IMFM Institute of Mathematics, Physics and Mechanics JADRANSKA 19, 1000 LJUBLJANA, SLOVENIA Preprint series Vol. 48 (2010), 1124 ISSN 2232-2094 TOWARDS THE COMPLETE CLASSIFICATION OF TENT MAPS INVERSE LIMITS Iztok Banic Matevz Crepnjak Matej Merhar Uros Milutinovic Ljubljana, August 04, 2010 Towards the complete classification of tent maps inverse limits m Iztok Banic, Matevz Crepnjak, Matej Merhar and Uros Milutinovic < C^ o o CSI I July 19, 2010 Abstract We study tent map inverse limits, i.e. inverse limits of inverse sequences of unit segments I with a tent map being the only bonding function. As the main result we identify an infinite family of curves in 12 such that if top points of graphs of tent maps belong to the same curve, the corresponding inverse limits are homeomorphic, and if thev 00 belong to different curves, the inverse limits are non-homcomorphic. The inverse limits corresponding to certain families of top points are explicitly determined, and certain properties of the inverse limit are proved in the case of (0,1) as the top point. CO ^ 1 Introduction Continua as inverse limits have been studied for a long time. One reason LLi for such intense research in this area is the fact that inverse sequences with very simple spaces and simple bonding maps can give extremely complicated continua as their inverse limits. The inverse limits may be both complicated and useful even in the case, when all the spaces a re unit intervals [0,1] and all the bonding functions are the same. Such inverse sequences and their inverse limits plav an important role in the continuum theory as well as in the theory of topological dynamical systems. They also appear in applications in such diverse areas as economy, mechanics of fluids, physics and more; see [34, 36, 37, 38, 40, 41]. 2010 Mathematics Subject Classification: Primary 54C60; Secondary 54B10. Key words and phrases: Continua, Inverse limits, Tent maps, Knaster continua. Even such a simple case when the graph of the bonding function is the union of two segments may be highly non-trivial. Such functions are called tent maps and they are the main object of our study. In Definition 1,1 we introduce the basic notation related to them, which we use later in the O paper. Definition 1.1. For any a,b G [0,1], the tent function f(a,b) : [0,1] ^ [0,1] is defined as the set-valued function with the graph r(f(a,b)) being the union of the segment (possibly degenerate) from (0, 0) to (a, b) and the segment (possibly degenerate) from (a,b) to (1, 0). The point (a,b) is called the top point of the graph r(f(a,b)), The inverse limit obtained from the inverse sequence of closed unit intervals [0,1] and the bonding function f(a,b) is denoted bv K(a,b) =Jjm{[0, 1], f(a,6)}~ 1. Note that f(a,b) is single-valued if and only if a G {0,1} or (a,b) G {(0,0), (1, 0)}. The first and the most famous example of continua K(ab) is K(i called the Brouwer-Janiszewski-Knaster continuum or sometimes just the Knaster continuum, 1 The whole family of continua - < 6 < 1, has been called Knaster 2 ' 2 continua and the famous Ingram conjecture, stated in 1992, claimed that all of them are pairwise non-homeomorphic. It generated a large number CSI CO CO of articles, such as Barge, Bracks, Diamond |6|, Barge, Diamond |8, 9| Barge, Jacklitch, Vago [10], Barge, Martin [11, 12, 13], Block, Jakimovik, Kailhofer, Keesling |14|, Block, Keesling, Raines, Stimac |15|, Bracks, Bruin |16|, Bracks, Diamond |17|, Brain 118, 19, 20|, Collet, Eckmann |22|, Good, Knight, Raines [26], Good, Raines [27], Kailhofer [32, 33], Raines [47], Raines, Stimac [48, 49], Stimac [50, 51, 52, 53], Swanson, Volkmer [54], and others, in which certain special cases of the conjecture were proved. Finally, the conjecture was proved in 2009 by M. Barge, H, Bruin and S, Stimac [7], In spite of such great effort and many obtained results the complete classification of all inverse limits K(ab) is still an open problem. In this paper we continue the study of inverse limits of tent maps f(a,b) and their classification, £ Note that in some cases the tent maps are not single-valued and therefore the concept of inverse limits of inverse sequences with upper semicontinuous set-valued bonding functions is needed. Such a generalization of the concept of inverse limits was introduced in [31, 39] by W, T. Ingram and W, S, Mahavier. They gave conditions under which the inverse limit of an inverse sequence of Hausdorff spaces with upper semicontinuous set-valued bonding functions is a Hausdorff continuum, provided some interesting examples of such inverse limits, and discussed their dimension. The concept of these generalized inverse limits has become very popular since their introduction and has been studied by many authors and many papers appeared; for examples see [1, 2, 3, 4, 5, 21, 23, 29, 30, 31, 35, 39, 45, 46, 55], M <| 2 Definitions and notation Our definitions and notation mostly follow [31] and [43], A continuum is a nonempty, compact and connected metric space. Lot W = {(#, sin G R2 | 0 < x < 1}. Any continuum homoomorphic to C\(W) is called a sin—continuum. A harmonic fan is anv continuum, homeomorphic to the continuum, defined as the union Kn U K, where for each n, Kn is the segment in the plane from (0,0) to (f, -), and I< is the segment from (0,0) to (1,0). o Let (Xn, dn) be a sequence of metric spaces, where all metrics are bounded bv 1, Then \ \ dn(xn,yn) D(x,y)= sup - CSI where x = (x1,x2,x3,...), y = (y1, y2, y3,...), will be used for the metric on the product space II Xn (it is well known that th e metric D induces the n= 1 product topology [24, p. 190], [42, p. 123]). If (X, d) is a compact metric space, then 2X denotes the set of all nonempty closed subsets of X, Let for each e > 0 and each A G 2X Nd(e, A) = {x G X | d(x, a) < e for some a G A}. We will always equip the set 2X with the Hausdorff metric Hd, which is •j-1 defined as CD Hd(H, K) = inf{e > 0 | H C Nd(e, K),K C Nd(e, H)}, for H, K G 2X. Then (2X, Hd) is a metric space, called the hyperspace of the space (X, d). For more details see [28, 43], Let X and F be compact metric spaces. A single-valued function f : X ^ 2y is also called a set-valued function f : X ^ Y. A set-valued function f : X — Y is upper semi-continuous (abbreviated u.s.c.) if for each open set V C Y the set {x G X | f (x) C V} is an open set in X, The graph T(f) of an u.s.c, set-valued function f : X — Y is the set of ^^ all points (x, y) G X x Y such that y G f (x), Ingram and Mahavier gave the following characterization of u.s.c. functions [31, p. 120]: Theorem 2.1. Let X and Y be compact metric spaces and f : X — Y a set-valued function. Then f is u.s.c. if and only if its graph r(f) is closed ^ in X x Y. In this paper we deal with inverse sequences {Xn, fn}c^=1j where Xn are compact metric spaces and fn : Xn+1 — Xn are u.s.c. set-valued functions. We denote {Xn, fnalso bv Xi £ X2 f2 X3 £ ... The inverse limit of an inverse sequence {Xn, fn}^=1 is defined to be the subspace of the product space n^L1 Xn of all x = (x1,x2,x3,...) G EIX^, such that xn G fn(xn+1) for each n. The inverse limit is denoted by £m{Xn, fn}^=1. The notion of the inverse limit of an inverse sequence with u.s.c. bonding functions was introduced by W. S, Mahavier in [39] and W. T. Ingram and W. S. Mahavier in [31]. For any compact metric space X we use dim(X) for the topological (covering) dimension of X (for the definition see [25, p. 385] or [44, p. 10]). HH For the reader's convenience we list the following well-known results that will be used later: CO CD $h CD CO XY that dim(X) = 0. Then dim(X x Y) = dim(Y). Theorem 2.3. [44, p. 15] Let {Xn}^c=1 be a sequence of compact subspaces of a metric space and let k be a nonnegative integer, such that dim(Xn) < k }h for all n. Then a dim Xn < k. n=1 3 Main results O In this section we formulate and prove the main results. First we introduce basic notation and facts. Let a G (0,1) and b G (0,1]. Note that f(a,b) is single-valued. Then f m-jzt ; if^M haM ) U*i(*-i) ; if t €[«,!]. 2 < CSI CSI £ CO CO u CD CO The point e = fiab)(b) = - plavs an important role in the studv of a — 1 K(a,b). The restrictions of f(a,b) mapping [0, a] onto [0, b], and [a, b] onto [e, b], respectively, are bijections. Therefore they have the inverse functions and we denote them bv L : [0,b] ^ [0, a^d R : [e,b] ^ [a,b], respectively. It is easy to see that Lit) = ft and R(t) = ^t + 1, For any point (x\,x2,x3,...) G K(a,b) and any positive integer n, xn = f(a,b)(Xn+ iMf xn+1 < a then xn+1 = L(xn^f xn+1 > a then xn+1 = R(xn), and finally if xn+1 = a then xn = ^d xn+1 = L(xn) = R(xn), Note that if xn+1 > a then xn+1 G [a,b], because xn+1 = f(a,b)(xn+2) < b. Therefore in that case xn G [a,e]. This fact was the main reason for the introduction of e and our choice of the restriction of f(a,b) in the definition of R, That means that xn+1 = R(xn) is possible only for xn > e, but note that for xn+1 = L(xn) there are no restrictions. We continue with the following lemma which will be used in the proof of Theorem 3,2, Lemma 3.1. Let X be a compact metric space and let A be a continuum and for each positive integer n, let An be an arc in X torn an to an+1 such that 1. for each positive integer n, An n An+1 = {an+1}, 2. Ai n Aj = 0 if and only if |i — j| < 1, 3. there is a point z G X \ ( I I An I such that lim An = {z} in 2X, l — I n^TO Vn=1 I a CD Then the subspace A of X is an arc. 4. A = |J Aj U {z}. n n=1 Proof. Lot for each positive integer n. fn : f2^-, ^-j-] —> An be a homeomor-phism such that fn(z=±) = an and fn(-^i) = • X be a function, defined by f(t) = fn(t) if t G for some positive integer n, and f(l) = z. Since = an+1 = fn+i(^i) for each positive integer n, it follows that f is continuous on [0,1) [24, p. 83], Let {ti}°=1 be a sequence in [0,1] such that lim ti = 1, Then it follows from 3, that lim f (ti) = z and therefore f is continuous at t = 1, It also follows from 4, that f is surjective and from 1,, 2,, and 3, that f is injective. Therefore f is a homeomorphism from [0, f] onto A. □ Theorem 3.2. Let a,b G (0,1), a < b and b < 1 — a. Then K(a,b) is an arc. Proof. Let e = f(a,b){b) = 1 f°U°ws from a < b < 1 and b < 1 — a i—l ,-H that a < e. Define A1 to be the set of all points (x1,x2,x3,...) G K(a,b) such that xi+1 = L(xi) ^^r each positive integer i. It follows that A1 is an arc from a1 = (0, 0, 0,...) to a2 = (b,a,L(a),L2(a),...), & 2 since one easily proves that t ^ (t,L(t),L2(t),.. ) is a homeomorphism torn [0, b] onto A1. Let A2 be the set of all p oints (x1,x2,x3,...) G K(ab) such th at x2 = CO R(x1) and xi+1 = L(xi) for each positive integer i > 2. It follows that A2 is an arc from a2 = (b, a, L(a), L2(a),...) to a3 = (e, b, a, L(a), L2(a),...), CO since t ^ (t, R(t), L(R(t)), L2(R(t)),...) is a homeomorphism from [e,b] i—i ^ onto A2, Also, for each positive integer n > 3 define the set An to be the set of all points (x1,x2 ,x3,...) G K(a, b) such th at xn = R(xn-1 ^d xi+1 = L(xi) for each positive integer i > n. It follows that An is an arc from from HH an =(fn-2(e),..., f2(e), f (e), e, b, a, L(a), L2(a), L3(a),...) "J^ t0 an+1 = (fn-3(e),..., f2(e), f (e), e, b, a, L(a), L2(a), L3(a),...), since t ^ (fn-2(t),..., f2(t), f (t), t, R(t), L(R(t)), L2(R(t)),...) is a home-omorphism from [e, b] onto An, Since R is an expansive map the only point of the form (t, R(t), R2(t),...) is obtained in the case when t = R(t). One easily checks that t = 1+bb_a ■ Let __h-__h— ) Vl+6-a' 1+6—a.' 1+6—a.' ■■■■>• Since a < e and L(b) = a, it follows that L(t) < L(b) = a < e for each t G [e, b], and therefore R(L(t)) is not defined. It follows that K(a,b) = oo \ U aA U {z}. Vn=l / One can easily check that An n An+1 = {an+1} for each positive integer n and that for positive integers ^^d n such that |m — n| > 1 it holds that Am n An = 0- The function f(a,b) is a contraction mapping on [e, b] with the contraction < factor M = j^- < 1. Using this we show that lim An = {z} in 2A(a>6), Take 1 a n^x any e > 0. Choose a positive integer k, such that j, < s. Let n0 be a positive integer such that Mn < ^ for each n > n0. Then for each n > n0 and for each t G [e, b], O 4J CO 2 CSI 0 o 1 CO ^ CO CO fn(t) — b 1 + b — a fn(t) — f b 1 + b — a b < Mn 1 + b-a n0 + k + 1 and for each x G An it holds that D(x, z) < e, Take any X = (xi,x2, . . . ,xfc, . . .) = (fn—2(t),..., fn—k—1(t),..., f2(t), f (t), t, R(t), L(R(t)), L2(R(t)),...) n-v-' k in An. Take arbitrary positive integer m. If m > k then m mk If m < k then n — m — 1 >n — k — 1 > n0, hence d(xm, TTT-7,) I f —^t)- h 1+b—a I < e. □ CO CD $H CD CO mm It follows that D(x, z) < e for each x G An and therefore HD(An, {z}) < e Using Lemma 3.1 it follows that K(a,b) is an arc. Definition 3.3. For any t G [1, to) let Ct = {(x, y) G [0,1] x [0,1] | xt+1 — xt = yt+1 — yt, 0 < x < y}. See Figure 1. U a CD U It is easy to see that C\ is the graph of the function / : (0, —> 1), f (x) = 1 — x. One can easily see that for each t G (1, to), Ct is a subset of [0,1] x [0,1] containing (|,y) for exactly one y G 1). O 4J CO 2 0 o 1 CO ^ CO CO CO CD $H CD CO u a CD U cu Figure 1: Ct for t = 1, 2, 3, 4. Theorem 3.4. Let n be a positive integer and (a, b), (c,d) G Cn. Then K(a,b) and K(c,d) are homeomorphic. Proof. In this proof for any closed interval L = [u, v] the corresponding o open interval (u, v) will be denoted bv L. Let e = f[aM(b) = and / = f{c4)(d) = Next define L1 : [0,b] — [0, a^d R1 : [e,b] — [a, b] with Li(t) = yt, i?i(t) = + bb and L2 : [0, d] — [0, c] and R2 : [f, d] — [c, d] with c c — 1 L2(t) = -t, R2(t) = ——t + 1. dd From (a,b),(c,d) G C„ it follows that (J)""^ = : (~)ne = (b\nb(b-l) __ ^ /(iy«-l f _ d{d—l) _ Myra f _ td\nd{d-l) \aJ \aJ a—i ' \CJ J \CJ c-1 \ c J J \CJ c-1 d Moreover 0 < e < f(a,b) (e) < e) < ■■■ fl(e) < e), and /(„b)(e) = )A'e for each A: = 0,1,..., /?.. Similarly 0 < f < f(c,d)(f) < f(2c,d) (f) < ■ ■ ■ < f^; (f) < fn,d)(f) a 2 O 4J m 2 0 o 1 CO ^ CO CO *nd f(c4)(f) = (iff for k = 0. 1. Let I0 = [0, e] and Ik f k-1 (a , n, b' a. k-1 for each k = 1,..., n. Since R1 (e) = b and R1(b) = a, it follows that R1 ([e,b]) = In and therefore R1(Ik) C In for each k = 1,..., n. Recall that R1 is not defined on Io \ {e}. Since L^fe) = (Jf~le it follows that Li(/k) = Ik-i for each k = 1,... ,n. Also L1 (Io) C Io. For any x G [0, b] let S1(x) = {Ik | k = 0,1, 2,..., n, x G Ik}. Obviously, is a singleton, except for x = )ke. k = 0,1,...,/?,— 1, when S\(x) = {Ik, Ik+1}- This notation will simplify the description of the dynamics of the mapping f(a,b) that will be the crucial part of the proof. Analogously, we define J0 = [0, f], Jk fM)(f ),f( (f ) k-1 Ms)' for each k = 1,..., n and prove that R2([f, d]) = Jn, L2(Jk) = Jk-1 for each k = 1, 2,..., n, and L2( J0) C J0, Also for any x G [0, d] we defi ne S2(x) = {Jk | k = 0,1, 2,...,n,x G Jk} which again turns out to be a singleton, except for x = f, h = 0,1,..., n - 1, when S2(x) = {Jk, Jk+1}. Now define the continuous piecewise linear increasing function ^ : [0, b] ^ [0, d], which maps each i nterval Ik afinelv on to Jk, fo r k = 0,1, 2,..., n. Explicitly is given bv CO CD U CD CO u a CD U p(t) k e i+ ad d d ad % [z$fAt-&<*) + &/ ** a ' a ' a ' a ' if t G Io if t G I1 if t G I2 if t G I3 We will prove that the function $ : K(a,b) ^ K(c,d) defined by (x1,x2,x3,...) 1—^ (^1,^2,^3,...), k e where CD C^ (3-1) yi = and CD /L2(yfc) ; if xfc+i = Li(xfc) (3-2) yfc+i = 4 R ( , R , , [R2 (yk) ; " xk+i = Ri (xfc) M for each positive integer k, is well-defined and that it is a homeomorphism. The well-definedness of $ follows from the following inductive argument. Let x = (xi, x2, x3,...) £ K(a,b) be arbitrary. By induetion on m we prove that for each m and for each j = 1, 2,..., m, yj is uniquely determined bv -H (3.1) and (3.2), and that o xj = 0 ^^ yj = 0, xj = b yj = d, as well as Ik £ Si(xj) ^^ Jk £ S2(yj), O for any k = 0,1,..., n. The claim is obviously true for m = 1, by (3,1) and the definition of p. Assume that the claim is true for a positive integer m. Now we distinguish several cases. Case 1. Si(xm+i) = {Ik}, for some k = 0,1,..., n. Subcase 1,1, xm+i = Ri(xm), Now xm+i £ In, i.e. k = n, xm+i = a therefore xm = b. It follows that xm+i = Li(xm) does not hold and therefore CO ym+i is uniquely determined as ym+i = R2(ym)- Sinee xm = b it follows that ym = d, hence ym+i = c. That means SKym+i) = {Jn}■ Note that xm+i = b implies xm+i = R^x^, for xm = e, and since Si(e) = {I0,Ii}, it follows by the induction assumption that S2(ym) = {J0, Ji} hence ym = /, and therefo re ym+i = R2(/) = d. o Subcase 1.2, xm+i = Li(xm), xm = 0 In this case xm+i £I^d k < n, ^ o o Therefo re xm £Ik+^. It follows t hat ym £Jk+i; sine e ym = ^d S2(ym) = o {Jk+i} by the induction assumption. It follows that ym+i = L2(ym) £ Jk and therefo re S2(ym+i) = {Jk}, Uniqueness of ym+i is clear, since it is not true that xm+i = Ri(xm)' Subcase 1.3, xm+i = Li(xm) xm = 0, Note that this is equivalent to xm+i = 0, By the induction assumption it follows that ym = 0, and therefore CD ym+i = £2(0) = 0 R2(0) is not defined and the uniqueness is therefore ^proved. Also, Si (xm+i) = {Io} and S2(ym+i) = {Jo}- 3 < CO CD $H Case 2, S1(xm+1) = {1k-1, }, for some k = 1,..., n, This is equivalent to xm+i = e (f)A'~1. Subcase 2.1. k < n. In this case xm+\ = Li(xm), where xm = e , and it is not true that xm+1 = R1(xm) Therefore S1(xm) = {Ik,ifc+1} and by the induction assumption S"2(ym) = {Jfc, Jfc+i}. It follows that ym = / Finally ym+i is uniquely determined as ym+\ = -^(ym) = f (~)k \ and ^ vcz Subcase 2.2. k = n. Now xm+1 = a, and therefore xm+1 = R1 (xm) = L^x^), for xm = 6. By the induction assumption ym = d, and it follows that ym+1 = R2(d) = L2(d) = c, proving the uniqueness part of the claim. Obviously S^ym+O = { Jn-1, Jn}■ The whole inductive proof is completed bv the following two observations. First, from ym+1 = 0 it follows that ym = 0 (since ym = f(c, d)(0) = 0), and by the induction assumption it follows that xm = 0, and therefore xm+1 = L1(0) = 0 (sinee R1(0) is not defined). Similarlv, from ym+1 = d it follows that ym = / (since ym = f(c , d)(d) = /), and bv the induction assumption it follows that xm = e (since from S2(ym) = {J0, J1} it follows that S1(xm) = {/0, I^), and therefore xm+1 = R1(e) = b (since xm+1 = L1(e) would implv ym+1 = L2 (/) = d). This proves that $ : K(ab) — K(c,d) is a well-defined function. But replacing a b e, fi, xk by c, d, /, fi-1, yk respectively, one obtains the proof of the well-definedness of the function ^ : K(c,d) — K(„,b), which is defined £ by (y1,y2,y3,...)1—> (x1,x2,x3,...), CO where and xfc+1 x1 = fi (yl), L1 (xfc) ; if yfc+1 = L2 (yk) R1(xfc) ; if yfc+1 = R>(yfc) k Obviouslv ^ o $ = 1 and $ o ^ = 1. Therefore both $ and ^ are bijections. It remains to be proved that $ and ^ are continuous functions. Let x e K(„,b) be an arbitrary point and let {xi}°=1 be any sequence in K(ab) converging to x. We shall prove that $(x*) converges to $(x), Coordinatewiselv it means that if for each positive integer j lim xJ = xj i^rx J for xi = (x1, x2, x3,.. .^d x = (x1, x2, x3,...), then V i ^ yi = yj , to each j, where o (y1 ,y2 ,y3,...) = $(x!,x2 ,x3,...) CO and 2 (y1,y2,y3,...) = $(x1,x2,x3,...). For each positive integer i we fa a sequence (N], N2, N3,...) of symbols L1; R1; such that for each positive integer k, it holds that xk+1 = Nk (xk). Then we introduce the sequences (O1, O2, O3,...) of symbols L2, R2, as follows: = L2 ^^ Nk = = R2 ^^ Nk = R1, to ea ch k and i, Bv the definiti on of $ it follows that 1 (N for ea ch k. First we show that lim y1 = y^ It follows from the delinition of $ that i—>- i0, x j+1 a ^ and hence Nj = L^, Therefore for all i > i0, Oj = L2, Obviously xj+1 = L1(x,), and hence y,+1 = L2(y,) as well. It follows from the definition ^ of $ and from the continuity of L2 that CD (3.3) lim yj+1 = lim Oj(yj) = lim L2(yj) = L2(lim yj) = L2(y,) = yj+1. i—n i—n i—n i—n CD If xj+1 > a then there is a positive integer i0 such that for all i > i0, xj+1 > a, and hence Nj = R^, Therefore for all i > i0, Oj = R2, Obviously xj+1 = R1(x, ), and hen ce yj+1 = R2(yj) as well. It follows from the definition ft of $ and from the continuity of R2 that CD (3.4) lim yj+1 = lim Oj (yj) = lim R2(yj) = R2(l—n yj) = R2(yj) = yj+1. i—n i—n i—n i—n If Xj+1 — a. then Nj — L1 may hold true for infinitely or finitelv many i, and also Nj — R1 may hold true for infinitely or fin itelv many i If Nj — L1 is true only for finitely many i, then (3.4) applies; if Nj — R1 is true only for finitely many i, then (3,3) applies, In case when both Nj — L1 and Nj — R1 hold true for infinitely many i, we apply (3.3) and (3.4) respectively for the two subsequences corresponding to the choice of ^ or R1 respectively, r-j Clearly in each of these cases we get lim yj+1 — yj+1, i—y^o j+ Therefore $ is continuous, Obviously the proof of continuity of ^ can be obtained from the proof above by appropriate replacements. □ Corollary 3.5. If (a, b) G Cm and (c, d) G Cn for some positive integers m, n > 2, m — n, then the continua A(„,b) and A(c,d) are not homeomorphic. o Proof. Let t\,t2 G 1) such that (|,ti) G Cm, , t2) G Cn- It follows from Theorem 3.4 that K(a,b) is homeomorphic to A'( i and A'(c,d) is homeomorphic to K^i tny Since A'(i and A'(i ^ are not homeomorphic, by the positive solution of the Ingram conjecture [7], it follows that A(„,b) and A(c,d) are not homeomorphic. □ o We have proved in Theorem 3.4 that A(„,b) and A(c,d) are homeomorphic to any (a,b), (c, d) G C1; but we are able to give more precise information about these continua, as shown in the following theorem. Theorem 3.6. If (a, b) G C\, then K^a,b) is a sin—continuum. £ CO Proof. It is easy to see that (a, b) G C1 if and only if 1 > b > a and b — 1 — a. Let Ao — {(t, 1 — t,t, 1 — t, . . .) | t G [a, b]}, ^ A1 — {(t,L(t),L2(t),L3(t),...) | t G [0,b]}, and for each positive integer n, A2n = {(t, 1 - t, t, 1 - t,..., t, 1 - t, L{ 1 - t), L2(l - t),...) | te [a, 6]}, v V y 2n A2n+1 — {(t, 1 — t,t, 1 — t,...,t, 1 — t,t,L(t),L2(t),...) | t G [a, b]}, N-v-' 2n+1 £ where L has the usual meaning (L : [0,6] —> [0,a], L(t) = f1 for any t). Note that in this case e — a and R(t) — 1 — t for each t G [a, b] making QJ the above formulas in coherence with what was said about elements of A(„,b) p , at the beginning of this section. xi > x2 > x3 > ... be a sequence in [0,1] converging to 0, Also 1 et T2n = (x2n, -1^d T2n+i = (x2n+i, 1), for all nonnegative integers n. Next, let B0 = {0} x [-1,1] be the arc from (0, — 1) to (0,1), and for each nonnegative integer n, CO 2 Btoi+1 = {(x,-+ 1) G [0, 1] x [-1, 1] I X G [x2ra+i, x2ra]} x2n+i — x2n be the arc from T2n to T2n+i, 2 B2n+2 = {O, -0-x2n+l) + l) G [0, 1]X[-1, 1] | x G [x2n+2,x2n+l]} x2n+i — x2n+2 be the arc from T2n+^o T2n+2, no. Then for each n > no fa the unique tn e [a, 6], such that < o (tn, 1 tn, tn, 1 tn . . .). lim zn = zo lim tn = to n^-x n^-x positive integer n > no, fi(zn) = fio(zn) e Bo and hence o lim fi(z„) = lim fio (tn, 1 — tn, tn, 1 — tn, . . .) n—rv^i rn—i. rv^ 2 = lim (0, --{tn -a)-I) n^x 6 — a = (0, j— (to — a) — 1) 6 — a = fio (to, 1 — to, to, 1 — to,...) = fi(zo). (b) Suppose there is a positi ve integer no, such that for each n > no, zn e Ao- If there is mo > no, such that for each n > mo, zn e A2k(n) for some positive integer k(n), then Zn (tn i 1 tn, tn, 1 tn, • • • , tn, 1 tn, L( 1 (1 • • •) 2k(n) for so me tn e [a, 6], Obviously lim tn = t^d lim xk(n) = 0, There- n^x n^x fore Jh lim fi(zn) = lim fi2fc(n)(zn) n^x n^x / x2fc(n) — x2fc(n)-1 / — 2 , n . n = lull (---(£„ - 6) + ^2fc(n)-l, -r(£n - o) + 1) n^x a — 6 a — 6 = fi(zo^ z If there is m0 > n0, such that for each n > m0, zn G A2k(n)+1 for some positive integer k(n), then zn (tn, 1 tn, tn, 1 tn, . . . , tn, 1 tn,tn, L (tn) , L (tn) , . . .) (c) If both zn G A^d zn G A0 hold true for infinitely many n, we apply 2fc(n)+1 4J for so me tn G [a, b], and therefore bv the same reasoning as above lim p(zn) = lim P2fc(n)+1(z„) n—n n—n — hm (---[tn - ci) + X'2k(n) i -j-{tn - a) - l) n—n b — a b — a 2 , = 0,--(to -a -1 b — a = P(z0^ If zn G A2k(n) for infinitely many n and zn G A2k(n)+1 for infinitely many n, then we apply calculations from the previous subcases respectively for the two subsequences corresponding to the choice of A2k(n) Oor A2fc(n)+^, Clearly in each of these cases we get lim p(zn) = p(z0), n—n (a) and (b) respectively for the two subsequences. Clearly in each of these cases we get lim p(zn) = p(z0), n—n We have shown that p is a continuous bijection from the compact space n n An onto the metric space Bn and therefore p is a homeomorphism, HH n=0 n=0 □ g 4 K(0,1) K(0,1) turns out to be a very complicated continuum. In this section we give a detailed description of K(0,1), which helps us to recognize some of its subcontinua as certain familiar continua. The continuum has already been studied in [5, 21], Let x = (x1 ,x2,x3,...) G K(0,1). Suppose there is an integer n such that xn G {0,1}. If xn = 0, then xn+1 G {0,1}. If xn = 1, then xn+1 = 0, In the case where xn = t G (0,1), one can easily see that xn+1 G {0,1 — t}. Let A = {(x1,x2,x3,x4, . . .) G K(0,1) | x1 = 0} C K(0,1) and B — {(X1,X2,X3,X4,...) G A(o,1) | X1 — 1} C A(o,1). If X — (x1, X2, X3,...) G A(0,1), then exactly one of the following is possible, o 1. X G A. 2. x G B. 3. There are an odd positive integer n and a G A such that x G An(a) = {(t,l-t,t,l-t,...,t, a) \ te (0,1)}. O 4, There are an even positive integer n and a G A such that X G An(a) — {(t, 1 — t, t, 1 — t,... , t, 1 — t, a) | t G (0,1)}. n n 5. X G A^ — {(t, 1 — t,t, 1 — t,...) | t G (0,1)}. One can easily see that C1(A^) — {(t, 1 — t, t, 1 — t,...) | t G [0,1]} is the arc from (0,1, 0,1,...) G A to (1, 0,1, 0,...) G B in A(0,1). Here and in the rest of this section by CI we denote the closure operator in the Hilbert cube. For each n, Cl(An(a)) is the arc {(t, 1 — t, t, 1 — t,..., t, a) | t G [0,1]} >- from (0,l,0,l,...,q, a) G A to (1,0,1,0,...,!, a) eB n n if n is odd, and the arc {(t, 1 — t, t, 1 — t,..., t, 1 — t, a) | t G [0,1]} from N-n-' (0,1, 0,1,..., 1, a) G A to (1, 0,1, 0,..., 0, a) G B n n CD if n is even. We will show that CD Z (41) A(o,1) — (U (lJCl(An(a)n ) ^ 1(Ato). \n=1 \aeA J ) For each a G A there are two possibilities: either a — (0, 0,...) or a — (0,1,...). In the first case there is an a0 G A such that a — (0,a0) and hence a G Cl(A1(a0)) ^n the second case a is of the form a — (0,1, 0,...) and therefore there is an ao e A such that a = (0,1,ao) and hence a e O Cl(A2(ao)), That proves A cf Q f|JCl(An(a))]] UCl(Ax). \n=1 VaeA / / For each 6 e B there is ao e A such that 6 = (1,ao), Therefore 6 e Cl(A1(ao)) and hence B C ^J^ QCl(An(a))jj UCl(Ax), and 4,1 follows. Using 4,1 we are able to prove some addition al properties of K(o,1) as ,_! follows, (a) A'(0,i) contains sin continua. For example, a proof similar to the proof of Theorem 3.6 can be obtained in order to prove that for a = (0,0,0,...) e A x U Cl(An(a)) U Cl(Ax) n=1 / is a sin i—continuum. See also |5|, (b) K(o,1) contains harmonic fans. For example, F1 = i\J Cl(A2n-1(a)^ UCl(Ax) Vn=1 CO where a = (0,1, 0,1, 0,1,...) e A, and F = Uci(A2n(a)) UC l(Ax), Vn=1 CO where a = (0,1, 0,1, 0,1,...) e A, are harmonic fans in K(o,1). (c) K(o,1) is one-dimensional. It is easy to see that A is a Cantor set and that for each positive integer n, CD Q Cl(An(a)) aeA is homeomorphic to the product A x [0,1], Since dim(A) = 0, it follows from Theorem 2,2 that ^ / x dim m Cl(An(a)) = 1. VaeA J A countable union of one-dimensional eompaeta is a one-dimensional eompaetum, see Theorem 2,3, therefore dim(K(oi)) = dim | (Q ( Q Cl(A„(a))) ) UC ) j =1. Vn=i VaeA See also |21|, (d) It has been proved in [21] that K(0)i) has trivial shape and is therefore tree-like, A question Unfortunately the techniques that were used in the proof of Theorem 3,4 do not work in general, i.e. using them one cannot prove that K(a>b) is homeomorphic to K(c,d) if (a, b), (c, d) G Ct for arbitrarv t G [1, to). Initially we conjectured such a result, but many unsuccessful attempts to prove it provided us with evidence of a very complicated behavior, and we are not so confident anvmore. Therefore we just pose the following question, o Question 5.1. 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Varagona, Inverse limits with upper semi-continuous bonding func- (D Hons and indecomposability, preprint (2010). £ Iztok Banic (1) Faculty of Natural Sciences and Mathematics, University of Maribor, CD Koroska 160, Maribor 2000, Slovenia p , (2) Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana 1000, Slovenia O E-mail: iztok,banic@uni-mb,si Matevz Crepnjak (1) Faculty of Natural Sciences and Mathematics, University of Maribor, Koroska 160, Maribor 2000, Slovenia CO (2) Faculty of Chemistry and Chemical Engineering, University of Maribor, M Smetanova 17, Maribor 2000, Slovenia ^ E-mail: matevz,crepnjak@uni-mb,si Matej Merhar Facultv of Natural Sciences and Mathematics, University of Maribor, O Koroska 160, Maribor 2000, Slovenia E-mail: matej .merharCiuni-mb,si Uros Milutinovic (1) Faculty of Nai Koroska 160, Maribor 2000, Slovenia (1) Facultv of Natural Sciences and Mathematics, Universitv of Maribor, o (2) Institute of Mathematics, Physics and Mechanics, Jadranska 19, Ljubljana 1000, Slovenia CO E-mail: uros,milutinovic@uni-mb,si CO CD CD CO u a CD U