Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs Auburn Unversty, Alabama 36849, USA E-mal: topolog@auburn.edu ISSN: 046-44 COPYRIGHT c by Topology Proceedngs. All rghts reserved.
TOPOLOGY PROCEEDINGS Volume 3 978 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 -+ -00) Let X be an nfnte compact metrc space, T an expansve
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
TOPOLOGY PROCEEDINGS Volume 3 978 3 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,
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
TOPOLOGY PROCEEDINGS Volume 3 978 5 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,
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), 386-39.. W. H. Gottschalk and G. A. Hedlund, Topologcal dynamcs, Amer. Math. Soc., provdence, Rhode Island, 955. 3. M. Morse and G. A. Hedlund, Symbolc dynamcs, Amer. J. Math. 60 (938), 85-866. 4. S. Schwartzman, Dssertaton, Yale Unversty, 95. 5. W. R. Utz, Unstable homeomorphsms, Proc. Amer. Math. Soc. (950), 769-774. Unversty of Mssour Columba, Mssour 650