arXiv:math/0608257v1 [math.DS] 10 Aug 2006 fepniecntnsdpnso h erc u h existence maps the but metric, the not. on does depends constant constants expansive of otnosmap continuous A [email protected] DOI: 304–305 (2006) 48 Vol. Stochastics & Series Dynamics Notes–Monograph Lecture IMS hni hudb,ads ulsigi nti etcrf sappropriate is Festschrift this in it publishing Theorem. so and be, should it than B. by recognized con was this S [1] of in of unaware theorem [2]. end technique Walters the been the the P. from have on follows at based may result curiosity theorem paper the “a the That that called they from of which follows authors theorem, also their The Propp t result J. [1]. of and The idea Geller, [6]. of basic W. Boyle, the M. that in by [3] 1980s appears Geller late proo the the in W. discovering discovered from After been learned new. not authors is the argument paper, our corre behind was idea increa The edition with reproved first been the has dissertation in it [5],[6],[7]. Yale years mistake See the 1952 (A proofs. Over his 10.30]. edition.) in second Theorem [8] the [4, is Schwartzman homeomorphism in S. tha expansive years by appears fifty positively proved than a first more supports for was known that been space has metric It reason. good with sition, Remark. finite. is map expansive positively oeie ald“oemrhssit. ehv rte h ro s proof the written have that We fact into.” “homeomorphisms called sometimes d oiieyexpansive positively c ( f nttt fMteaia ttsis 2006 Statistics, Mathematical of Institute ewrsadphrases: and Keywords xmlsicuealoesddsit n l xadn endomorphism expanding all and shifts one-sided all include Examples 1 oehls,tefc htti euthsasml ro ean les remains proof simple a has result this that fact the Nonetheless, of proof simpler, even another, give to is paper this of purpose The M 00sbetclassifications: subject 2000 AMS k eateto ahmtc,Wsea nvriy Middleto University, Wesleyan Mathematics, of Department ( 10.1214/074921706000000310 x z ) f f , 7→ Abstract: 2 vr opc ercsaethat space metric compact Every f k = uprsapstvl expansive positively a supports z u ttmn osntasm httempi no uhmp ar maps Such onto. is map the that assume not does statement Our ( soet-n sue nyonce. only used is one-to-one is y n vr opc ercsaeta uprsacniuu,one- continuous, a supports that space metric compact Every f )) n , ◦ ǫ > 0 6= f oemrhs sfinite is homeomorphism egv ipepofo h title. the of proof simple a give We t.Aysuch Any etc. , ; ta .Coven M. Ethan f [email protected] o some for , fadol fteeexists there if only and if ± : X oooia yais xasv homeomorphisms. expansive dynamics, topological ,o h ntcrl.Nn fteemp soet-n,and one-to-one, is maps these of None circle. unit the of 1, → X k elynUiest,USA University, Wesleyan ≥ facmatmti pc,wt metric with space, metric compact a of rmr 70;scnay37B25. secondary 37B05; primary .Hr xoetaindntsrpae compo- repeated denotes exponentiation Here 0. > ǫ 1 scle an called is 0 304 n ihe Keane Michael and > ǫ uhta if that such 0 xasv constant expansive n T049 S,e-mail: USA, 06459, CT wn, 1 vr compact every t fa expansive an of fti work.” this of s .Byn and Bryant F. igysimpler singly ie nthis in given f x well-known s epofhad proof he hi proof Their . h proof The . euneof sequence nt.This finite. . hsresult. this d 6= htthe that o ,eg,the e.g., s, scalled is , h set The . cin3 ection y tdin cted to-one, then , e Support of a positively expansive homeomorphism is finite 305

Proof. Let X be a compact metric space with metric d, let f be a continuous, one- to-one, positively expansive map of X into (not necessarily onto) itself, and let ǫ> 0 be an expansive constant. Consider the following condition (∗) there exists n ≥ 1 such that if d(f i(x),f i(y)) ≤ ǫ for i = 1, 2,...,n, then d(x, y) ≤ ǫ, too.

Suppose that (∗) is not true. Then for every n ≥ 1, there exist xn,yn such that i i d(f (xn),f (yn)) ≤ ǫ for i =1, 2,...,n, but d(xn,yn) >ǫ. Choose convergent subse- i i quences xnk → x and ynk → y. Then x 6= y and for every i ≥ 1, d(f (xnk ),f (ynk )) → i i i i i i d(f (x),f (y)). Now d(f (xnk ),f (ynk )) ≤ ǫ for i =1, 2,...,nk, so d(f (x),f (y)) ≤ ǫ for all i ≥ 1. But f is one-to-one, so f(x) 6= f(y). This contradicts positive ex- pansiveness (with f(x) and f(y) in place of x and y), so the condition holds. Fix k ≥ 0, and apply (∗) consecutively for j = k, k − 1,..., 1, 0, with f j (x) and f j (y) in place of x and y. We get (∗∗) for every k ≥ 0, if d(f i(x),f i(y)) ≤ ǫ for i = k +1, k +2,...,k + n, then d(f i(x),f i(y)) ≤ ǫ for i =0, 1,...,k, too. Now cover X by finitely many, say N, open sets of the form

U(z) := {x ∈ X : d(f i(x),f i(z)) ≤ ǫ/2 for i =1, 2,...,n}, where n is as in (∗). If X contains more then N points, consider a finite subset containing N+1 points. k k For every k ≥ 0, there exist xk 6= yk in this subset such that f (xk) and f (yk) lie i i in the same set U(z). Then by (∗∗), d(f (xk),f (yk)) ≤ ǫ for i = 0, 1,...,k + n. Since there are only finitely many pairs of these N + 1 points, there exist x 6= y such that d(f i(x),f i(y)) ≤ ǫ for infinitely many i ≥ 0, and hence for all i ≥ 0. This contradicts positive expansiveness.

References

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