On expansive homeomorphisms on manifolds
Masaharu Kouno · Journal of the Mathematical Society of Japan · 1981
Introduction.$X$ will be a metric space with a metric $d$ .A homeomorphism $f$ of $X$ onto itself is expansive if there exists a positive number $C$ (called expansive constant) such that for each pair $(x, y)$ of distinct points of $X$ , there is an integer $n$ for which $d(f^{n}(x), f^{n}(y))>C$ .There is a question what manifolds admit such homeomorphisms.Several examples of existence and non-existence of expansive homeomorphisms are known.An open interval, a 1-sphere and a closed 2-disk do not admit expansive homeo- morphisms (Bryant [1], Jakobsen and Utz [2]).An open $2n$ -ball $(n\geqq 1)$ and an r-dimensional torus $(r\geqq 2)$ admit expansive homeomorphisms (Reddy [3]).In this paper, we prove the followings.THEOREM 1.Let $M$ be a closed n-manifold $(n\geqq 1)$ , and $J$ be an open interval.Then there exists an expansive homeomorPhism of $M\times J$ .THEOREM 2. If $M$ is a closed n-manifold $(n\geqq 1)$ , there exist an expansive homeomorphism of Int ( $M^{*}$ {point}).Where $P^{*}Q$ is the join of $P$ and $Q$ , and Int $M$ is the interior of $M$ .COROLLARY.There exists an expansive homeomorphism of an open n-ball $(n\geqq 2)$ .The auther thanks Prof. K. Kobayashi for his helpful advices. Proof of Theorem 1.Let $M$ be a closed n-manifold with a metric $d$ .$J=(O, 2)$ and $R^{n}$ be an open interval with a standard metric $d_{1}$ and an n-dimensional Euclidean space with a standard metric $d_{n}$ , respectively.And put $U(x, \epsilon)=\{y\in M|d(y, x)<\epsilon\}$ , $U_{n}(z, \delta)=\{y\in R^{n}|d_{n}(y, z)<\delta\}$ .We define the metric $\rho$ of $M\times J$ to be $d\times d_{1}$ (where $d\times d_{1}((x,$ $t),$ $(y,$ $s))=d(x,$ $y)+d_{1}(t,$ $s)$ and $x,$ $y\in M$ and $t,$ $s\in J$ ), and $I_{k}$ $(k\geqq 0)$ to be $I_{k}=[\frac{1}{k+1}$ , $\frac{1}{k}](k\in N)$ and $I_{0}=[1,2$ ).Put $A_{k}=M\times I_{k}$ .First, we define several homeomorphisms of $A_{1}$ .We will use these homeo- morphisms for constructing an expansive homeomorphism of $M\times J$ .For any