The nonexistence of expansive homeomorphisms of chainable continua
The nonexistence of expansive homeomorphisms of chainable continua
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DOI：
10.4064/fm-149-2-119-126
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发表时间：
1996
影响因子：
0.6
通讯作者：
H. Kato
中科院分区：
文献类型：
--
作者：
H. Kato
设$(X,d)$为紧致度量空间，同胚$f:X\to X$称为是扩张的，如果存在$c>0$，使得对于任意$x,y\in X$且$x\neq y$，存在整数$n\in\mathbb{Z}$，使得$d(f^n(x),f^n(y))>c$。在本文中，我们证明如果连续统$X$的同胚$f:X\to X$可提升为伪弧$P$上的满射$h:P\to P$，那么$f$不是扩张的。作为推论，我们证明链状连续统上不存在扩张同胚。这是对威廉姆斯的一个猜想的肯定回答。
A homeomorphism f : X ! X of a compactum X with metric d is expansive if there is c > 0 such that if x,y 2 X and x 6 y, then there is an integer n2 Z such that d(f n (x),f n (y)) > c. In this paper, we prove that if a homeomorphism f : X ! X of a continuum X can be lifted to an onto map h : P ! P of the pseudo- arc P , then f is not expansive. As a corollary, we prove that there are no expansive homeomorphisms on chainable continua. This is an armative answer to one of Williams' conjectures.