Various Properties of Sturmian Words P

Czech Technical University in Prague Acta Polytechnica Vol. 45 No. 5/2005 Various Properties of Sturmian Words P. Baláži

This overview paper is devoted to Sturmian words. The first part summarizes different characterizations of Sturmian words. Besides the well known theorem of Hedlund and Morse it also includes recent results on the characterization of Sturmian words using return words or palindromes. The second part deals with substitution invariant Sturmian words, where we present our recent results. We generalize one-sided Sturmian words using the cut-and-project scheme and give a full characterization of substitution invariant Sturmian words.

Keywords: Sturmian words, mechanical words, 2-interval exchange map, palindromes, return words, substitutions.

1 Introduction on a substitution. In the last section we present some open problems related to generalizations of Sturmian words. In recent years, the combinatorial properties of finite and infinite words have become significantly important in fields of 2 Sturmian words physics, biology, mathematics and computer science. One of An infinite one-sided word the first impulses for extensive research in this field was the ww==()Î wwwK , discovery of quasi-crystals. nnN0 012 Normal crystal structures show rotational and translation- is a sequence of letters from a finite set A which is called an al symmetry. In 1982, however, Dan Shechtman discovered an alphabet. We use the notation N for the set of integers and =È = K aperiodic structure (which was formed by rapidly-quenched NN0 {}0 .Afinitewordvvvv01n - 1is a finite string = aluminum alloys) that has a perfect long-range order, but no of letters from A and vnis the length of v.Thesetofall three-dimensional translational periodicity (see e.g. [1] or finite words over the alphabet A is denoted by A*. A finite Ï ££ [2]). Since then many stable and unstable aperiodic structures word uA* is a factor of w if there exist 0 klsuch that = K e have been discovered. They are now known as quasi-crystals. uwkl w. The empty word is denoted by .Thesetoffac- tors of w of length n is written L (w) and the set of all factors of The problematics of aperiodic structures has been studied n w is denoted by L(w). The set L(w) is often called the language. from various points of view and there are numerous relations with other applications (besides solid-state physics), such as An infinite word w is ultimately periodic if there exist a = w w pseudo-random number generators [3], pattern recognition word u and a word v such that wuvwhere v is the infinite and symbolic dynamical systems. Early results are reported in concatenation of the word v.Itisperiodicifu is the empty [4] or [5]. word. If the infinite word w does not have any of the previous This paper is devoted to Sturmian words. Sturmian words forms we say that it is aperiodic. We say that u is a prefix (resp. = K Î ££ - + suffix) of vvvv01n - 1, vLwn (), if there exists01ln are infinite words over a binary alphabet with exactly n 1fac- = K = K ³ such that uv0 vl (resp. uvvll+-11 v n). The infinite tors of length n for each n 0. They represent the simplest ³ family of quasi-crystals. The history of Sturmian words dates word w, l 0, is a suffix of w. back to the astronomer J. Bernoulli III [6]. He considered the An infinite word w is uniformly recurrent if for every inte- sequence []()nn++112aa -[] + 12,wheren ³ 1anda is a ger k there exists an integer l such that each word of Lwk() occurs in every word of length l. positive irrational. Based on the continued fraction expan- The usual way of defining Sturmian words is via the sion of a, he gave (without proof) an explicit description of complexity function. Let w be an infinite word and Cw be a the terms of the sequence. There also exist some early works ® mapping NN0 ,suchthatCnw()is the number of different by Christoffel and Markoff. = factors of length n of w;i.e.Cnwn() # Lw (). Cw is called A. A. Markoff was the first to prove the validity of Ber- the complexity and we say that the word w is Sturmian if noulli’s description. He did that in his work [7], where he =+ = Cnw() n1for all n.SinceCw()12,Sturmianwordsarede- described the terms of the sequence [][][]()nn+-1 aaa -, = = ³ fined over a binary alphabet, say A {,}01. Clearly Cw()01 n 1 (later known as a mechanical sequence). for any w. The first detailed investigation of Sturmian words is due There have been some attempts to extend Sturmian words to Hedlund and Morse [4], who studied such words from the to words over alphabets with more than two letters, for in- point of view of symbolic dynamics and, in fact, introduced stance [8] or [9], but none of these constructions show such the term „Sturmian“; named after the mathematician Charles nice properties as Sturmian words. However, the approach of Francois Sturm. Arnoux and Rauzy presented in [8] resulted in another inter- It appears that there are several equivalent ways of con- esting family of words called Arnoux-Rauzy sequences. structing Sturmian words. We will describe some of them and Another combinatorial definition of Sturmian words is show the relationship with other notions from the combina- based on the distribution of letters in the word. Let w be an in- torics on words such as palindromes and return words. Then finite word, uv,()Î Lw be two factors of the same length we make some notes on an extension of Sturmian words using uv= and function d be defined the cut-and-project scheme. The next section is devoted to d(,)uv=- u v , the important question of the invariance of Sturmian words 00

where u denotes the number of occurrences of 0 in u.Wesay Txa()=+-+=+ xaaëû x { x a }, "Îx [,)01. 0 d £ Î that w is balanced if and only if (,)uv 1 for all uv,() Lw For the n-th iteration of the 2-interval exchange map Ta = with uv. Note that the structure of a balanced word over we obtain = the alphabet A {,}01 is formed, either by a block of 0’s be- ()n TxTxnxaa()==+ () {a }. tween two consecutive 1’s, or by a block of 1’s between two n consecutive 0’s. It is easy to see that the length of the blocks of Since a is irrational, Ta does not have any fixed point. It is 0’s between two consecutive 1’s (resp. blocks of 1’s between not difficult to check that two consecutive 0’s) differs at most by 1 in a balanced word. As ()n ëû()nxnxTx++=101aaëû +Ûa ()[, Î- a) we will see balanced words are equivalent to Sturmian words. In the literature one can find several constructions of which implies that ì ()n binary words. We start with the construction presented by 001if Ta ()ba)Î- [, , a b snab()= í HedlundandMorsein[4].Let , be real numbers, where , ()n ba)Î- a Î b Î = = î111if Ta () [ , . (,)01 and [,)01.Thenssnab,, ab(), ssnab,, ab()are sequences defined by, Let us show the most famous example of Sturmian words, =++-ab ab + ³ snab, ():(ëûëû n1 ) n ,forn 0, the Fibonacci word. =++-ab ab + ³ Example 2.1: Let j be the map given by j:,00110aawith snab, ():(éùéù n1 ) n ,forn 0. jjj= These words are usually referred to as mechanical words the property ()uv ()() u v for any finite words u, v over and a is called the slope and b the intercept. Mechani- the alphabet {0, 1}. Define the n-th finite Fibonacci word fn in = ³ the following way: f0 0 and for all n 0, cal words of s ab, resp. sab, have a nice geometrical 2 = j interpretation. Consider the integer lattice Z , the straight ffnn+1 (). line yx=+ab, x ³ 0, and two sequences of integer points = j =+ab =+ab Since f0 0 and (0) starts with 0, then fn is the prefix of Xnnn (),ëûand Ynnn (),éù. The elements of ³ fn +1 for each n 0. The Fibonacci word is defined as the limit Xn cover integer points of the lattice just below the line ofthesequenceofwordsf and thus yx=+aband the elements of Y cover points just above the n n lim f = 010010100100101001010010010K line. If snab()= 0 then the points X and X + lie on the hori- n , n n 1 n ®¥ zontal line, and if snab()=1 then they lie on the diagonal , Note that the further applications of the map fn do not line. The same holds for snab()and Y . In fact, the sequences , n change the Fibonacci word. Observe that the length fn is the s ab, sabare the coding of a discretization of a straight line. =+ , , n-th element of the Fibonacci sequence Fn,(FFnn--12 F n; a = = ³ Note that if is irrational then s ab, and sab, differ by at F0 1, F1 2 ), for all n 0.Itcanbeshownthatthe most for two values of n. Clearly, this can happen only if complexity of the Fibonacci word equals to n+1, thus the nab+ is an integer for some n. If we consider b = 0 and n = 0, word is Sturmian. The Fibonacci word also coincides with the = = 2 then s a,0()00and sa,0()01and we obtain the important mechanical word with slope 1 t and zero intercept, where special case of mechanical words t =+121() 5 is the golden mean. = = ³ sca,0 0 a , scaa,0 1 ; n 0. We finalize this part with a theorem by Hedlund and The infinite word ca is called the characteristic sequence Morse [4] which states that Sturmian, balanced and mechani- of a. cal words are indeed equivalent. Crispetal.study,intheirwork[10],anotherwayof Theorem 2.1: (Hedlund, Morse). Let w be an infinite word over constructing characteristic sequences. Consider again the in- the alphabet A = {,}01. The following conditions are equivalent: 2 = r r teger lattice Z and the straight line yx,where is a 1. w is Sturmian; positive irrational and x ³ 0. We label the intersections of the 2. w is balanced and aperiodic; line yx= r with verticals of the grid using 0, and we label by 1 3. there exist an irrational a, a Î(,)01 and a real b Î[,)01 such that the intersections of yx= r with horizontals. The sequence = abor = ab,forall ³ . of labels forms the so called cutting sequence and is denoted ws, ws, n 0 by Sr.Itcanbeshownthatcar= S if and only if ra=-()1 a Thereexistseveralproofsofthetheorem.Theorigi- (see e.g. [10]). nal proof [4] is of combinatorial nature, while the other One can often encounter the term r-interval exchange by Lunnon and Pleasants [14] is based on geometrical map in connection with infinite words. Properties of words considerations. generated by the r-interval exchange map (called the coding of the r-interval exchange map) are studied from different as- 2.1 Other characteristics of Sturmian words pects in [11], [12] or [13]. Let us closely mention the 2-interval We have mentioned several equivalent definitions of exchange map and its reference to Sturmian words. Sturmian words as those with minimal complexity, balanced Let a Î(,)01 be an irrational number and x Î[,)01.Let aperiodic sequences and mechanical words. In the past few =-a =-a years, there have been successful attempts to find a new I1 [,01 ), I2 [,)11be the decompositions of the interval [,)01.ThemapTa,givenbytherule characterization of Sturmian words. The first one, which ì xx+Î-aa),[,,for 01 we will describe uses return words, while the second uses Txa()= í palindromes. î xx+-aa)111,[,,for Î- Let w be a one-sided infinite word and u afactorofw.We is called the 2-interval exchange map. It can be written in a say that a finite word v isthereturnwordoveru if vu is a factor more complex way of w, u is a prefix of vu and there are exactly 2 occurrences of u

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S bb-= =++ be h| Î in vu. In other words, the return word v over u starts with the eh, (,]{:1 xnnnZn ëû }, occurrence of u and ends just before the next occurrence of u. S bb- thus the elements of eh, (,]1 form an increasing se- Example 2.2: Let 0100101001001010010100100101… be quence ()xnnZÎ . We can compute the lengths between two the Fibonacci word. The set of return words over 0101 con- consecutive points of ()xnnZÎ as tains words 01010010 and 01010. For clarity, here is the xxn+ -=ëûëû() ++1 eb - n eb + + h. Fibonacci word with indicated return words over 0101 : nn1 0100101001001010010100100101… The term ëûëû()nn++-1 eb eb +takes only two values 0 Î Vuillon [15] observed that the number of return words in- and 1 for each nZ. Thus there are only two distances be- S bb- + h h dicates whether a word is Sturmian or not. He showed the tween neighbors in eh, (,]1 ,namely1 and .Letus following theorem. define a sequence ()wnnZÎ of 0’s and 1’s by ì0 if xx+ -=h, Theorem 2.2: A binary infinite word w is Sturmian if and only if = í nn1 wn -=+h the set of return words over u has exactly two elements for every non î11if xxnn+1 . empty word u. The sequence ()w Î is a pointed bidirectional infinite Note that the proof of the necessary condition includes a nnZ word nice application of Rauzy graphs, which are often used for in- =KK vestigation of growth of the complexity in infinite words [8]. ()wwwwwwnnZÎ--21012 | , Let us focus on palindromes. A palindrome is a finite word where | denotes a delimiter. Since the distances between the S bb- that reads the same backwards as forwards. For instance, these consecutive points in eh, (,]1 are ordered as the me- are the first palindromes of the Fibonacci word: e,0,1,00, chanical word s eb, then w (i.e. the infinite word to the right 010, 101, 1001, 00100, 01010,… In [16], Droubay and Pirillo from the delimiter) is the Sturmian word. From the construc- K showed the characterization of Sturmian words by observing tion it is clear that the language of www012 isthesameas K the number of palindromes of even and odd length. the language of ww--21. The complexity function Cw has Theorem 2.3: An infinite word is Sturmian if and only if, for each been defined for the right sided infinite words, however nonnegative integer n, there is exactly one palindrome of length n, if the definition can be naturally extended to left sided infi- ¢ =K = ¢ n is even, and there are exactly two palindromes of length n, if n nite words, say www--21as : Cnwn¢() # Lw ( ).Since = ¢ == K is odd. Cnwn¢() # Lw ( ) # Lw nw () Cn () the word ww--21is The mapping that assigns to an integer n the number of also Sturmian. Since an arbitrary shift of the whole interval (,]bb-1 (i.e. we are keeping the unit length of the interval) palindromes of length n in a word is called the palindromic S bb- complexity. does not change the set eh, (,]1 , it just shifts the position of the delimiter, we can conclude that the word ()w Î is the In the context of palindromes and Sturmian words, let us nnZ pointed bidirectional infinite Sturmian word. From now on draw attention to paper [17]. The authors proved that the we will consider only bidirectional infinite words. number of palindromes in a word n of length n is less or is S bb- equal to n+1. Note that this holds for any kind of words (not Note that the set eh, (,]1 is called the cut-and-project bb- necessary Sturmian) over arbitrary alphabets. However in the set and the interval (,]1 is an acceptance window. case of Sturmian words it was shown in [17] that the number There are a couple of papers dealing with cut-and-project of all palindromes in a factor of length n is equal to n+1. The sets. In [18] it is shown that a cut-and-project set has either reader may like to try finding words of length n over a 2-letter two or three distances between adjacent points; two distances alphabet with the number of palindromes less than n+1; this correspond to the case of the unit length of the acceptance is not as trivial as it may appear to be. window and the distances form a Sturmian word. On the other hand, the words corresponding to the cut-and-project 2.2 Bidirectional Sturmian words sets with three distances are exactly those which arise from the coding of the 3-interval exchange map, and vice versa. In In the previous sections we have outlined several charac- [19], [20] the authors study substitution properties and the teristics of Sturmian words. However we have limited our- substitutivity of cut-and-project sets. selves, from the very first definitions, to one-sided infinite words. This restriction is typical for a large number of papers, in spite the fact that the definitions of notions like balanced 3 Sturmian words and substitutions words, mechanical words etc. can be extended, very naturally, to both sides. The question is, whether the above listed theo- Let us take a look at Sturmian words from a different point rems still hold for bidirectional infinite words. Let us show a of view. One can see in Example 2.1 that there exist Sturmian way of generating bidirectional infinite words from which it words which are generated by certain maps (here we will call will be clear that such a generalization is possible. them substitutions). In fact, the mentioned Fibonacci word is a fixed point of the map j. The question is whether this is a Consider h, e fixed positive irrational numbers, he¹-, general property of all Sturmian words, or whether there ex- b Î[,)01 and the set ists a class of Sturmian words invariant under substitutions. S bb-=+ hb -<-£ eb Î eh, (,]{|11mn mn ,,}đ mnZ. Let us now state basic definitions and then we give an over- It follows that m has to satisfy view of the most interesting results. be1+-<£+nmn be, A morphism j is a map of A* into itself satisfying jjj= Î hence mn=+ëûbe, i.e. any element of Seh(,]bb-1 has the ()uv ()() u v for each uv,* A . The morphism is called , j form ëûbe++nn h,forsomenZÎ .Usingthisfactwecan non-erasing if ()ai is not an empty word for any Î j write aAi . A non-erasing morphism is called a substitution.

22´ The action of j can be extended to bidirectional infinite fined for every fixed Sturm number a amatrixMa ÎQ words that is called the generating matrix of a and is closely related ()wwwwwwÎ--=KK | to the smallest solution of a Pell equation. They showed that a nnZ 21012 a b Î Sturmian sequence s ab, ( Sturm number, [,)01)isafixed by point of a substitution with substitution matrix A if and only if KKjj jjj l ³ (ww--21012 )( )|( www )( )( ) A has the form Ma,forsomel 1. We complete this overview by giving a few notes on paper We say that the word ()wnnZÎ is invariant under the substitution j (or is a fixed point of j)if [27]. The authors have completely solved the question of the substitution invariance of pointed bidirectional Sturmian KKKKjj jjj = (ww--21012 )( )|( www )( )( ) wwwww -- 21012 | words. The main theorem shown in the paper is the following. Suppose that we have a substitution j and there exist let- Theorem 3.1: Let a be an irrational number, a Î(,)01, b Î[,)01. ters aa, Î A,suchthatj()aua= and j()aav= for some ij ii jj The pointed bidirectional Sturmian word with slope a and intercept b non-empty words uv,*Î A .Thenbyrepeatedapplicationof j is invariant under a non-trivial substitution if and only if the follow- on the pair aaij| of letters separated by the delimiter | we jj()n ()n Î ing three conditions are satisfied: obtain words ()|aai (j ), nNof increasing length. a Clearly 1. is a Sturm number, -- 2. baÎQ(), jj()n ()|aaua ()n ( )= j (n 11 ) ()| j (n ) ( av ) , i jn i jn 3. ab¢ £ ¢ £-1 a¢ or 1 - aba¢ £ ¢ £ ¢,whereb¢ is the image of b Î jj()n ()n under the Galois automorphism of the quadratic field Q()a . for certain words uvnn,* A.Thelimitof ()|aai (j ) ®¥ for n is an infinite bidirectional word ()wnnZÎ and we Note that this result is analogous to those derived in [25] j say that generates the word ()wnnZÎ . and [26] for the one-sided case. Let us define a weaker notion of a substitutive word. The proof presented in [27] is constructive based on =KKthe cut-and-project scheme that was sketched in the para- We say that ()wwwwwwnnZÎ--21012 | is a substitutive word, if there exists a substitution j on an alphabet B graph devoted to bidirectional words. This approach has not =KKbeen used yet in the study of substitution invariant Sturmian with a fixed point ()vvvvvvnnZÎ--21012 | and a map c a = c Î words. It turns out to be a good choice because the proof is :BAsuch that wvnn(), for each nZ.Notethatall fixed points of a substitution are substitutive. The opposite is simple. One of the advantages of the cut-and-project scheme not true. isthatthemoredifficultpartsoftheproofcanbeillus- Let Aa= {,K , a }be an alphabet. To a substitution j one trated on vivid examples, which makes the whole paper more 1 k ´ comprehensible. may assign a substitution matrix A Î Nkkin the following way: We believe that methods similar to those in [27] can solve the question of the substitution invariance of words over a A :()= number of lettersaa in the word j . ij j i 3 letter alphabet, which arise from the coding of the 3-interval The problem of invariance under a substitution (or the exchange map. weaker notion of substitutivity) has motivated many papers. There are some partial results, where authors consider only one sided Sturmian words or characterize substitution invari- 4 Open problems ant bidirectional Sturmian words depending on the slope a if the intercept b = 0. We have summarized several interesting properties of Crispetal.[10]carriedontheworkofBrown[21]and Sturmian words and here we would like to highlight how the studied substitution invariant cutting sequences. They proved results about Sturmian words can help in further advances in that the cutting sequence Sr (resp. the mechanical sequence the field of aperiodic words. As Sturmian words are the sim- ca is substitution invariant if and only if the continuous frac- plest aperiodic structures, the question of generalisation of tion expansion of r (resp. a) has a certain form. The author in resultsobtainedforSturmianwordsisparticularlyinterest- [22] used some of their results and simplified their condition ing. The direct generalization of Sturmian words are infinite on the invariance of the characteristic sequence ca under a words which arise from the coding of a 3-interval (resp. r-in- substitution. He showed that ca, a Î(,)01, is invariant under a terval) exchange map. We believe that results and techniques substitution j if and only if a is a quadratic irrational with used in the study of Sturmian words can be applied with conjugate a¢ Ï(,)01.Sucha is called a Sturm number. This success to open problems connected with words coding a 3-in- result was shown independently by Allauzen, [23]. terval exchange map. Below you may find a list of the most Parvaix [24] proved that bidirectional non-pointed interesting issues. Sturmian words with b ¹ 0 are invariant under a substitution 1. A necessary and sufficient condition on the substitution if and only if a is a Sturm number and the intercept b belongs invariance of the words coding a 3-interval exchange to the quadratic field Qa()=+ { a ba | a,bQ Î }. map is still not known. The problem of finding a similar A complete characterization of infinite one-sided substitu- condition to that of Theorem 3.1 is still open, but the tech- tion invariant Sturmian words was done by Yasutomi [25]. niques used in [27] may bring some answers. Berthé et al. [26] studied infinite words which arise from the 2. Giving a description of substitution matrices of substi- coding of the 2-interval exchange map and gave an alterna- tutions which generate the words coding a 3-interval ex- tive proof of Yasutomi’s result using Rauzy fractals associated change map is a problem closely related to the previous with invertible primitive substitutions. The authors also de- one. The question of finding generating matrices of such

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substitutions seems to be very challenging. This issue was [14] Lunnon, W. F., Pleasants, P. A. B.: “Characterization of solved in [26] for Sturmian words. Two-Distance Sequences.” J. Austral. Math. Soc., Vol. 53 3. Although a description of palindromes in the words cod- (1992), p. 198–218. ing a 3-interval exchange map is already known [17], we [15] Vuillon, L.: “A Characterization of Sturmian Words by believe that the use of the cut-and-project scheme can Return Words.” European J. Combin., Vol. 22 (2001), give an alternative proof of this result. Nevertheless the p. 263–275. palindromic complexity is described only for a 2-interval [16] Droubay, X., Pirillo, G.: “Palindromes and Sturmian exchange map (see theorem 2.3) and a 3-interval ex- Words.” Theoret. Comput. Sci., Vol. 223 (1999), p. 73–85. change map [28], but we believe that a general description [17] Droubay, X. et al.: “Episturmian Words and some Con- canbeobtainedbyconsiderationsbasedonthecut-and- structions of de Luca and Rauzy.” Theoret. Comput. Sci., -project scheme. Vol. 225 (2001), p. 539–553. [18] Guimond, L. S. et al.: “Combinatorial Properties of Infi- 5 Acknowledgment nite Words Associated with Cut-and-Project Sequences.” The author acknowledges partial funding from the Czech J. de Th. des Nom. de Bordeaux,Vol.15 (2003), p. 697–725. Science Foundation GA, ČR 201/05/0169. The author is also [19] Baláži, P.: “Infinite Words Coding 3-Interval Ex- very obliged to Edita Pelantová and Zuzana Masáková for change.” Diploma work, 2003. their help and discussions on the paper. [20] Masáková, Z. et al.: “Substitution Rules for Aperiodic Se- quences of Cut and Project Type.” J. of Phys. A: Math. References Gen., Vol. 33 (2000), p. 8867–8886. [21] Brown, T. C.: “A Characterization of the Quadratic Irra- [1] Shechtman, D. et al.: “Metallic Phase with Long-Range tionals.” Canad. Math. Bull., Vol. 34 (1991), p. 36–41. Orientational Order and No Translational Symmetry.” [22] Baláži, P.: “Jednoduchá charakteristika Sturmových čí- Phys.Rev.Lett.Vol.53 (1984), p. 1951–1953. siel.” Proc. of 9th Math. Conf. of Tech. Uni., 2001, [2] Bombieri, E., Taylor, J. E.: “Which Distributions of Mat- p. 5–15. ter Diffract? An Initial Investigation.” J. Phys. Colloque [23] Allauzen, C.: “Une charactérisation simple des nombres C3,Vol.14 (1986), p. 19–28. de Sturm.” J. de Th. des Nom. de Bordeaux,Vol.10 (1998), [3] Guimond,L.S.,Patera,J.:“ProvingtheDeterministic p. 237–241. Period Breaking of Linear Congruential Generators Us- [24] Parvaix, B.: “Substitution Invariant Sturmian Bise- ing Two Tile Quasicrystals.” Math. of Comput.,Vol.71 quences.” J. de Th. des Nom. de Bordeaux,Vol.11 (1999), (2002), p. 319–332. p. 201–210. [4] Hedlund, G., Morse, M.: “Symbolic Dynamics II: Sturm- [25] Yasutomi, S.: “On Sturmian Sequences which are Invari- ian Sequences.” Amer.J.Math., Vol. 61 (1940), p. 1–42. ant under some Substitutions.” Number Theory and its Ap- [5] Venkov, B. A.: Elementary Number Theory, Groningen: plications,Vol.2 (1999), p. 347–373. Wolters-Noordhoff, 1970. [26] Berthé, V. et al.: Invertible Substitutions and Sturmian [6] Bernoulli, J.: “Sur une nouvelle espece de calcul.” Receuil Words: An Application Of Rauzy Fractals, preprint pour les astronomes,Vols.1, 2, Berlin, 1772. 2003. [7] Markoff, A. A.: “Sur une question de Jean Bernoulli.” [27] Baláži, P. et al.: “Complete Characterization of Sub- Math. Ann.,Vol.19 (1882), p. 27–36. stitution Invariant Sturmian Sequences.” To appear in [8] Arnoux, P., Rauzy, G.: “Représentation géométrique Integers, 2005. de suites de complexité 2n+1.” Bull. Soc. Mat. France, [28] Damanik D., Zamboni L. Q: Arnoux-Rauzy Subshifts: Lin- Vol. 119 (1991), p. 199–215. ear Recurrences, Powers, and Palindromes,arXiv:math. [9] Coven, E. M.: “Sequences with Minimal Block Growth CO/0208137v1. II.” Math.Sys.Theory,Vol.8 (1973), p. 376–382. [29] Berstel, J.: “Recent Results on Extensions of Sturmian [10] Crips, D. et al.: “Substitution Invariant Cutting Se- Words.” Internat.J.AlgebraComput.,Vol.12 (2002), quences.” J. de Th. des Nom. de Bordeaux,Vol.5 (1993), p. 371–385. p. 123–137. [11] Adamczewski, B.: “Codages de rotations et phénomPnes d’autosimilarité.” J. de Th. des Nom. de Bordeaux,Vol.14 Ing. Peter Baláži (2002), p. 351–386. e-mail: [email protected] [12] Alessandri, B., Berthé, V.: “Three Distance Theorems Department of Mathematics and Combinatorics on Words.” L’Enseignement Mat., Vol. 44 (1998), p. 103–132. Czech Technical University in Prague [13] Ferenczi, S. et al.: “Structure of Three Interval Exchange Faculty of Nuclear Sciences and Physical Engineering Transformations I: an Arithmetic Study Ann. Inst. Fou- Trojanova 13 rier,Vol.51 (2001), p. 861–901. 120 00 Prague 2, Czech Republic