TREE-LIKE CONTINUA AS LIMITS OF CYCLIC GRAPHS

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1 Volume 4, 1979 Pages TREE-LIKE CONTINUA AS LIMITS OF CYCLIC GRAPHS by Lex G. Oversteege ad James T. Rogers, Jr. Topology Proceedigs Web: Mail: Topology Proceedigs Departmet of Mathematics & Statistics Aubur Uiversity, Alabama 36849, USA ISSN: COPYRIGHT c by Topology Proceedigs. All rights reserved.

2 TOPOLOGY PROCEEDINGS Volume TREE-LIKE CONTINUA AS LIMITS OF CYCLIC GRAPHS Lex G. Oversteege ad James T. Rogers, Jr. A cotiuum X is a compact, coected metric space with more tha oe poit. A cotiuum is tree-like if it is homeomorphic to the limit of a iverse sequece of trees. A map is a cotiuous fuctio. A map f: X ~ X has a fixed poit if there exists a poit x i X such that f(x) x. A cotiuum X has the fixed-poit-property if each map f: X ~ X has a fixed poit. R. H. Big [2] has raised the followig questio: Questio. Does each tree-like cotiuum have the fixedpoit-property? Big's questio has recetly bee aswered by David Bellamy [1]. Theopem 1. Thepe exists a tpee-like cotiuum X without the fixed-poit-ppopepty. Here is the gist of Bellamy's proof: Take a certai soleoid. Remove a arc ad recompactify with the suspesio of a ifiite set. The example X is the orbit space of a certai ivolutio o this modified soleoid. This example has sparked a reewed iterest i a secod, veerable questio. Questio. Does there exist a plaar tree-like cotiuum without the fixed-poit-property?

3 508 Oversteege ad Rogers Bellamy's cotiuum X is ot plaar. If we apply the Fugate-Mohler techique [3] to X, we get a tree-like cotiuum Y = lim{x }, where each factor space X is a copy of X. The embeddig properties of Yare ot obvious. Hece we are iterested i the followig questio: Questio. How ca we fid a geometric descriptio of a tree-like cotiuum without the fixed-poit-property, so we ca study embeddigs? Before cosiderig possible aswers to this questio, let us formulate the followig defiitio: Defiitio (Mioduszewski). Let {E } be a sequece of positive umbers with E + O. Suppose X is the iverse limit of the sequece (X,o). Let (Y : X + X ) be a se quece of maps such that each diagram X X m (1) jy jym ~E X X m is -commutative. Defie y: X + X by y(x l,x 2,x, ) 3 (Yl'Y2'Y3' )' where m Yk = lim ok Ym(x m ) m+ oo It was proved by Mioduszewski [4] that the map Y is well defied ad cotiuous. The map Y is said to be almost a simple iduced map. (Observe if E o for each, the Y is a (regular) simple iduced map.) Oe obstructio to obtaiig a iverse limit (or, equivaletly, a tree-chai) descriptio of a fixed-poit-free

4 TOPOLOGY PROCEEDINGS Volume map o a tree-like cotiuum is the ext theorem. Theopem 2. Let the cotiuum X be the limit of the ivepse sequece (X,O)3 whepe each factop space X has the fixed-poit-ppopepty. The each almost simple iduced map has a fixed poit. Ppoof. Suppose the almost simple iduced map y: X ~ X has o fixed poit ad let {y } ad {E }, E ~ 0, be as i the defiitio of y. There exists a E > 0 such that d (x,y (x» > E for each x i X. Recall that where we assume that diam(x ) < 1 ( = 1,2,---). Hece there exists a such that: di(xi'yi) (1) L < E/2 ad i= 2 i (2) E < E/2. Let x be a fixed poit of Y: X ~X. Let x = (x l,x 2,---, x,---) be ay poit of X whose th coordiate is x ad let y(x) = Y = (y l,y 2,---). It follows from the E-commutativity i (1) that for 1 < k < ad m > m m m dk(xk,ok Ym(x m» = dk(ok Y (x m ), ok Ym(x m» < e: Hece, by defiitio of Yk' dk(xk'yk) < E Hece dk(xk,y k ) d(x,y(x» L k k=l 2 dk(xk,y k ) dk(xk,y k ) L + < k L k k=l 2 k=+l 2 E E L...!-+ < E. k=l 2 k 2

5 510 Oversteege ad Rogers This is a cotradictio. I spite of the fact that each almost simple iduced map has a fixed poit, oe would like to cosider maps that, i some sese, preserve the idex, for otherwise it is more difficult to verify that a map is fixed-poit-free. Oe possibility is to drop the restrictio that each X has the fixed-poit-property. Let us see how we are led to this choice i aother way. Let I be the iterval [0,6]. Let g: I ~ I be the rooftop fuctio whose graph is pictured i Figure 1, ad let f: I ~ I be the three-rooftops fuctio whose graph is pictured i Figure 1. 6 o 3 6 o Graph of 9 Graph of f Figure 1 Note the followig facts about the maps f ad g: Fact 1. The maps f ad g commute. Fact 2. The set of fixed poits of g is {0,4}. Fact 3. The set f({0,4}) = {Ole It is a corollary to Theorem 2 ad Fact 1 that the map goo: X ~ X defied o the arc-like cotiuum X = lim(i,f) by the diagram

6 TOPOLOGY PROCEEDINGS Volume I~I~I +-- (2) gl gl gl I~I~I... ~ x has a fixed poit. What is it? It follows from Facts 2 ad 3 that (0,0, ) is the oly fixed poit of g. A perso uaware of Theorem 2 might be tempted to modify Diagram 2 by removig a small arc cotaiig a ad replacig x it with a V. Such a perso would the have a diagram of trees (3) 6 I OL OR yl 6 1 OL OR OL OR where the maps a ad y satisfy the coditios ad TI 0 a(x,y) = f(y) 2 TI 0 y(x,y) = g(y) 2 There will ow be a left zero ad a right zero, deoted OL ad OR respectively, ad the map y will iterchage OL ad OR. Ay map iduced by the y's will iterchage the poits (OL,OL, > ad (OR,OR, ), ad hece will have o fixed poit. Those of us i the kow o Theorem 2 will seek the part

7 512 Oversteege ad Rogers of the hypothesis of Theorem 2 that has bee violated. Ideed, it is the commutativity of the diagram. Cosider 0(4). We must decide whether to defie 0(4) OL or 0(4) = OR. Without loss of geerality, defie 0(4) OLe The y 00(4) = OR, while 0 0 y(4) = 0(4) = OLe There seems to be o alterative other tha to have two poits called "4," amely 4L ad 4R. Itroducig two 14's" will cause two problems. First, the factor space X, pictured below, is o loger acyclic. 4L 4R OL OR Hece, there appears the ew problem of provig that the iverse limit space is tree-like. Secodly, a check of the commutativity of the diagram for the six poits i 0-1 (4) reveals that because of the two 14's," it will be ecessary to istall two 12's." The two 12's" force us to split still other poits ito left ad right compoets, ad commutativity of the square is ever attaied. I spite of these problems, the followig theorem is proved i [5]. Theorem 3. There exists a iverse sequece of plae curves (X,o) whose iverse Limit X is tree-like, ad a sequece {y : X ~ X } of mappigs that defies a

8 TOPOLOGY PROCEEDINGS Volume fixed-poit-free map y: X + X. The map y is "almost" a simple iduced map with respect to {y}. X has the proper ties that (1) X is atriodic (2) Each proper subcotiuum is a arc (3) The complemet of ay oempty ope set ca be embedded i the plae. Alteratively, if oe is willig to forego the property of preservig idices, the the followig theorem ca be proved (see [6] ) : Theorem 4. There exists a iverse sequece (T,a) of T} of maps makig parallelo trees ad a sequece {Y: T + l + grams commute Tl~ T T3~ ;/r/ Tl~T2~T3~ x such that the sequece {y } iduces a fixed-poit-free map y: X + X o the tree-like cotiuum X = lim (T,a). The apparet simplicity of this last theorem is somewhat deceivig, for the price for commutativity is "highpowered" bodig maps, which make it ecessary to develop a apparatus similar to the proof of the previous theorem i order to prove that the map y has o fixed poit. Bibliography 1. D. Bellamy, A tree-like cotiuum without the fixed poit property (preprit).

9 514 Oversteege ad Rogers 2. R. H. Big, The elusive fixed poit property, Amer. Math. Mothly 76 (1969), J. B. Fugate ad L. Mohler, A ote o fixed poits i tree-like cotiua, Topology Proceedigs 2 (1977), J. Mioduszewski, Mappigs of iverse limits, Co11oquim Math. 10 (1963), L. G. Oversteege ad J. T. Rogers, Jr., A iverse limit descriptio of a atriodic~ tree-like cotiuum ad a iduced map without a fixed poit, HOllsto J. of Math. (to appear). 6., Fixed-poit-free maps o tree-like ootiua (preprit). Tulae Uiversity New Orleas, Louisiaa The secod author was partially supported by a research grat from the Tulae Uiversity Seate Committee o Research.