TREE-LIKE CONTINUA AND SIMPLE BONDING MAPS

Transcription

1 Voume 7, 198 Pges TREE-LIKE CONTINUA AND SIMPLE BONDING MAPS by Sm W. Young Topoogy Proceedings Web: Mi: Topoogy Proceedings Deprtment of Mthemtics & Sttistics Auburn University, Abm 36849, USA E-mi: ISSN: COPYRIGHT c by Topoogy Proceedings. A rights reserved.

2 TOPOLOGY PROCEEDINGS Voume TREE-LIKE CONTINUA AND SIMPLE BONDING MAPS Sm W. Young 1. Introduction nd Definitions The purpose of this pper is to demonstrte css of mppings J, ced simpe fods, on trees such tht every tree-ike continuum hs n inverse imit representtion for which every bonding mp is in the css J. Furthermore, J is, in sense, the simpest such css. In terms of the structure of finite covers of tree-ike continu, it coud be sid tht this pper concerns the subject of "mgmtions." See for exmpe [L] nd [M]. A continuum is compct connected metric spce. A tree is non-degenerte, connected, finite cycic grph. If P is point of the tree T, then order(p) = the number of components of T - {P}. A mpping is continuous function nd is considered to be onto uness otherwise indicted. A mpping f of tree T onto tree T is ced p.z. if nd ony if there exists finite V c T such tht if C is component of T 1 - V, then e is n rc nd fie is homeomorphism. Two mppings, f: X ~ Y nd g: X ~ Yre ced topoogicy equivent if there exist homeomorphisms h: X ~ X nd k: Y ~ Y such tht kfh = g. If T is tree, then the continuum M is T-ike if nd ony if for ech > 0, there exists n -mp TI: M ~ T. The continuum M is tree-ike if nd ony if for ech > 0, there exists tree T nd n -mp TI: M ~ T.

3 18 Young. Simpe Fods Definition. Suppose tht ech of T nd T is tree, P E T T nd T re subtrees of T T U T = T nd, b, b T n T = {P. If B: T ~ T is not homeomorphism but b ech restriction BIT B is ced fod. nd SIT b is homeomorphism, then If furthermore, one of the trees T or T b is the cosure of component of T - {P}, then B is ced simpze fozd. We mke the foowing observtions concerning the foregoing definitions: obs. 1) If B: T ~ T is p.., then B is fod if nd ony if there is ony one point of T ocy 1-1. t which B is not obs. ) If B: T ~ T is fod nd order(p), then B is simpe fod. obs. 3) If B: T ~ T is simpe fod, then on n open set contining P, B identifies two components nd ony two components of T n O~{p} nd order(b(p)) = order(p) - 1. The exmpes in digrm 1 serve to iustrte some of the types of fods. The mppings 6 1 nd 6 re the "simpest" of the simpe fods nd wi be referred to gin ter. The mpping 6 1 is simpe fod of triod onto n rc nd 6 is simpe fod of n rc onto triode with the proper choice of T nd T 6 is fod but cnnot be simpe b, 3 fod. 6 4 is simpe fod. 3. The Fctoriztion Theorem In this section we wi prove gener theorem bout the fctoriztion of mppings' by fods nd then mke some

4 TOPOLOGY PROCEEDINGS Voume observtions bsed upon the proof. Theorem 1. Suppose ech of T nd T is tree nd f is ight mpping of T onto T which is not homeo morphism. Then there exists tree T3~ mp : T + T nd 3 fod B: T + T such tht f = B. 3 Proof. Let nd b be two points of T such tht f() = f(b). Since f is ight, there exists point P which seprtes from b in T nd such tht f(p) 1 f(). Let T nd T be two trees which contin the points nd b b respectivey nd such tht T U T = T nd T n T = {p}. b b Now ech of f(t ) nd f(t ) is nondegenerte subtree of b T nd f(t ) U f(t ) = T. b We construct the tree T s the union of homeomorph 3 of f(t ) nd homeomorph of f(t ) joined ony t the point b f(p). Thus we define homeomorphisms B nd B such tht b S: f(t ) + Bf(T ) C T 3 nd B b : f(t b ) + Bbf(T b ) C T 3, Sf(T ) U Bbf(T b ) = T 3 nd Bf(T ) n Bbf(T b ) = {Bf(P) Bbf (P) } The mpping : T + T is defined by 3 (x) = Sf(X) for x E T Sbf(x) for x E T b The continuity of is ssured since T n T = {p} nd b Sf(P) = Bbf(P). The mpping B: T 3 + T is defined by s~1(x) for x E Sf(T ) S (x) -1 Sb (x) for x E Sbf(T b ) Likewise, 8 is continuous since Bf(T ) n Sbf(Tb) {Sf(P) = Bbf(P)}.

5 184 Young It is esy to check now tht S(x) = f(x) for x E T. It remins now to show tht S is fod. S is obviousy the union of two homeomorphisms joined t point. Now consider the points (), (b) E T. We hve 3 () = Sf() E Sf(T ) nd (b) = Sbf(b) E Sbf(Tb). not possibe for () = (b) since this woud impy tht It is () = Sf(P) which is the ony point in Sf(T) n Sbf(T ). b If so, then f() = S() wy tht P ws chosen. S Sf(P) = f(p) contrry to the We now hve () ~ (b) but S() = S(b). Since S is not homeomorphism, S stisfies the definition of fod nd the proof is compete. The proof of Theorem 1 is rey quite simpe. The mpping spits the imge of f into two prts nd the mpping S mends the prts bck together. It is hoped, however, tht the more detied rgument given wi fciitte our mking few more observtions bsed upon the proof. obs: 4 ) The mpping is homeomorphism if nd ony if ech of fit nd ft b is homeomorphism. obs. 5) If f is p.., then ech of ex, nd S is p.. obs. 6) Let [,b] denote the subrc of T joining nd b. The point P coud hve been chosen so tht f(p) is n end point of the subtree f([,b]) nd f(p) is different from f (). In such cse, f is not ocy 1-1 t P. obs. 7) If order(p) = nd f is not ocy 1-1 t P, then is ocy 1-1 t P. obs. 8) If f is fod, then f is finite composition of simpe fods. To prove this, we choose P to be the

6 TOPOLOGY PROCEEDINGS Voume unique point t which f is not ocy 1-1 nd ppy Theorem 1 to obtin f S. In this cse, wi be fod nd so if either or S is not simpe fod, then Theorem 1 cn be ppied gin to using the point P or to S using the point {P) nd so on. If f ocy mps m rcs onto n rcs, then ech simpe fod fctor merey spits one identifiction. The resut wi be composition of m - n simpe fods. We now hve sight improvement upon Theorem 1. Theorem. Suppose ech of T nd T is tree nd f is ight mpping of T onto T which is not homeo morphism. Then there exist tree T3~ mp : T + T nd 3 simpe fod S: T 3 + T such tht f = S. Proof. The proof foows from Theorem 1 nd obs. 8. Theorem 3. If f is p.. mpping of tree onto tree then f is either homeomorphism or is finite composition of simpe fods. Proof. We wi first show tht f is finite composition of fods. If f: T + T is p.. nd not homeomorphism, then we choose point P of T t which f is not ocy 1-1 (s in obs. 6) nd ppy Theorem 1 to obtin f S where S is fod nd is p.. (obs. 5). If is homeomorphism, then f = S is fod. And if is not homeomorphism, then one of two cses must occur: First, performs t est one identifiction of pir of subrcs in neighborhood of P but fewer such identifictions thn f does, or second, is ocy 1-1 t P but fis to be ocy 1-1 t some other point.

7 186 Young In the first cse we cn choose the point P gin nd ppy Theorem 1 to the mp. In the second cse, we choose nother point Q ~ P such tht fis to be ocy 1-1 t Q nd ppy Theorem 1. We obtin 13 ' ' where 13' is fod nd ' is p.. We ppy Theorem 1 to ' nd continue in this mnner. Since f fis to be 1-1 t ony finitey mny points nd mkes ony finitey mny identifictions t ech such point, this process must terminte. Now tht f is composition of finitey mny fods, we cn fctor ech fod into finitey mny simpe fods (obs. 8). This competes the proof. Ech simpe fod in the fctoriztion of f cn be thought of s performing one of the oc identifictions done by f. The fctoriztion obtined wi ony depend upon the order in which the identifictions re to be removed. Thus if f performs n oc identifictions, then the number of possibe fctoriztions of f into simpe fods woud seem to be n! In gener, however, some dupictions wi occur. It shoud so be pointed out tht simpe fods re not "indivisibe." The simpe fod 6 (see digrm 1) is the composition of two simpe fods. Let 13 = where 13 is topoogicy equivent to 13 nd 13 identifies prts of the two bottom egs of the triode 4. Appictions In this section we give some ppictions to inverse imit spces. We begin with corory to Theorem 3.

8 TOPOLOGY PROCEEDINGS Voume Corory 1. If f is mp of tree T onto tree T nd E > 0, then there exist simpe fods S,S,.,6 such n tht d(f,6 S Sn) < E. Proof. The proof foows from Theorem 3 nd the fct tht ny mp cn be uniformy pproximtey by p.. mps. Corory. If M is tree-ike continuum, then M is the imit of n inverse ~m~t system T + T + T + M 3 where ech of the bonding mps is simpe fod. Proof. It foows from the bsic theory contined in [M-S], the pproximtion theorem of [B] nd Corory 1 tht every tree-ike continuum M hs n inverse imit represent. f f f 3 ton T + T + T + M where ech bonding mp is 3 finite composition of simpe fods. The inverse imit spce is the sme fter tking s bonding mps the individu simpe fod fctors. Even though the bonding mps in the representtion given by Corory re retivey simpe, there is not, in gener, ny contro over the compexity of the trees which pper s fctor spces. Of course the number of endpoints of the trees must be unbounded if M is infinitey brnched [Y ]. In cse M is rc-ike, we cn offer the foowing: Theorem 4. There exists simpe fod 6 1 : triod ~ rc nd simpe fod 6 : rc 7 triod such tht if M is n rc ike continuum, then M is the inverse imit of the inverse imit system T + T + T + M such tht for ech 3 i ~ 1, Si is topoogicy equivent to nd is 6 6 i + topozogiczy equivent to 6 1 "

9 188 Young dig.rm 1. Ppoof. The mppings 8 nd 8 re the sme s in 1 In Corory 3.7 of [Y ] it is shown tht there re two mps of the unit interv onto itsef, Lnd Z such tht every mp of the unit interv onto itsef cn be uniformy pproximted by composition of mps ech of which is either L or Z or homeomorphism. In other words, every np of the unit interv onto itsef cn be uniformy pproximted by composition of mps ech of which is topoogicy equivent to either L or Z. See digrm for description of the mps Lnd Z. As ws pointed out in the proof of Corory, it foows tht every rc-ike continuum M hs n inverse imit representtion f f f3 rc +- rc +- rc +- M where ech bonding mp is topoogicy equivent to either L or Z. Now the mp Z is the composition of 8 foowed by 8 if 8 is crefuy\pppied so tht the eg 1 1 of the triod which receives the int~7ior point of the rc is foded down into one of the other two egs. is so the composition of 8 foowed 'QY 8 1. The mp L Tke one of the egs of the triod which receives n endpoint of the rc nd ppy 8 1 the other end point. to fod tht eg into the eg which received So if M is n rc-ike continuum, then M hs n inverse imit representtio~ 81. S 83. d rc +- tr~od +- rc +- tr~o +-. M where for ech i, Si+ is topoogicy equivent to 8 1 nd i is topoogicy equivent to 8 This competes the proof.

10 TOPOLOGY PROCEEDINGS Voume DIAGRAM I 8 1 : ---L~ t ( C ) P B : Ip ~...L I ~ I DIAGRAM II L:, ~ t «) z: I ~, ( <.),)

11 190 Young References [B] M. Brown, Some ppictions of n pproximtion theorem for inverse imits, Proc. Amer. Mth. Soc. 11 (1960), [L] W. Lewis, Stbe homeomorphisms of the pseudo-rc, Cn. J. Mth. XXXI, No. (1979), [M] E. E. Moise, A note on the pseudo-rc, Trns. Amer. Mth. Soc. 64 (1949), [M-S] S. Mrdesic nd J. Seg, -mppings onto poyhedr, Trns. Amer. Mth. Soc. 109 (1963), [Y-] S. W. Young, Finitey generted semigroups of continuous functions on [0,1], Fund. Mth. LXVIII (1970), [Y-], On the retive compexity of tree-ike continuum nd its proper subcontinu, Topoogy Symposium, Univ. of Texs Press (to pper). Auburn University Auburn, Abm 36849