. Then some composition of members of F has a fixed point. We also show that the above does hold for a (continuous) commuting F containing only two maps. We conjecture that it holds for commuting F of any finite size.
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