HENNING FERNAU. by appropriately chosen IFS both in terms of Hausdorff distance and of

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1 INFINITE ITERATED FUNCTION SYSTEMS HENNING FERNAU LEHRSTUHL INFORMATIK F UR INGENIEURE UND NATURWISSENSCHAFTLER UNIVERSIT AT KARLSRUHE (TH) GERMANY June 8, 1994 Abstract: We examine iterated function systems consisting of a countably innite number of contracting mappings (IIFS). We state results analogous to the well-known case of nitely many mappings (IFS). Moreover, we show that IIFS can be approximated by appropriately chosen IFS both in terms of Hausdorff distance and of Hausdorff dimension. Comparing the descriptive power of IFS and IIFS as mechanisms dening closed and bounded sets, we show that IIFS are strictly more powerful than IFS. On the other hand, there are closed and bounded non-empty sets not describable by IIFS. Keywords: Fractal geometry, iterated function systems, complete metric spaces, Baire space, Hausdorff measure, Hausdorff dimension, selfsimilarity. AMS classication: 28A80, 54E50, 54E52, 28A78, 54F Introduction and Main Definitions IFS theory, starting out from Hutchinson's paper [14], gained more and more interest. Several books on this topic are available [3, 7, 5, 18, 19] which have become popular even among non-mathematicians. In those references, mostly the case of systems consisting of nitely many mappings has been treated. In this paper, we elaborate the case of countable many mappings, contrasting it with the well-known results on nite IFS. 1 Especially in [11, 17, 2, 21, 1, 16], the case of innitely many mappings has already been considered. We shortly review the main notions and denotations basic to IFS. Throughout this paper, we only consider complete metric spaces (X; ). A function f : X! X is called a contraction (with contractivity r 2 [0; 1)) i, for all x; y 2 X, (f(x); f(y)) r(x; y). By Banach's xed point theorem, a contraction f has a unique xed point. A similitude is a contraction with the property 8x; y 2 X, (f(x); f(y)) = r f (x; y). r f is called similarity ratio. The set of all similitudes on a non-empty complete space (X; ) will be denoted by S(X; ). Important examples for complete metric spaces are closed subsets of the n- dimensional Euclidean space, e.g. (R n ; E ) with E (x; y) = q P n i=1 (x i? y i ) 2, and the collection K(X; ) of all non-empty compact sets of a metric space (X; ) together with 1 Most of the results in this paper are contained in the PhD thesis [9] of the author which is available as a book [10]. 1

2 the Hausdorff metric H dened by H (A; B) = inffr : A B r (B); B B r (A)g with B r (A) = fx 2 X j (9a 2 A)((a; x) < r)g. H is also a metric on the collection of all non-empty closed and bounded sets CB(X; ). 1.1 Denition Let N be some index set. An iterated function system (on (X; )) is a mapping F :! S(X; ) such that the corresponding sequence of similarity ratios R F : N! [0; 1); n 7! 8 < : r F (n) if n 2 0 if n 62 converges to zero. We call an iterated function system nite (IFS) or innite (IIFS) i is nite or innte, respectively. 2 An iterated function system is called strict i R F (n) 6= 0 for every n 2. Equivalently, we can specify an IFS by a list F = (F (1); : : : ; F (n)) of similitudes. 1.2 Remark We may identify an IFS with an IIFS F with the property (9n 2 N)(9x 2 X)(8i n)(f (i) = fxg): Without loss of generality, we may use N as index set for IIFS. Consider now an IIFS F. Let A 2 CB(X; ). As with IFS, F (n)(a) 2 CB(X; ). But contrary to IFS, in the case of IIFS, S n2n F (n)(a) 2 CB(X; ) does not necesssarily hold. 1.3 Example F (n)(x) = 1? 1 2 n? n x. Obviously, F 2 (S(R; E )) N is an IIFS with S n2n F (n)([0; 1]) = S n2n F (n)([0; 1)) = [0; 1) 62 K(R; E ). We can overcome this diculty if we dene F (A) to be the closure of S n2n F (n)(a). Thus, [0; 1] becomes a xed point of F in the above example. 1.4 Denition A non-empty set A of a metric space (X; ) is called a xed point of the given IIFS F :! S(X; ), i [ (1) F (A) = cl F (n)(a) = A: n2 The intuition behind this denition of IIFS should be clear from IFS: Any fractal set dened in such a way may have an innite number of subsets which are themselves similar to the whole structure. What about the existence of xed points for IIFS? Before addressing this question, we turn to a similar notion following Ellis and Branton [6]. 1.5 Denition Let be an arbitrary index set, and F :! X X, where (X; ) is some metric space. A pseudo-attractor of F is a closed set A 6= ; such that F (n)(a) A for each n 2. A pseudo-attractor of F not properly containing another pseudoattractor is called attractor of F. The notion of a xed point may be carried over to arbitrary function families F using Equation (1). 1.6 Remark Let (X; ) be some metric space. Let F :! X X be a function family. A xed point A of F is a pseudo-attractor of F. 2 The reader may spell these abbreviations also as `(in)nite iterated function schemes' as suggested by Falconer [7].

3 If A is a pseudo-attractor of F, then cl S n2 F (n)(a) is a pseudo-attractor, too. Thus, an attractor A of F is a xed point of F. If (X; ) is compact, then F has an attractor. (Apply Zorn's lemma to the family of all pseudo-attractors.) 1.7 Example (A pseudo-attractor chain without attractor) Consider Baire's space X = N N (which is not compact) and the function family F : N! X X of continuous functions dened by F (n)(f(m)) = 8 < : f(m); if f(m) > n n + 1 otherwise for n; m 2 N and f : N! N. Obviously, cl S n2n F (n)(x) = F (1)(X) = f2; 3; : : : gn, and [ cl F (n)(fk; k + 1; : : : g N ) = F (k)(fk; k + 1; : : : g N ) = fk + 1; k + 2; : : : g N : n2n K = n fk; k + 1; : : : g N j k 2 No is a linearly ordered set of pseudo-attractors. But the intersection of the members of K is empty, preventing us from applying Zorn's lemma. Banach's xed point theorem leads to a characterization of xed points for IIFS. Compare with [21, Theorem 3.1.1]. 1.8 Theorem (Fixed Point Theorem) Let (X; ) be a complete and bounded metric space. Let F be an IIFS. Then F : CB(X; )! CB(X; ) is a contraction with contractivity r = sup n2n R F (n) < 1 on the complete and bounded metric space (CB(X; ); H ). Hence, F possesses a unique xed point, abbreviated as A F. The above theorem could also be stated using contractions which are not necessarily similitudes. Moreover, the requirement that R F is a zero sequence (also implicitly contained in the denition of IIFS) may be weakened as indicated in [2]. We need both properties when dealing with dimensions below. A similar result may be obtained using the space K(X; ) instead of CB(X; ). Moreover, in compact spaces, A is a xed point of the IIFS F i A is an attractor of F. 2. Properties and Applications of the Sequence Space Let R : N! [0; 1) be some zero sequence. Let R = fn 2 N j R(n) 6= 0g. We dene the metric R on the sequence space N R as follows: 3 Y R (v; w) = n inf n2n f R(v(i)) j (81 i n)(v(i) = w(i))g: i=1 The most commonly known (ultra)metric on N R is Baire's metric B given by B (v; w) = 1 inf n2n fn j v(n) 6= w(n)g : 2.1 Theorem Let R; S : N! [0; 1) be two dierent zero sequences with j R j < 1 and R = S. 3 Note that empty products deliver 1. Observe that N R = ; i R is constant.

4 R, S and B are uniformly equivalent on ( N R ; R). The sequence space ( N R ; R) is compact. ( N R ; R), ( N R ; S) and ( N R ; B) are pairwise Lipschitz-inequivalent. Proof. We only prove that R and S are Lipschitz-inequivalent. Assume that R, S are Lipschitz-equivalent. Hence, there are r s > 0 satisfying s f R(x; y) S (x; y) j (x; y 2 N R) ^ (x 6= y)g r: Since R 6= S, there is an i 2 R such that R(i) 6= S(i). Moreover, we nd a j 2 R rfig. Dene x m : N! R ; k 7! 8 < : i if k < m j if k m : In a notation borrowed from formal languages, we may write x m = i m?1 j!. Then, R (x m1 ; x m2 ) = (R(i)) minfm 1;m 2 g?1 : Without loss of generality, assume R(i) > S(i). The set is unbounded, hence f( R(i) S(i) )m j m 2 N 0 g = f R(x m1 ; x m2 ) S (x m1 ; x m2 ) j m 1; m 2 2 N; m 1 6= m 2 g f R(x; y) S (x; y) j (x; y 2 N R ) ^ (x 6= y)g is unbounded, too. A similar construction shows that S and B are Lipschitzinequivalent. The picture changes when considering sequence spaces N R summarize the properties in the following theorem. with j Rj = 1. We 2.2 Theorem Let R; S : N! [0; 1) be two zero sequences with j R j = 1 and R = S. The sequence space ( N R ; R) is complete, separable, and bounded but not totally bounded and hence not compact. R and S are uniformly equivalent on ( N R ; R). ( N R ; R) and ( N R ; B) are homeomorphic, but R and B are not uniformly equivalent. How can we dene iterated function systems on a sequence space? Let R : N! [0; 1) be some zero sequence. Have a look at the continuous shift map : N R! N R, (a)(n) = a(n + 1): It has j R j right inverses. Dene for m 2 R (?1 (a)(n) m if n = 1 m = a(n? 1) if n > 1 : R denes the syntactical IIFS F R : R! S( N R ; R); m 7!?1 m. Obviously, N R is the attractor of F R. We summarize the main connection between sequence spaces and IIFS in the following theorem.

5 2.3 Theorem Let (X; ) be a bounded complete metric space and F : N! S(X; ) be an IIFS on X. Let ( N R F ; RF ) be the corresponding sequence space. For each w 2 N R F, n 2 N, x 2 X, dene (w; n; x) = F (w(1))(f (w(2))(: : : (F (w(n))(x)) : : : )) 2 X; and : N R F! A F by (w) = lim n!1 (w; n; x). (considered as a map ( RF ; RF )! (A F ; )) is well-dened and continuous. We call model map. Fix some x 0 2 A F. The mappings m : N R F! A F given by w 7! (w; m; x 0 ) are continuous and converge uniformly to : N R F! A F. Moreover, is a Lipschitz mapping, and the following diagram commutes (for n 2 N) (2) ( N R F ; RF ) F RF (n)? y ( N R F ; RF )???!???! (X; )? y F (n) (X; ) and is the only continuous function N R F is called an address of (w) 2 A F.! X satisfying the above diagram. w 2 N R F The proof is literally the same as e.g. [21, Section 3.3] or [3, Chapter 4.2]. There is one subtle dierence between the IFS and the IIFS case: The model map of IFS is onto, which need not be the case for IIFS. 4 This statement is trivial in the case RF 6= N. 2.4 Lemma Let (X; ) be a bounded complete metric space and F : N! S(X; ) be a strict IIFS on X. Both of the [ trivial inclusions (3) ( f1; : : : ; ng N ) (N N ) A F could be strict. n2n Proof. Consider, for the rst inclusion, the address space of the syntactical IIFS F R itself (where R is a zero sequence with R = N). equals the identity on N N. Hence, the identity on N is contained in (N N ) = N N. But id N is not contained in any of the sets f1; : : : ; ng N. For the second inclusion, consider Example 1.3. Assume that there is a w 2 N N such that (w) = 1. By denition, (w) = lim (w(1)w(2)w(3) ; m; x) m!1 = lim F (w(1))(f (w(2))(: : : (F (w(m))(x)) : : : )) m!1 = F (w(1))( lim F (w(2))(: : : (F (w(m))(x)) : : : )) m!1 = F (w(1))( lim (w(2)w(3) ; m; x)) m!1 = F (w(1))((w(2)w(3) )); since each F (n) is continuous on ([0; 1]; E ). On the other hand, for any n 2 N, we know that E (F (n)(w); 1) > 1 2 n > 0 for any w 2 N N. But if there were a w with (w) = 1, 4 This contrasts the corresponding statement for IFS and for the function systems considered by Wicks.

6 there should be a v 2 N N and an n 2 N such that w(1) = n and w(m) = v(m? 1) for m > 1 and (w) = F (n)((v)), as derived above. Hence, 1 62 (N N ). 5 On the other hand, by taking closures, we obtain equality in chain (3). 2.5 Lemma Let F be a strict IIFS over the complete bounded space (X; ). Then, cl ( S n2nf1; : : : ; ng N ) = cl (N N ) = A F : Let R be a zero sequence such that R 6= ;. Following usual notations in formal language theory, the semigroup generated by R is denoted by + R, and the semigroup operation is denoted by a dot `'. We extent a?1 by uv?1 = u?1 v?1. Thus, a formal language L + R describes an IIFS 6 F R;L :! S( N R ; R) called syntactical IIFS with respect to L, given a suitable bijective enumeration e :! L. Moreover, the zero sequence R may be inherited by some other IIFS F, i.e. R = R F. Considering the map F : R! S(X; ) as a semigroup morphism F : ( + R ; )! (S(X; ); ), a formal language L + describes the IIFS R F F;R;L :! S(X; ), n 7! F (e(n)), given a suitable bijective enumeration e :! L. 2.6 Remark (Properties) Let F be some IIFS on (X; ). Abbreviate R = R F. A FR;L = cl L! = adh(l ) 7 A FF;R;L = (cl L! ) = cl (L! ) Since formal language theory provides us with many dierent ways to describe formal languages over nite alphabets (i.e. j R j < 1 in our case), we obtain several methods to describe innite iterated function systems by nite means. In [14, Theorem 3.1(3)], another characterization of the xed point of an IFS F was given, namely as the closure of the periodic points of F. We transfer this result to IIFS. If F :! S(X; ) is an IIFS, any xed point of a composition F (v 1 ) F (v m ) is called a periodic point of F. From the address space point of view, we consider the points V = f(v 1 v m )! j v i 2 g. Since any v 2 N with v i = v(i) may be approximated by the sequence (v 1 v m )!, cl V = N. Since is continuous, (cl V ) cl (V ). Therefore, A F = cl (cl V ) is contained in the closure of the periodic points of F. Wicks proved a similar result in [21, Proposition 3.4.3]. 2.7 Theorem Let F be an IIFS on the complete bounded metric space (X; ). The closure of the periodic points of F equals the xed point A F of F. 3. Approximating IIFS by IFS In the rst section, we learned about a method to `construct' attractors of IIFS's via Banach's xed point theorem. But contrary to the situation found in (nite) IFS's, this approach does not provide a feasible computational method. We can overcome this diculty by showing that any IIFS can be approximated by an IFS. 3.1 Denition Let F : N! S(X; ) be an IIFS. The corresponding IFS of degree n is given by F n = (F (1); : : : ; F (n)). Analogously, if R : N! [0; 1) is a zero sequence, then R n = (R(1); : : : ; R(n)). 5 This part of the proof is due to my colleague Thomas Worsch. 6 In this paper, we only touch issues from formal language theory. Refer e.g. to [9] for more details. 7 The!-power and adh-operator are dened as usual in formal language theory, see e.g. [15, 4].

7 This ts into our previous consideration of formal languages considering the language f1; : : : ; ng N. Hence, the model map of the strict IIFS F can be interpreted as the model map : f1; : : : ; ng N! A Fn of the IFS F n = F F;RF ;f1;:::;ng. From f1g N f1; 2g N f1; 2; 3g N [ f1; : : : ; ng N N N ; we obtain for a strict IIFS F over a complete bounded [ metric space (4) A F1 A F2 A F3 A Fn A F : Compare this with Equation (3). Combining these observations with [12, page 150], we can prove: Theorem (Approximation Theorem) Let F be an IIFS on the compact space (X; ). Then, lim n!1 H (A Fn ; A F ) = 0. The corresponding statement for complete and bounded spaces is false in general Example Let R be some strict ratio sequence. We consider the space (CB(N N ; R ); H ). For any w 2 f1; : : : ; ng N, R ( n+1(w);?1 w) = 1, hence, H (f1; : : : ; ng N ; f1; : : : ; n + 1g N ) = 1. Therefore, the increasing chain of xed points of the syntactical IIFS (F R ) n of degree n is not a Cauchy sequence and is therefore not converging. The approximation theorem provides us with a method for calculating xed points of IIFS. Choose some of the well-known methods for calculating IFS fractals to generate A F1, A F2, A F3, : : : until some sucient precision is reached. We shall now present some sort of shortcut of this procedure. Its idea originated from the consideration of the following diagram describing the straightforward deterministic method more elaborately. Let F denote some IIFS on a compact metric space (X; ) and A some compact set. A F 1 (A) F1 2 (A) : : : lim m!1 F m 1 (A) = A F1 = A F 2 (A) F2 2 (A) : : : lim m!1 F m 2 (A) = A F2 = = S A n2n F n (A) n2n Fn(A) 2 : : : = = = = A (N)(A) (N 2 )(A) : : : (lim m!1 N m )(A) = (N! ) = n2n n2n S n2n lim m!1 F m n (A) = S n2n A Fn A F (A) F 2 (A) : : : lim m!1 F m (A) = A F Instead of the indicated double limit process, the sequence F 1 1 (A), F 2 2 (A), F 3 3 (A), : : : lying on the diagonal of the diagram also tends towards A F. But it is necessary to calculate many points which in some sense belong to earlier steps in the process, 8 By our argument, we obtain the convergence theorem only for strict IIFS. But this theorem holds in the non-strict case, too. 9 The statement in [8] is false, too

8 e.g. for some x 2 A, F (1)(F (3)(F (2)(x))) and F (1)(F (2)(F (2)(x))) are computed in the third step of the process; for the calculation of the latter point, we can use F (2)(F (2)(x)) calculated in the second step, whereas for the rst one no such re-use of previously calculated results is possible. In order to reduce the number of calculations, A should contain as few points as possible. Hence, we start with e.g. A = fxg, where x is the xed point of F (1). We now turn to some address space argument explaining the above method and its improvement. The xed point of F (1) has an address 1!. In the second step, 1!, 121!, 21! and 221! are generated, and so on. More precisely, let B 1 =?1 1 f1! g = f1! g, B 1, : : :. Obviously, the B k form B 2 = S i;j=1;2?1 i?1 j B 1, B 3 = S i;j;k=1;2;3?1 i?1 j?1 k an increasing sequence of compact sets, and cl S k B k = N N, since any prex v 1 v k of length k is considered in step maxfv 1 ; : : : ; v k ; kg. This very argument holds for the sequence C 1 =?1 1 f1! g, C 2 = S i=1;2?1 i C 1, C 3 = S i=1;2;3?1 i C 2, : : :, too. Any prex v 1 v k of length k is considered in step k + maxfv 1 ; : : : ; v k g. This shows cl S n2n C n = N N. As a continuous image of a compact set, (C n ) is compact, too. Since the underlying space (X; ) is compact, lim n!1 H ((C n ); A F ) = 0 by [12, page 150]. Note that this argument does not depend on the special way the C n are dened, since any increasing chain of compact sets whose union is dense in N N will suce. We summarize our observations in the following theorem which embraces the approximation theorem as a special case. 3.4 Theorem (Diagonal Method Theorem) Let F be an IIFS on the compact metric space (X; ). Let (C n ) be an increasing chain of compact subsets of (N N ; RF ) with cl S n2n C n = N N. Then, lim n!1 H ((C n ); A F ) = 0. Especially, this is true for the diagonal sequence (B n ) as dened above. For practical purposes, the increase of the C n should be `rapid' in some sense. Furthermore, we must take care of texture eects when drawing pictures. The following procedure called adaptive diagonal method (cf. [13]) is of this type. In order to measure the growth of the C n somehow, we require that F is strict and that R F is weakly decreasing. (1) Choose some n 2 N. (2) Calculate the m 2 N with R F (m + 1) < 2?n and R F (m) 2?n. (3) Calculate the nite set L of words v 1 v k over the alphabet f1; : : : ; mg such that, for some j 2 f1; : : : ; mg, (2 n R F (j))?1 < R F (v 1 ) R F (v k ) 2?n. (4) Let C n = Lf1! g. For IFS, it is known that there is a unique number s 0 satisfying P n i=1 (R F (i)) s = 1, where the IFS is given by the list F = (F (1); : : : ; F (n)). This s is termed similarity dimension of F or R F, written dim S (R F ). The theory of IFS tells us that, under certain circumstances, s equals the Hausdorff dimension of the xed point of F. We derive a similar result for IIFS in this paper. 3.5 Denition Let R : N! [0; 1) be a zero sequence. We call R (s) = P i2n(r(i)) s the ratio sum of R. Its similarity dimension is given by inffs j R (s) 1g and is denoted by dim S (R) Sometimes we write e.g. F, if an IIFS instead of a ratio sequence is given.

9 3.6 Example The simplistic-looking sequence R(n) = (n + 1)?1 is connected to the famous Riemann -function by R (s) = (s)? 1. A table-lookup shows that its similarity dimension is slightly greater than 1: Theorem Let R : N! (0; 1) be a zero sequence. Then, dim S (R) = lim n!1 dim S (R n ). Proof. (dim S (R n )) n is an increasing sequence of positive real numbers, obviously bounded by S = dim S (R). Let s = lim n!1 dim S (R n ). We want to show that s = S. We distinguish two cases. (1) Assume that there exists a t < S such that R (t) < 1. Hence, R (t) > 1. Since the decreasing continuous functions Rn (x) converge uniformly to R (x) in the interval (t; 1), there is an m with dim S (R m ) 2 (t; 1) and R (dim S (R m )) < 1, showing s = S in this case. (2) Assume s < S and, for any t 2 [s; S), R (t) = 1. Fix some t 2 [s; S). Hence, the sequence of numbers ( Rm (t)) m is not bounded, contradicting Rm (t) Rm (s) Rm (dim S (R m )) = 1 for every m Example The second case in the proof is possible, as shown by [17, Example 4.5] (s) = 1X n=1 2 n2?1 1X 2 = s(n2 +n) n=1 2 n2 (1?s)?ns?1 : In the following, we discuss the relation of similarity dimension and Hausdorff dimension for sequence spaces and Euclidean spaces in detail. 3.9 Theorem Let R : N! (0; 1) be a zero sequence. Then, dim H ( S n2nf1; : : : ; ng N ) = dim H (N N ) = dim S (R) in the space (N N ; R ). Proof. We may restrict ourselves to covers by balls of the form C n = f?1 w (N N ) j w 2 N n g. For any " > 0 there is an n 0 such that, for any n n 0 and any C 2 C n, diam(c) ". Moreover, X X (diam(c)) s = ( R(m) s ) n = ( R (s)) n : C2C n m2n For s > dim S (R), inf n2n ( R (s)) n = 0, hence dim H (N N ) dim S (R). The analogue of this theorem for IFS is well-known [5, p.142f.]. By dim H (f1; : : : ; ng N ) = dim S (R n ) dim H (N N ) for every n, we see dim H (N N ) dim S (R). Note that with the assumption of the last theorem, the dim S (R)-dimensional Hausdorff measure of S n2nf1; : : : ; ng N vanishes. We can show that the dim S (R)- dimensional Hausdorff measure of N N equals 1 if R (dim S (R)) = 1, but this need not be the case, as an example similar to Ex. 3.8 shows. Since the model map is Lipschitz continuous, we nd: 3.10 Theorem Let (X; ) be a bounded complete metric space. Let F be an IIFS on X. Then, dim S (R F ) is an upperbound of the Hausdorff dimension of ( N R F ). 11 The idea of the part (2) of this proof is due to Dr. Staiger. In previous papers, we isolated property (1) using the term `nite summability' in that case.

10 By [14], for Euclidean spaces X R m, we know a criterion called open set condition proving the coincidence of Hausdorff dimension and similarity dimension for IFS F = (F (1); : : : ; F (n)): Assume there exists a non-empty bounded open test set V such that S n i=1 F (i)(v ) V and (8i; j 2 f1; : : : ; ng)(i 6= j =) F (i)(v ) \ F (j)(v ) = ;): We transfer this result to IIFS Theorem Let (X; E ) be some Euclidean space. Let F be an IIFS on X such that all corresponding IFS F n satisfy some open set condition. Then, the Hausdorff dimension of ( N R F ) equals the similarity dimension s of R F. If the s-dimensional Hausdorff measure of A F r( N R F ) is bounded, then s is also the Hausdorff dimension of A F Remark Instead of presuming an individual open set condition for each F n, we could have presupposed one open set condition for the IIFS F, assuming the existence of a non-empty bounded open test set V such that S 1 i=1 F (i)(v ) V and (8i; j 2 N)(i 6= j =) F (i)(v ) \ F (j)(v ) = ;): It is clear that such a test set V for F might also serve as a test set for each F n. It is not clear to us whether our condition is really weaker than the open set condition for IIFS. 4. On the Descriptive Power of IIFS 4.1 Lemma In a separable metric space (X; ), every closed set is a xed point of an IIFS. Proof. Let fx n j n 2 Ng be dense in the subspace (A; ) (X; ). Dene F F (n)(x) = x n. Then, A F = A. 4.2 Lemma There is a closed and bounded subset of a complete metric space that is a xed point of a strict IIFS but not of an IFS. Proof. (N N ; R ) is the xed point of the syntactical IIFS F R. Since (N N ; R ) is not compact, it is not a xed point of an IFS. 4.3 Lemma There is a compact subset of a compact metric space that is not xed point of a strict IIFS. Proof. Consider f1! ; 2! g f1; 2g N. Instead of taking the nite set f1! ; 2! g in the proof of the preceding lemma, it is also possible to take the innite compact set f1! g [ f2; 3g! f1; 2; 3g N. 4.4 Lemma Every strict IFS F = (F (1); : : : ; F (n)), n > 1, containing pairwise different maps can be substituted by a strict IIFS G with pairwise dierent maps such that A F = A G and dim S (R F ) = dim S (R G ). Proof. Consider the language L = f1; : : : ; n? 1g [ f1; : : : ; n? 1g fng +. Dene G = F F;RF ;L. Since cl L! = f1; : : : ; ng!, A F = A G. Obviously, P n?1 i=1 (R F (i)) s G (s) = : 1? R F (n) s Hence, G (s) = 1 () F (s) = P n i=1 R F (i) s = 1. by

11 We can use a language L f1; : : : ; ng + in order to cut o pieces (cl L! ) of the attractor A F of a strict IFS F = (F (1); : : : ; F (n)). 4.5 Lemma If the strict IFS F = (F (1); : : : ; F (n)) satises an open set condition with test set V, then V can also serve as a test set for F F;RF ;L provided L is prexfree. Interestingly, the sets cl L! with L prex-free are known in the theory of!-languages as `strongly connected' [20]. This notion is equivalent to the irreducibility of the state matrix. Also refer to [2]. Acknowledgements I thank L. Staiger, Th. Worsch and the unknown referee for their helpful comments on earlier versions of this paper. This paper will appear in `Mathematische Nachrichten', Volume 169, References 1. L. M. Andersson. Recursive Construction of Fractals. PhD thesis, Helsinki: Suomalainen Tiedeakatemia, Aug Annales Academiae Scientiarum Fennicae Series A, I. Mathematica, Dissertationes, C. Bandt. Self-similar sets. I. Topological Markov chains and mixed self-similar sets. Mathematische Nachrichten, 142:107{123, M. F. Barnsley. Fractals Everywhere. Boston: Academic Press, L. Boasson and M. Nivat. Adherences of languages. Journal of Computer and System Sciences, 20:285{309, G. A. Edgar. Measure, Topology, and Fractal Geometry. Undergraduate Texts in Mathematics. New York: Springer, D. B. Ellis and M. G. Branton. Non-self-similar attractors of hyperbolic iterated function systems. In J. C. Alexander, editor, Dynamical Systems, volume 1342 of LNM, pages 158{171. New York: Springer, K. J. Falconer. Fractal Geometry. Chichester: John Wiley & Sons, H. Fernau. IIFS and codes. To appear in the Proceedings of the IMYCS'92, H. Fernau. Varianten Iterierter Funktionensysteme und Methoden der Formalen Sprachen. PhD thesis, Universitat Karlsruhe (TH) (Germany), H. Fernau. Iterierte Funktionen, Sprachen und Fraktale. Mannheim: BI-Verlag, M. Hata. On some properties of set-dynamical systems. Proceedings Japan Acad., Serie A, 61:99{ 102, F. Hausdor. Mengenlehre, volume 7 of Goschens Lehrbucherei: Reine Mathematik. Berlin: de Gruyter, D. Hepting, P. Prusinkiewicz, and D. Saupe. Rendering methods for iterated function systems. In H.-O. Peitgen, J. M. Henriques, and L. F. Penedo, editors, Fractals in the Fundamental and Applied Sciences (Proceedings of the Conference Fractals'90), pages 183{224. IFIP, Elsevier/North- Holland, J. Hutchinson. Fractals and self-similarity. Indiana University Mathematics Journal, 30:713{747, R. Lindner and L. Staiger. Algebraische Codierungstheorie; Theorie der sequentiellen Codierungen, volume 11 of Elektronisches Rechnen und Regeln. Berlin: Akademie-Verlag, R. D. Mauldin and M. Urbanski. Dimensions and measures in innite iterated function systems. Unpublished Manuscript, December R. D. Mauldin and S. C. Williams. Random recursive constructions: Asymptotic geometric and topological properties. Transactions of the American Mathematical Society, 295(1):325{346, 1986.

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