the only compact metric spaces that admit a positively expansive homeomorphism are discrete spaces
Lemma 1. If is a compact metric space and there is an expansive homeomorphism such that every point is Lyapunov stable, then every point is asymptotically stable.
Proof. Let be the expansivity constant of . Suppose some point is not asymptotically stable, and let be such that implies for all . Then there exist , a point with , and an increasing sequence such that for each By uniform expansivity, there is such that for every and such that there is with such that . Choose so large that . Then there is with such that . But since , this contradicts the choce of . Hence every point is asymptotically stable.
Lemma 2 If is a compact metric space and is a homeomorphism such that every point is asymptotically stable and Lyapunov stable, then is finite.
Proof. For each let be a closed neighborhood of such that for all we have . We assert that . In fact, if that is not the case, then there is an increasing sequence of positive integers , some and a sequence of points of such that , and there is a subsequence converging to some point .
From the Lyapunov stability of , we can find such that if , then for all . In particular if is large enough. But also if is large enough, because . Thus, for large , we have . That is a contradiction from our previous claim.
Now since is compact, there are finitely many points such that , so that . To show that , suppose there is such that . Then there is such that for but since for some , we have a contradiction.
Proof of the theorem. Consider the sets for and , where is the expansivity constant of , and let be the mapping given by . It is clear that is a homeomorphism. By uniform expansivity, we know that for each there is such that for all , there is such that .
We will prove that for each , there is such that for all . This is equivalent to say that every point of is Lyapunov stable for , and by the previous lemmas the proof will be completed.
Let , and let . Since is compact, the minimum distance is reached at some point of ; i.e. there exist and such that . Since is injective, it follows that and letting we have .
Given , there is and some such that , and for . Also, there is with such that . Hence , and ; On the other hand, . Therefore , and inductively for any . It follows that for each as required.
|Title||the only compact metric spaces that admit a positively expansive homeomorphism are discrete spaces|
|Date of creation||2013-03-22 13:55:11|
|Last modified on||2013-03-22 13:55:11|
|Last modified by||Koro (127)|