Approximation of the attractor of a countable iterated function system 1

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1 General Mathematics Vol. 17, No. 3 (2009), Approximation of the attractor of a countable iterated function system 1 Nicolae-Adrian Secelean Abstract In this paper we will describe a construction of a sequence of sets which is converging, with respect to the Hausdorff metric, to the attractor of a countable iterated function system on a compact metric space. The importance of that method consists in the fact that the approximation sequence can be constructed of finite sets, hence it is very useful for computer graphic representation Mathematics Subject Classification: Primary 28A80, Secondary 65D05. Key words and phrases: Hausdorff metric, countable iterated function system, countable fractal interpolation 1 Received 25 July, 2009 Accepted for publication (in revised form) 26 August,

2 222 N.A. Secelean 1 Introduction In the famous paper [4], J.E. Hutchinson proves that, given a set of contractions (IFS) (ω n ) n=1 in a complete metric space X, there exists a unique nonempty compact set A X, named the attractor of IFS. This attractor is, generally, a fractal set. These ideas has been extended to infinitely many contractions, a such generalization can be founded in [5] for countable iterated function systems (CIFS) on a compact metric space. The approximation of the attractor of a IFS has been studied by S. Dubuc, A. Elqortobi, P.M. Centore, E.R. Vrscay, E. de Ámo, I. Chiţescu, C. Díaz, N.A. Secelean (see [1]) and many others. If we consider a CIFS (ω n ) n 1 whose attractor is A, then A can by approximated (see [5]) by the attractors A of the partial IFS (ω n ) n=1, = 1, 2,. However, these attractors are, generally, infinite sets so it cannot be represented by using the computer. Here, we will construct a sequence of finite sets (which can be subsets of A, hence we use not one point outward of A) converging with respect to the Hausdorff metric to A. As a particular case, we will approximate, by finite subsets, the graph of the countable interpolation function associated of a countable system of data and a corresponding CIFS. Finally, an example in R 2 which use this method is given.

3 Approximation of the attractor Preliminary Facts 2.1 Iterated Function Systems, Countable Iterated Function Systems In this subsection we give some well nown aspects on Fractal Theory used in the sequel (more complete and rigorous treatments may be found in [4], [3], [5], [7]). Let (X, d) be a complete metric space and K(X) be the class of all compact non-empty subsets of X. The function δ : K(X) K(X) R +, δ(a, B) = max{d(a, B), d(b, A)}, ( where d(a, B) = sup inf d(x, y)), for all A, B K(X), is a metric, namely x A y B the Hausdorff metric. The set K(X) is a complete metric space with respect to this metric δ. Theorem 1. [5, Th. 1.1] Let (A n ) n be an increasing sequence of compact sets in a complete metric space such that the set A = A n is relatively compact. Then A = lim A n, the limit being taen with respect to the Hausdorff metric, n the bar means the closure. A set of contractions (ω n ) n=1, 1, is called an iterated function system (IFS). Such a system of maps induces a set function S : K(X) K(X), S (E) = ω n (E) which is a contraction on K(X) with contraction ratio n=1 r max r n, r n being the contraction ratio of ω n, n = 1,...,. According 1 n to the Banach contraction principle, there is a unique set A K(X) which is invariant with respect to S, that is A = S (A ) = ω n (A ). We say n=1 n=1

4 224 N.A. Secelean that the set A K(X) is the attractor of IFS (ω n ) n=1. Now, we suppose further that (X, d) is a compact metric space. The (K(X), δ) is also a compact metric space. A sequence of contractions (ω n ) n 1 on X whose contraction ratios are, respectively, r n > 0, such that sup r n < 1 is called a countable iterated function system n (CIFS). If we consider the CIFS (ω n ) n 1, then the set function S : K(X) K(X), given by (1) S(E) = n 1 ω n (E) (the bar means the closure of the respective set) is a contraction map on (K(X), h) with contraction ratio r sup r n. Thus, there exists a unique n non-empty compact set A X invariant for the family (ω n ) n 1, that is A = S(A) = n 1 ω n (A). Further, if B K(X), then, by a successive approximation process, (2) S p (B) p A (with respect to the Hausdorff metric) where S p := S} {{ S}. The set A p times invariant under the set function S is called the attractor of CIFS (ω n ) n 1 and it can be obtained as limit, with respect to the Hausdorff metric, of sequence of attractors (A ) 1 of partial IFS (ω n ) n=1, = 1, 2,... (see [5]). 2.2 Countable fractal interpolation Now we will describe an extension of the fractal interpolation to the case of the countable system of data (more details can be found in [7]).

5 Approximation of the attractor Let (Y, d Y ) be a compact metric space. A countable system of data is a set of points having the form := {(x n, F n ) R Y : n = 0, 1,... } where the sequence (x n ) n 0 is strictly increasing and bounded and (F n ) n 0 is convergent. Denote a = x 0, b = lim x n and X = [a, b] Y. n An interpolation function corresponding to this system of data is a continuous map f : [a, b] Y such that f(x n ) = F n for n = 0, 1,.... The points (x n, F n ) R 2, n 0, are called the interpolation points. It can construct a CIFS on X which is associated with. Theorem 2. [7, Th.2] There exists an interpolation function f corresponding to the considered countable system of data such that the graph of f is the attractor A of the associated CIFS. That is A = { (x, f(x)) : x [a, b] }. 3 Approximation of the attractor of a countable iterated function system Lemma 1. Let us consider two families (A i ) i I, (B i ) i I of compact subsets of the metric space (X, d) such that the both sets A i and B i are compact. Then δ( i I A i, B i ) sup δ(a i, B i ). i i I Proof. To establish the inequality from statement, it is enough, because of symmetry, to prove d( i I A i, B i ) sup d(a i, B i ). i i I i I i I

6 226 N.A. Secelean Let be x A i. There is i x I such that x A ix. Therefore and hence i I inf d(x, y) inf d(x, y) d(a ix, B ix ) sup d(a i, B i ) A i y S i y B i x i d( A i, ( B i ) = sup inf i I i I x S A i y S d(x, y) ) sup d(a i, B i ). B i i i i The following lemma describes a standard topological fact: Lemma 2. If (E i ) i I E i = E i. i I i I is a family of subsets of a topological space, then Let (B ) be a sequence of compact nonempty sets on the compact metric space (X, d) converging (with respect to the Hausdorff metric δ) to the compact set B X, B. We also consider a CIFS (ω n ) n on X and denote by A its attractor. Theorem 3. Under the above context, A can be approximated by the sequence of compact nonempty sets (S p (B )) p,. More precisely, we have lim p lim S p (B ) = A, the limiting process being taen with respect to the Hausdorff metric and S p := S S }{{}. p times Proof. For each p = 1, 2,..., we denote ω i1 i 2...i p := ω i1 ω ip, where i 1,..., i p are positive integers. Obviously, ω i1 i 2...i p is a contraction with contraction ratio r i1... r ip, r n assigning the contraction ratio of ω n.

7 Approximation of the attractor Let be p 1 and, for each 1, N := {1, 2,..., }. First, we prove that (3) δ ( S p (B ), S p (B) ) 0. (4) One has δ ( S p (B ), S p (B) ) δ ( S p (B ), S p (B)) + δ ( S p (B), Sp (B) ), = 1, 2,. It is simple to verify that S p (E) = (E) for any arbitrary i 1,...,i p N set E X. Now, in view of Lemma 1, (5) δ ( S p (B ), S p (B)) = δ ( Since, for all = 1, 2,..., i 1,...,i p N (B ), sup δ ( (B ), (B) ). i 1,...,i p N d ( (B ), (B) ) = sup x ω i1...ip (B ) i 1,...,i p N (B) ) inf d(x, y) = y ω i1...ip (B) = sup a B inf b B d( (a), (b) ) r n sup a B inf b B d(a, b) = r i 1 r i2... r ip d(b, B) and, analogously, d ( (B), (B ) ) r i1 r i2... r ip d(b, B), one obtain δ ( (B), (B ) ) r i1 r i2... r ip δ(b, B) δ(b, B), i 1,..., i p N,

8 228 N.A. Secelean so, from (5), δ ( S p (B ), S p (B)) δ(b, B),, p 1. Thus, taing into account the hypothesis, (6) δ ( S p (B ), S p (B)) 0, p = 1, 2,. Afterwards, we observe that (7) S p (B) = i 1,...,i p 1 (B). Indeed, we can proceed by induction. Thus, if we suppose that (7) is true for p 1. In view of Lemma 2 and by using the continuity of the functions ω n, we have ( S p+1 (B) = S ( ω i i=1 i 1,...,i p 1 i 1,...,i p 1 ) (B) = ) (B) = ω i ( i=1 ( ω i i=1 i 1,...,i p 1 ( ) Since the sequence of sets (B) i 1,...,i p N we can apply Theorem 1 and obtain, by using (7), i 1,...,i p 1 ) (B) (B)) = S p+1 (B). 1 is clearly increasing, (8) lim S p = lim (B) = i 1,...,i p N ( ) (B) = i 1,...,i p N =1 = (E) = S p (B). i 1,...,i p 1 Now, from (6) and (8) becomes (3), so lim S p (B ) = S p (B).

9 Approximation of the attractor Finally, from (7) and (2), it follows lim p lim S p (B ) = lim p S p (B) = A, completing the proof. Remar 1. By taing in the preceding theorem B := {e 1, e 2,..., e }, = 1, 2,..., e n being the fixed point of ω n for every n 1, one obtain an increasing sequence of subsets of A converging, with respect to the Hausdorff metric, to B := B = {e 1, e 2,... }. Thus, the attractor A of CIFS (ω n ) n 1 can be approximated from inside, because S p (B ) A, from all, p 1. Remar 2. If (B ) are finite sets, then ( S p (B ) ) are finite sets too. As p, follows, the attractor A can be approximated by using a sequence of finite sets. This fact is very useful for the computer graphic representation of the CIFS s attractor in R 2. Remar 3. Let us consider a countable system of data = (x n, F n ) n 1 R Y (see section 2.2) and let be B := {(x n, F n ); n = 1, 2,..., }. Then B A for any and lim B =, the convergence being considered with respect to the Hausdorff metric. It follows that the graph of the countable interpolation function is approximated from inside by a sequence of finite sets. Finally, we give an example which shows some progressive steps to approximate an attractor of Sierpinsi-infinite type (see [5]), from inside, by some finite sets.

10 230 N.A. Secelean Example On the space X = {(x, y) R 2 : 0 x 1, 0 y 1 x} ( 1 we consider the sequence of contractions ω ij (x, y) = 2 x + (j 1) 1 i 2, 1 i 2 y + i (2 i j 1) 1 ) for any i = 1, 2,..., j = 1, 2,..., 2i 1. Three steps on the 2 i 2 1 attractor s approximation process are represented as follows.

11 Approximation of the attractor References [1] E. de Ámo, I. Chiţescu, M. Díaz Carrillo, N.A. Secelean, A new approximation procedure for fractals, Journal of Computational and Applied Mathematics, vol. 151, Issue 2, 2003, [2] M.F. Barnsley, Fractal Functions and Interpolations, Constructive Approximation 2, 1986, [3] M.F. Barnsley, Fractals everywhere, Academic Press, Harcourt Brace Janovitch, 1988 [4] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math. 30, 1981, [5] N.A. Secelean, Countable Iterated Fuction Systems, Far East Journal of Dynamical Systems, Pushpa Publishing House, vol. 3(2), 2001, [6] N.A. Secelean, Measure and Fractals, University Lucian Blaga of Sibiu Press, 2002, [7] N.A. Secelean, The fractal interpolation for countable systems of data, Beograd University, Publiacije, Electrotehn., Fa., ser. Matematia, 14, 2003, Department of Mathematics Lucian Blaga University of Sibiu Romania address: nicolae.secelean@ulbsibiu.ro