INFINITE ITERATED FUNCTION SYSTEMS HENNING FERNAU LEHRSTUHL INFORMATIK F?UR INGENIEURE UND NATURWISSENSCHAFTLER UNIVERSIT?AT KARLSRUHE -LRB- TH -RRB- GERMANY June 8 , 1994 Abstract : We examine iterated function systems consisting of a countably infinite number of contracting mappings -LRB- IIFS -RRB- . We state results analogous to the well-known case of finitely many mappings -LRB- IFS -RRB- . 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 defining 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 . AMS classification : 28A80 , 54E50 , 54E52 , 28A78 , 54F45 . 1 . Introduction and Main Definitions IFS theory , starting out from Hutchinson 's paper -LSB- 14 -RSB- , gained more and more interest . Several books on this topic are available -LSB- 3 , 7 , 5 , 18 , 19 -RSB- which have become popular even among non-mathematicians . In those references , mostly the case of systems consisting of finitely many mappings has been treated . In this paper , we elaborate the case of countable many mappings , contrasting it with the well-known results on finite IFS .1 Especially in -LSB- 11 , 17 , 2 , 21 , 1 , 16 -RSB- , the case of infinitely 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 -LRB- X ; ae -RRB- . A function f : X ! X is called a contraction -LRB- with contractivity r 2 -LSB- 0 ; 1 -RRB- -RRB- iff , for all x ; y 2 X , ae -LRB- f -LRB- x -RRB- ; f -LRB- y -RRB- -RRB- < = r?ae -LRB- x ; y -RRB- . By Banach 's fixed point theorem , a contraction f has a unique fixed point . A similitude is a contraction with the property 8x ; y 2 X , ae -LRB- f -LRB- x -RRB- ; f -LRB- y -RRB- -RRB- = rf ? ae -LRB- x ; y -RRB- . rf is called similarity ratio . The set of all similitudes on a non-empty complete space -LRB- X ; ae -RRB- will be denoted by S -LRB- X ; ae -RRB- . Important examples for complete metric spaces are closed subsets of the ndimensional Euclidean space , e.g. -LRB- Rn ; aeE -RRB- with aeE -LRB- x ; y -RRB- = qPni = 1 -LRB- xi ? yi -RRB- 2 , and the collection K -LRB- X ; ae -RRB- of all non-empty compact sets of a metric space -LRB- X ; ae -RRB- together with 1Most of the results in this paper are contained in the PhD thesis -LSB- 9 -RSB- of the author which is available as a book -LSB- 10 -RSB- . 2 Henning Fernau the Hausdorff metric aeH defined by aeH -LRB- A ; B -RRB- = inffr : A ? Br -LRB- B -RRB- ; B ? Br -LRB- A -RRB- g with Br -LRB- A -RRB- = fx 2 X j -LRB- 9a 2 A -RRB- -LRB- ae -LRB- a ; x -RRB- < r -RRB- g. aeH is also a metric on the collection of all non-empty closed and bounded sets CB -LRB- X ; ae -RRB- . 1.1 Definition Let ? ? N be some index set . An iterated function system -LRB- on -LRB- X ; ae -RRB- -RRB- is a mapping F : ? ! S -LRB- X ; ae -RRB- such that the corresponding sequence of similarity ratios RF : N ! -LSB- 0 ; 1 -RRB- ; n 7 ! 8 < : rF -LRB- n -RRB- if n 2 ? if n 62 ? converges to zero . We call an iterated function system finite -LRB- IFS -RRB- or infinite -LRB- IIFS -RRB- iff ? is finite or infinte , respectively .2 An iterated function system is called strict iff RF -LRB- n -RRB- < > for every n 2 ? . Equivalently , we can specify an IFS by a list F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- of similitudes . 1.2 Remark We may identify an IFS with an IIFS F with the property -LRB- 9n 2 N -RRB- -LRB- 9x 2 X -RRB- -LRB- 8i > = n -RRB- -LRB- F -LRB- i -RRB- = fxg -RRB- : ? Without loss of generality , we may use N as index set for IIFS . Consider now an IIFS F . Let A 2 CB -LRB- X ; ae -RRB- . As with IFS , F -LRB- n -RRB- -LRB- A -RRB- 2 CB -LRB- X ; ae -RRB- . But contrary to IFS , in the case of IIFS , S n2N F -LRB- n -RRB- -LRB- A -RRB- 2 CB -LRB- X ; ae -RRB- does not necesssarily hold . 1.3 Example F -LRB- n -RRB- -LRB- x -RRB- = 1 ? 1 2n ? 1 + 12n ? x. Obviously , F 2 -LRB- S -LRB- R ; aeE -RRB- -RRB- N is an IIFS with S n2N F -LRB- n -RRB- -LRB- -LSB- 0 ; 1 -RSB- -RRB- = S n2N F -LRB- n -RRB- -LRB- -LSB- 0 ; 1 -RRB- -RRB- = -LSB- 0 ; 1 -RRB- 62 K -LRB- R ; aeE -RRB- . We can overcome this difficulty if we define F -LRB- A -RRB- to be the closure of S n2N F -LRB- n -RRB- -LRB- A -RRB- . Thus , -LSB- 0 ; 1 -RSB- becomes a fixed point of F in the above example . 1.4 Definition A non-empty set A of a metric space -LRB- X ; ae -RRB- is called a fixed point of the given IIFS F : ? ! S -LRB- X ; ae -RRB- , iff F -LRB- A -RRB- ? = cl -LSB- n2 ? F -LRB- n -RRB- -LRB- A -RRB- = A : -LRB- 1 -RRB- The intuition behind this definition of IIFS should be clear from IFS : Any fractal set defined in such a way may have an infinite number of subsets which are themselves similar to the whole structure . What about the existence of fixed points for IIFS ? Before addressing this question , we turn to a similar notion following Ellis and Branton -LSB- 6 -RSB- . 1.5 Definition Let ? be an arbitrary index set , and F : ? ! XX , where -LRB- X ; ae -RRB- is some metric space . A pseudo-attractor of F is a closed set A < > ; such that F -LRB- n -RRB- -LRB- A -RRB- ? A for each n 2 ? . A pseudo-attractor of F not properly containing another pseudoattractor is called attractor of F . The notion of a fixed point may be carried over to arbitrary function families F using Equation -LRB- 1 -RRB- . 1.6 Remark Let -LRB- X ; ae -RRB- be some metric space . Let F : ? ! XX be a function family . ffl A fixed point A of F is a pseudo-attractor of F . 2The reader may spell these abbreviations also as ` -LRB- in -RRB- finite iterated function schemes ' as suggested by Falconer -LSB- 7 -RSB- . IIFS 3 ffl If A is a pseudo-attractor of F , then cl S n2 ? F -LRB- n -RRB- -LRB- A -RRB- is a pseudo-attractor , too . ffl Thus , an attractor A of F is a fixed point of F . ffl If -LRB- X ; ae -RRB- is compact , then F has an attractor . -LRB- Apply Zorn 's lemma to the family of all pseudo-attractors . -RRB- ? 1.7 Example -LRB- A pseudo-attractor chain without attractor -RRB- Consider Baire 's space X = NN -LRB- which is not compact -RRB- and the function family F : N ! XX of continuous functions defined by F -LRB- n -RRB- -LRB- f -LRB- m -RRB- -RRB- = 8 < : f -LRB- m -RRB- ; if f -LRB- m -RRB- > n n + 1 otherwise for n ; m 2 N and f : N ! N. Obviously , cl S n2N F -LRB- n -RRB- -LRB- X -RRB- = F -LRB- 1 -RRB- -LRB- X -RRB- = f2 ; 3 ; : : : gN , and cl -LSB- n2N F -LRB- n -RRB- -LRB- fk ; k + 1 ; : : : gN -RRB- = F -LRB- k -RRB- -LRB- fk ; k + 1 ; : : : gN -RRB- = fk + 1 ; k + 2 ; : : : gN : K = n fk ; k + 1 ; : : : gN j k 2 N o 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 fixed point theorem leads to a characterization of fixed points for IIFS . Compare with -LSB- 21 , Theorem 3.1.1 -RSB- . 1.8 Theorem -LRB- Fixed Point Theorem -RRB- Let -LRB- X ; ae -RRB- be a complete and bounded metric space . Let F be an IIFS . Then F : CB -LRB- X ; ae -RRB- ! CB -LRB- X ; ae -RRB- is a contraction with contractivity r ? = supn2N RF -LRB- n -RRB- < 1 on the complete and bounded metric space -LRB- CB -LRB- X ; ae -RRB- ; aeH -RRB- . Hence , F possesses a unique fixed point , abbreviated as AF . ? The above theorem could also be stated using contractions which are not necessarily similitudes . Moreover , the requirement that RF is a zero sequence -LRB- also implicitly contained in the definition of IIFS -RRB- may be weakened as indicated in -LSB- 2 -RSB- . We need both properties when dealing with dimensions below . A similar result may be obtained using the space K -LRB- X ; ae -RRB- instead of CB -LRB- X ; ae -RRB- . Moreover , in compact spaces , A is a fixed point of the IIFS F iff A is an attractor of F . 2 . Properties and Applications of the Sequence Space Let R : N ! -LSB- 0 ; 1 -RRB- be some zero sequence . Let ? R = fn 2 N j R -LRB- n -RRB- < > g . We define the metric aeR on the sequence space ? NR as follows :3 aeR -LRB- v ; w -RRB- ? = inf n2Nf nY i = 1 R -LRB- v -LRB- i -RRB- -RRB- j -LRB- 81 < = i < = n -RRB- -LRB- v -LRB- i -RRB- = w -LRB- i -RRB- -RRB- g : The most commonly known -LRB- ultra -RRB- metric on ? NR is Baire 's metric aeB given by aeB -LRB- v ; w -RRB- ? = 1 infn2Nfn j v -LRB- n -RRB- < > w -LRB- n -RRB- g : 2.1 Theorem Let R ; S : N ! -LSB- 0 ; 1 -RRB- be two different zero sequences with j?Rj < 1 and ? R = ? S. 3Note that empty products deliver 1 . Observe that ? NR = ; iff R is constant . 4 Henning Fernau ffl aeR , aeS and aeB are uniformly equivalent on -LRB- ? NR ; aeR -RRB- . ffl The sequence space -LRB- ? NR ; aeR -RRB- is compact . ffl -LRB- ? NR ; aeR -RRB- , -LRB- ? NR ; aeS -RRB- and -LRB- ? NR ; aeB -RRB- are pairwise Lipschitz-inequivalent . Proof . We only prove that aeR and aeS are Lipschitz-inequivalent . Assume that aeR , aeS are Lipschitz-equivalent . Hence , there are r > = s > satisfying s < = f aeR -LRB- x ; y -RRB- aeS -LRB- x ; y -RRB- j -LRB- x ; y 2 ? NR -RRB- ^ -LRB- x < > y -RRB- g < = r : Since R < > S , there is an i 2 ? R such that R -LRB- i -RRB- < > S -LRB- i -RRB- . Moreover , we find a j 2 ? Rrfig . Define xm : N ! ? R ; k 7 ! 8 < : i if k < m j if k > = m : In a notation borrowed from formal languages , we may write xm = im ? 1j ! . Then , aeR -LRB- xm1 ; xm2 -RRB- = -LRB- R -LRB- i -RRB- -RRB- minfm1 ; m2g ? 1 : Without loss of generality , assume R -LRB- i -RRB- > S -LRB- i -RRB- . The set f -LRB- R -LRB- i -RRB- S -LRB- i -RRB- -RRB- m j m 2 N0g = f aeR -LRB- xm1 ; xm2 -RRB- aeS -LRB- xm1 ; xm2 -RRB- j m1 ; m2 2 N ; m1 < > m2g is unbounded , hence f aeR -LRB- x ; y -RRB- aeS -LRB- x ; y -RRB- j -LRB- x ; y 2 ? NR -RRB- ^ -LRB- x < > y -RRB- g is unbounded , too . A similar construction shows that aeS and aeB are Lipschitzinequivalent . ? The picture changes when considering sequence spaces ? NR with j?Rj = 1 . We summarize the properties in the following theorem . 2.2 Theorem Let R ; S : N ! -LSB- 0 ; 1 -RRB- be two zero sequences with j?Rj = 1 and ? R = ? S . ffl The sequence space -LRB- ? NR ; aeR -RRB- is complete , separable , and bounded but not totally bounded and hence not compact . ffl aeR and aeS are uniformly equivalent on -LRB- ? NR ; aeR -RRB- . ffl -LRB- ? NR ; aeR -RRB- and -LRB- ? NR ; aeB -RRB- are homeomorphic , but aeR and aeB are not uniformly equivalent . ? How can we define iterated function systems on a sequence space ? Let R : N ! -LSB- 0 ; 1 -RRB- be some zero sequence . Have a look at the continuous shift map oe : ? NR ! ? NR , oe -LRB- a -RRB- -LRB- n -RRB- = a -LRB- n + 1 -RRB- : It has j?Rj right inverses . Define for m 2 ? R oe ? 1 m -LRB- a -RRB- -LRB- n -RRB- ? = -LRB- m if n = 1 a -LRB- n ? 1 -RRB- if n > 1 : R defines the syntactical IIFS FR : ? R ! S -LRB- ? NR ; aeR -RRB- ; m 7 ! oe ? 1 m . Obviously , ? NR is the attractor of FR . We summarize the main connection between sequence spaces and IIFS in the following theorem . IIFS 5 2.3 Theorem Let -LRB- X ; ae -RRB- be a bounded complete metric space and F : N ! S -LRB- X ; ae -RRB- be an IIFS on X. Let -LRB- ? NRF ; aeRF -RRB- be the corresponding sequence space . For each w 2 ? NRF , n 2 N , x 2 X , define OE -LRB- w ; n ; x -RRB- = F -LRB- w -LRB- 1 -RRB- -RRB- -LRB- F -LRB- w -LRB- 2 -RRB- -RRB- -LRB- : : : -LRB- F -LRB- w -LRB- n -RRB- -RRB- -LRB- x -RRB- -RRB- : : : -RRB- -RRB- 2 X ; and OE : ? NRF ! AF by OE -LRB- w -RRB- = limn ! 1 OE -LRB- w ; n ; x -RRB- . OE -LRB- considered as a map -LRB- ? RF ; aeRF -RRB- ! -LRB- AF ; ae -RRB- -RRB- is well-defined and continuous . We call OE model map . Fix some x0 2 AF . The mappings OEm : ? NRF ! AF given by w 7 ! OE -LRB- w ; m ; x0 -RRB- are continuous and converge uniformly to OE : ? NRF ! AF . Moreover , OE is a Lipschitz mapping , and the following diagram commutes -LRB- for n 2 N -RRB- -LRB- ? NRF ; aeRF -RRB- OE ???! -LRB- X ; ae -RRB- FRF -LRB- n -RRB- ??? y ??? yF -LRB- n -RRB- -LRB- ? NRF ; aeRF -RRB- ???! OE -LRB- X ; ae -RRB- -LRB- 2 -RRB- and OE is the only continuous function ? NRF ! X satisfying the above diagram . w 2 ? NRF is called an address of OE -LRB- w -RRB- 2 AF . ? The proof is literally the same as e.g. -LSB- 21 , Section 3.3 -RSB- or -LSB- 3 , Chapter 4.2 -RSB- . There is one subtle difference 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 < > N. 2.4 Lemma Let -LRB- X ; ae -RRB- be a bounded complete metric space and F : N ! S -LRB- X ; ae -RRB- be a strict IIFS on X. Both of the trivial inclusions OE -LRB- -LSB- n2N f1 ; : : : ; ngN -RRB- ? OE -LRB- NN -RRB- ? AF -LRB- 3 -RRB- could be strict . Proof . Consider , for the first inclusion , the address space of the syntactical IIFS FR itself -LRB- where R is a zero sequence with ? R = N -RRB- . OE equals the identity on NN . Hence , the identity on N is contained in OE -LRB- NN -RRB- = NN . But idN is not contained in any of the sets f1 ; : : : ; ngN . For the second inclusion , consider Example 1.3 . Assume that there is a w 2 NN such that OE -LRB- w -RRB- = 1 . By definition , OE -LRB- w -RRB- = lim m ! 1 OE -LRB- w -LRB- 1 -RRB- w -LRB- 2 -RRB- w -LRB- 3 -RRB- ? ? ? ; m ; x -RRB- = lim m ! 1F -LRB- w -LRB- 1 -RRB- -RRB- -LRB- F -LRB- w -LRB- 2 -RRB- -RRB- -LRB- : : : -LRB- F -LRB- w -LRB- m -RRB- -RRB- -LRB- x -RRB- -RRB- : : : -RRB- -RRB- = F -LRB- w -LRB- 1 -RRB- -RRB- -LRB- lim m ! 1F -LRB- w -LRB- 2 -RRB- -RRB- -LRB- : : : -LRB- F -LRB- w -LRB- m -RRB- -RRB- -LRB- x -RRB- -RRB- : : : -RRB- -RRB- = F -LRB- w -LRB- 1 -RRB- -RRB- -LRB- lim m ! 1 OE -LRB- w -LRB- 2 -RRB- w -LRB- 3 -RRB- ? ? ? ; m ; x -RRB- -RRB- = F -LRB- w -LRB- 1 -RRB- -RRB- -LRB- OE -LRB- w -LRB- 2 -RRB- w -LRB- 3 -RRB- ? ? ? -RRB- -RRB- ; since each F -LRB- n -RRB- is continuous on -LRB- -LSB- 0 ; 1 -RSB- ; aeE -RRB- . On the other hand , for any n 2 N , we know that aeE -LRB- F -LRB- n -RRB- -LRB- w -RRB- ; 1 -RRB- > 12n > for any w 2 NN . But if there were a w with OE -LRB- w -RRB- = 1 , 4This contrasts the corresponding statement for IFS and for the function systems considered by Wicks . 6 Henning Fernau there should be a v 2 NN and an n 2 N such that w -LRB- 1 -RRB- = n and w -LRB- m -RRB- = v -LRB- m ? 1 -RRB- for m > 1 and OE -LRB- w -RRB- = F -LRB- n -RRB- -LRB- OE -LRB- v -RRB- -RRB- , as derived above . Hence , 1 62 OE -LRB- NN -RRB- .5 ? On the other hand , by taking closures , we obtain equality in chain -LRB- 3 -RRB- . 2.5 Lemma Let F be a strict IIFS over the complete bounded space -LRB- X ; ae -RRB- . Then , cl OE -LRB- S n2Nf1 ; : : : ; ngN -RRB- = cl OE -LRB- NN -RRB- = AF : ? Let R be a zero sequence such that ? R < > ; . 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 oe ? 1 a by oe ? 1 u?v = oe ? 1 u ffi oe ? 1 v . Thus , a formal language L ? ? + R describes an IIFS6 FR ; L : ? ! S -LRB- ? NR ; aeR -RRB- 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 = RF . Considering the map F : ? R ! S -LRB- X ; ae -RRB- as a semigroup morphism F : -LRB- ? + R ; ? -RRB- ! -LRB- S -LRB- X ; ae -RRB- ; ffi -RRB- , a formal language L ? ? + R describes the IIFS FF ; R ; L : ? ! S -LRB- X ; ae -RRB- , n 7 ! F -LRB- e -LRB- n -RRB- -RRB- , given a suitable bijective enumeration e : ? ! L. 2.6 Remark -LRB- Properties -RRB- Let F be some IIFS on -LRB- X ; ae -RRB- . Abbreviate R = RF . ffl AFR ; L = cl L ! = adh -LRB- L ? -RRB- 7 ffl AFF ; R ; L = OE -LRB- cl L ! -RRB- = cl OE -LRB- L ! -RRB- Since formal language theory provides us with many different ways to describe formal languages over finite alphabets -LRB- i.e. j?Rj < 1 in our case -RRB- , we obtain several methods to describe infinite iterated function systems by finite means . In -LSB- 14 , Theorem 3.1 -LRB- 3 -RRB- -RSB- , another characterization of the fixed 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 -LRB- X ; ae -RRB- is an IIFS , any fixed point of a composition F -LRB- v1 -RRB- ffi ? ? ? ffi F -LRB- vm -RRB- is called a periodic point of F . From the address space point of view , we consider the points V = f -LRB- v1 ? ? ? vm -RRB- ! j vi 2 ? g . Since any v 2 ? N with vi = v -LRB- i -RRB- may be approximated by the sequence -LRB- v1 ? ? ? vm -RRB- ! , cl V = ? N . Since OE is continuous , OE -LRB- cl V -RRB- ? cl OE -LRB- V -RRB- . Therefore , AF = cl OE -LRB- cl V -RRB- is contained in the closure of the periodic points of F . Wicks proved a similar result in -LSB- 21 , Proposition 3.4.3 -RSB- . 2.7 Theorem Let F be an IIFS on the complete bounded metric space -LRB- X ; ae -RRB- . The closure of the periodic points of F equals the fixed point AF of F . ? 3 . Approximating IIFS by IFS In the first section , we learned about a method to ` construct ' attractors of IIFS 's via Banach 's fixed point theorem . But contrary to the situation found in -LRB- finite -RRB- IFS 's , this approach does not provide a feasible computational method . We can overcome this difficulty by showing that any IIFS can be approximated by an IFS . 3.1 Definition Let F : N ! S -LRB- X ; ae -RRB- be an IIFS . The corresponding IFS of degree n is given by Fn = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- . Analogously , if R : N ! -LSB- 0 ; 1 -RRB- is a zero sequence , then Rn = -LRB- R -LRB- 1 -RRB- ; : : : ; R -LRB- n -RRB- -RRB- . 5This part of the proof is due to my colleague Thomas Worsch . 6In this paper , we only touch issues from formal language theory . Refer e.g. to -LSB- 9 -RSB- for more details . 7The ! - power and adh-operator are defined as usual in formal language theory , see e.g. -LSB- 15 , 4 -RSB- . IIFS 7 This fits into our previous consideration of formal languages considering the language f1 ; : : : ; ng ae N. Hence , the model map OE of the strict IIFS F can be interpreted as the model map OE : f1 ; : : : ; ngN ! AFn of the IFS Fn = FF ; RF ; f1 ; : : : ; ng . From f1gN ae f1 ; 2gN ae f1 ; 2 ; 3gN ae ? ? ? ae -LSB- n2N f1 ; : : : ; ngN ae NN ; we obtain for a strict IIFS F over a complete bounded metric space AF1 ? AF2 ? AF3 ? ? ? ? ? -LSB- n2N AFn ? AF : -LRB- 4 -RRB- Compare this with Equation -LRB- 3 -RRB- . Combining these observations with -LSB- 12 , page 150 -RSB- , we can prove : 8 3.2 Theorem -LRB- Approximation Theorem -RRB- Let F be an IIFS on the compact space -LRB- X ; ae -RRB- . Then , limn ! 1 aeH -LRB- AFn ; AF -RRB- = 0 . ? The corresponding statement for complete and bounded spaces is false in general .9 3.3 Example Let R be some strict ratio sequence . We consider the space -LRB- CB -LRB- NN ; aeR -RRB- ; aeH -RRB- . For any w 2 f1 ; : : : ; ngN , aeR -LRB- oe ? 1 n +1 -LRB- w -RRB- ; w -RRB- = 1 , hence , aeH -LRB- f1 ; : : : ; ngN ; f1 ; : : : ; n + 1gN -RRB- = 1 . Therefore , the increasing chain of fixed points of the syntactical IIFS -LRB- FR -RRB- n of degree n is not a Cauchy sequence and is therefore not converging . ? The approximation theorem provides us with a method for calculating fixed points of IIFS . Choose some of the well-known methods for calculating IFS fractals to generate AF1 , AF2 , AF3 , : : : until some sufficient 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 -LRB- X ; ae -RRB- and A some compact set . A F1 -LRB- A -RRB- F 21 -LRB- A -RRB- : : : limm ! 1 Fm1 -LRB- A -RRB- = AF1 = ? ? ? ? A F2 -LRB- A -RRB- F 22 -LRB- A -RRB- : : : limm ! 1 Fm2 -LRB- A -RRB- = AF2 = ? ? ? ? ... ... ... ... ... ... = ? ? ? ? A S n2N Fn -LRB- A -RRB- S n2N F 2n -LRB- A -RRB- : : : S n2N limm ! 1 Fmn -LRB- A -RRB- = S n2N AFn = = = = ? ? A OE -LRB- N -RRB- -LRB- A -RRB- OE -LRB- N2 -RRB- -LRB- A -RRB- : : : OE -LRB- limm ! 1 Nm -RRB- -LRB- A -RRB- = OE -LRB- N ! -RRB- = ? ? ? ? A F -LRB- A -RRB- F 2 -LRB- A -RRB- : : : limm ! 1 Fm -LRB- A -RRB- = AF Instead of the indicated double limit process , the sequence F 11 -LRB- A -RRB- , F 22 -LRB- A -RRB- , F 33 -LRB- A -RRB- , : : : lying on the diagonal of the diagram also tends towards AF . But it is necessary to calculate many points which in some sense belong to earlier steps in the process , 8By our argument , we obtain the convergence theorem only for strict IIFS . But this theorem holds in the non-strict case , too . 9The statement in -LSB- 8 -RSB- is false , too 8 Henning Fernau e.g. for some x 2 A , F -LRB- 1 -RRB- -LRB- F -LRB- 3 -RRB- -LRB- F -LRB- 2 -RRB- -LRB- x -RRB- -RRB- -RRB- and F -LRB- 1 -RRB- -LRB- F -LRB- 2 -RRB- -LRB- F -LRB- 2 -RRB- -LRB- x -RRB- -RRB- -RRB- are computed in the third step of the process ; for the calculation of the latter point , we can use F -LRB- 2 -RRB- -LRB- F -LRB- 2 -RRB- -LRB- x -RRB- -RRB- calculated in the second step , whereas for the first 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 fixed point of F -LRB- 1 -RRB- . We now turn to some address space argument explaining the above method and its improvement . The fixed point of F -LRB- 1 -RRB- has an address 1 ! . In the second step , 1 ! , 121 ! , 21 ! and 221 ! are generated , and so on . More precisely , let B1 = oe ? 1 1 f1!g = f1!g , B2 = S i ; j = 1 ; 2 oe ? 1 i oe ? 1 j B1 , B3 = S i ; j ; k = 1 ; 2 ; 3 oe ? 1 i oe ? 1 j oe ? 1 k B1 , : : : . Obviously , the Bk form an increasing sequence of compact sets , and cl S k Bk = NN , since any prefix v1 ? ? ? vk of length k is considered in step maxfv1 ; : : : ; vk ; kg . This very argument holds for the sequence C1 = oe ? 1 1 f1!g , C2 = S i = 1 ; 2 oe ? 1 i C1 , C3 = S i = 1 ; 2 ; 3 oe ? 1 i C2 , : : : , too . Any prefix v1 ? ? ? vk of length k is considered in step k + maxfv1 ; : : : ; vkg . This shows cl S n2N Cn = NN . As a continuous image of a compact set , OE -LRB- Cn -RRB- is compact , too . Since the underlying space -LRB- X ; ae -RRB- is compact , limn ! 1 aeH -LRB- OE -LRB- Cn -RRB- ; AF -RRB- = by -LSB- 12 , page 150 -RSB- . Note that this argument does not depend on the special way the Cn are defined , since any increasing chain of compact sets whose union is dense in NN will suffice . We summarize our observations in the following theorem which embraces the approximation theorem as a special case . 3.4 Theorem -LRB- Diagonal Method Theorem -RRB- Let F be an IIFS on the compact metric space -LRB- X ; ae -RRB- . Let -LRB- Cn -RRB- be an increasing chain of compact subsets of -LRB- NN ; aeRF -RRB- with cl S n2N Cn = NN . Then , limn ! 1 aeH -LRB- OE -LRB- Cn -RRB- ; AF -RRB- = 0 . Especially , this is true for the diagonal sequence -LRB- Bn -RRB- as defined above . ? For practical purposes , the increase of the Cn should be ` rapid ' in some sense . Furthermore , we must take care of texture effects when drawing pictures . The following procedure called adaptive diagonal method -LRB- cf. -LSB- 13 -RSB- -RRB- is of this type . In order to measure the growth of the Cn somehow , we require that F is strict and that RF is weakly decreasing . -LRB- 1 -RRB- Choose some n 2 N. -LRB- 2 -RRB- Calculate the m 2 N with RF -LRB- m + 1 -RRB- < 2 ? n and RF -LRB- m -RRB- > = 2 ? n. -LRB- 3 -RRB- Calculate the finite set L of words v1 ? ? ? vk over the alphabet f1 ; : : : ; mg such that , for some j 2 f1 ; : : : ; mg , -LRB- 2nRF -LRB- j -RRB- -RRB- ? 1 < RF -LRB- v1 -RRB- ? ? ? RF -LRB- vk -RRB- > = 2 ? n. -LRB- 4 -RRB- Let Cn = Lf1!g . ? For IFS , it is known that there is a unique number s > = satisfying Pni = 1 -LRB- RF -LRB- i -RRB- -RRB- s = 1 , where the IFS is given by the list F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- . This s is termed similarity dimension of F or RF , written dimS -LRB- RF -RRB- . The theory of IFS tells us that , under certain circumstances , s equals the Hausdorff dimension of the fixed point of F . We derive a similar result for IIFS in this paper . 3.5 Definition Let R : N ! -LSB- 0 ; 1 -RRB- be a zero sequence . We call ? R -LRB- s -RRB- = P i2N -LRB- R -LRB- i -RRB- -RRB- s the ratio sum of R. Its similarity dimension is given by inffs j ? R -LRB- s -RRB- < = 1g and is denoted by dimS -LRB- R -RRB- .10 10Sometimes we write e.g. ? F , if an IIFS instead of a ratio sequence is given . IIFS 9 3.6 Example The simplistic-looking sequence R -LRB- n -RRB- = -LRB- n + 1 -RRB- ? 1 is connected to the famous Riemann ? - function by ? R -LRB- s -RRB- = ? -LRB- s -RRB- ? 1 . A table-lookup shows that its similarity dimension is slightly greater than 1:7 . ? 3.7 Theorem Let R : N ! -LRB- 0 ; 1 -RRB- be a zero sequence . Then , dimS -LRB- R -RRB- = limn ! 1 dimS -LRB- Rn -RRB- . Proof . -LRB- dimS -LRB- Rn -RRB- -RRB- n is an increasing sequence of positive real numbers , obviously bounded by S = dimS -LRB- R -RRB- . Let s = limn ! 1 dimS -LRB- Rn -RRB- . We want to show that s = S . We distinguish two cases . -LRB- 1 -RRB- Assume that there exists a t < S such that ? R -LRB- t -RRB- < 1 . Hence , ? R -LRB- t -RRB- > 1 . Since the decreasing continuous functions ? Rn -LRB- x -RRB- converge uniformly to ? R -LRB- x -RRB- in the interval -LRB- t ; 1 -RRB- , there is an m with dimS -LRB- Rm -RRB- 2 -LRB- t ; 1 -RRB- and ? R -LRB- dimS -LRB- Rm -RRB- -RRB- < 1 , showing s = S in this case . -LRB- 2 -RRB- Assume s < S and , for any t 2 -LSB- s ; S -RRB- , ? R -LRB- t -RRB- = 1 . Fix some t 2 -LSB- s ; S -RRB- . Hence , the sequence of numbers -LRB- ? Rm -LRB- t -RRB- -RRB- m is not bounded , contradicting ? Rm -LRB- t -RRB- < = ? Rm -LRB- s -RRB- < = ? Rm -LRB- dimS -LRB- Rm -RRB- -RRB- = 1 for every m. 11 ? 3.8 Example The second case in the proof is possible , as shown by -LSB- 17 , Example 4.5 -RSB- ? -LRB- s -RRB- ? = 1 X n = 1 2n2 ? 1 2s -LRB- n2 + n -RRB- = 1 X n = 1 2n2 -LRB- 1 ? s -RRB- ? 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 ! -LRB- 0 ; 1 -RRB- be a zero sequence . Then , dimH -LRB- S n2Nf1 ; : : : ; ngN -RRB- = dimH -LRB- NN -RRB- = dimS -LRB- R -RRB- in the space -LRB- NN ; aeR -RRB- . Proof . We may restrict ourselves to covers by balls of the form Cn = foe ? 1 w -LRB- NN -RRB- j w 2 Nng . For any '' > there is an n0 such that , for any n > = n0 and any C 2 Cn , diam -LRB- C -RRB- < = '' . Moreover , X C2Cn -LRB- diam -LRB- C -RRB- -RRB- s = -LRB- X m2N R -LRB- m -RRB- s -RRB- n = -LRB- ? R -LRB- s -RRB- -RRB- n : For s > dimS -LRB- R -RRB- , infn2N -LRB- ? R -LRB- s -RRB- -RRB- n = 0 , hence dimH -LRB- NN -RRB- < = dimS -LRB- R -RRB- . The analogue of this theorem for IFS is well-known -LSB- 5 , p. 142f . -RSB- . By dimH -LRB- f1 ; : : : ; ngN -RRB- = dimS -LRB- Rn -RRB- < = dimH -LRB- NN -RRB- for every n , we see dimH -LRB- NN -RRB- > = dimS -LRB- R -RRB- . ? Note that with the assumption of the last theorem , the dimS -LRB- R -RRB- - dimensional Hausdorff measure of S n2Nf1 ; : : : ; ngN vanishes . We can show that the dimS -LRB- R -RRB- - dimensional Hausdorff measure of NN equals 1 if ? R -LRB- dimS -LRB- R -RRB- -RRB- = 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 find : 3.10 Theorem Let -LRB- X ; ae -RRB- be a bounded complete metric space . Let F be an IIFS on X. Then , dimS -LRB- RF -RRB- is an upperbound of the Hausdorff dimension of OE -LRB- ? NRF -RRB- . ? 11The idea of the part -LRB- 2 -RRB- of this proof is due to Dr. Staiger . In previous papers , we isolated property -LRB- 1 -RRB- using the term ` finite summability ' in that case . 10 Henning Fernau By -LSB- 14 -RSB- , for Euclidean spaces X ? Rm , we know a criterion called open set condition proving the coincidence of Hausdorff dimension and similarity dimension for IFS F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- : Assume there exists a non-empty bounded open test set V such that Sni = 1 F -LRB- i -RRB- -LRB- V -RRB- ? V and -LRB- 8i ; j 2 f1 ; : : : ; ng -RRB- -LRB- i < > j =-RRB- F -LRB- i -RRB- -LRB- V -RRB- \ F -LRB- j -RRB- -LRB- V -RRB- = ;-RRB- : We transfer this result to IIFS . 3.11 Theorem Let -LRB- X ; aeE -RRB- be some Euclidean space . Let F be an IIFS on X such that all corresponding IFS Fn satisfy some open set condition . Then , the Hausdorff dimension of OE -LRB- ? NRF -RRB- equals the similarity dimension s of RF . If the s-dimensional Hausdorff measure of AFrOE -LRB- ? NRF -RRB- is bounded , then s is also the Hausdorff dimension of AF . ? 3.12 Remark Instead of presuming an individual open set condition for each Fn , 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 S1i = 1 F -LRB- i -RRB- -LRB- V -RRB- ? V and -LRB- 8i ; j 2 N -RRB- -LRB- i < > j =-RRB- F -LRB- i -RRB- -LRB- V -RRB- \ F -LRB- j -RRB- -LRB- V -RRB- = ;-RRB- : It is clear that such a test set V for F might also serve as a test set for each Fn . 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 -LRB- X ; ae -RRB- , every closed set is a fixed point of an IIFS . Proof . Let fxn j n 2 Ng be dense in the subspace -LRB- A ; ae -RRB- ? -LRB- X ; ae -RRB- . Define F by F -LRB- n -RRB- -LRB- x -RRB- = xn . Then , AF = A. ? 4.2 Lemma There is a closed and bounded subset of a complete metric space that is a fixed point of a strict IIFS but not of an IFS . Proof . -LRB- NN ; aeR -RRB- is the fixed point of the syntactical IIFS FR . Since -LRB- NN ; aeR -RRB- is not compact , it is not a fixed point of an IFS . ? 4.3 Lemma There is a compact subset of a compact metric space that is not fixed point of a strict IIFS . Proof . Consider f1 ! ; 2 ! g ? f1 ; 2gN . ? Instead of taking the finite set f1 ! ; 2 ! g in the proof of the preceding lemma , it is also possible to take the infinite compact set f1!g -LSB- f2 ; 3g ! ? f1 ; 2 ; 3gN . 4.4 Lemma Every strict IFS F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- , n > 1 , containing pairwise different maps can be substituted by a strict IIFS G with pairwise different maps such that AF = AG and dimS -LRB- RF -RRB- = dimS -LRB- RG -RRB- . Proof . Consider the language L = f1 ; : : : ; n ? 1g -LSB- f1 ; : : : ; n ? 1g ? fng + . Define G = FF ; RF ; L . Since cl L ! = f1 ; : : : ; ng ! , AF = AG . Obviously , ? G -LRB- s -RRB- = Pn ? 1 i = 1 -LRB- RF -LRB- i -RRB- -RRB- s 1 ? RF -LRB- n -RRB- s : Hence , ? G -LRB- s -RRB- = 1 -LRB- -RRB- ? F -LRB- s -RRB- = Pni = 1 RF -LRB- i -RRB- s = 1 . ? IIFS 11 We can use a language L ? f1 ; : : : ; ng + in order to cut off pieces OE -LRB- cl L ! -RRB- of the attractor AF of a strict IFS F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- . 4.5 Lemma If the strict IFS F = -LRB- F -LRB- 1 -RRB- ; : : : ; F -LRB- n -RRB- -RRB- satisfies an open set condition with test set V , then V can also serve as a test set for FF ; RF ; L provided L is prefixfree . ? Interestingly , the sets cl L ! with L prefix-free are known in the theory of ! - languages as ` strongly connected ' -LSB- 20 -RSB- . This notion is equivalent to the irreducibility of the state matrix . Also refer to -LSB- 2 -RSB- . 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 , 1994 . References 1 . L. M. Andersson . Recursive Construction of Fractals . PhD thesis , Helsinki : Suomalainen Tiedeakatemia , Aug. 1992 . Annales Academiae Scientiarum Fennicae Series A , I. Mathematica , Dissertationes , 86 . 2 . C. Bandt . Self-similar sets . I. Topological Markov chains and mixed self-similar sets . Mathematische Nachrichten , 142:107 -LCB- 123 , 1989 . 3 . M. F. Barnsley . Fractals Everywhere . Boston : Academic Press , 1988 . 4 . L. Boasson and M. Nivat . Adherences of languages . Journal of Computer and System Sciences , 20:285 -LCB- 309 , 1980 . 5 . G. A. Edgar . Measure , Topology , and Fractal Geometry . Undergraduate Texts in Mathematics . New York : Springer , 1990 . 6 . D. B. Ellis and M. G. Branton . Non-self-similar attractors of hyperbolic iterated function systems . In J. C. 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O B I I O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

complete metric spaces
fractal geometry
hausdorff dimension
hausdorff measure
iterated function systems
selfsimilarity