ITERATED FUNCTION SYSTEMS Communicated by N

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Asian-European Journal of Mathematics Vol. 6, No. 4 (2013) 1350055 (12 pages) c World Scientific Publishing Company
DOI: 10.1142/S1793557113500551

ON (n, m)-ITERATED FUNCTION SYSTEMS

Rinju Balu∗ and Sunil Mathew† Department of Mathematics National Institute of Technology Calicut – 673601, India ∗[email protected] †[email protected]

Communicated by N.-C. Wong Received November 28, 2012 Revised September 18, 2013 Published November 28, 2013 One of the most common way to generate a fractal is by using an iterated function system (IFS). In this paper, we introduce an (n, m)-IFS, which is a collection of n IFSs and discuss the attractor of this system. Also we prove the continuity theorem and collage theorem for (n, m)-IFS. Keywords: Fractals; iterated function system; attractor; hausdorff metric; contraction mapping; contractivity. AMS Subject Classification: 28A78, 28A80

1. Introduction Mandelbrot called fractals as sets with Hausdorff dimension strictly greater than the topological dimension [9]. Such sets with some additional properties will occur as exact, quasi or statistically self-similar. Mandelbrot and others have been extensively used this to model various physical phenomena [8]. Mathematically fractals can be defined by transformations. Huchinson [5] proved that, given a finite set of contraction maps S = {S1 , S2 , . . . , Sn } on a complete metric space X, there exit a unique non-empty compact set K ⊂ X such that
n K = i=1 Si (K). This finite set of contraction mappings together with a metric space X forms an iterated function system (IFS) and K is called the attractor of this IFS [1, 3]. This attractor is generally a fractal. There are several generalizations of IFSs in the literature, some of them are recurrent IFS, directed graph IFS, infinite IFS, countable IFS and random IFS [2, 7, 10, 11, 12]. Most of the results in the literature are based on a single IFS. A collection of IFSs called as an (n, m)-IFS providing a unique attractor is defined in this paper. Barnsely [1] proved a theorem to generate the attractor of an IFS, which is 1350055-1

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mentioned in Sec. 2. This result is generalized for an (n, m)-IFS in Sec. 3. Some examples for an (n, m)-IFS and their attractors are also given. Mihail [11] defined recurrent IFS as a family of contractions fk : X × X → X, k = 1, 2, . . . , n, where X is a complete metric space and he proved some of its properties. This paper introduces a collection of IFSs with contraction mappings Tij : Xi → Xi , i = 1, 2, . . . , n, j = 1, 2, . . . and discuss the attractor A of this (n, m)-IFS which is a subset of X1 × X2 × · · · × Xn . The continuity theorem and collage theorem for the attractor of an IFS is stated in preliminaries and these theorems are generalized in Sec. 4. 2. Preliminaries This section include basic definitions and results from fractal theory which are found in [3, 4, 6, 11, 13–15]. Let (X, d) be a complete metric space with metric d. A mapping f : X → X is called a contraction if there is a number c with 0 < c < 1 such that d(f (x), f (y)) ≤ c d(x, y) for all x, y in X. If d(f (x), f (y)) = c d(x, y), then f is a similarity transformation. Clearly any contraction is a continuous mapping. Let f1 , f2 , . . . , fn be contraction mappings. A subset A of X is said to be self-similar if A=

n 

fi (A).

i=1

It is observed that some of the self-similar sets are fractals. For examples Cantor set, Sierpinski gasket, Koch curve etc. are self-similar fractals. 2.1. Hausdorff metric Let (X, d) be a complete metric space. Then H(X) denotes the space whose points are the non-empty compact subset of X. The distance between a point and a set and the distance between two sets are defined as follows [1]. Definition 2.1. Let (X, d) be a metric space, x ∈ X and B ∈ H(X). Define d(x, B) = min{d(x, y) : y ∈ B} d(x, B) is called the distance from the point x to the set B. Definition 2.2. Let A, B ∈ H(X). The distance from the set A to the set B is defined as d(A, B) = max{d(x, B) : x ∈ A}. Note that d(A, B) = d(B, A). This distance also possess the following properties. If A, B and C ∈ H(X) then (i) B ⊂ C ⇒ d(A, C) ≤ d(A, B). (ii) d(A ∪ B, C) = d(A, C) ∨ d(B, C), where x ∨ y = max{x, y}. 1350055-2

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Since d : H(X) → H(X) is not a metric, there defines a Hausdorff metric h on H(X) as h(A, B) = d(A, B) ∨ d(B, A). Some of the results on the Hausdorff metric h are as follows. Result 2.1. For all B, C, D and E, in H(X) h(B ∪ C, D ∪ E) ≤ h(B, D) ∨ h(C, E). This result is generalized in the following lemma. Lemma 2.1. Let (X, d) be a complete metric space. If {Ai }ni=1 and {Bi }ni=1 are two finite sequence from H(X), then   n n   Ai , Bi ≤ max h(Ai , Bi ). h i=1

i=1

1≤i≤n

Barnsley referred (H(X), h) as the “space of fractals”. He established that the space of fractals (H(X), h) is a complete metric space and characterized the convergent sequence in H(X) in the theorem “The completeness of the space of fractals” which is stated below. Theorem 2.1. Let (X, d) be a complete metric space. Then (H(X), h) is a complete metric space. Moreover, if {An ∈ H(X)}∞ n=1 is a cauchy sequence, then A = limn→∞ An = {x ∈ X: there is a cauchy sequence {xn ∈ An } that converges to x} ∈ H(X). 2.2. Iterated function system Definition 2.3. An IFS consist of a complete metric space (X, d) together with a finite set of contraction mappings fi : X → X, with respective contractivity factor ti for i = 1, 2, . . . , n. The notation for the IFS is {X; fi , i = 1, 2, . . . , n} and its contractivity factor is t = max1≤i≤n {ti }. Barnsley has given a theorem to obtain a set A using an IFS and the set A is called the attractor of the IFS. It is stated as follows: Theorem 2.2. Let {X; fi , i = 1, 2, . . . , n} be an IFS with contractivity factor t.
Then the transformation W : H(X) → H(X) defined by W (B) = ni=1 fi (B) for all B ∈ H(X), is a contraction mapping on the complete metric space (H(X), h) with contractivity factor t. Its unique fixed point, A ∈ H(X), exist and is given by A = limn→∞ W on (B) for any B ∈ H(X). Example 2.1. Let (R, d) be the complete metric space. Consider the IFS {[0, 1]; f1 (x) = x3 , f2 (x) = x3 + 23 }. The iteration is shown in Fig. 1. The attractor of the IFS is the classical Cantor set. 1350055-3

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Fig. 1.

Fig. 2.

Cantor set.

Iteration of sierpinski triangle.

Example 2.2. Let (R2 , d) be the complete metric space. Consider the IFS {R2 : ( x2 , y2 ), ( x2 + 12 , y2 ), ( x2 + 14 , y2 + 12 )}. The attractor of the IFS is the sierpinski triangle (Fig. 2). The attractor of an IFS is a continuous function of the parameters of the IFS. That is, if we vary the collection of IFS in a continuous manner, then the attractor will change continuously. In particular, the induced mapping on H(X) is composed of several individual mapping and this makes the situation more complicated. The below definition gives a representation for the collection of contraction mappings with the contractivity factor atmost s. Definition 2.4. Let 0 ≤ s < 1. Define Cons (X) = {f : X → X : d(f (x), f (y)) ≤ s d(x, y)}

for all x, y ∈ X.

The metric on Cons (X) is   d(f, g) = max 1, sup d(f (x), g(x)) . x∈X

Thus Cons (X) is the collection of contraction mappings with contractivity factor s. It is easy to show that (Cons (X), d) is a complete metric space. The following theorem states the continuity of the fixed point of f ∈ Cons (X). Theorem 2.3. The fixed point of f ∈ Cons (X) is a continuous function of f . Since an IFS is a finite collection of contraction mappings, not just a single contraction, we can think of it as an IFS in a finite product of Cons (X). Consider 1350055-4

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Conns (X) = Cons (X) × Cons (X) × · · · × Cons (X) where n ∈ N . The metric on Conns (X) is given by d(F, G) = max d(fi , gi ), 1≤i≤n

where F = (f1 , f2 , . . . , fn ) and G = (g1 , g2 , . . . , gn ). Also (Conns (X), d) is a complete metric space. The following theorem is the continuity of the attractor of an IFS. Theorem 2.4. The function A : Conns (X) → H(X) that maps an IFS to its attractor is continuous. In other words, if S = (X; fk , k = 1, 2, . . . , n) and S  = (X; gk , k = 1, 2, . . . , n) are the IFSs with contractivity factors s and s , respectively, then h(A(S), A(S  )) ≤ max

1≤k≤n

d(fk , gk ) , 1 − min(s, s )

where A(S) and A(S  ) are the attractors. One of the important theorem in fractal theory is the collage theorem. The collage theorem gives an approximation for the question how to design an IFS whose attractor resembles the given set. The idea of the theorem is to “collage” the image with reduced copies of the entire image. It is stated below. Theorem 2.5. Let L ∈ H(X) and
> 0 be given. Choose an IFS {X; wi , i = 1, 2, . . . , n} with contractivity factor 0 ≤ s < 1, so that   n  h L, wn (L) ≤
. i=1

Then h(L, A) ≤


, (1 − s)

where A is the attractor of the IFS. 3. An (n, m)-IFS and Its Attractor In this section, we define an (n, m)-IFS using n complete metric spaces and m contraction mappings. Definition 3.1. Let X1 , X2 , . . . , Xn be a finite sequence of complete metric spaces. Let Tij be a contraction mapping on Xi with contractivity factor tij for i = 1, 2, . . . , n and j = 1, 2, . . . , p. Consider the IFS {Xi : Tij , j = 1, 2, . . . , p} for each i = 1, 2, . . . , n. This collection of n IFSs is said to be an (n, m)-IFS and its contractivity factor is given by t = max1≤i≤n max1≤j≤p {tij }. 1350055-5

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In an (n, m)-IFS, the first coordinate n stands for number of IFSs and the second coordinate m stands for the number of contraction mapping involved in the whole iteration system. Note that p may vary for each IFS. An (n, m)-IFS is denoted by (X; T ), where X = {X1 , X2 , . . . , Xn } and T is the collection of contraction mappings {Tij }. Suppose that number of function in each IFS is same. Then an (n, m)-IFS is said to be an n2 -IFS. Remark 3.1. A (1, n)-IFS coincide with the usual IFS on X. Consider H(Xi ), the space consisting of non-empty compact subset of Xi , provided with hausdorff metric hi . To describe an attractor for (n, m)-IFS, we define n a Hausdorff metric on i=1 H(Xi ) as follows. Definition 3.2. Let (X; T ) be an (n, m)-IFS with contractivity factor t. Let  h be  = n H(Xi ) defined as the metric on H i=1  h((A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn )) = max hi (Ai , Bi ) 1≤i≤n

 for all (A1 , A2 , . . . , An ), (B1 , B2 , . . . , Bn ) ∈ H.  Clearly  h is a hausdorff metric on H.   Define W : H → H by      (A1 , A2 , . . . , An ) =  T1j (A1 ), T2j (A2 ), . . . , Tnj (An ) W j

j

(3.1)

(3.2)

j

 for all (A1 , A2 , . . . , An ) ∈ H. Note that j may vary in each co-ordinate of the above n-tuple in right-hand side of (3.2), so there is no certainty in specifying a particular limit for j. The above is a contraction mapping, which is proved in the following lemma. defined W is a contraction mapping on H  with contractivity factor t. In Lemma 3.1. W :H  →H  obeys the condition other words W  (A), W (B)) ≤ t  h(W h(A, B), where t = max1≤i≤n maxj {tij }. Proof. Let (A1 , A2 , . . . , An ) and (B1 , B2 , . . . , Bn ) Consider       (A), W (B)) =  h(W h  T1j (A1 ), T2j (A2 ), . . . , Tnj (An ), j

j

j

      T1j (B1 ), T2j (B2 ), . . . , Tnj (Bn ) j

j

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j

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       = max hk  Tkj (Ak ), Tkj (Bk )  1≤k≤n  j

j

≤ max max{hk (Tkj (Ak ), Tkj (Bk ))} 1≤k≤n

j

≤ max max {tkj hk ((Ak ), (Bk ))} 1≤k≤n

j

since Tkj is a contraction mapping with contractivity factor tkj ,  (A), W (B)) ≤ t max {hk ((Ak ), (Bk ))} h(W 1≤k≤n

= h(A, B). (A), W (B)) ≤ t   Thus  h(W h(A, B) for all A, B ∈ H. This completes the proof. This contraction mapping lighten a way to the existence of the fixed point for the (n, m)-IFS, which is shown below. Theorem 3.1. Let (X; T ) be an (n, m)-IFS with contractivity factor t. Then there is a unique non-empty compact fixed point for the (n, m)-IFS. Proof. Consider the metric space H(Xi ), the space consisting of non-empty compact subset of Xi , provided with hausdorff metric hi for all i = 1, 2, . . . , n. Then  is a space whose points are the non-empty compact sets. By the comclearly H pleteness of the space of fractals, we have H((Xi ), hi ) is a complete metric space.  is complete metric Since finite product of complete metric space is complete, H space. :H  →H  by Consider the function W      (A1 , A2 , . . . , An ) =  T1j (A1 ), T2j (A2 ), . . . , Tnj (An ) W j

j

j

 for all (A1 , A2 , . . . , An ) ∈ H. We have continuous image of a compact set is compact. Therefore for each i,
Tij (Ai ) is compact. Since finite union of compact sets is compact, j Tij (Ai ) is compact for each i = 1, 2, . . . , n. Thus by Tychonoff theorem (arbitrary product is well (A1 , A2 , . . . , An ) is compact. That means W of compact set is compact), W defined. is a contraction mapping. By contraction mapping theorem, By Lemma 3.1, W a contraction mapping on a complete non-empty metric space has a unique fixed has a unique fixed point. point. Thus W 1350055-7

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With the help of Lemma 3.1 and Theorem 3.1, we get an idea about the attractor of (n, m)-IFS. The following theorem describes the attractor of an (n, m)-IFS. Theorem 3.2. Let (X; T ) be an (n, m)-IFS with contractivity factor t. Then the :H  →H  defined above is a contraction mapping on the mettransformation W    obeys A = W (A) and is given by ric space (H, h). Its unique fixed point A ∈ H

on (B) for any B ∈ H.  A=W

 is the attractor of the (n, m)Note that the above mentioned fixed point A ∈ H IFS. Suppose that A = (A1 , A2 , . . . , An ) and p = n. This means that the number of contraction mappings in each IFS of this (n, m)-IFS are equal. If these unique vectors of compact sets (A1 , A2 , . . . , An ) mentioned in Theorem 3.2 satisfies the condition n  Tij (Aj ) i = 1, 2, . . . , n Ai = j=1

and A=

n 

Ai .

i=1

Then A resembles a recurrent self-similar fractal. Also if A1 = A2 = · · · = An and
n Ai is an exact self-similar fractal, then A = i=1 Ai resembles an exact self-similar fractal. Example 3.1. Let X1 = X2 = R. Consider the IFS {R; x3 , x3 + 23 }. Clearly the attractor of the given IFS is a Cantor Set. Suppose that the first and second IFS are same. So here we have a (2, 4)-IFS. So the attractor of the (2, 4)-IFS is (A1 , A1 ) (Fig. 3), where A1 is the usual Cantor set.

Fig. 3.

Cantor set × Cantor set. 1350055-8

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Example 3.2. Let X1 = R, X2 = R2 . Consider the (2, 5)-IFS   x x 2 R; , + 3 3 3       x y x 1 y x 1 y 1 , + , + , + , , . R2 : 2 2 2 2 2 2 4 2 2 The attractor (A1 ) of first IFS is a Cantor set and the attractor (A2 ) of second IFS is a sierpinski triangle. Therefore the attractor of (2, 5)-IFS is (A1 , A2 ). Example 3.3. Let X1 = R, X2 = R, X3 = R2 . Let the three IFSs be   x x 1 R; , + 2 2 2   x x 2 R; , + 3 3 3       x y x 1 y x 1 y 1 2 , + , + , + , , . R : 2 2 2 2 2 2 4 2 2 This is a (3, 7)-IFS. The attractor of this (3, 7)-IFS is (A1 , A2 , A3 ) where A1 is [0, 1], A2 is the classical Cantor set and A3 is the sierpinski triangle. 4. The Continuity Theorem and the Collage Theorem for (n, m)-IFS There exists many important mathematical results concerning the attractors of an IFS. Among them we are interested in the continuity of the attractor of an IFS and the collage theorem for an IFS, which are discussed in the preliminaries. Now we will generalize these concepts in terms of our (n, m)-IFS. Our (n, m)-IFS is the collection of n IFSs, in which each IFS is acting on different complete spaces. As per Theorem 3.2, (n, m)-IFS has a unique attractor. Here, we are trying to design an (n, m)-IFS whose attractors are close to the given set, using the continuity theorem of the attractor of (n, m)-IFS and the collage theorem for (n, m)-IFS. First we describe the continuity theorem. Theorem 4.1. Let (X; T ) and (X; T  ) be two (n, m)-IFS with contractivity factor t and t respectively. Then  h(F, G) ≤

 max1≤k≤n maxj d(Tkj , Tkj )

1 − min(t, t )

,

where F and G are the attractors of (X; T ) and (X; T  ) respectively. Proof. Let F = (F1 , F2 , . . . , Fn ) and G = (G1 , G2 , . . . , Gn ). 1350055-9

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By definition of  h  h(F, G) = max hk (Fk , Gk ) 1≤k≤n

 maxj d(Tkj , Tkj )  1≤k≤n 1 − min(tk , tk )

≤ max

(applying continuity theorem for hk (fk , Gk ))  h(F, G) ≤

 ) max1≤k≤n maxj d(Tkj , Tkj 1 − max1≤k≤n min(tk , tk )

we have t = max max tij = max ti . 1≤i≤n

j

1≤i≤n

Therefore 1 − max min(tk , tk ) ≥ 1 − min( max tk , max tk ) 1≤k≤n

1≤k≤n

1≤k≤n

= 1 − min(t, t ). Thus  h(F, G) ≤

 max1≤k≤n maxj d(Tkj , Tkj ) . 1 − min(t, t )

Next we discuss the collage theorem for an (n, m)-IFS. This theorem tells us to find an (n, m)-IFS, whose attractors are n sets, which are closed to the given set. The distance between the two set is measured using the Hausdorff metric  h.  and
> 0 be given. Theorem 4.2 (Collage theorem for (n, m)-IFS). Let L ∈ H Choose an (n, m)-IFS, (X; T ) with contractivity factor t, so that  (L)) ≤
, h(L, W  Then where  h is the hausdorff metric on H.  h(L, F ) ≤


, 1−t

where F is the attractor of the (n, m)-IFS. Proof. Let F = (F1 , F2 , . . . , Fn ) and (L = L1 , L2 , . . . , Ln ). By definition of  h  h(F, L) = max hi (Fi , Li ). 1≤i≤n

Applying the collage Theorem 2.5 for ith IFS in (n, m)-IFS we get,
hi (Li , ni=1 Tij (Li )) . hi (Li , Fi ) ≤ 1 − maxj tij 1350055-10

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Thus
n  hi (Li , i=1 Tij (Li )) 1≤i≤n 1 − maxj tij
n max1≤i≤n hi (Li , i=1 Tij (Li )) ≤ 1 − max1≤i≤n maxj tij

 h(F, L) ≤ max



 (L)) h(L, W 1−t
≤ . 1−t ≤

Hence the proof.

5. Conclusion

n In this paper, we have defined a Hausdorff metric on i=1 Xi and generalized the attractor of an IFS to the higher dimension. Some of the major results in fractal geometry which are true for an (n, m)-IFS are established in this paper. More properties of (n, m)-IFS and their differences from other generalized IFSs will be discussed in the forthcoming papers.

References 1. M. F. Barnsley, Fractals Everywhere (Academic Press, Harcourt Brace Janovitch, 1988). 2. M. F. Barnsley, J. Elton and D. Hardin, Recurrent iterated function systems, Constr. Approx. 5 (1989) 3–31. 3. G. Edgar, Measure, Topology and Fractal Geometry (Springer-Verlag, New York, 1990). 4. K. J. Falconer, Fractal Geometry-Foundations and Applications (Wiley, 1990). 5. J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981) 713–747. 6. H. Kunze, D. La Torre, F. Mendivil and E. R. Vrscay, Fractal-Based Methods in Analysis (Springer, 2012). 7. K. Le´sniak, Infinite iterated function systems: A multivalued approach, Bull. Polish Acad. Sci. Math. 52 (2004) 1–8. 8. B. Mandelbrot, Fractals, Form, Chance, and Dimension (Freeman, San Francisco, 1977). 9. B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Co., San Francisco, 1982). 10. R. D. Mauldin and S. C. Williams, Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc. 309 (1988) 811–829. 11. A. Mihail, Recurrent iterated function systems, Rev. Roumaine Math. Pures Appl. 53 (2008) 43–53. 12. N. A. Secelean, Countable iterated fuction systems, Far East J. Dyn. Syst. 3(2) (2001) 149–167. 1350055-11

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13. N. A. Secelean, Approximation of the attractor of a countable iterated function system, Gen. Math. 17 (2009) 221–231. 14. N. A. Secelean, Generalized countable iterated fuction systems, Filomat 25(1) (2011) 21–36. 15. R. S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc. 357 (2004) 571–615.

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