EXPANSIVE MAPPINGS. by W. R. Utz

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1 Volume 3, 978 Pages 6 EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: Mal: Topology Proceedngs Department of Mathematcs & Statstcs Auburn Unversty, Alabama 36849, USA E-mal: topolog@auburn.edu ISSN: COPYRIGHT c by Topology Proceedngs. All rghts reserved.

2 TOPOLOGY PROCEEDINGS Volume EXPANSIVE MAPPINGS w. R. Utz. Introducton In ths paper we ndcate, n Secton, how certan prmary results for expansve homeomorphsms follow from a theorem proved by modfyng the proof of a known theorem. In Secton 3 we compare the property of a compact dsconnected space carryng an expansve homeomorphsm to ts beng, topologcally, a subset of the Cantor dscontnuum. Fnally, n Secton 4 we prove a theorem gvng a suffcent condton for a homeomorphsm to be expansve on a space f t s expansve on each of two subsets formng a decomposton of the space. A self-homeomorphsm, T, of a metrc space X s sad to be expansve ([5], wheren the orgnal term "unstable" was used) f there exsts a o(x,t) > 0 such that correspondng to dstnct x,y E X, there s an nteger n(x,y) for whch p (Tn (x),tn (y)) > o.. If X s a metrc space and f T s a self-homeomorphsm of X, then the orbts of the dstnct ponts x,y E X are sad to be postvely (negatvely) asymptotc under T f lm P(Tn(X),Tn(y)) = 0 n -+ (n ) Let X be an nfnte compact metrc space, T an expansve

3 utz self-homeomorphsm of X, A c X be closed and nvarant under T and let X' be the derved set of X. Theorem. If A c X' ~, then there exsts a pont of A whose orbt s asymptotc n at Zeast one sense to an orbt of a pont of x. Ths theorem ncludes Theorem. of [5] (let A = X), Lemma.3 of [5] (A s a fxed pont) and Theorem.4 of [5] (A s a perodc orbt). In both [5] and Sol Schwartzman's mprovement [, p. 87] of Theorem. of [5] t s assumed that X s dense-n-tself. That X only be a non-trval compact metrc space was assumed n Schwartzman's thess [4]. Schwartzman's theorem gves no advantage, however, n examples such as the followng one wheren one may take A = {O}, for nstance. ExampZe. Let X = {O} U {I} U {x. Ix. 4' =,,3, } U {x. Ix. = ( _l)/,,3,4, } where T: X -+ X s defned as T(O) 0, T(l), T(x) s a shft to the rght by one element of X f x ~ 0, (0 = for any > 0). One may take A as anyone of the three sets {O}, {I}, {O,l}. Proof of the Theorem. The proof may be had by mod fyng the proof of Theorem. [5] as follows. x = X x X becomes P = X x A and D = {zlz = x x x,x E X} becomes D = {zlz = a x a,a E A}. The sequence of ponts {z} c P - D for whch z -+ zed may be had as follows. Let a E A n X' and consder a sequence of dstnct elements {x} C X for

4 TOPOLOGY PROCEEDINGS Volume whch x ~ a. Then, snce each x ~ a, x x a ~ D and one has z = x x a ~ a x a E D where each z E P - D (t s possble, of course, to have x E A). 3. It s well-known [3] that subsets of the symbol space E(C) based on a fnte set of symbols, C, s homeomorphc to the Cantor dscontnuum. It s also well-known [5] that the shft homeomorphsm on E(C) s expansve. In the theorem of ths secton we show a close relatonshp between the property of a compact dsconnected space carryng an expansve homeomorphsm and ts beng a subset of the Cantor dscontnuum. Theorem. If X s a compact metrc space~ T(X) = X s an expansve homeomorphsm wth expanson constant 0 and f X can be expressed as a fnte sum of dsjont closed sets each of dameter less than o~ then X s topologcally contaned n the Cantor dscontnuum. Proof. Suppose that X s the unon of the dsjont, closed sets A,A,,A Consder the space, E(C), of all l k. mappngs of the ntegers, I, nto the set, C, of symbols {,,,k} wth metrc I T(f,g) l+max[mlf() g() for I < m] for f, g E E(C), m E I+, E I. Ths space s homeomorphc to the Cantor dscontnuum [3]. We wll now map X homeomorphcally nto E(C). If x E X, defne $: x ~ f E E(C) by the requrement that f(s) =, s E I, E C, f TS(x) E A.. To see that $ s one-to-one,

5 4 Utz we notce that f x ~ y, x,y E X, then, by the expansveness of T, p(tn(x),tn(y» > 0 for some n and so ~(x) ~ ~(y). If x E X and ~(x) f E E(C), let N, a postve nteger, be gven. Snce X s the unon of a fnte number of closed sets, A' each of these sets s also open. For any j E {O,±,±,,±N}, f Tj(x) E A. for some E {,,,k}, t s clear that there exsts a neghborhood V. of x such ] j that T (V.) c A.. Now, let E be a postve number such that J U(X,E) = {y: p(x,y) < E} c n{v.: -N < j ~ N} ] If p(x,y) < E and f ~(y) = g, then f(j) g(j) for all j = O,±,±,,±N and so a(f,g) < N+I. Thus, ~ s contnuous. Snce X s compact, ~ s a homeomorphsm. Ths completes the proof of the theorem. 4. In the next theorem we show that f a homeomorphsm s expansve on each of two subsets of a metrc space and f one of the subsets conssts of a fnte number of orbts, then the homeomorphsm s expansve on the entre space. An example s gven to show the strength of the requrement that the orbts be fnte n number n one of the subsets. B. F. Bryant [] has gven a smlar theorem. Theorem 3. Let B 3 B be nvarant dsjont subsets of B B U B under the homeomorphsm T(B) = B. Suppose that l T s expansve on B l and 3 als0 3 expansve on B. If B con ssts of the orbts of a fnte number of ponts 3 expansve on B. then T s

6 TOPOLOGY PROCEEDINGS Volume Proof. Let 0 and 0' respectvely, be expanson con $,tants for T onb and B. Let e = /3. Suppose that B l conssts of orbts of the s ponts x,x,---,x. l s nteger n the nterval < Y E B l such that p (T m (x. ),_T m (y.» < e l. l. For each < s there can be at most one for all mel. To see ths, suppose that Y and z are two dfferent ponts of B l wth ths property. Then, m' m p (T (y), T (z» m m. p (T (y), T (x» m m + p (T (x)' T (z» < for all mel whch contradcts the expansveness of T on B l. For each nteger n the nterval.. s, select a such that a < a otherwse select a = /3. a < S < mn[mn(a ), ]. < P(x,Y ) where a companon Y exsts; Let S be any number for whch We wll now show that S s an expanson constant for T on all of B. Ths s certanly clear for x,y E B or l x,y E B Suppose x E B and Y E B. There exsts a q E I. l for whch Tq(x) = x. for some on the nterval < < s. l. If p(tq(x»,tq(y» > S, the proof s complete. If p(tq(x), Tq(y» = P(x,Tq(y». S, then because of the choces of a and S we have S < a and so Tq(y) ~ Y. Thus for some N, for M = N + q to complete the proof of the theorem. Theorem 3 s not vald f B contans an nfnte number of orbts. To see ths consder the followng subset B of E 3, let B be the famly of rays z = 0, y x for x > 0, l = ''3'--- Let B be the rays z =, y = x for x > 0,

7 6 Utz '3'. Let T be a shft outward on each ray such that f R denotes the orgn (0,0,0) and f P E B, then p(r,t(p» = p(r,p). Regardless of the 0 selected, there wll be rays of B l and B (near the x-axs) for whch ponts are not separated by o. References. B. F. Bryant, Expansve self-homeomorphsms of a compact metrc space, Amer. Math. Monthly 69 (96), W. H. Gottschalk and G. A. Hedlund, Topologcal dynamcs, Amer. Math. Soc., provdence, Rhode Island, M. Morse and G. A. Hedlund, Symbolc dynamcs, Amer. J. Math. 60 (938), S. Schwartzman, Dssertaton, Yale Unversty, W. R. Utz, Unstable homeomorphsms, Proc. Amer. Math. Soc. (950), Unversty of Mssour Columba, Mssour 650