summary:In this paper we will demonstrate that, in some conditions, the attractor of a countable iterated function system is a parameterized curve. This fact results by generalizing a construction of J. E. Hutchinson [Hut81]

Secelean, Nicolae-Adrian

English

Institute of Mathematics AS CR, v. v. i.

Archivum MathematicumNicolae-Adrian SeceleanParameterized curve as attractors of some countable iterated function systemsArchivum Mathematicum, Vol. 40 (2004), No. 3, 287--293Persistent URL: http://dml.cz/dmlcz/107911Terms of use:© Masaryk University, 2004Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czARCHIVUM MATHEMATICUM (BRNO)Tomus 40 (2004), 287 – 293PARAMETERIZED CURVE AS ATTRACTORS OF SOMECOUNTABLE ITERATED FUNCTION SYSTEMSNICOLAE-ADRIAN SECELEANAbstract. In this paper we will demonstrate that, in some conditions, theattractor of a countable iterated function system is a parameterized curve.This fact results by generalizing a construction of J. E. Hutchinson [3].1. Preliminary factsWe will present some notions and results used in the sequel (more complete andrigorous treatments may be found in [2], [1]).1.1. Hausdorff metric. Let (X, d) be a complete metric space and K(X) be theclass of all compact non-empty subsets of X .If we define a function δ : K(X) ×K(X) −→ R+,δ(A, B) = max{d(A, B), d(B, A)} ,whered(A, B) = supx∈A( infy∈Bd(x, y)) , for all A, B ∈ K(X) ,we obtain a metric, namely the Hausdorff metric.The set K(X) is a complete metric space with respect to this metric δ.1.2. Parameterized curve in the case of iterated function systems. In thissection, we will present the iterated functions system (abbreviated IFS) and theHutchinson’s construction of a continuous function f defined to [0, 1] such thatIm(f) (the image of f) is the attractor of some IFS (for details see [3]).Let (X, d) be a complete metric space. A set of contractions (ωn)Nn=1 , N ≥ 1, iscalled an iterated function system (IFS), according to M. Barnsley. Such a systemof maps induces a set function S : K(X) −→ K(X),S(E) =N⋃n=1ωn(E)2000 Mathematics Subject Classification: Primary 28A80, Secondary 37C70, 47H10.Key words and phrases: Hausdorff metric, countable iterated function system, attractor.Received September 10, 2002.288 N.-A. SECELEANwhich is a contraction on K(X) with contraction ratio r ≤ max1≤n≤Nrn, rn beingthe contraction ratio of ωn, n = 1, . . . , N . According to the Banach contractionprinciple, there is a unique set A ∈ K(X) which is invariant with respect to S,that isA = S(A) =N⋃n=1ωn(A) .The set A ∈ K(X) is called the attractor of IFS (ωn)Nn=1 .Suppose that (ωn)Nn=1 has the property thata = e1 is the fixed point of ω1 ,b = eN is the fixed point of ωN ,ωi(b) = ωi+1 if 1 ≤ i ≤ N − 1 .Fix 0 = t1 < t2 < · · · < tN+1 = 1. Define gi : [ti, ti+1] → [0, 1] for 1 ≤ i ≤ N bygi(x) =x − titi+1 − ti.LetF = {f : [0, 1] −→ X : f is continuous, and obeys f(0) = a, f(1) = b} .Define S(f) for f ∈ F byS(f)(x) = ωi ◦ f ◦ gi(x) for x ∈ [ti, ti+1], 1 ≤ i ≤ N .Theorem 1. Under the above hypotheses, there is a unique f ∈ F such thatS(f) = f . Furthermore Im(f) = A.1.3. Countable iterated function systems. In this section, we will present thecompact set invariant with respect to a sequence of contraction maps (for detailssee [4]).Suppose that (X, d) is a compact metric space.A sequence of contractions (ωn)n≥1 on X whose contraction ratios are, respec-tively, rn, rn > 0, such that supnrn < 1 is called a countable iterated functionsystem, for simplicity CIFS.Let (ωn)n≥1 be a CIFS.We define the set function S : K(X) −→ K(X) byS(E) =⋃n≥1ωn(E) ,where the bar means the closure of the corresponding set. Then, S is a contractionmap on (K(X), δ) with contraction ratio r ≤ supnrn. According to the Banachcontraction principle, it follows that there exists a unique non-empty compact setA ⊂ X which is invariant for the family (ωn)n≥1 , that isA = S(A) =⋃n≥1ωn(A) .The set A is called the attractor of CIFS (ωn)n≥1 .PARAMETERIZED CURVE AS ATTRACTOR 2892. Parameterized curveLet (X, d) be a compact and connected metric space and (ωn)n≥1 be a sequenceof contraction maps on X whose contraction ratios are, respectively, rn, rn > 0,such that supnrn < 1 having the following properties:a) rnn−→ 0;b) (en)n is a convergent sequence, we denote by b = limnen (en is the uniquefixed point of the contraction map ωn, n ∈ N∗);c) if we denote by a = e1, then ωn(b) = ωn+1(a), ∀n ≥ 1.We note that there exists a sequence of contractions as above, this fact resultsfrom the example which is presented in the sequel.We shall show that, in the above conditions, there exists a continuous functionh : [0, 1] −→ X with Im(h) = A, A being the attractor of CIFS (ωn)n≥1 , where wedenote by Imh = h([0, 1]) the image of h.We consider a sequence of real numbers (tn)n such that0 = t1 < t2 < · · · < tn < tn+1 < · · · < 1 and limntn = 1 .We define, for each n ≥ 1, gn : [tn, tn+1] −→ [0, 1],gn(x) =x − tntn+1 − tn.We denote byF = {f : [0, 1] −→ X : f is continuous, and obeys f(0) = a, f(1) = b}and by P the uniform metric on F , P(f1, f2) = supx∈[0,1]d(f1(x), f2(x)).It is a standard fact that (F ,P) is a complete metric space.For every f ∈ F , we define S(f) byS(f)(x) ={ωn ◦ f ◦ gn(x) , x ∈ [tn, tn+1] , n = 1, 2, . . .b , x = 1 .Proposition 1. The application S : F −→ F is well-defined and S is a contrac-tion map with respect to the metric P.Proof. First we observe that gn(tn) = 0, gn(tn+1) = 1 for all n.Next, if x = tn+1, thenωn ◦ f ◦ gn(x) = ωn(f(1)) = ωn(b)c)= ωn+1(a)= ωn+1 ◦ f ◦ gn+1(tn+1) = ωn+1 ◦ f ◦ gn+1(x) ,thus S(f) is uniquely defined.I We shall show that S(f) ∈ FWe consider x0 ∈ [0, 1] and we will demonstrate that S(f) is continuous in x0.If x0 ∈ (tn, tn+1), the assertion is obvious since S(f) is a composition of threecontinuous functions.290 N.-A. SECELEANIf x0 = tn, n ≥ 2, thenlimxրx0S(f)(x) = limxրx0ωn−1 ◦ f ◦ gn−1(x) = ωn−1(f(1)) = ωn−1(b)c)= ωn(a) ,limxցx0S(f)(x) = S(f)(x0) = ωn ◦ f ◦ gn(tn) = ωn(f(0)) = ωn(a) .We suppose that x0 = 1.Let (xk)k ⊂ [0, 1], xk ր 1. For each k ∈ N∗, there exists nk ∈ N∗ such thatxk ∈ [tnk , tnk+1].Let ε > 0. Then there exists kε ∈ N, such that, for all k ≥ kε, we havernk · diam(X) <ε2(by a));(1)andd(enk , b) <ε2,(2)where diam(X) = supx,y∈Xd(x, y) is the diameter of the set X .Thus, for all k ≥ kε, one hasd(S(f)(xk), b) = d(ωnk ◦ f ◦ gnk(xk), b)≤ d(ωnk ◦ f ◦ gnk(xk), ωnk(enk)) + d(ωnk(enk), b)≤ rnkd(f(gnk(xk)), enk) + d(enk , b)≤ rnkdiam(X) + d(enk , b) <ε2+ε2= ε .It follows that S(f)(xk) −→ b = S(f)(1), hence S(f) is continuous.It is clearly that S(f)(0) = a, S(f)(1) = b. Thus S(f) ∈ F .II S is a contraction map with respect to P:Choose f1, f2 ∈ F and x ∈ [0, 1].If x = 1, then, it is evident that d(S(f1)(x),S(f2)(x)) = d(b, b) = 0.Assume that x ∈ [tn, tn+1], n ∈ N∗. Thend(S(f1)(x),S(f2)(x)) = d(ωn ◦ f1 ◦ gn(x), ωn ◦ f2 ◦ gn(x))≤ rnd(f1(gn(x)), f2(gn(x))) ≤ rnP(f1, f2) .Thus P(S(f1),S(f2)) ≤ rP(f1, f2), where r = supnrn < 1(by a)).Theorem 2. In the above context, there is a unique function h ∈ F such thatS(h) = h. Further Im(h) = A.Proof. The existence and uniqueness result by the contraction principle.The second assertion follows by equalityImS(f) =⋃n≥1ωn(Imf) , ∀ f ∈ F .(3)We will demonstrate that equality by using the double inclusion.PARAMETERIZED CURVE AS ATTRACTOR 291“⊂”Choose y ∈ ImS(f). Then there exists x ∈ [0, 1] such that S(f)(x) = y.A. If x = 1, then S(f)(x) = S(f)(1) = b = y.For each b ∈ Imf , we have the relations:ωn(b) ∈∞⋃n=1ωn(Imf) , ∀n ∈ N ;d(ωn(b), en) = d(ωn(b), ωn(en)) ≤ rnd(b, en) .Henced(ωn(b), b) ≤ d(ωn(b), en) + d(en, b) ≤ (rn + 1)d(b, en) n−→ 0(by using b)).It follows that ωn(b) −→ b ∈⋃n≥1ωn(Imf). Thusy ∈⋃n≥1ωn(Imf) .B. If x ∈ [0, 1), then there exists n ≥ 1 such that x ∈ [tn, tn+1], henceωn ◦ f ◦ gn(x) = y. It follows thaty = ωn(f(gn(x))) ∈ ωn(Imf)which implies y ∈⋃n≥1ωn(Imf).“⊃”Choose y ∈ S(Imf) =⋃n≥1ωn(Imf).Then there exists (yk)k ⊂∞⋃n=1ωn(Imf), yk −→ y.For every fixed k ∈ N∗, one has:∃nk ∈ N∗ such that yk ∈ ωnk(Imf) hence zk ∈ Imf with yk = ωnk(zk) .Thus there exists xk ∈ [0, 1] such thatf(xk) = zk .If xk = 1, it follows thatzk = f(xk) = b = S(f)(xk) ∈ ImS(f) .Assume that xk ∈ [0, 1). Then there is x′k ∈ [tnk , tnk+1] such thatgnk(x′k) = xk .We deduce thatωnk ◦ f ◦ gnk(x′k) = ωnk(f(xk)) = ωnk(zk) = yk ∈ ImS(f) .Thus (yk)k ⊂ ImS(f) = S(f)([0, 1]), the set S(f)([0, 1]) being compact.Thus y = limkyk ∈ ImS(f).292 N.-A. SECELEANSince h = S(h) by using (3), it followsImh = ImS(h) =⋃n≥1ωn(Imh) = S(Imh)and we conclude that A = Imh is the attractor of CIFS (ωn)n≥1 .ExampleWe consider CIFS von-Koch-infinite given as follows (see [4]).Let X = [0, 1] × [0, 1] ⊂ R2 and we consider the contraction maps which aredefined as follows: for every n ∈ N∗, there exists an uniquely p ∈ {0, 1, . . .},k ∈ {1, 2, 3, 4} such that n = 4p + k. Thenωn(x, y) :=12p+1(13x + 2p+1 − 2, 13y), if k = 1;12p+1(16x −√36y + 2p+1 − 53,√36x +16y), if k = 2;12p+1(16x +√36y + 2p+1 − 32,−√36x +16y +√36), if k = 3;12p+1(13x + 2p+1 − 43,13y), if k = 4.The attractor of that CIFS is the content of Fig. 1.Now we shall show that CIFS von-Koch-infinite verifies the conditions a), b),c). Thusa) Clearly rnn−→ 0 (since n → ∞ ⇔ p → ∞);b) It is, also, immediate that∀x, y ∈ [0, 1] , ωn(x, y) n−→ (1, 0)thus b = (1, 0);c) ω1(x, y) = (16x,16y), hence e1 = a = (0, 0).We will prove that ωn(b) = ωn+1(a), ∀n ≥ 1.If p ≥ 0 and k ∈ {1, 2, 3}, it can prove, most difficulty, thatω4p+k(1, 0) = ω4p+k+1(0, 0) .Next, if p ≥ 0 and k = 4, we haveω4p+4(1, 0) =12p+1(13+ 2p+1 − 43, 0) =12p+2(2p+2 − 2, 0) = ω4(p+1)+1(0, 0) .PARAMETERIZED CURVE AS ATTRACTOR 293Fig. 1. The attractor von-Koch-infiniteReferences[1] Barnsley, M.F., Fractals everywhere, Academic Press, Harcourt Brace Janovitch, 1988.[2] Falconer, K. J., The Geometry of Fractal Sets, Cambridge University Press, 85, 1985.[3] Hutchinson, J., Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), 713–747.[4] Secelean, N.A., Countable Iterated Fuction Systems, Far East J. Dyn. Syst., Pushpa Pub-lishing House, vol. 3(2) (2001), 149–167.Department of Mathematics“Lucian Blaga” University of SibiuRomaniaE-mail: secelean@mail.ulbsibiu.ro

Parameterized curve as attractors of some countable iterated function systems

http://dml.cz/bitstream/handle/10338.dmlcz/107911/ArchMathRetro_040-2004-3_9.pdf

Parameterized curve as attractors of some countable iterated function systems

Abstract

Similar works

Full text

Available Versions

Institute of Mathematics AS CR, v. v. i.