A direct proof that each Peano continuum with a free arc admits no expansive homeomorphisms
A direct proof that each Peano continuum with a free arc admits no expansive homeomorphisms
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DOI：
10.21099/tkbjm/1496160848
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发表时间：
1988-12
影响因子：
0.7
通讯作者：
K. Kawamura
中科院分区：
文献类型：
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作者：
K. Kawamura
（*）对于\(X\)中任意一对不同的点\(x\)，\(y\)，存在一个整数\(n\)，使得\(d(f^{n}(x), f^{n}(y))>c\)，其中\(d\)是\(X\)的一个度量。在紧致度量空间中，扩张性不依赖于度量的选择。一个紧致连通度量空间被称为连续统。一个佩亚诺连续统是指局部连通的连续统。在一个连续统\(X\)中，端点为\(\{a,b\}\)的弧\(A\)记为\([a,b]\)。\(bd A\)表示\(\{a,b\}\)，\(int A = A - bdA\)。\(X\)中的一条弧\(A\)如果\(int A\)在\(X\)中是开集，则被称为自由弧。设\((X,d)\)是一个连续统。对于\(X\)中的一个点\(x\)以及\(\epsilon>0\)，\(U(x,\epsilon)\)表示\(x\)的\(\epsilon\) -邻域。豪斯多夫度量记为\(d_{H}\)。在本文中，我们直接证明以下定理，它是平出（Hiraide）[2]中命题\(C\)的一个推论。
$(*)$ for each pair $x,$ $y$ of distinct points of $X$, there exists an integer $n$ such that $d(f^{n}(x), f^{n}(y))>c$ , where $d$ is a metric for $X$. Expansiveness does not depend on the choice of metrics for compact metric spaces. A compact connected metric space is called a continuum. A Peano continuum means a locally connected continuum. An arc $A$ in a continuum $X$ with end points $\{a, b\}$ is denoted by $[a, b]$ . $bd$ $A$ means $\{a, b\}$ and int $A=A-bdA$ . An arc $A$ in $X$ is called a free arc if int $A$ is open in $X$. Let (X, d) be a continuum. For a point $x\in X$ and $\epsilon>0,$ $U(x, \epsilon)$ denotes the $\epsilon$ -neighbourhood of $x$ . The Hausdorff metric is denoted by $d_{H}$ . In this paper, we give a direct proof of the following theorem, which is a consequence of Proposition $C$ in Hiraide [2].