Directive words of episturmian words: equivalences and normalization

Directiv e w ords of episturmian w ords: equiv alences and normalizatio n Am y Glen † Florence Lev ´ e ‡ Gw ´ ena ¨ el Ric homme ‡ F eb ruary 26 , 2008 Abstract Episturmian morphisms constitute a powerful to ol to study episturmia n words. Indeed, any episturmian word can be infinitely decomp osed ov er the s et of pure episturmian mo rphisms. Thu s, an episturmian w or d can b e defined b y one of its mo rphic decomp ositions o r , equiv alent ly , by a certain directive word. Here we characterize pair s of words dir ecting a common episturmian word. W e also prop ose a way to uniquely define any e pisturmian word through a normaliza tio n of its directiv e w ords. As a consequence o f these r esults, we characterize episturmian words having a unique directive word. Keyw ords : episturmian w ord; Sturmian w ord; Arnoux-Rauzy sequence ; episturmian mor- phism; dire ctive word. MSC (20 00): 68 R1 5. 1 In tro duction Since th e seminal works of Morse and Hedlun d [21], Sturmian wor ds hav e b een widely studied and their b eautiful prop erties are related to many fields like Nu mber Theory , Geometry , Dynam- ical Systems, and Combinatorics on W ord s (see [1, 20, 23, 3] for recent surveys). These infinite wo rd s , which are defin ed on a binary alphab et, hav e numerous equiv alent d efinitions and c harac- terizatio ns. No wada ys most wo rks deal with generalizations of S turmian wo rds to arbitrary fin ite alphab ets. T wo v ery interesting generalizatio ns are very close : th e Arno ux-R auzy se q u enc es (e.g., see [2, 14, 23, 30]) and episturmian wor ds (e.g., see [5, 13, 15]). The firs t of these t wo families is a particular su b class of the second one. More precisely , the family of episturmian words is composed of the Ar noux-Rauzy sequ en ces, images of the Ar noux-Rauzy sequ en ces by episturmian morp hisms , and certain p erio dic infi nite wo rd s . In the binary case, Arn oux-Rauzy sequences are exactly the Sturmian wo rd s wh ereas episturmian wo rds include all recur r ent b alanc e d words, that is, p erio dic balanced w ords and Sturmian w ords (see [10, 22, 29] for recen t results relating episturmian w ords to the balanced prop erty). See also [9 ] for a recen t survey on episturmian theory . Episturmian morph isms p la y a central r ole in the study of these words. Introduced fir st as a generalizat ion of Stur m ian morphisms, Justin and Pir illo [13] show ed that they are exac tly the morphisms that preserve the ap erio dic epistu r mian words. They also prov ed that an y episturmian † Am y Glen LaCIM, Universit ´ e du Qu´ ebe c ` a Mo nt r´ eal, C.P . 8888 , s ucc ursale Cent re-v ille, Montr ´ ea l, Qu´ ebec, CANAD A, H3C 3P8 E-mail: am y.gle n@gmai l.com (with the supp or t of CRM-ISM-LaCIM) ‡ F. Lev´ e, G. Ric homme Univ ers it´ e de Picardie Jules V erne, Lab oratoir e MIS (Mo d´ elisation, Information, Syst` emes) 33, Rue Sain t Leu, F-80039 Amiens cedex 1, FRANCE E-mail: {f loren ce.lev e, gwe nael. richomme}@u-picardie.fr 1 wo rd is the image of another episturmian w ord by some so-called pur e episturmian morphism . Eve n more, any epistur mian word can b e infi nitely decomp osed o ver the s et of pur e episturm ian morphisms. This last prop erty allows an episturmian word to b e defi ned by one of its morphic decomp ositions or, equiv alen tly , by a certain dir e ctive wor d , which is an in finite sequence of ru les for d ecomp osing the giv en episturmian word by morphisms. In consequence, many pr op erties of episturmian words can b e deduced from prop erties of epistu r mian morph ism s. This approach is used for instance in [4, 8, 16, 28, 29, 30] and of course in the pap ers of Justin et al. In S ection 2, we recall u seful resu lts on episturmian wo rds and explain the vision of morp hic decomp ositions and directiv e words int ro duced by Justin and Pirillo in [13]. An episturm ian word can hav e several d irectiv e words. T he question: “When do tw o words direct a common episturmian word?” was considered in [15]. Usin g a blo ck-e quiv alence notion for directiv e words, Justin and Pirillo provided several results to answer this question in most cases (see Section 3). In Section 4, we state a complete result characte rizing the form of words d irecting a common epistur mian wo rd, without using blo ck-equiv alence. In [4], Berth´ e, Holton, and Zamb oni sho w that any S turmian word has a u nique directive word with some p articular prop erties. I n [18], the second and third authors rephrased this resu lt and used it to c haracterize all quasip eriod ic Sturmian words. In Section 5, we extend this result to all episturmian words by introducing a way to n orm alize the directive words of an episturmian word so that any episturmian word can b e defin ed u niquely by its normalize d dir e ctive wor d , d efi ned by some factor av oidance (T h eorem 5.2). Th is result was previously presented at the Sixth International Conference on W ord s [17] to c haracterize all quasip er io dic episturmian words (see also [11]). As an application of the previous results, we end this pap er with a charac terization of epistur mian wo rd s h a ving a un ique directiv e word. 2 Episturmian w ords and morphisms W e assume the reader is f amiliar w ith com binatorics on words and m orp hisms (e.g., see [19, 20]). In this section, we recall some basic definitions and pr op erties relating to episturmian words which are needed later in the p ap er. F or the most part, w e follo w the notation and terminology of [5, 13, 15, 10]. 2.1 Notation and terminology Let A denote a finite alphab et . A finite wor d o ve r A is a finite sequence of letters from A . The empty wor d ε is the empty sequence. Und er the op eration of concatenation, the set A ∗ of all finite wo rd s o ver A is a fr e e monoid with identit y element ε and s et of generators A . T he set of non-empty wo rd s ov er A is the fr e e semigr oup A + = A ∗ \ { ε } . Giv en a finite word w = x 1 x 2 · · · x m ∈ A + with eac h x i ∈ A , the length of w is | w | = m . T h e length of the empty word is 0. By | w | a we denote the number of o ccurrences of the letter a in the wo rd w . If | w | a = 0, th en w is said to b e a -fr e e . F or an y integer p ≥ 1, the p -th p ow er of w is the wo rd w p obtained by concate nating p o ccurrences of w . A (right) i nfinite wor d x is a s equence in dexed by N + with v alues in A , i.e., x = x 1 x 2 x 3 · · · with eac h x i ∈ A . Th e set of all infinite words o ve r A is den oted by A ω . Given a n on-empty fin ite wo rd v , we denote by v ω the infinite word obtained by concate nating v with itself infi nitely many times. F or easier reading, infin ite wo rds are hereafter typed in b oldface to distinguish them fr om finite words. Giv en a set X of w ords, X ∗ (resp. X ω ) is the set of all finite (r esp. infinite) words that can be obtained by concatenating words of X . The empty word ε b elongs to X ∗ . A finite word w is a factor of a finite or infinite word z if z = uw v f or some words u , v (where v is infin ite iff z is infi nite). F u rther, w is called a pr efix (resp. suffix ) of z if u = ε (resp. v = ε ). W e use the notation p − 1 w (resp. w s − 1 ) to indicate th e remov al of a pr efix p (resp. suffix s ) of the 2 wo rd w . The alphab et of a word w , denoted by Alph( w ) is the set of letters o ccurring in w , and if w is infinite, we d en ote by Ult( w ) the set of all letters o ccurring infinitely often in w . 2.2 Episturmian w ords In th is pap er, our vision of episturmian words will b e the c haracteristic prop erty stated in The- orem 2.1 b elo w. Neve rtheless, to give an idea of wh at an episturmian word is, let us give one of the equiv alent defi nitions of an episturmian word provided in [5]. Before doing so, we recall that a factor u of an infinite word w ∈ A ω is right (resp . left ) sp e cial if ua , ub (resp. au , bu ) are factors of w for s ome letters a , b ∈ A , a 6 = b . W e recall also that the r eversal e w of a finite word w is its mirror image: if w = x 1 . . . x m − 1 x m , then e w = x m x m − 1 · · · x 1 . An infin ite wo rd t ∈ A ω is episturmian if its set of factors is closed under reversal and t has at most one right (or equiv alently left) sp ecial factor of each length. Moreov er, an episturmian word is standar d if all of its left sp ecial factors are prefixes of it. In the initiating pap er [5 ], episturmian words were defined in tw o steps. Standard episturmian wo rd s were fir st introduced and stu d ied as a generalization of standard S turmian words. (Note that in the rest of this paper, w e refer to a stand ard episturmian word as an epistand ar d wor d , for simplicit y). Th en an epistur mian word was defi n ed as an infin ite word h a ving exactly the same set of factors as some epistandard word. Moreo ve r, it was prov ed in [5] that an y episturmian word is r e curr ent , that is, all of its factors occur infin itely often (actually episturmian words are uniformly recurrent b ut this w ill not b e needed here). An ultimately p erio dic infi n ite word is a word that can b e written as uv ω = uv v v · · · , for some u , v ∈ A ∗ , v 6 = ε . If u = ε , then such a word is p erio dic . S ince they are recur r ent, all ultimately per io dic episturmian words are p eriod ic. Let us recall that an infin ite word that is n ot ultimately p erio dic is said to b e ap e rio dic . 2.3 Episturmian morphisms T o s tudy epistur mian words, Justin and P irillo [13] intro d uced episturmian morphisms . In p artic- ular th ey pro ved that these m orphisms (defined b elow) are p recisely th e morphisms that pr eserve the set of ap erio dic episturmian words. Let us recall th at given an alphab et A , a morp hism f on A is a map from A ∗ to A ∗ such that f ( uv ) = f ( u ) f ( v ) for any words u , v o ve r A . A morphism on A is entirely defined by the images of letters in A . All m orphisms considered in this pap er will b e n on -erasing: th e image of any non-empty word is never empty . Hence the action of a morphism f on A ∗ can b e naturally extended to infi nite wo rds; that is, if x = x 1 x 2 x 3 · · · ∈ A ω , then f ( x ) = f ( x 1 ) f ( x 2 ) f ( x 3 ) · · · . In what follo ws, we will denote the comp osition of morphisms by juxtap osition as for concate- nation of words. Episturmian morph isms are the compositions of the p ermutation morph isms (the morphism s f such that f ( A ) = A ) and th e morphisms L a and R a where, for all a ∈ A : L a :  a 7→ a b 7→ ab , R a :  a 7→ a b 7→ ba for all b 6 = a in A . Here we will work only on pur e episturm ian morphisms, i.e., morph isms obtained by comp osition of elemen ts of the sets: L A = { L a | a ∈ A} and R A = { R a | a ∈ A} . Note. In [13], the morphism L a (resp. R a ) is denoted by ψ a (resp. ¯ ψ a ). W e adopt the current notation to emphasize the action of L a (resp. R a ) wh en applied to a word, which consists in placing an o ccurrence of the letter a on th e l eft (r esp . right) of each o ccurrence of any letter different from a . 3 Epistandar d morphisms are th e morphisms obtained by concatenation of morph isms in L A and p ermutatio ns on A . Likewise, the pur e episturmian morph isms (resp. pur e epistandar d morphisms ) are the morphism s obtained b y concate nation of morphism s in L A ∪ R A (resp. in L A ). Note that the episturmian morph isms are exactly the Sturmian morph isms when A is a 2-letter alphab et. All episturmian m orp hisms are injectiv e on b oth finite and infin ite words. The monoid of episturmian morp hisms is left c anc el lative (see [26, Lem. 7.2]) which means that for any episturmian morphisms f , g , h , if f g = f h then g = h . Note that this fact, which is a by-product of th e injectivity , can also b e seen as a consequence of the in vertibilit y of these morphism s (see [7, 12, 26, 32 ]). 2.4 Morphic decomp osition of episturmian w ords Justin and Pirillo [13] prov ed the follo wing ins ightful characteriza tions of ep istand ard and epis- turmian words (see Theorem 2.1 b elow), whic h show that any ep isturmian word can b e infinitely de c omp ose d o ver the set of p ure episturmian morph isms. The statement of Theorem 2.1 needs some extra d efinitions and notation. First we define th e following new alphab et, ¯ A = { ¯ x | x ∈ A} . A letter ¯ x is considered to b e x with spin R , whilst x itself h as spin L . A fi nite or infin ite word o ver A ∪ ¯ A is called a spinne d wo rd . T o ease the r eading, we sometimes call a letter with spin L (resp. spin R ) an L -sp inned (resp. R -spinned ) letter. By extension, an L -spinned (resp. R -sp inned) word is a wo rd having only letters with sp in L (resp. sp in R ). The opp osite ¯ w of a finite or infinite sp inned word w is obtained f rom w b y exchanging all spins in w . F o r instance, if v = ab ¯ a , then ¯ v = ¯ a ¯ ba . Wh en v ∈ A + , then its opp osite ¯ v ∈ ¯ A + is an R -spinn ed word and we set ¯ ε = ε . Note th at, giv en a finite or infi nite word w = w 1 w 2 . . . ov er A , we sometimes denote ˘ w = ˘ w 1 ˘ w 2 · · · any spinned word s u ch that ˘ w i = w i if ˘ w i has spin L and ˘ w i = ¯ w i if ˘ w i has spin R . Such a wo rd ˘ w is called a spinne d version of w . Note. In Justin and Pirillo’s original pap ers, spins are 0 and 1 instead of L and R . It is conv enient here to change this vision of the spins b ecause of the r elationship with episturmian morphisms, which we now recall. F or a ∈ A , let µ a = L a and µ ¯ a = R a . This op erator µ can b e n aturally extended (as d one in [13]) to a morph ism from the free monoid ( A ∪ ¯ A ) ∗ to a pure epistu rmian morph ism: for a sp inned finite word ˘ w = ˘ w 1 . . . ˘ w n o ver A ∪ ¯ A , µ ˘ w = µ ˘ w 1 . . . µ ˘ w n ( µ ε is the identit y morphism). W e sa y that the word w dir e cts or is a dir e ctiv e wor d of th e m orp hism µ w . The following r esult extends the notion of d ir ectiv e words to infinite episturm ian w ords. Theorem 2.1. [13] i ) An infinite wor d s ∈ A ω is epistandar d if and only if ther e exist an infinite wor d ∆ = x 1 x 2 x 3 · · · over A and an infinite se quenc e ( s ( n ) ) n ≥ 0 of infinite wor ds such that s (0) = s and for al l n ≥ 1 , s ( n − 1) = L x n ( s ( n ) ) . ii ) An infinite wor d t ∈ A ω is episturmian if and only if ther e exist a spinne d infinite wor d ˘ ∆ = ˘ x 1 ˘ x 2 ˘ x 3 · · · over A ∪ ¯ A and an infinite se quenc e ( t ( n ) ) n ≥ 0 of r e curr ent infinite wor ds such that t (0) = t and for al l n ≥ 1 , t ( n − 1) = µ ˘ x n ( t ( n ) ) . F or any epistand ard word (resp. episturmian word) t and L -spinn ed infinite word ∆ (resp. spinned infinite word ˘ ∆) satisfying the conditions of the abov e th eorem, we say th at ∆ (resp. ˘ ∆) is a (spinne d) dir e ctive wor d for t or that t is dir e cte d by ∆ (resp. ˘ ∆). Notice that this directiv e word is exactly the one that arises f rom the equiv alent definition of epistandard words that uses p alindr omic closur e [5, 9 , 13] and, in the b in ary case, it is related to the con tinued fraction of the slop e of the straight line represented by a standard wo rd (see [20 ]). It follo ws immediately from Theorem 2.1 that, with the notation of case ii ), each t ( n ) is an episturm ian w ord directed by ˘ x n +1 ˘ x n +2 · · · 4 The natural question: “Does any sp in ned infi nite word direct a uniqu e episturm ian word?” is answered in [13]: Prop osition 2.2. [13, Prop. 3.11] 1. Any spinne d infinite wor d ˘ ∆ having infinitely many L - spinne d letters dir e cts a uniqu e epis- turmian wor d b e ginning with the left-most letter having spin L in ˘ ∆ . 2. Any R -spinne d infinite wor d ˘ ∆ dir e cts exactly | Ult(∆) | episturmian wor ds. 3. L et ˘ ∆ b e an R -spinne d i nfinite wor d, and let a b e a letter such that ¯ a ∈ Ult( ˘ ∆) . Then ˘ ∆ dir e cts exactly one episturmian wor d starting with a . Note. In [13], item 3 was stated in the more general case where ˘ ∆ is ultimately R -spinn ed. In th is case, ˘ ∆ still directs exactly one epistu rmian word for each letter ¯ a in Ult( ˘ ∆), but contrary to what is written in [13], nothing can b e said on its first letter. As a consequen ce of the p revious prop osition and part i ) of Theorem 2.1, any L -sp inned infin ite wo rd directs a un ique epistandard word. The following imp ortant remark links th e tw o parts of Theorem 2.1. Remark 2.3. [13] If ˘ ∆ is a sp inned version of an L -spinned word ∆ and if t is an ep istu rmian wo rd d irected by ˘ ∆, then the set of factors of t is exactly the set of factors of the epistandard word s directed by ∆. Moreo ve r (with the same notatio n as in the previous remark): Remark 2.4. The ep istu rmian word t is p erio dic if and only if the epistandard word s is p erio dic, and this holds if an d only if there is only one letter occurrin g infin itely often in ∆, that is, | Ult(∆) | = 1 (see [13, Prop. 2.9]). More pr ecisely , a p erio dic epistur mian word tak es the form ( µ ˘ w ( x )) ω for some fin ite sp inned word ˘ w and letter x . Note. S turmian words are precisely the ap erio dic episturmian words on a 2-letter alphab et. When an ep istu rmian word is ap erio dic, we hav e the follo wing fu ndamental link b etw een the wo rd s ( t ( n ) ) n ≥ 0 and th e spinn ed in finite word ˘ ∆ o ccur ring in Theorem 2.1: if a n is the fir st let- ter of t ( n ) , then µ ˘ x 1 ... ˘ x n ( a n ) is a prefix of t and the sequence ( µ ˘ x 1 ... ˘ x n ( a n )) n ≥ 1 is not u ltimately constan t (since ˘ ∆ is not ultimately constant), then t = lim n →∞ µ ˘ x 1 ··· ˘ x n ( a n ). This fact is a slight generalizat ion of a result of Risley and Zamboni [30, Pr op. I I I.7] on S-adic r epr esentations for c haracteristic Arnoux-Rauzy sequences. See also the recent pap er [4] for S-adic representations of Sturmian wo rd s . Note that S -adic dynamic al systems were introd uced by F erenczi [6] as minimal dynamic al systems (e.g., see [23]) generated by a finite number of sub stitutions. In the case of episturmian words, the notion itself is actually a reformulation of the well -known R auzy rules , as studied in [25]. T o anticipate next sections, let us also obs er ve: Remark 2.5. [13] If an ap eriod ic epistur m ian w ord is d irected by tw o spinn ed words ∆ 1 and ∆ 2 , then ∆ 1 and ∆ 2 are spin ned v ersions of a common L -spinned word. This is n o longer true for perio dic episturmian wo rd s ; for instance ab ω and ¯ ba ω direct the same episturmian word ( ab ) ω = ababab · · · . 3 Kno wn results on directiv e-equiv alen t w ords W e h av e jus t seen an examp le of a p erio dic episturmian word that is d irected by t wo d ifferent spinned infinite words. This situation holds also in the ap eriodic case (see [13, 15]). F or example, 5 the T rib onac ci wor d (or R auzy wor d [24]) is directed by ( abc ) ω and also by ( abc ) n ¯ a ¯ b ¯ c ( a ¯ b ¯ c ) ω for eac h n ≥ 0, as well as infinitely many other spin ned words. More generally , by [13], any epistandard wo rd has a un ique L -sp inned directive word but also has other directive w ords (see also [15] and Theorem 4.1). W e no w consider in detail the following tw o questions: When do t wo fi n ite spinned words direct a common epistur m ian morphism? When do tw o spinned infi nite words direct a common unique episturmian word? W e say that that tw o finite (resp. infinite) sp inned words are dir e ctive-e quivalent wo rd s if they direct a common episturmian morphism (resp. a common episturmian word). In S ection 3.1, we recall the charact erizations of d irectiv e-equiv alent finite spinned words. In Section 3.2, we recall known resu lts ab out directiv e-equiv alent infinite wo rds . Section 4 will present a new characte rization of these words. 3.1 Finite directive -equiv alen t words: presen t ation ve rsus blo ck-equ iv alence Generalizing a study of the monoid of Stur mian morphism s by S´ e ´ eb old [31], the th ird author [26] answered the question: “When do tw o spinn ed finite words direct a common episturmian morphism?” by giving a p resentat ion of the monoid of episturmian morph ism s. This result was reformulate d in [27] u sing another set of generators and it was ind ep en dently and differently treated in [15]. As a dir ect consequence, one can see that the monoid of pu re epistand ard morph isms is a free m on oid and one can obtain the follo wing p resentat ion of the monoid of pur e epistu rmian morphisms: Theorem 3.1. (direct consequence of [27, Prop. 6.5]; reform ulation of [15, Th. 2.2]) The monoid of pur e episturmian morphisms with { L α , R α | α ∈ A} as a set of gener ators has the fol lowing pr esentation: R a 1 R a 2 . . . R a k L a 1 = L a 1 L a 2 . . . L a k R a 1 wher e k ≥ 1 is an inte g er and a 1 , . . . , a k ∈ A with a 1 6 = a i for al l i , 2 ≤ i ≤ k . This result m eans that t wo different compositions of morp hisms in L A ∪ R A yield a common pure epistu r mian morphism if and only if one comp osition can b e deduced fr om the other one in a r ewriting sys tem, called the blo c k -e qu i valenc e in [15]. Although Th eorem 3.1 allo ws us to show that m any prop erties of epistu r mian words are linked to prop erties of episturmian morphisms, it will b e conv enient for us to hav e in mind the b lock-equiv alence that we now recall. A word of the form xv x , wh ere x ∈ A and v ∈ ( A \ { x } ) ∗ , is called a ( x -based) blo c k . A ( x - based) blo ck- tr ansforma tion is the r ep lacemen t in a spinned word of an o ccurrence of xv ¯ x (where xv x is a blo ck) by ¯ x ¯ v x or vice-versa. Two fin ite spinned words w , w ′ are said to b e blo ck - e qui v alent if w e can pass f rom one to the other by a (p ossibly empty) chain of b lock-transformations, in which case we write w ≡ w ′ . F or example, ¯ b ¯ ab ¯ cb ¯ a ¯ c and babc ¯ b ¯ a ¯ c are b lo ck-equiv alen t b ecause ¯ b ¯ ab ¯ cb ¯ a ¯ c → ba ¯ b ¯ cb ¯ a ¯ c → babc ¯ b ¯ a ¯ c and vice-v ersa. T he blo ck- equiv alence is an equiv alence r elation ov er spinned words, and moreo v er one can observ e that if w ≡ w ′ then w and w ′ are spinned versions of a common word o ver A . Theorem 3.1 can b e reformulated in terms of blo ck-equiv alence: Theorem 3.1. L et w , w ′ b e two spinne d wor ds over A ∪ ¯ A . Then µ w = µ w ′ if and only if w ≡ w ′ . 3.2 Infinite directive- equiv alen t words: previous results The question: “When do tw o spinned infinite words direct a common un ique episturmian wo rd ? ” wa s tac kled by Justin and Pirillo in [15] for bi-infinite episturmian wor ds , that is, epistu r mian words with letters indexed by Z (and not by N as considered until no w). Let us recall relations betw een right- infin ite episturmian words and bi-infinite ep istu rmian words (see [15, p. 332] and [9] f or more details). 6 First we observe that a righ t-infinite episturmian w ord t can b e prolonged infinitely to the left with the same set of factors. Note also that the d efinition of episturmian wo rds considered in Section 2.2 (us ing reversal and sp ecial factors) can b e extended to bi-infinite words (see [15]). F urthermore, the charac terization (Theorem 2.1) of righ t-infinite episturmian words by a sequence ( t ( i ) ) i ≥ 0 extends to b i-infinite episturmian words, with all the t ( i ) now bi-infinite epistu rmian words. That is, as for righ t-infinite episturmian words, we hav e b i-infinite words of the form l ( i ) . r ( i ) where l ( i ) is a left-infinite episturmian wo rd and r ( i ) is a r ight-infinite epistu r mian word. Moreov er, if the bi-infinite episturmian word b = l . r is directed by ˘ ∆ with asso ciated bi-infinite episturmian words b ( i ) = l ( i ) . r ( i ) , then r is directed by ˘ ∆ with asso ciated righ t-infinite epistur mian wo rd s r ( i ) . As a consequence of what p recedes, Justin and Pirillo’s results ab out spinned words directing a common bi-infinite episturmian w ord are still v alid for words d irecting a common (righ t-infinite) episturmian word. W e summarize now these results, which will b e h elpf ul for the p roof of our m ain theorem (Theorem 4.1, to follo w). First of all, Justin and Pirillo charac terized pairs of wo rd s directing a common episturmian wo rd in the case of wavy directiv e wo rds , that is, sp inned in finite wo rds conta ining infin itely many L -spinned letters and infi nitely many R -sp inned letters. This characterizati on uses the f ollo wing extension of the b lock-equiv alence ≡ for infinite words. Let ∆ 1 , ∆ 2 b e sp inned v ersions of ∆. W e write ∆ 1 ∆ 2 if there exist infinitely many pr efixes f i of ∆ 1 and g i of ∆ 2 with the g i of strictly increasing lengths, and such that, for all i , | g i | ≤ | f i | and f i ≡ g i c i for a su itable spinned word c i . Infin ite words ∆ 1 and ∆ 2 are said to b e blo ck-e qui valent (denoted by ∆ 1 ≡ ∆ 2 ) if ∆ 1 ∆ 2 and ∆ 2 ∆ 1 . Theorem 3.2. [15, T h. 3.4, Cor. 3.5] L et ∆ 1 and ∆ 2 b e wavy spinne d versions of ∆ ∈ A ω with | Ult(∆) | > 1 . Then ∆ 1 and ∆ 2 dir e ct a c ommon (unique) episturmian wor d if and only if ∆ 1 ≡ ∆ 2 . Mor e over when ∆ 1 and ∆ 2 do not have any c ommon pr efix mo dulo ≡ , and when ther e exists a letter x such that ∆ 1 and ∆ 2 b e gin with x and ¯ x r esp e ctiv e ly, if ∆ 1 ≡ ∆ 2 , then ∆ 1 = x Q n ≥ 1 v n ˘ x n , ∆ 2 = ¯ x Q n ≥ 1 ¯ v n ˆ x n for an L -spinne d letter x , a se q u enc e ( v n ) n ≥ 1 of x -fr e e L -spinne d wor ds, and se quenc es of spinne d letters ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 in { x, ¯ x } such that ( ˘ x n ) n ≥ 1 c ontains infinitely many times the R -spinne d letter ¯ x , and ( ˆ x n ) n ≥ 1 c ontains infinitely many times the L -spinne d letter x . The relati on (and hence the blo ck-e quiv alence ≡ for infin ite words) is rather in tricate to understand . So in some wa y the forms of ∆ 1 and ∆ 2 at the en d of Theorem 3.2 are, although tec hnical, easier to und erstand. Theorem 4.1, w h ic h refin es the end of the previous result and prov es the con verse, d escrib es all possible forms for pairs of d irectiv e-equiv alent words without an y use of notations and ≡ . When one of the tw o considered directive w ords is n ot wa vy , Justin and Pirillo established: Prop osition 3.3. [15, Prop . 3.6] L e t ∆ 1 and ∆ 2 b e spinne d versions of a c ommon wor d such that ∆ 1 is wavy and letters of ∆ 2 ar e ultimately of spin L (r esp. ultimately of spin R ). If ∆ 1 and ∆ 2 ar e dir e ctiv e-e qu ivalent, then ∆ 1 ∆ 2 . Mor e over ther e exist spinne d wor ds w 1 , w 2 , an L -spinne d letter x , and L -spinne d x -fr e e wor ds ( v i ) i ≥ 1 such that µ w 1 = µ w 2 , ∆ 1 = w 1 ¯ x Q i ≥ 1 ¯ v i x and ∆ 2 = w 2 x Q i ≥ 1 v i x (r esp. ∆ 1 = w 1 x Q i ≥ 1 v i ¯ x and ∆ 2 = w 2 ¯ x Q i ≥ 1 ¯ v i ¯ x ). With the next tw o resu lts, they consid er ed the remaining cases of w ords directing aperio dic episturmian wo rds. In the fir st one, the spins of th e lette rs in eac h of the tw o directive words are ultimately L or ultimately R . The second result shows that if one of the dir ective words has the spins of its letters ultimately L (resp . ultimately R ), th en the other directiv e word cannot h av e the spins of its letters ultimately R (resp. ultimately L ). Prop osition 3.4. [15, Prop. 3.7] L et ∆ 1 and ∆ 2 b e spinne d versions of a c ommon wor d ∆ ∈ A ω with | Ult (∆) | > 1 . If ther e exist spinne d wor ds w 1 , w 2 and an L -spinne d infinite wor d ∆ ′ such that ∆ 1 = w 1 ∆ ′ and ∆ 2 = w 2 ∆ ′ (r esp. ∆ 1 = w 1 ¯ ∆ ′ and ∆ 2 = w 2 ¯ ∆ ′ ), then ∆ 1 , ∆ 2 ar e dir e ctive- e quivalent if and only if µ w 1 = µ w 2 . 7 Prop osition 3.5. [15, Pr op. 3.9] L et ∆ b e an L -spinne d infinite wor d. Then ∆ and ¯ ∆ do not dir e ct a c ommon right-infinite episturmian wor d. Actually the previous statement is a corollary of P rop osition 3.9 in [15 ] w h ic h considers more generally words directing episturmian words d iffering only by a shift. Justin and Pir illo also discus s ed in [15] th e perio dic case and pr ov ed: Prop osition 3.6. [15, Prop. 3.10] Supp ose that ∆ 1 = ˘ w ˘ y a ω and ∆ 2 = ˆ w ˆ y ¯ a ω , wher e ˘ w and ˆ w (r esp. ˘ y and ˆ y ) ar e spinne d versions of a c ommon wor d and a is an L - spinne d letter. Then ∆ 1 and ∆ 2 ar e dir e ctive-e quivalent if and only if ther e exist se quenc es of letters (˘ a n ) n ≥ 1 and (ˆ a n ) n ≥ 1 such that ˘ w ˘ y Q n ≥ 1 ˘ a n ≡ ˆ w ˆ y Q n ≥ 1 ˆ a n . W e will see in Th eorem 4.1 that other cases can o ccur for p eriod ic episturm ian w ords. 4 Directiv e-equiv alen t w ords: a c haracterization As shown in the previous section, Ju stin and Pirillo provided quite complete r esults ab out directive- equiv alent in fi nite words. Nev ertheless they did not systematically p rovi de the relative forms of t wo directive- equiv alen t words. The following characterizat ion does it, moreov er w ith ou t the u se of relations and ≡ . T his result also fully solv es th e p erio dic case, which wa s only partially solved in [15]. Theorem 4.1. Give n two spinne d infinite wor ds ∆ 1 and ∆ 2 , the fol lowing assertions ar e e quivalent. i) ∆ 1 and ∆ 2 dir e ct a c ommon right-infinite e pisturmian wor d; ii) ∆ 1 and ∆ 2 dir e ct a c ommon bi-infinite e pisturmian wor d; iii) One of the fol lowing c ases holds for some i, j such that { i, j } = { 1 , 2 } : 1. ∆ i = Q n ≥ 1 v n , ∆ j = Q n ≥ 1 z n wher e ( v n ) n ≥ 1 , ( z n ) n ≥ 1 ar e spinne d wor ds such that µ v n = µ z n for al l n ≥ 1 ; 2. ∆ i = w x Q n ≥ 1 v n ˘ x n , ∆ j = w ′ ¯ x Q n ≥ 1 ¯ v n ˆ x n wher e w , w ′ ar e spinne d wor ds such that µ w = µ w ′ , x is an L -spinne d letter, ( v n ) n ≥ 1 is a se quenc e of non-empty x -fr e e L -spinne d wor ds, and ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 ar e se quenc es of non-e mpty spinne d wor ds over { x, ¯ x } such that, for al l n ≥ 1 , | ˘ x n | = | ˆ x n | and | ˘ x n | x = | ˆ x n | x ; 3. ∆ 1 = w x and ∆ 2 = w ′ y wher e w , w ′ ar e spinne d wor ds, x and y ar e letters, and x ∈ { x, ¯ x } ω , y ∈ { y , ¯ y } ω ar e spinne d infinite wor ds such that µ w ( x ) = µ w ′ ( y ) . Note. F or a, b, c three different letters in A , the spinn ed infi nite words ∆ 1 = a ( bc ¯ a ) ω and ∆ 2 = ¯ a ( ¯ b ¯ c ¯ a ) ω direct a common epistu rmian word that starts w ith the letter a . Indeed, these tw o d irectiv e wo rd s fulfill item 2 of T heorem 4.1 with w = w ′ = ε , x = a , and for all n , v n = bc and ˘ x n = ˆ x n = ¯ a . Moreo ve r the fact that ∆ 1 starts with the L -sp inned letter a sh o ws that the word it directs starts with a . S im ilarly ∆ ′ 1 = ¯ ab ( ca ¯ b ) ω and ∆ ′ 2 = ¯ a ¯ b (¯ c ¯ a ¯ b ) ω direct a common episturmian word starting with the letter b . Since ∆ 2 = ∆ ′ 2 , this shows that the relation “dir ect a common episturmian word” o ver spinned infinite word is not an equiv alence relation. Items 2 and 3 of Theorem 4.1 show that any episturmian w ord is d ir ected by a spinned infi nite wo rd having infinitely many L -spinned letters, but also by a spinn ed word having b oth infi nitely many L -spinn ed letters and infinitely many R -spinned letters (i.e., a wa vy word). T o emphasize the imp ortance of th ese f acts, let us recall from Proposition 2.2 th at if ˘ ∆ is a sp inned in fi nite word o ver A ∪ ¯ A with infin itely many L -spinned letters, then there exists a u nique episturmian word t directed by ˘ ∆. Unicit y comes from the f act that the first letter of t is fi xed by the fir st L -spinn ed letter in ˘ ∆. Before proving Theorem 4.1, let u s make two more r emarks . 8 Remark 4.2. In items 1 and 2 of Theorem 4.1 , the tw o considered directive words are sp inned ve rsions of a common L -spinn ed wo rd. T his do es not hold in item 3, wh ic h deals only with p erio dic episturmian words. T h is is consisten t w ith Remark 2.5. As an example of item 3, one can consid er the w ord ( ab ) ω = L a ( b ω ) = R b ( a ω ) whic h, as already said at the en d of Section 2.4 , is directed by ab ω and by ¯ ba ω ( L a ( b ) = ab = R b ( a )). Note also that ( ab ) ω is dir ected by ( a ¯ b ) ω , underlinin g the fact that x and y can b e equal in item 3 of Theorem 4.1. Remark 4.3 . If an episturm ian word t has tw o directive words satisfying items 2 or 3, then t has in finitely many directiv e wo rds. Ind eed, if item 2 is satisfied and ¯ x o ccurs in ˘ x p ( p ≥ 1), then by Theorem 3.1 , x  Q p − 1 k =1 v n ˘ x n  v p ˘ x ′ p ¯ x ≡ ¯ x  Q p − 1 k =1 ¯ v n ˘ x n  ¯ v p ˘ x ′ p x wh er e ˘ x ′ p is such that ˘ x p ≡ ¯ x ˘ x ′ p . Thus t is also dir ected by w ¯ x  Q p − 1 k =1 ¯ v n ˘ x n  ¯ v p ˘ x ′ p x Q n ≥ p +1 v n ˘ x n . S im ilarly if item 2 is satisfied an d x o ccurs in ˘ x p ( p ≥ 1), then t is also directed by w ′ x  Q p − 1 k =1 v n ˆ x n  v p ˆ x ′ p ¯ x Q n ≥ p +1 ¯ v n ˆ x n where ˆ x ′ p is such that ˆ x p ≡ x ˆ x ′ p . If item 3 is satisfied, then t is p erio dic and d irected by w x where x is any spinned version of x ω . The rest of this section is d edicated to the proof of Theorem 4.1. Pr o of of The or em 4.1. W e h a ve i ) ⇔ ii ) by the r emarks on bi-infin ite words at the b eginn in g of Section 3.2. iii ) ⇒ i ). Assu me first th at ∆ 1 = Q n ≥ 1 v n and ∆ 2 = Q n ≥ 1 z n for spinned w ords ( v n ) n ≥ 1 , ( z n ) n ≥ 1 such that µ v n = µ z n for all n ≥ 1. F r om the latter equality and T heorem 3.1 , ∆ 1 has infi nitely many L -spinn ed letters if and only if ∆ 2 has infin itely many L -spinn ed letters. Let us first consider the case when b oth ∆ 1 and ∆ 2 hav e infinitely many L -sp inned letters. Without loss of generalit y we can assum e that for all n ≥ 1, v n and z n con tain at least one L -spinned letter. Now we n eed to d efine some more n otatio ns. Let t 1 and t 2 b e the episturmian words directed by ∆ 1 and ∆ 2 , resp ectively (these episturmian words exist and are unique by Prop osition 2.2). F or n ≥ 0, let t ( n ) 1 and t ( n ) 2 b e the episturmian words as in ii ) of T heorem 2.1 and let a n and b n b e their resp ectiv e first letters. Fin ally , for n ≥ 1, set p n = Q n i =1 v i and q n = Q n i =1 z i . The words µ p n ( a | p n | ) (resp. µ q n ( b | q n | )) are prefixes of t 1 (resp. of t 2 ). The letter a | p n | (resp. b | q n | ) is the fir s t letter of µ v n +1 ( t ( m ) 1 ) (resp. µ z n +1 ( t ( m ) 2 )) with m = P n +1 i =1 | v i | = P n +1 i =1 | z i | . Sin ce v n +1 (resp. z n +1 ) contains at least one L -sp inned letter, a | p n | (resp. b | q n | ) is the firs t letter of µ v n +1 ( w ) (resp . µ z n +1 ( w )) for any word w . F rom µ v n +1 = µ z n +1 , we hav e a | p n | = b | q n | and so µ p n ( a | p n | ) = µ q n ( b | q n | ) for all n ≥ 1. If the sequence ( µ p n ( a | p n | )) n ≥ 1 is not ultimately constant, then from t 1 = lim n →∞ µ p n ( a | p n | ) and t 2 = lim n →∞ µ q n ( b | q n | ), w e deduce that t 1 = t 2 . I f ( µ p n ( a | p n | )) n ≥ 1 is ultimately constant, th en necessarily th ere exists a letter a and an integ er m s uch that for all n > m , v n and z n b elong to { a } ∗ . Th en t 1 = µ v 1 ...v m ( a ω ) = µ z 1 ...z m ( a ω ) = t 2 . No w, with the same n otations as in the abov e case, we consider the case when the letters of ∆ 1 and ∆ 2 are ultimately R -spin n ed. By Th eorem 3.1, any equality µ v = µ z (for some different spinned words v and z ) implies th at v and z b oth contai n at least one L -spinned letter and one R -spinn ed letter. Hence, in our current case, there exists an intege r m such that v n = z n for all n > m . Let t b e an epistur mian word dir ected by Q n>m v n = Q n>m z n (such an episturmian word exists by Proposition 2.2 ). Then µ p m ( t ) = µ q m ( t ) and ∆ 1 and ∆ 2 are directive-e quiv alen t. No w consid er item 2 of part iii ). W e assume that ∆ 1 = w x Q n ≥ 1 v n ˘ x n and ∆ 2 = w ′ ¯ x Q n ≥ 1 ¯ v n ˆ x n where w , w ′ are s p inned words such that µ w = µ w ′ , x is an L -spinned letter, ( v n ) n ≥ 1 is a sequ ence of non-empty x -free L -spinned words, and ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 are non-empty spinn ed words ov er { x, ¯ x } such that, for all n ≥ 1, | ˘ x n | = | ˆ x n | and | ˘ x n | x = | ˆ x n | x . By injectivit y of the morphisms µ w = µ w ′ , ∆ 1 and ∆ 2 are directiv e-equiv alent if and only if w − 1 ∆ 1 and w ′− 1 ∆ 2 are directiv e-equiv alent . So, from now on, w e assume without loss of generality that w = w ′ = ε . By Prop osition 2.2, there exist un ique episturmian words t 1 and t 2 starting with x directed by the resp ectiv e words ∆ 1 and ∆ 2 (observe that if ˆ x n ∈ ¯ x + for all n ≥ 1, then ¯ x ∈ Ult(∆ 2 )). F or 9 i ≥ 1, let ∆ ( i ) 1 = x Q n ≥ i v n ˘ x n and ∆ ( i ) 2 = ¯ x Q n ≥ i ¯ v n ˆ x n and let t [ i ] 1 and t [ i ] 2 b e the words beginnin g with x and directed by the r esp ectiv e words ∆ ( i ) 1 and ∆ ( i ) 2 . (The ep isturmian words t [ i ] 1 and t [ i ] 2 exist by Proposition 2.2.) F or i ≥ 1 we also defin e α i := | ˘ x i | x = | ˆ x i | x and β i := | ˘ x i | ¯ x = | ˆ x i | ¯ x . Assume fir st that α i 6 = 0. Then ¯ x ¯ v i ˆ x i ≡ ¯ x ¯ v i xx α i − 1 ¯ x β i ≡ xv i ¯ xx α i − 1 ¯ x β i ≡ xv i x α i − 1 ¯ x β i ¯ x and xv i ˘ x i ≡ xv i x α i − 1 ¯ x β i x . L et p i = xv i x α i − 1 ¯ x β i . F rom w hat precedes we deduce that ∆ ( i ) 1 and p i ∆ ( i +1) 1 are directiv e-equiv alent, as ∆ ( i ) 2 and p i ∆ ( i +1) 2 are directiv e-equiv alent. By the choice of words t [ i ] 1 and t [ i ] 2 , we dedu ce that t [ i ] 1 = µ p i ( t [ i +1] 1 ) and t [ i ] 2 = µ p i ( t [ i +1] 2 ) and eac h of these words starts with µ p i ( x ). No w let us consider the case wh en α i = 0. Then ˘ x i = ˆ x i = ¯ x β i . W e hav e xv i ˘ x i ≡ ¯ x ¯ v i ¯ x β i − 1 x and ¯ x ¯ v i ˆ x i = ¯ x ¯ v i ¯ x β i − 1 ¯ x . T aking p i = ¯ x ¯ v i ¯ x β i − 1 , we reach the same conclusion as in the case w hen α i 6 = 0. It follo ws from what p recedes that t 1 and t 2 b oth start with µ p 1 ...p i ( x ) for all i ≥ 1. Sin ce v i 6 = ε , the sequence ( µ p 1 ...p i ( x )) i ≥ 1 is n ot u ltimately constan t; whence t 1 = t 2 = lim i →∞ µ p 1 ...p i ( x ). Lastly , assume that ∆ 1 = w x and ∆ 2 = w ′ y for some sp in ned words w , w ′ , some letters x and y , and some spinn ed in finite wo rds x ∈ { x, ¯ x } ω , y ∈ { y , ¯ y } ω such that µ w ( x ) = µ w ′ ( y ). The w ord ∆ 1 (resp. ∆ 2 ) directs the episturmian word µ w ( x ω ) = ( µ w ( x )) ω (resp. µ w ′ ( y ω ) = ( µ w ′ ( y )) ω ). Hence ∆ 1 and ∆ 2 are directive-e quiv alen t. i ) ⇒ iii ). Su pp ose ∆ 1 and ∆ 2 direct a common (right-infinite) epistur mian word t . Let us first assume th at t is ap erio dic. Then, by Remark 2.5, ∆ 1 and ∆ 2 are spinned ve rsions of a common infinite word ∆ ∈ A ω . W e n o w show that item 1 or item 2 holds usin g r esults of Justin an d Pirillo in [15]. First consider the case when b oth ∆ 1 and ∆ 2 are wa vy . Supp ose th ere exist a sequen ce of prefixes ( p n ) n ≥ 0 of ∆ 1 and a sequence of prefixes ( p ′ n ) n ≥ 0 of ∆ 2 such that for all n ≥ 0, µ p n = µ p ′ n . Without loss of generalit y we can assum e that p 0 = p ′ 0 = ε and the sequence ( | p n | ) n ≥ 0 is strictly increasing. F or n ≥ 1, let v n , z n b e such that p n = p n − 1 v n , p ′ n = p ′ n − 1 z n ; that is ∆ 1 = Q n ≥ 1 v n and ∆ 2 = Q n ≥ 1 z n . Let u s prov e by indu ction that µ v n = µ z n for all n ≥ 1. First µ v 1 = µ p 1 = µ p ′ 1 = µ z 1 . F or n ≥ 2, since µ p n = µ p n − 1 µ v n , µ p ′ n = µ p ′ n − 1 µ z n , µ p n = µ p ′ n and µ p n − 1 = µ p ′ n − 1 , we hav e µ p n − 1 µ v n = µ p n − 1 µ z n and s o µ v n = µ z n by left cancellativit y of the monoid of episturmian morphisms. So item 1 is satisfied in this case. No w assume that previous sequ en ces ( p n ) n ≥ 0 and ( p ′ n ) n ≥ 0 do not exist. Let w and w ′ b e the longest pr efixes of the resp ectiv e spinn ed words ∆ 1 and ∆ 2 such that µ w = µ w ′ . F urther, let ∆ ′ 1 and ∆ ′ 2 b e the s pinned words such th at ∆ 1 = w ∆ ′ 1 and ∆ 2 = w ′ ∆ ′ 2 . T hen, by in j ectivit y of µ w , the wo rd s ∆ ′ 1 and ∆ ′ 2 are directive-e quiv alen t and ha ve n o prefixes with equal images by µ . By Theorem 3.2, ther e exists a letter x in A , a sequence of non-empty x -free words ( v n ) n ≥ 1 o ver A , and t wo sequences of non-empty words ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 o ver { x, ¯ x } s uch that ∆ ′ i = x Q n ≥ 1 v n ˘ x n and ∆ ′ j = ¯ x Q n ≥ 1 ¯ v n ˆ x n for some integers i, j such that { i, j } = { 1 , 2 } . W e hav e to prov e that for all n ≥ 1, | ˘ x n | = | ˆ x n | and | ˘ x n | x = | ˆ x n | x . W e use indu ction on n and prov e also that for all n ≥ 0, the words ∆ ( n +1) i = x Q m ≥ n +1 v m ˘ x m and ∆ ( n +1) j = ¯ x Q m ≥ n +1 ¯ v m ˆ x m are directiv e-equiv alent. Let n ≥ 1 b e an integer. By defin ition of ∆ (1) i = ∆ ′ i and ∆ (1) j = ∆ ′ j (when n = 1) and by th e indu ction hypothesis (when n ≥ 2), we kn o w that the words ∆ ( n ) i = x Q m ≥ n v m ˘ x m and ∆ ( n ) j = ¯ x Q m ≥ n ¯ v m ˆ x m are directiv e-equiv alent. Assume first that ˆ x n con tains at least one o ccurrence of x . Th en, with α n = | ˆ x n | x and β n = | ˆ x n | ¯ x , we ha ve ¯ x ¯ v n ˆ x n ≡ ¯ x ¯ v n xx α n − 1 ¯ x β n ≡ xv n x α n − 1 ¯ x β n ¯ x . By injectivit y of the mor- phism µ xv n we d educe that th e words ˘ x n Q m ≥ n +1 v m ˘ x m = ˘ x n v n +1 ˘ x n +1 Q m ≥ n +2 v m ˘ x m and x α n − 1 ¯ x β n ¯ x Q m ≥ n +1 ¯ v m ˆ x m = x α n − 1 ¯ x β n +1 ¯ v n +1 ˆ x n +1 Q m ≥ n +2 ¯ v m ˆ x m direct a common episturmian wo rd t n . The word v n +1 is n ot empty . Let c be its first letter, let D = c − 1 v n +1 ˘ x n +1 Q m ≥ n +2 v m ˘ x m and let D ′ = ( ¯ c ) − 1 ¯ v n +1 ˆ x n +1 Q m ≥ n +2 ¯ v m ˆ x m : the word t n is d irected by ˘ x n cD and by x α n − 1 ¯ x β n +1 ¯ cD ′ . Since ∆ j is wa vy , D ′ is also wa vy . So x occurs in D ′ (among the ˆ x n ) and the word directed by D ′ starts w ith x . Consequently t n starts w ith µ x α n − 1 ¯ x β n +1 ¯ c ( x ) = x α n cx β n +1 . The words 10 v m are n on-empty , thus there exists a letter d 6 = x th at o ccurs in the w ord d irected by D ′ . Consequently cx α n + β n d is the smallest factor of t n b elonging to c { x } ∗ d . Since t n is also di- rected by ˘ x n cD , it follows th at t n starts with x | ˘ x n | x c and the smallest factor of t n b elong- ing to c { x } ∗ d is cx | ˘ x n | d . Hence | ˘ x n | x = α n = | ˆ x n | x and | ˘ x n | = α n + β n = | ˆ x n | . Conse- quently ˘ x n ≡ x α n ¯ x β n ≡ x α n − 1 ¯ x β n x . The injectivit y of the morph ism µ x α n − 1 ¯ x β n implies that ∆ ( n +1) i = x Q m ≥ n +1 v m ˘ x m and ∆ ( n +1) j = ¯ x Q m ≥ n +1 ¯ v m ˆ x m are directiv e-equiv alent. When ˘ x n con tains at least one o ccurrence of ¯ x , we similarly r eac h the same conclusion. No w we show that it is imp ossible that ˘ x n ∈ x + and ˆ x n ∈ ¯ x + . Assume these relations hold and let k b e the least integ er s tr ictly greater than n su c h that x ∈ Alph( ˆ x k ) (such an integer exists since ∆ j is wa vy). Let α k = | ˆ x k | x and β k = | ˆ x k | ¯ x . Since all of the words ˆ x n , . . . , ˆ x k − 1 b elong to ¯ x + , we hav e ¯ x ¯ v n ˆ x n ¯ v n +1 . . . ¯ v k ˆ x k ≡ ¯ x ¯ v n ˆ x n ¯ v n +1 . . . ¯ v k x α k ¯ x β k ≡ xv n ˆ x n v n +1 . . . v k ¯ xx α k − 1 ¯ x β k . Then by in jectivit y of the morphism µ xv n , there exists an episturmian word d irected by b oth ∆ = ˘ x n Q m ≥ n +1 v m ˘ x m and ∆ ′ = ˆ x n v n +1 . . . v k x α k − 1 ¯ x β k ¯ x Q m ≥ k + 1 ¯ v m ˆ x m . But th is is imp ossible s in ce ∆ directs a w ord starting with x (recall th at ˘ x n ∈ x + ) and ∆ ′ directs a word starting with the first letter of v n +1 (recall that ˆ x n ∈ ¯ x + ). Let us now consider the case when one of the t wo words ∆ 1 , ∆ 2 is w avy and the other has all of its sp ins ultimately L or ultimately R . T hen item 2 is ve rified by Prop osition 3.3. Supp ose n o w that b oth ∆ 1 and ∆ 2 hav e all spin s ultimately L (resp . ultimately R ). Th en by Remark 2.5, ∆ 1 and ∆ 2 are spinned versions of a common word. Hence ∆ 1 = w ∆ and ∆ 2 = w ′ ∆ (resp. ∆ 1 = w ¯ ∆ and ∆ 2 = w ′ ¯ ∆) for some spinned words w , w ′ of the same length and an infin ite L -spinned word ∆ (resp. R -spinned word ∆). Since ∆ 1 and ∆ 2 are directive-e quiv alen t, µ w = µ w ′ by Prop osition 3.4, and further m ore ∆ 1 and ∆ 2 hav e infi nitely many p r efixes whose images are equal by µ . Therefore, as already seen, this situation satisfies item 1. W e hav e now end ed the s tu dy of th e ap eriodic case, since by Prop osition 3.5, ∆ 1 and ∆ 2 cannot direct a common ap eriod ic episturmian word if one of them has all s pins ultimately L and the other has all sp ins ultimately R . Finally we come to the p eriod ic case: ∆ 1 = w x an d ∆ 2 = w ′ y for s ome spinn ed wo rds w , w ′ , letters x and y , and spinned infi nite words x ∈ { x, ¯ x } ω , y ∈ { y , ¯ y } ω . In this case, the episturmian wo rd d irected by ∆ 1 and ∆ 2 is µ w ( x ) ω = µ w ′ ( y ) ω , wh ic h imp lies that µ w ( x ) | y | = µ w ′ ( y ) | x | . Then (see [19 ] for instance) there exists a primitive word z such th at µ w ( x ) and µ w ′ ( y ) are p o wers of z (let us r ecall that a word w is primitive if it is not an intege r p ow er of a shorter word, i.e., if w = u p with p ∈ N , then p = 1 and w = u ). One can quite easily verify th at an y episturmian morphism maps any primitive word to another primitive word (see also [13, Prop. 2.8, Pr op. 3.15]). Since any letter constitutes a p rimitive word, b oth µ w ( x ) and µ w ′ ( y ) are primitive. Thus µ w ( x ) = z = µ w ′ ( y ). 5 Normalized directiv e w ord of an episturmian w ord In the previous section we hav e seen that any episturm ian word t h as a d ir ectiv e word w ith infinitely many L -spinn ed letters. T o work on Stur mian words, Berth´ e, Holton, and Z amboni recentl y p rov ed that it is alw ays p ossible to choose a particular directiv e word: Theorem 5.1. [4] Any Sturmian wor d w over { a, b } has a unique r epr e se ntation of the form w = lim n →∞ L d 1 − c 1 a R c 1 a L d 2 − c 2 b R c 2 b . . . L d 2 n − 1 − c 2 n − 1 a R c 2 n − 1 a L d 2 n − c 2 n b R c 2 n b ( a ) wher e d k ≥ c k ≥ 0 for al l inte ger k ≥ 1 , d k ≥ 1 for k ≥ 2 and if c k = d k then c k − 1 = 0 . In other words, any Stur mian word has a u nique directive word ov er { a, b, ¯ a , ¯ b } conta ining infinitely many L -spinned letters bu t no factor of the form ¯ a ¯ b n a or ¯ b ¯ a n b with n an integer. Actually this result is quite natur al if one thinks ab out th e pr esent ation of the monoid of Stu r mian morph isms (see [31]). Usin g Theorems 3.1 and 4.1, we generalize Th eorem 5.1 to episturmian words: 11 Theorem 5.2. A ny episturmian wor d t has a spinne d dir e ctive wor d c ontaining infinitely many L -spinne d letters, but no factor in S a ∈A ¯ a ¯ A ∗ a . Such a dir e ctive wor d is unique if t is ap erio dic. The example given in R emark 4.2 sho ws that unicit y does not necessarily hold for p erio dic episturmian words. A dir ectiv e word of an ap eriod ic episturmian wo rd t with th e ab o ve prop erty is called the normalize d dir e ctive wor d of t . W e extend this defin ition to m orphisms: a fin ite sp inned wo rd w is said to b e a normalize d dir e ctive wor d of the morphism µ w if w has no factor in S a ∈A ¯ a ¯ A ∗ a . One can observe that, by Theorem 3.1, for any morp hism in L a L ∗ A R a , we can find another decomp osition of the morp h ism in the set R a R ∗ A L a . Equiv alen tly , for any spin n ed word in a A ∗ ¯ a , there exists a wo rd w ′ in ¯ a ¯ A ∗ a su c h that µ w = µ w ′ . This is the main idea used in the pro of of the lemma b elow. Th e pro of of Theorem 5.2 is based on an extension of this lemma to in finite words. Lemma 5.3. Any pur e episturmian morphism has a unique normalize d dir e ctive wor d. Pr o of. Existenc e of the norma lize d dir e ctive wor d : Let w = ( w i ) 1 ≤ i ≤| w | b e a spinned word ov er A ∪ ¯ A . W e construct by induction on | w | a normalized d irectiv e word of µ w . If | w | = 0, there is nothing to do: ε is a normalized directive word of the empty morphism. Assume we hav e constru cted a normalized directive wo rd w ′ = ( w ′ i ) 1 ≤ i ≤ k of the morp hism µ w ′ = µ w . Let ¯ x be a letter in ¯ A . Then , by n orm alizatio n of w ′ , the wo rd w ′ ¯ x has no factor in ∪ a ∈A ¯ a ¯ A ∗ a . Moreo ve r since µ w = µ w ′ , we hav e µ w ¯ x = µ w ′ ¯ x : th e word w ′ ¯ x is a normalized directive wo rd of µ w ¯ x . No w let x b e a letter in A . The word w ′ x can h a ve factors in ∪ a ∈A ¯ a ¯ A ∗ a , bu t only as su ffixes. If this do es not hold, as in the previous case, the word w ′ x is a n ormalized dir ectiv e word of µ w x . E lse w ′ = p ¯ x ¯ u 1 ¯ x ¯ u 2 . . . ¯ x ¯ u k for an integer k ≥ 1, some L -spin ned x -free words ( u i ) 1 ≤ i ≤ k and a spinn ed wo rd p having no suffix in ¯ x ¯ A ∗ . T he wo rd w ′′ ¯ x where w ′′ = p xu 1 ¯ xu 2 . . . ¯ xu k con tains no factor in ∪ a ∈A ¯ a ¯ A ∗ a . Moreov er T heorem 3.1 implies µ w ′ x = µ w ′′ ¯ x . Hence w ′′ ¯ x is a normalized directive word of µ w x . Let us mak e a r emark on the indu ctive constru ction presented in this pro of: Remark 5.4. Let u, v , u ′ , v ′ b e four spinn ed wo rds such that u ′ (resp. v ′ ) is the normalized directive wo rd obtained b y the ab ov e construction from u (resp . v ). If u is a prefix of v and if p is a p refix of u ′ ending by an L -spinned letter, then p is also a prefix of v ′ . Unicity : Assume by wa y of contradict ion that w and w ′ are t wo different spin ned normalized wo rd s su c h that µ w = µ w ′ . By left cancellativit y of the mon oid of episturmian morphisms, we can assume that w and w ′ start with d ifferent letters. Moreov er it follows from Theorem 3.1 that w and w ′ are spin ned versions of a common word. Without loss of generalit y , we can assume that w b egins with a letter a ∈ A and w ′ b egins with ¯ a and so for any word z , µ w ( z ) = µ w ′ ( z ) begins with a . Hence w ′ must start with ¯ a ¯ v a for a word v ∈ A ∗ . This contradicts its normalization. Example 5 .5. Let f b e the p ure episturmian morphism with directiv e word ¯ a ¯ bc ¯ ba ¯ b ¯ a ¯ c ¯ b ¯ a ¯ ca . By Theorem 3.1, µ ¯ a ¯ c ¯ b ¯ a ¯ ca = µ ¯ a ¯ c ¯ bac ¯ a = µ acb ¯ ac ¯ a and hence µ ¯ a ¯ bc ¯ ba ¯ b ¯ a ¯ c ¯ b ¯ a ¯ ca = µ ¯ a ¯ bc ¯ ba ¯ bacb ¯ a c ¯ a and ¯ a ¯ bc ¯ ba ¯ bacb ¯ ac ¯ a is the normalized d irectiv e word of f . No w we provide the Pr o of of The or em 5.2. Existenc e of the normalize d dir e ctive wor d : Let ∆ = ( w i ) i ≥ 1 b e a sp inned directive word of an episturm ian word t (with w i ∈ A ∪ ¯ A ). F rom Theorem 4.1, we can assume that ∆ has infin itely many L -spinned letters. By Lemma 5.3, for any n ≥ 1, th e morphism µ w 1 ...w n has a u nique n ormalized directiv e word ( w ( n ) i ) 1 ≤ i ≤ n . (It follows fr om the p roof of Lemma 5.3 that w i and w ( n ) i are spinned ve rsions of a common letter). 12 Let p n b e the longest prefi x of w ( n ) 1 . . . w ( n ) n that b elongs to ( A ∪ ¯ A ) ∗ A . Let i n ≤ n b e the integer such that p n = w ( n ) 1 . . . w ( n ) i n , and let π n b e the word π n = µ w ( n ) 1 ...w ( n ) i n − 1 ( w ( n ) i n ). Since the morphism s µ w 1 ...w n and µ w ( n ) 1 ...w ( n ) n are equal, t h as the directive word ( w ( n ) 1 , . . . , w ( n ) n , w n +1 , w n +2 , . . . ), so π n is a p r efix of t . By Remark 5.4, for any n ≥ 1, p n is a pr efix of p n +1 , and since ∆ cont ains infin itely many L -spinned letters, for an y n ≥ 1, ther e exists an m > n s uch that | p m | > | p n | . If | Ult(∆) | = 1, then th ere exists a letter a and an integer m suc h that t = µ p m ( a ω ) and p m a ω is a normalized dir ectiv e word of t . If | Ult(∆) | > 1, the sequence ( π n ) n ≥ 1 is not ultimately constan t, and lim n →∞ π n = t . In this case t is directed by the sequence lim n →∞ p n which is normalized by construction (indeed otherwise one of the p refixes p n wo uld n ot b e normalized). Unicity of the norma lize d dir e ctive wor d : Assume by wa y of con tradiction that an ap eriod ic episturmian word t has tw o different nor- malized sp inned directive wo rds ∆ 1 = ( w n ) n ≥ 1 and ∆ 2 = ( w ′ n ) n ≥ 1 (with w n and w ′ n ∈ A ∪ ¯ A f or all n ). Let i ≥ 1 be the smallest inte ger such that w i 6 = w ′ i (and for all j < i , w j = w ′ j ). By Theorem 4.1, ∆ 1 and ∆ 2 are spinned versions of the same word (see Remark 4.2). T hus, with ou t loss of generality , we can assume that w i = ¯ x and w ′ i = x for some letter x . Let t ′ ( i ) b e the episturmian word with (normalized) d irectiv e wo rd ( w ′ n ) n ≥ i (by Prop osition 2.2 this word is un ique), then t ′ ( i ) starts with x sin ce w ′ i = x . Since the word ( w n ) n ≥ 1 has in fi nitely many L -spinned letters, there exists an integer j > i such that w j = y for a letter y ∈ A and w ℓ ∈ ¯ A for eac h ℓ , i < ℓ < j . Let t ( i ) b e the word w ith n ormalized directiv e word ( w n ) n ≥ i , then t ( i ) has the wo rd µ w i ...w j − 1 ( y ) as prefix since w j = y and so t ( i ) starts with y since w i . . . w j − 1 ∈ ¯ A ∗ . W e hav e t = µ w 1 ...w i − 1 ( t ( i ) ) = µ w ′ 1 ...w ′ i − 1 ( t ′ ( i ) ). By choice of i , w 1 . . . w i − 1 = w ′ 1 . . . w ′ i − 1 . Consequent ly , since episturmian morp hisms are in jectiv e on infinite w ords, t ( i ) = t ′ ( i ) and so x = y . But s in ce w i = ¯ x , w i +1 . . . w j − 1 ∈ ¯ A ∗ , and w j = x , we reac h a contradiction to the normalization of ( w n ) n ≥ 1 . 6 Episturmian w ords ha ving a unique directiv e w ord In S ection 4 we ha ve c haracterized pairs of wo rds d irecting a common epistu rmian word. In Section 5 we ha ve prop osed a wa y to uniquely define any episturmian word through a normaliza tion of its dir ectiv es w ords (as mentioned in th e in tro duction, see [4, 18 , 17, 11] for some u ses of this normalization). Using these results we now c haracterize episturm ian words ha ving a unique d irectiv e wo rd . Theorem 6.1. A n episturmian wor d has a uniqu e dir e c tiv e wor d if and only if its (normalize d) dir e ctive wor d c ontains 1) infinitely many L -spinne d letters, 2) infinitely many R -spinne d letters, 3) no factor in S a ∈A ¯ a ¯ A ∗ a , 4) no factor in S a ∈A a A ∗ ¯ a . Such an episturmian wor d is ne c essarily ap erio dic. Pr o of. Assume first that an episturmian word t has a uniqu e sp in ned directiv e word ∆. By The- orem 5.2, ∆ is normalized and so cont ains infin itely many L -spin ned letters and n o factor in S a ∈A ¯ a ¯ A ∗ a . By item 3 of Th eorem 4.1 and by Remark 2.4, t cann ot b e p eriod ic. By item 2 of Theorem 4.1, ∆ also contains infin itely many R -spinned letters, and hence is wa vy (otherwise one can construct another directive word of t – the fact that t is ap erio dic is imp ortant for ha ving the ( v n ) n ≥ 1 non-empty in this construction). Finally Theorem 3.1 im p lies the n on-existence of a factor in S a ∈A a A ∗ ¯ a (otherwise, one can again construct another directiv e word for t ). Let us no w prov e that the four conditions (giv en in the statement of the th eorem) are sufficien t. Arguing by contradict ion, we assume that an epistu rmian word t is directed by tw o spin ned infin ite wo rd s ∆ 1 and ∆ 2 , b oth fulfi lling the four giv en conditions. W e observe that if ∆ 1 or ∆ 2 is ultimately written ov er { x, ¯ x } for a letter x (wh ic h can o ccur only if t is p eriod ic), then at least one of the conditions is not fulfilled. Thus the tw o w ords ∆ 1 and ∆ 2 should verify one of the t wo first items 13 in part iii ) of Theorem 4.1 (item 3 do es not apply since t is ap eriod ic). But the h yp otheses on ∆ 1 and ∆ 2 imply that only item 1 can b e verified so that ∆ 1 = Q n ≥ 1 v n , ∆ 2 = Q n ≥ 1 z n for sp inned wo rd s ( v n ) n ≥ 1 , ( z n ) n ≥ 1 such that µ v n = µ z n for all n ≥ 1. Now by T heorem 3.1 and by the fact that wo rd s ( v n ) n ≥ 1 and ( z n ) n ≥ 1 hav e n o f actor in S a ∈A ¯ a ¯ A ∗ a nor S a ∈A a A ∗ ¯ a , we must hav e v n = z n for all n ≥ 1. Th us ∆ 1 = ∆ 2 . As an example, a particular f amily of episturmian w ords h a ving unique dir ectiv e w ords consists of those directed by r e gular wavy wor ds , i.e., spinn ed in fi nite words h a ving b oth infi n itely many L -spinned letters and infinitely many R -spinned letters such that eac h letter occurs with the s ame spin eve rywh ere in the directive word. More formally , a sp inned version ˘ w of a fin ite or in fi nite wo rd w is said to b e r e gu lar if, for each letter x ∈ Alph ( w ), all occurren ces of ˘ x in ˘ w ha ve the same spin ( L or R ). F or example, a ¯ baa ¯ c ¯ b and ( a ¯ bc ) ω are regular, wh ereas a ¯ ba ¯ a ¯ cb and ( a ¯ b ¯ a ) ω are n ot regular. In the S turmian case, we ha ve: Prop osition 6.2. Any Sturmian wor d has a uni q ue spinne d dir e ctiv e wor d or infinitely many spinne d dir e ctive wor ds. Mor e over, a Sturmian wor d has a unique dir e c tiv e wor d if and only i f its (normalize d) dir e ctive wor d is r e g ular wavy. Pr o of. Let ∆ b e the normalized directiv e word of a Stu rmian w ord t o ver { a, b } . T hen ∆ con tains no factor b elonging to ¯ a ¯ b ∗ a ∪ ¯ b ¯ a ∗ b (wh ere α ∗ = { α } ∗ for any letter α ). Assume fi rst that ∆ contains infin itely many factors in ab ∗ ¯ a ∪ ba ∗ ¯ b . Then ∆ = p S n ≥ 1 x n y n for some spinned w ords p and ( x n , y n ) n ≥ 1 such that, for all n ≥ 1, x n ∈ ab ∗ ¯ a ∪ ba ∗ ¯ b and y n ∈ { a, b, ¯ a , ¯ b } ∗ . I n th is case, ∆ has infin itely many directive words; indeed, the s pinned words ( p [ S k − 1 n =1 x n y n ] ¯ x k y k S n ≥ k +1 x n y n ) k ≥ 1 are (b y Theorem 3.1) pairwise different directiv e words for t . No w assume that ∆ con tains only fi nitely many f actors in ab ∗ ¯ a ∪ ba ∗ ¯ b . S ince ∆ contains no factor in ¯ a ¯ b ∗ a ∪ ¯ b ¯ a ∗ b , it is ultimately regular wa v y . More precisely ∆ is regular wa vy and either ∆ b elongs to { a, ¯ b } ω ∪ { ¯ a, b } ω , or ∆ b elongs to one of the following sets of infi nite words: S 1 = { a, b, ¯ a , ¯ b } ∗ a { ¯ a, b } ω , S 2 = { a, b, ¯ a , ¯ b } ∗ b { a, ¯ b } ω , S 3 = { a, b, ¯ a , ¯ b } ∗ ¯ a { a, ¯ b } ω or S 4 = { a, b, ¯ a , ¯ b } ∗ ¯ b { ¯ a, b } ω . Assume ∆ ∈ S 1 . Since any Stu rmian word is ap erio dic, ∆ is not ultimately constant (see Remark 2.4). Thus ∆ = pa S n ≥ 1 x n ¯ a with x n ∈ b ∗ for all n ≥ 1. O nce again in this case, t has infin itely many directive wo rd s since the words ( p [ S k − 1 n =1 ¯ a ¯ x n ] a S n ≥ k x n ¯ a ) k ≥ 1 are pairwise different directive words for t . The cases when ∆ ∈ S 2 or ∆ ∈ S 3 or ∆ ∈ S 4 are similar. W e end with the case w hen ∆ is regular wa vy . I n this case, ∆ conta ins infinitely many L -spinned letters, infin itely many R -spinned letters, no factor in ab ∗ ¯ a ∪ ba ∗ ¯ b , and n o f actor in ¯ a ¯ b ∗ a ∪ ¯ b ¯ a ∗ b . Hence by T heorem 6.1, t has a u nique directive w ord. Prop osition 6.2 shows a great d ifference b etw een Stu rmian words and episturmian words con- structed ov er alphab ets with at least three letters. Indeed, when consid er in g words o ver a ternary alphab et, one can find episturmian words h a ving exactly m dir ectiv e words for any m ≥ 1. F or instance, th e episturmian word t directed by ∆ = a ( b ¯ a ) m − 1 b ¯ c ( ab ¯ c ) ω has exactly m directive words, namely ( ¯ a ¯ b ) i a ( b ¯ a ) j b ¯ c ( ab ¯ c ) ω with i + j = m − 1. Notice that the s u ffix b ¯ c ( ab ¯ c ) ω of ∆ is regular wa vy , and the other m − 1 spin ned versions of ∆ that also direct t arise from the m − 1 words that are block-equiv alent to the prefix a ( b ¯ a ) m − 1 . Ac kno wledgemen t . 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