the only compact metric spaces that admit a positively expansive homeomorphism are discrete spaces

Theorem. Let $(X,d)$ be a compact metric space. If there exists a positively expansive homeomorphism $f\colon X\to X$ , then $X$ consists only of isolated points, i.e. $X$ is finite.

Lemma 1. If $(X,d)$ is a compact metric space and there is an expansive homeomorphism $f\colon X\to X$ such that every point is Lyapunov stable, then every point is asymptotically stable.

Proof. Let $2c$ be the expansivity constant of $f$ . Suppose some point $x$ is not asymptotically stable, and let $\delta$ be such that $d(x,y)<\delta$ implies $d(f^{n}(x),f^{n}(y))<c$ for all $n\in\mathbb{N}$ . Then there exist $\epsilon>0$ , a point $y$ with $d(x,y)<\delta$ , and an increasing sequence $\{n_{k}\}$ such that $d(f^{n_{k}}(y),f^{n_{k}}(x))>\epsilon$ for each $k$ By uniform expansivity, there is $N>0$ such that for every $u$ and $v$ such that $d(u,v)>\epsilon$ there is $n\in\mathbb{Z}$ with $|n|<N$ such that $d(f^{n}(x),f^{n}(y))>c$ . Choose $k$ so large that $n_{k}>N$ . Then there is $n$ with $|n|<N$ such that $d(f^{n+n_{k}}(x),f^{n+n_{k}}(y))=d(f^{n}(f^{n_{k}}(x)),f^{n}(f^{n_{k}}(y)))>c$ . But since $n+n_{k}>0$ , this contradicts the choce of $\delta$ . Hence every point is asymptotically stable.

Lemma 2 If $(X,d)$ is a compact metric space and $f\colon X\to X$ is a homeomorphism such that every point is asymptotically stable and Lyapunov stable, then $X$ is finite.

Proof. For each $x\in X$ let $K_{x}$ be a closed neighborhood of $x$ such that for all $y\in K_{x}$ we have $\lim_{n\to\infty}d(f^{n}(x),f^{n}(y))=0$ . We assert that $\lim_{n\to\infty}\operatorname{diam}(f^{n}(K_{x}))=0$ . In fact, if that is not the case, then there is an increasing sequence of positive integers $\{n_{k}\}$ , some $\epsilon>0$ and a sequence $\{x_{k}\}$ of points of $K_{x}$ such that $d(f^{n_{k}}(x),f^{n_{k}}(x_{k}))>\epsilon$ , and there is a subsequence $\{x_{k_{i}}\}$ converging to some point $y\in K_{x}$ .

From the Lyapunov stability of $y$ , we can find $\delta>0$ such that if $d(y,z)<\delta$ , then $d(f^{n}(y),f^{n}(z))<\epsilon/2$ for all $n>0$ . In particular $d(f^{n_{k_{i}}}(x_{k_{i}}),f^{n_{k_{i}}}(y))<\epsilon/2$ if $i$ is large enough. But also $d(f^{n_{k_{i}}}(y),f^{n_{k_{i}}}(x))<\epsilon/2$ if $i$ is large enough, because $y\in K_{x}$ . Thus, for large $i$ , we have $d(f^{n_{k_{i}}}(x_{k_{i}}),f^{n_{k_{i}}}(x))<\epsilon$ . That is a contradiction from our previous claim.

Now since $X$ is compact, there are finitely many points $x_{1},\dots,x_{m}$ such that $X=\bigcup_{i=1}^{m}K_{x_{i}}$ , so that $X=f^{n}(X)=\bigcup_{i=1}^{m}f^{n}(K_{x_{i}})$ . To show that $X=\{x_{1},\dots,x_{m}\}$ , suppose there is $y\in X$ such that $r=\min\{d(y,x_{i}):1\leq i\leq m\}>0$ . Then there is $n$ such that $\operatorname{diam}(f^{n}(K_{x_{i}}))<r$ for $1\leq i\leq m$ but since $y\in f^{n}(K_{x_{i}})$ for some $i$ , we have a contradiction.

Proof of the theorem. Consider the sets $K_{\epsilon}=\{(x,y)\in X\times X:d(x,y)\geq\epsilon\}$ for $\epsilon>0$ and $U=\{(x,y)\in X\times X:d(x,y)>c\}$ , where $2c$ is the expansivity constant of $f$ , and let $F\colon X\times X\to X\times X$ be the mapping given by $F(x,y)=(f(x),f(y))$ . It is clear that $F$ is a homeomorphism. By uniform expansivity, we know that for each $\epsilon>0$ there is $N_{\epsilon}$ such that for all $(x,y)\in K_{\epsilon}$ , there is $n\in\{1,\dots,N_{\epsilon}\}$ such that $F^{n}(x,y)\in U$ .

We will prove that for each $\epsilon>0$ , there is $\delta>0$ such that $F^{n}(K_{\epsilon})\subset K_{\delta}$ for all $n\in\mathbb{N}$ . This is equivalent to say that every point of $X$ is Lyapunov stable for $f^{-1}$ , and by the previous lemmas the proof will be completed.

Let $K=\bigcup_{n=0}^{N_{\epsilon}}F^{n}(K_{\epsilon})$ , and let $\delta_{0}=\min\{d(x,y):(x,y)\in K\}$ . Since $K$ is compact, the minimum distance $\delta_{0}$ is reached at some point of $K$ ; i.e. there exist $(x,y)\in K_{\epsilon}$ and $0\leq n\leq N_{\epsilon}$ such that $d(f^{n}(x),f^{n}(y))=\delta_{0}$ . Since $f$ is injective, it follows that $\delta_{0}>0$ and letting $\delta=\delta_{0}/2$ we have $K\subset K_{\delta}$ .

Given $\alpha\in K-K_{\epsilon}$ , there is $\beta\in K_{\epsilon}$ and some $0<m\leq N_{\epsilon}$ such that $\alpha=F^{m}(\beta)$ , and $F^{k}(\beta)\notin K_{\epsilon}$ for $0<k\leq m$ . Also, there is $n$ with $0<m<n\leq N_{\epsilon}$ such that $F^{n}(\beta)\in U\subset K_{\epsilon}$ . Hence $m<N_{\epsilon}$ , and $F(\beta)=F^{m+1}(\alpha)\in F^{m+1}(K_{\epsilon})\subset K$ ; On the other hand, $F(K_{\epsilon})\subset K$ . Therefore $F(K)\subset K$ , and inductively $F^{n}(K)\subset K$ for any $n\in\mathbb{N}$ . It follows that $F^{n}(K_{\epsilon})\subset F^{n}(K)\subset K\subset K_{\delta}$ for each $n\in\mathbb{N}$ as required.

Title	the only compact metric spaces that admit a positively expansive homeomorphism are discrete spaces
Canonical name	TheOnlyCompactMetricSpacesThatAdmitAPositivelyExpansiveHomeomorphismAreDiscreteSpaces
Date of creation	2013-03-22 13:55:11
Last modified on	2013-03-22 13:55:11
Owner	Koro (127)
Last modified by	Koro (127)
Numerical id	10
Author	Koro (127)
Entry type	Theorem
Classification	msc 37B99