Episturmian words: a survey

EPISTURMIAN W ORDS: A S UR VEY ∗ Am y Glen † Jacques Justin ‡ Submitted: Decem b er 11, 2007; Revised: Septem b er 16, 2008 Abstract In this pap er, we survey the rich theory of inﬁnite e pisturm ian wor ds which generalize to any ﬁnite alphabet, in a r ather resembling way , the well-known family of St urmian wor ds on t wo letters. After recalling deﬁnitions and basic prop erties, w e co nsider epi st urmian mor- phisms that allow for a deeper study of these words. Some prop erties of factors a re described, including factor complexity , palindromes, fractional p o wers, frequenc ie s , and retur n words. W e also consider lexicogr aphical pr oper ties of episturmian w or ds, as well as their connection to the ba lance prop erty , a nd rela ted notions such as ﬁnite episturmia n words, Arnoux- Rauzy sequences, a nd “episkew words” that g e neralize the s k ew words of Morse a nd Hedlund. Keyw ords: combinatorics o n w ords ; episturmian w o r ds; Arno ux-Rauzy sequences; Sturmian words; episturmian mo r phisms. MSC (20 00): 68R15. 1 In tro duction 1.1 F rom Sturmian to episturmian Most reno wned amongst the bran ches o f combinato rics on w ords is the theory of inﬁnite binary sequences c alled Sturmian wor ds , whic h are fascinating in many resp ects, h a ving b een studied fr om com binatorial, algebraic, and geometric p oint s of view. Their b eautiful prop erties are related to many ﬁ elds such as Number Theory , Geometry , Symb olic Dynamical Systems, Theoretical Ph ysics, and Theoretical Computer Science (see [7, 83, 96] for recen t surv eys). Since the seminal w orks of Morse and Hedlund [91], Stu rmian words ha ve b een shown to admit n umerous e qu iv a lent deﬁnitions and c haracterizatio ns . F or instance, it is w ell kno wn that an inﬁnite word w ov er { a, b } is S turmian if and only if w is ap erio dic and b alanc e d : for a ny t wo factors u , v of w of the s ame length, the num b er of a ’s in eac h of u and v d iﬀers by at most 1. Sturmian w ord s are also c haracterized b y their factor c omplexity function (w h ic h counts the n umb er of distinct facto rs of eac h length): they h a v e exactly n + 1 distinct fact ors of length n for eac h n . In this sens e, Sturmian words are precisely the ap eriod ic inﬁ nite wo r ds of minimal factor complexit y since, as is well kno wn, an inﬁ nite w ord is ultimately p erio dic if and only if it has le ss th an n + 1 factors of length n f or some n (see [3 7 ]). Man y int eresting p rop erties of ∗ This paper grew out of an invited lecture given by th e second author at the Sixth I n ternational Conference on W ords, Marseill e, F rance, September 17–21, 2007. † Corresponding author: LaCIM, Universit ´ e du Qu´ ebec ` a Montr ´ eal, C.P . 8888, succursale Centre-ville, Montr ´ eal, Qu´ eb ec, H3C 3P8 , CANADA \ The Ma th ematics In stitute, Reyk ja vik Universit y , Kringlan 1, IS-103 R eykja vik, ICELAND ( amy.glen@g mail.com ). ‡ LIAF A, Universi t´ e Par is Diderot - Paris 7, Case 7014, 75205 Paris Cedex 13, FRANCE ( jacjustin@free.fr ). 1 Sturmian words can b e attributed to their lo w co mp lexit y , which induces certain regularities in suc h words without, ho w ever, making them p erio dic. S turmian w ords can also b e geo metrically realized as cutting se quenc es by considering the sequence of ‘cuts’ in an inte ger grid made b y a line of irrational slop e (see for in stance [38, 13]). They also p r o vide a symb olic co ding of the orbit of a point on a circle with resp ect to a rotation by an irr atio n al num b er (see [9 1 , 4]). All of the abov e c h aracte ristic p r op erties of Sturmian w ord s le ad to natural generaliza tions on arbitrary ﬁnite alphab ets. In one direction, the balance prop erty naturally extends to an alphab et with m ore than t wo letters (e.g., see [68, 110, 115]) as do es the follo wing g e ner alize d b alanc e pr op erty that also c haracterizes Sturm ian words (see [49, 1]): the diﬀerence b et w een the n umb er of o ccurrences of a wo rd u in any p air of factors of the same length is at most 1. In another direction, we could consider relaxing the minimalit y condition for th e factor complexit y p ( n ). F or example, quasi-Sturmian wor ds are inﬁnite words for which there exist t w o p ositiv e int egers N and c such that n + 1 ≤ p ( n ) ≤ n + c for all n ≥ N . Th is generalizatio n was in tro du ced in [5] w h en studying th e transcendence of certain con tinued fraction expansions. See also [31, 36, 66, 105] for similar extensions of Sturmian w ords with resp ect to facto r complexit y . F rom the geometric p oin t of view, cutting sequences naturally generalize to tra jectories in the hyper cu b e billiard (e.g., see [25]), a n d co dings o f rotatio n al orbits carry o ver to co dings of in terv al exc h an ge transformations (e.g., see [18]). Tw o other very inte resting natural generaliz ations of Sturmian words are A rnoux-R auzy se- quenc es [12, 9 7 ] and episturmian wor ds [43, 73 ], whic h w e will now deﬁne. F rom the factor complexit y of Sturmian w ords, it im m ediately follo ws that any Sturmian word is o v er a 2-letter alph ab et and h as exact ly o n e left sp e cial f acto r of eac h lengt h . A factor u of a ﬁnite or inﬁnite w ord w is said to b e left sp e cial (resp. right sp e cial ) in w if there exists at least t wo distinct letters a , b su c h that au and bu (resp . ua , ub ) are factors of w . Extendin g the left sp ecial prop ert y of Sturmian w ord s, a recurrent inﬁ nite w ord w ov er a ﬁnite alphab et A is said to b e an Arnoux-R auzy se quenc e (or a strict episturmian wor d ) if it has exactly one left sp ecial factor a nd one righ t sp ecial factor of eac h length, and fo r ev ery left (resp. righ t) sp ecial factor u of w , xu (r esp . ux ) is a factor of w for all letters x ∈ A . A noteable prop erty that is shared b y Sturmian w ords and Ar nouxy-Rauzy sequences is their closure under reversal, i.e., if u is a factor of such a w ord, then its reversal is also a f acto r. This n ice p rop ert y inspired Droub a y , Justin, and Pirillo’s generaliz ation of Sturmian w ord s in [43]: an inﬁnite w ord is episturmian if it is closed under rev ersal a nd has at most one left special factor o f eac h length. Sturmian, A r noux-Rauzy , and episturmian words all hav e standar d (or char acteristic ) e lements, wh ich are those having all of their left sp ecial factors as p reﬁxes. Within th ese families of w ord s, standard wo r ds are go od represent ativ es in the sens e that an inﬁnite wo rd b elongs to one such family if and only if it h as the s ame set of f actors as some standard word in th at family . F rom the deﬁnitions, it is clear that the family of Arnoux-Rauzy sequences is a p articular sub class of the family of epistur mian words. More p recisely , episturmian words are comp osed of the Arn oux-Rauzy sequences, images of the Arnoux-Rauzy s equ ences by episturmian morphisms , and certain p erio dic inﬁn ite w ords (see Section 5 ). In the 2-lette r case, Arnoux-Rauzy sequences are exactly the Stur mian words whereas episturmian words include all r ecurren t b alance d w ords , i.e., p erio dic balanced words and Sturmian words. The study of epistur mian w ords and Arnoux-Rauzy sequ ences has enjo yed a great deal of p opularit y in recent times, owing mostly to the many prop erties that they share with Stur mian w ords. In th is pap er w e surv ey th e p urely com binatorial w ork on episturmian words, b eginning with their deﬁnition and b asic p r op erties in Section 2 . Th en, in Section 3 , we recall epistur m ian morphisms w h ic h allo w for a d eeper s tudy of episturmian wo rd s. In p articular, an y epistur mian 2 w ord is the image of another episturmian word b y some so-called pur e episturmian morphism . Ev en more, any epistur m ian w ord can b e inﬁnitely decomp osed ov er the set of pur e episturmian morphisms. This last prop ert y all ows an episturmian word to be deﬁned by one of its morphic decomp ositions or, equiv alently , by a certain dir e ctive wor d , whic h is an inﬁnite sequence of rules for d ecomp osin g th e giv en episturmian wo rd by morp h isms. In Section 4 w e consider n otions suc h as shifts , spins , and blo ck-e quivalenc e in connection with directiv e words, w hic h allo w us to study when t wo diﬀeren t spinne d inﬁnite wor ds d irect the sa me episturmian w ord. W e also consider p erio dic and purely morphic episturm ian words. In Section 5, our discussion brieﬂy tur ns to Arnoux-Rauzy sequ ences and ﬁnite e pisturmian wor ds . F ollo wing this, w e stud y in Section 6 some p rop erties of factors of episturmian w ords (and Arnoux-Rauzy s equ ences), including factor complexit y , palindromes, fractional p ow ers, frequencies, and retur n w ords. Lastly , we consider more recen t w ork inv olving lexic o gr aphic or der and the balance prop erty (including F r aenkel’s c onje ctur e ). 1.2 Notation & terminology W e assume the reader is familiar with com binatorics on w ords and morph isms (e.g., see [82, 83]). In this sectio n, we recall some basic deﬁnitions and p rop erties r elat ing to episturmian words whic h are needed throughout the pap er. F or the most part, we follo w the notation and terminology of [43, 73, 75, 62]. Let A d enote a ﬁnite alphab et , i.e., a non-empty ﬁ nite s et o f sym b ols called letters . A ﬁnite wor d ov er A is a ﬁn ite sequence of letters from A . The empty wor d ε is the empt y sequence. Under the op eration of concatenation, the set A ∗ of all ﬁnite wo rd s o v er A is a fr e e monoid with identit y elemen t ε and set of generators A . The set of non-empty words ov er A is the fr e e semigr oup A + := A ∗ \ { ε } . A right-inﬁnite (resp. left- i nﬁnite , bi- inﬁnite ) word o ver A is a sequence indexed by N + (resp. Z \ N + , Z ) with v alues in A . F or instance, a left-inﬁnite word is r epresen ted by u = · · · b − 2 b − 1 b 0 and a r igh t-inﬁnite w ord by v = b 1 b 2 b 3 · · · where b i ∈ A . The concatenatio n of u and v g ives the bi-inﬁnite wo rd u . v = · · · b − 2 b − 1 b 0 .b 1 b 2 b 3 · · · with a dot written b et wee n b 0 and b 1 to a v oid am biguity . F or easier r eadin g, inﬁnite words are h ereafter t ypically typed in b oldface to distinguish them from ﬁnite words. The shift map T is deﬁn ed for bi-inﬁ nite words b = ( b i ) i ∈ Z b y T( b ) = ( b i +1 ) i ∈ Z and its k -th iteration is denoted by T k . Th is extends to right -inﬁ nite wo rd s for k ≥ 0 and left-inﬁnite wo rd s for k ≤ 0. F or ﬁnite words w ∈ A ∗ , the shift map T acts circularly , i.e., if w = xv wh ere x ∈ A , then T( w ) = v x . The set of all righ t-inﬁnite words o ve r A is d enoted by A ω , and w e deﬁne A ∞ := A ∗ ∪ A ω . An ultimately p erio dic r igh t-inﬁnite word can b e wr itten 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 . A righ t-inﬁn ite word that is not ultimately p erio dic is said to b e ap e rio dic . Giv en a ﬁnite w ord w = x 1 x 2 · · · x m ∈ A ∗ with eac h x i ∈ A , the length of w , denoted by | w | , is equal to m . By conv entio n , the empty w ord ε is the unique w ord of length 0. The n umber of o ccurrences of a letter a in w is d enoted by | w | a . If | w | a = 0, then w is said to b e a -fr e e . The r ev ersal e w of w is its mirror image: e w = x m x m − 1 · · · x 1 , and i f w = e w , then w is called a p alindr ome . The rev ersal op er ator naturally e xtend s to bi-inﬁ nite wo rd s; that is, the reversal of the b i-inﬁnite wo rd b = l . r , with l left-inﬁn ite and r right -inﬁ nite, is given by e b = e r . e l . A ﬁ nite wo rd w is a factor of a ﬁ nite or inﬁnite word z if z = uwv for s ome words u , v (whic h are ﬁnite or inﬁnite dep end ing on z ). In the sp ecial case u = ε (resp. v = ε ), we call w a pr eﬁx 3 (resp. suﬃx ) of z . W e use the notation p − 1 w (resp. w s − 1 ) to indicate the r emov al of a pr eﬁx p (resp. suﬃx s ) of a ﬁnite wo rd w . Note that a preﬁx or suﬃx u of a ﬁnite wo rd w is said to b e pr op er if u 6 = w . A factor u of a ﬁn ite or inﬁnite w ord w is right (resp. left ) sp e cial if ua , ub (resp. au , bu ) are factors of w for some letters a , b ∈ A , a 6 = b . F or an y ﬁnite or inﬁnite w ord w , F ( w ) denotes the set of all its factors. Moreo ve r, the alpha b et of w is Alph ( w ) := F ( w ) ∩ A and, if w is inﬁn ite, we denote b y Ult( w ) the set of all letters o ccurrin g inﬁnitely often in w . Any t wo inﬁnite w ords x , y are said to b e factor-e quivalent if F ( x ) = F ( y ), i.e., if x and y ha ve the s ame set of f acto rs. A factor of an inﬁn ite w ord x is r e curr ent in x if it o ccurs in ﬁnitely often in x , and x itself is said to b e r e curr ent if all of its factors are recurrent in it. F or a bi-inﬁn ite wo rd to b e recurrent, an y f acto r must o ccur inﬁn itely often to the left and to the right. An in ﬁnite w ord is said to b e uniformly r e curr ent if any factor o ccurs inﬁ nitely man y times in it with b ounded gaps [37]. A morphism ϕ on A is a map from A ∗ to A ∗ suc h that ϕ ( uv ) = ϕ ( u ) ϕ ( v ) for any w ords u , v ov er A . A morphism on A is entirely deﬁned by the image s of letters in A . All m orp hisms considered in this p ap er will b e non-er asing : the image of an y non-empty word is never empt y . Hence the action of a morph ism ϕ on A ∗ can b e naturally extended to inﬁn ite words; that is, if x = x 1 x 2 x 3 · · · ∈ A ω , then f ( x ) = f ( x 1 ) f ( x 2 ) f ( x 3 ) · · · . An inﬁnite w ord x can therefore b e a ﬁxe d p oint of a morph ism ϕ , i.e., ϕ ( x ) = x . If ϕ is a (non-erasing) morph ism su c h that ϕ ( a ) = aw for so me letter a ∈ A and w ∈ A + , then ϕ n ( a ) is a pr op er pr eﬁx of t h e wo rd ϕ n +1 ( a ) for eac h n ∈ N , and the limit of th e sequence ( ϕ n ( a )) n ≥ 0 is the uniqu e in ﬁ nite w ord: w = lim n →∞ ϕ n ( a ) = ϕ ω ( a ) (= aw ϕ ( w ) ϕ 2 ( w ) ϕ 3 ( w ) · · · ) . Clearly , w is a ﬁxed p oint of ϕ and we say that w i s gener ate d by ϕ . F u rthermore, an inﬁn ite w ord generated by a morphism is said to b e pur ely morphic . In what follo ws, we w ill denote the comp osition of morp h isms by juxtap osition as for concate- nation of w ord s. 2 Deﬁnitions & basic pr op erties In the initiating pap er [43], episturm ian words were deﬁned as an extension of standar d epis- turmian wor ds , w hic h we re ﬁrst in tro du ced as a generalization of standar d (or char acteristic ) Sturmian words using iterated palind romic clo su re (a construction due to d e Luca [41]). Here w e choose instead to b egin with the follo wing deﬁnition for deriving the main basic prop erties of episturmian w ords. Deﬁnition 2.1. [43] An inﬁnite wor d t ∈ A ω is episturmian if F ( t ) is close d under r eversal and t has at most one left sp e cial factor (or e qui v alently, right sp e cial f actor) of e ach leng th. M or e over, an episturmian wor d is stand ard if a l l of its left sp e cial factors ar e pr e ﬁxes of it. Note. W e can equiv alen tly consid er left or righ t sp ecial factors in the ﬁrst p art of the ab o v e deﬁnition since, by cl osur e under rev ers al, a factor is left (resp. righ t) sp ecial if and only if its rev ersal is right (resp. left) sp ecial. Remark 2.2. When |A| = 2, Deﬁnition 2.1 giv es th e (ap eriod ic) Sturmian words, as we ll as the p erio dic balanced inﬁnite wo rd s (also kno wn as the p erio dic Sturmian wor ds ). See for instance [62] or Section 7.1. 4 The follo wing theorem collects together some usefu l c h aracteristic prop erties of standard epis- turmian words. Before stating it, let us ﬁ rst recall the some deﬁn itions. Giv en t w o palindr omes p , q , we s a y that q is a c entr al factor of p if p = w q e w for some w ∈ A ∗ . The p alindr omic right-closur e w (+) of a ﬁnite word w is the (unique) shortest palindrome having w as a p reﬁx (see [41]). T h at is, w (+) = w v − 1 e w where v is the longest palindromic suﬃx of w . F or example, ( r ace ) (+) = r ace car . The iter ate d p alindr omic closur e fu nction [71], denoted b y P al , is deﬁned recursiv ely as fol lows. S et P al ( ε ) = ε and, for an y word w and letter x , deﬁne P al ( wx ) = ( P al ( w ) x ) (+) . F or instance, P al ( abc ) = ( P al ( ab ) c ) (+) = ( abac ) (+) = abacaba . (See Sections 4.1 and 6.2.1 for fu rther insigh t ab out palindromic closure.) Theorem 2.3. F or an inﬁnite wor d s ∈ A ω , the fol lowing pr op erties ar e e qu ivalent. i) s is standar d episturmian. ii) Any ﬁrst o c curr enc e of a p alindr ome in s is a c entr al factor of some p alindr omic pr eﬁx of s (pr op erty Pi). iii) If w is a pr eﬁx of s , then w (+) is also a pr eﬁx of s (pr op erty Al). iv) Ther e exists an inﬁnite wor d ∆ = x 1 x 2 · · · ( x i ∈ A ) , c al le d the directiv e word of s , such that s = lim n →∞ P al ( x 1 · · · x n ) . Remark 2.4. Th e palindromes P al ( x 1 · · · x n ) are ve ry often denoted by u n +1 in t h e literature (and w e will sometimes u se th e latter nota tion when con venien t). By constru ction, these palin- dromes are exactly the palindromic preﬁx es of s . Moreo v er, s is uniquely determined by the directiv e word ∆. Pr o of of The or e m 2.3. i ) ⇒ ii ): L et s = up t , u ∈ A ∗ , t ∈ A ω sho wing the ﬁrst o ccurrence of some palindrome p in s . Sup p ose p is not the cen tral factor of a palindromic preﬁx. T hen we ha v e s = vxw p ˜ w y t ′ , x 6 = y ∈ A . By the rev ers al pr op ert y , y w p ˜ w x ∈ F ( s ), th us w p ˜ w is left sp ecial, hence is a preﬁx of s . Thus p has another o ccurrence strictly on the left of the considered one, a con tradiction. i ) ⇒ iii ): If iii ) is false, let w = ux , with u ∈ A ∗ and x ∈ A , b e the sh ortest p reﬁx of s su c h that w (+) is not a preﬁx of s . Thus u (+) is a preﬁx of s . If u were not a palindrome then w wo u ld b e a p reﬁx of u (+) ; whence w (+) = u (+) , a con tradiction. Thus u is a palindr ome. No w let q b e the longest p alindromic suﬃx of w . Then w (+) = w 1 q ˜ w 1 = w ˜ w 1 where w = w 1 q , and w (+) = w 1 q f y g and w 1 q f z is a preﬁx of s f or some y 6 = z ∈ A , f , g ∈ A ∗ . Hence y ˜ f q ∈ F ( ˜ w ) ⊂ F ( s ) and z ˜ f q ∈ F ( s ). Therefore ˜ f q is a left sp ecial p r eﬁx of s . As q f is a pr eﬁx of ˜ w = xu , x − 1 q f is a preﬁx o f u , hence x − 1 q f α is a preﬁx of u for some letter α . S o we h a v e x − 1 q f α = ˜ f q , whence α = x and q f x = x ˜ f q . This wo r d is a palindrome and, as it is a su ﬃx of w , this con tradicts the minimalit y of | q | . iii ) ⇒ iv ): T rivial. A t this stage, we h a v e pro ve d that standard epistur mian w ords satisfy ii ) , iii ) , iv ). T he equiv- alence of these thr ee prop erties is pro ve d in [43, Theorem 1]. Finally , if s satisﬁes them, then F ( s ) is closed un der reve rs al and by [43, Proposition 5] all of its left sp ecial factors are preﬁxes of it, thus s is standard episturmian. ⊔ ⊓ Remark 2.5. Hereafter, we adopt “epistandard” as a shortcut for “standard episturmian”, as in [64, 99, 101]. Also, u nless stated otherwise, the notation ∆ = x 1 x 2 x 3 · · · ( x i ∈ A ) will remain for the d irectiv e w ord of an epistandard word s . 5 Example 2.6. The epistandard w ord d irected by ∆ = ( abc ) ω is known as the T rib onac ci wor d (or R auzy wor d [97]); it b egins in the follo wing w ay: r = a bacabaabacababacabaabacabacabaabaca · · · , where eac h p alindromic preﬁx P al ( x 1 · · · x n ) is follo we d by an u n derlined letter x n . More generally , for k ≥ 2, the k -b onac ci wor d is the epistandard word ov er { a 1 , . . . , a k } directed by ( a 1 a 2 · · · a k ) ω (e.g., see [59]). Note. F or rece nt studies of t h e p r op erties of T rib onacci w ord , s ee for instance [57, 107] and the c hapter by Allouc he and Berth ´ e in [84]. 2.1 Equiv alence classes In [43], an inﬁn ite word t ∈ A ω w as s aid to b e ep istu rmian if F ( t ) = F ( s ) for some epistandard w ord s . This deﬁnition is equiv alen t to Deﬁnition 2.1 b y Theorem 5 in [43]. Moreo ver, it w as pro ved in [43] that epistu r mian w ords are u niformly recurrent , b y sh o wing that this nice prop erty is im p lied by iv ) of Th eorem 2.3. Th u s, u ltimate ly p erio dic epistur mian words are (pur ely) p erio dic. The ap erio dic episturm ian words are exactly those episturm ian words with exactly on e left sp ecial factor of eac h length. In eac h e quivalenc e class of episturmian words (i.e., same set of f actors), there is one epistan- dard w ord in the ap erio dic case and t wo in the p eriod ic ca se, except if this word is a ω with a a letter. F or example, s 1 = ( abac ) ω has directiv e w ord ∆ 1 = abc ω and s 2 = ( acab ) ω is d irected b y ∆ 2 = acb ω . Bo th s 1 and s 2 are standard with the same factors. Theorem 4.8 in Sect ion 4.3 demonstrates w h y th is is tr ue in general (see also Remark 4.10). 2.2 Bi-inﬁnite episturmian words Deﬁnition 2.1 can b e extended to bi-inﬁnite words, in which case w e m ust assume they are recur- ren t. (As is well kno wn , recurr ence follo ws automaticall y from closure un der reversal in the case of righ t-inﬁn ite w ords ; s ee for ins tance [29] for a pro of of this f act.) Bi-inﬁnite w ord s are sometimes more natural b ecause in particular they can b e shifted in b oth directions, allo wing for simpler form ulations. More sp eciﬁcally , a (r ight-inﬁnite) epistu r mian w ord t can b e prolonged inﬁn itely to the left with the same set of factors, i.e., r emaining in the same equiv alence class. There are sev eral or one such pr olongat ion a ccording t o wh ether or not t = T i ( s ), w ith s epistandard and i ≥ 0 (see [73, 75]). Note. Hereafter, ‘inﬁnite word’ shou ld b e tak en to m ean a righ t-inﬁnite w ord, whereas left-inﬁnite and b i-inﬁnite w ord s will b e explicitly referred to as s u c h. 2.3 Strict episturmian words An epistandard w ord s ∈ A ω , or an y factor-equiv alen t (episturmian) w ord t , is said to b e B -strict (or k - strict if |B | = k , or strict if B is u ndersto o d) if Alph (∆) = Ult(∆) = B ⊆ A . That is, an episturmian word is s trict if ev ery letter in its alphab et o ccurs inﬁ nitely often in its directiv e w ord. The k -strict epistur mian words are precisely the episturmian words t h a ving exactly one left sp ecial factor of eac h length and for wh ic h any left sp ecial factor u in t has k = |A| diﬀerent left extensions in t (i.e., xu is a factor of t for all letters x in the k -letter alphab et A ). As a consequence, k -strict episturmian wo rd s ha v e factor complexit y ( k − 1) n + 1 for eac h n ∈ N 6 (see [43, Theorem 7]); s u c h w ord s are exactly the k -letter Arnoux-R auzy se quenc es , the study of whic h b egan in [12] (see also [74, 105] for example). I n particular, the 2-strict episturm ian wo r ds corresp ond to the (ap erio dic) Sturmian wo rd s. Arnoux-Rauzy sequen ces will b e discussed further in Section 5. Remark 2.7. A notew orth y f act is that an epistur mian word is p erio dic if and only if | Ult (∆) | = 1 (see [73, Prop osition 2.9]). The exact form of a p erio dic epistur mian word is giv en b y Theorem 4.15 in Section 4.4. W e ﬁrst n eed to consider episturmian morphisms . 3 Episturmian morphisms F rom Lemm a 4 in [43], if s is epistandard with ﬁ rst letter a = x 1 , then a is separating for s and its factors, i.e., an y factor of s of length 2 con tains t h e letter a . An y epistu rmian wo rd t that is factor-equiv alen t to s also has s ep arating letter a , and hence can b e factorized with a co de: ( { a } ∪ a ( A \ { a } ) if t b egins with a, { a } ∪ ( A \ { a } ) a otherwise . This leads to episturmian morphisms , w hic h w ere i ntrod uced by Ju stin and P ir illo [73] in ord er to study deep er prop erties of epistu r mian w ords. As w e s hall see in S ectio n 3.2, episturmian morphisms are pr ecisely the morphisms that p reserv e the set of ap erio dic episturmian w ords (i.e., the morph ism s that map aper io dic e p istu rmian words on to aperio dic ep isturmian words). Suc h morphisms naturally generalize to any ﬁnite alphab et the Sturmian morphisms on t wo letters. A morp hism ϕ is said t o b e Sturmian if ϕ ( s ) is Sturmian f or an y Sturmian wo r d s . T h e s et of Sturmian morph isms o v er { a, b } is closed und er comp osition, and consequen tly it is a su bmonoid of the endomorp h isms of { a, b } ∗ . Moreo ver, it is w ell kno wn that th e monoid of Sturmian morp hisms is generated b y the three morph isms: ( a 7→ ab, b 7→ a ), ( a 7→ ba.b 7→ a ), ( a 7→ b, b 7→ a ) and that Sturmian morphisms are precisely the morphisms that map Sturmian words on to Sturm ian w ords (see [16, 87]). 3.1 Generators & monoids By deﬁn ition (see [43, 73]), the monoid of all episturmian morphisms E is generated, u nder comp osition, by all th e morphism s: • ψ a : ψ a ( a ) = a , ψ a ( x ) = ax for an y letter x 6 = a ; • ¯ ψ a : ¯ ψ a ( a ) = a , ¯ ψ a ( x ) = xa for an y letter x 6 = a ; • θ ab : exc hange of letters a and b . Note. This system of generators is f ar from minimal, e.g., ψ a = θ ab ψ b θ ab , but giv es simpler form ulae. Moreo v er, the monoid of so-called epistandar d morphisms S is generated by all the ψ a and the θ ab , and the monoid of pur e epist urmian morphisms E p (resp. pur e epistandar d morp hisms S p ) is generated by the ψ a and ¯ ψ a only (resp. the ψ a only). Th e monoid P of the p ermutation morphisms (i.e., the morphisms ϕ su ch that ϕ ( A ) = A ) is g enerated by all the θ ab . Th e imp ortance of th e monoid of pure episturm ian morphisms will b ecome clearer in th e next section where we shall see that su c h morph isms are strongly linked to spinne d dir e ctive wo r ds of ep isturmian words, which 7 can b e viewed as inﬁn ite sequences of r u les for decomp osing epistur mian w ords by morphisms (see Theorems 3.1 and 3.3, to follo w). In particular, an y episturm ian word is the image of another episturmian w ord by some pu re ep isturmian morph ism. The follo win g d iagram illustrates th e inclus ions b et ween the m onoids deﬁned ab o ve . { Id } S p P E p S E Semidirect pro ducts: S = S p ⋊ P , E = E p ⋊ P W e note in particular that the monoid E is a semidir ect pr od uct of the su bmonoids of its pure m orphisms and of its p erm utations. Consequ en tly , any episturmian morphism ϕ ∈ E can b e expressed in a u nique wa y as ϕ = π µ = µ ′ π , where µ , µ ′ are pur e episturmian morp hisms and π is a p ermutat ion. Note. Th e episturmian morphisms are exactly the S turmian morp hisms when |A| = 2. Clearly , all episturmian morph isms on A can b e viewed as automorphism s of the fr ee group generated by A (e.g., see [57, 65, 99, 116]) and it follo ws that they are injectiv e and that the monoids E and S are left c anc el lative (see [99, L emma 7.2]) whic h means that for an y episturmian morphisms f , g , h , if f g = f h then g = h . Other fu n damen tal prop erties of episturmian morph isms will b e discussed in the n ext section and in Section 4. F or an in-depth study of some fu rther prop erties of these m orp hisms, the intereste d reader is referred to Ric h omme’s pap er [99], in which he considers inv ertibility , p resen tation, cancellativit y , unitarit y , c haracterizatio n b y conjugacy , and so on. Most of the results in [99] naturally generalize those already known for Stu rmian morph isms, but some n ew ones are also prov ed , suc h as a c haracterization of epistu r mian morphisms that preserve palindromes. In [100 , 103], R ichomme also c haracterized the epistu r mian morp hisms that preserv e ﬁ nite and inﬁnite Lyndon wor ds and those that pr eserv e a lexicographic o r d er on w ords. 3.2 Relation with episturmian words W e now state tw o insigh tfu l characte rizations of epistand ard and epistur mian w ords, whic h s h o w that an y episturmian wo rd can b e inﬁnitely de c omp ose d o ve r the set of pur e episturmian mor- phisms. In the ‘standard’ case: Theorem 3.1. [73, Corollary 2.7] An i nﬁnite wor d s ∈ A ω is epistanda r d i f a nd only if ther e exists an inﬁnite wor d ∆ = x 1 x 2 · · · over A and a se q uenc e ( s ( i ) ) i ≥ 0 of r e curr ent inﬁnite wor ds such that s (0) = s and s ( i − 1) = ψ x i ( s ( i ) ) for i > 0 . ⊔ ⊓ 8 In [73], Justin and Pirillo sh o w ed that the inﬁnite w ord ∆ app earing in the ab o v e theorem is exactly the directiv e w ord of s that arises fr om th e equ iv alen t d eﬁnition of epistandard w ords giv en in Theorem 2.3. In the binary case, the d irectiv e w ord ∆ is related to the con tin ued fr action expansion of the slop e of the s tr aigh t line represen ted b y a standard w ord (see Chapter 2 in [83]). Example 3.2. Recall the T rib onacci w ord r , wh ic h has d irectiv e word ∆ = ( abc ) ω . W e hav e r = ψ a ( r (1) ), wh ere r (1) is directed by T(∆) = ( bca ) ω . Notice that r (1) = π ( r ) with π = ( abc ); a v ery particular case. More generally , the follo wing result (Theorem 3.3) extends the notion of a direct ive word to all episturm ian words. Before stating the theorem, w e need to in tro du ce some more notation. First w e deﬁne a new alphab et, ¯ A := { ¯ x | x ∈ A} . A lette r ¯ x ∈ ¯ A is c onsid ered t o b e x w ith spin 1, whilst x itself has spin 0. The notion of a spin pro vides a co nv enien t wa y to call up on the elemen tary pure epistur mian morphisms ψ x and ¯ ψ x . Moreo ve r, as w ell shall see in S ecti on 4 , it a llows us to deriv e man y prop erties of episturmian w ords from episturmian morphisms (a s a consequence of the next theorem). This approac h is used for instance in [23, 60, 81, 101, 102, 105] and of course in th e pap ers of Ju s tin et al. A ﬁnite or inﬁnite w ord o ver A ∪ ¯ A is said to be a sp inne d w ord. Given a ﬁnite or in ﬁnite w ord w = x 1 x 2 · · · o ver A , we sometimes denote b y ˘ w = ˘ x 1 ˘ x 2 · · · a ny spinned word suc h that ˘ x i = x i if x i has spin 0 and ˘ x i = ¯ x i if x i has spin 1. S uc h a word ˘ w is called a spinne d version of w . Theorem 3.3. [73, T h eorem 3.10] An inﬁnite wor d t ∈ A ω is episturmian if and only if ther e exists a spinne d inﬁnite wor d ˘ ∆ = ˘ x 1 ˘ x 2 ˘ x 3 · · · over A ∪ ¯ A and an inﬁnite se quenc e ( t ( i ) ) i ≥ 0 of r e curr ent inﬁnite wor ds such that t (0) = t and t ( i − 1) = ψ x i ( t ( i ) ) or t ( i − 1) = ¯ ψ x i ( t ( i ) ) for al l i > 0 , ac c or ding to the spin 0 or 1 of ˘ x i , r esp e ctively. F or any epistandard w ord (resp. epistu rmian w ord) t and inﬁnite word ∆ (resp . spinned inﬁn ite w ord ˘ ∆) satisfying th e conditions of the Theorem 3.1 (resp . Theorem 3.3), w e sa y that ∆ (resp. ˘ ∆) is a dir e ctive wor d (resp. a (spinne d) dir e ctive wor d ) for t or t is dir e cte d by ∆ (resp . ˘ ∆). Remark 3.4. It follo ws immediately from Theorem 3.3 that if t is an ep istu rmian word dir ecte d b y a sp inned inﬁ nite w ord ˘ ∆, then eac h t ( n ) (as deﬁ n ed in th e theorem) is an episturmian word directed by T n ( ˘ ∆) = ˘ x n +1 ˘ x n +2 ˘ x n +3 · · · . The follo win g imp ortan t fact links Theorems 3.1 and 3.3. Remark 3.5. [73] If t is an epistur mian w ord directed by a spinned v ersion ˘ ∆ of an inﬁn ite word ∆ o ver A , then t is f acto r-equiv alen t to the (u nique) epistandard w ord s directed by ∆. Moreo v er, with the same n ota tion as in the ab o ve remark, the epistu rmian w ord t is p erio dic if and only if the epistandard w ord s is perio dic, and t h is h olds if and only if | Ult(∆) | = 1 (see Remark 2.7 or T heorem 4.15 later). Example 3.6. Consider the epistu r mian w ord m = baabacabab · · · directed b y ˘ ∆ = ¯ ab ¯ c ( abc ) ω . Observe that m is factor-equiv alen t to th e T rib onacci word r , and we ha ve m = ¯ ψ a ( m (1) ) = ¯ ψ a ψ b ( m (2) ) = ¯ ψ a ψ b ¯ ψ c ( m (3) ) , where m (3) is directed by T 3 ( ˘ ∆) = ( abc ) ω , i.e., m (3) = r . 9 Example 3.7. W e now consider an example where the condition that t h e t ( i ) in Theorem 3. 3 are recurrent is not satisﬁed. Let t = d r = dabacabaabacaba · · · where r is the T rib onacci w ord and d is a letter. Then t = ¯ ψ a ( t (1) ), t (1) = ¯ ψ b ( t (2) ), t (2) = ¯ ψ c ( t (3) ), and so on; how ever, these t ( i ) are not recur r en t (and t is n ot episturmian). Th e inﬁn ite word t = d r is actually an example of an episkew wor d , i.e., a non-recur ren t inﬁ n ite word ha ving episturmian factors. S uc h wo rd s are discussed in more detail in S ectio n 7.2. Remark 3.8. Let u s p oint out that th e construction of epistandard words by palind romic closure (giv en in Theorem 2.3) extends to all epistur mian w ords: w h en ˘ x n = ¯ x n write x n on the left and use p alindr omic left-closur e . Here m (from the ab o v e example) app ears step b y step on the righ t: a · a · b a abac a · ba abaca · baa bacaba When an epistur mian w ord is ap erio dic, we ha ve the follo wing f undamen tal link b et w een the w ords ( t ( n ) ) n ≥ 0 and th e spinned inﬁn ite wo rd ˘ ∆ o ccurring in Theorem 3.3: if a n is the ﬁrs t letter of t ( n ) , then µ ˘ x 1 ··· ˘ x n ( a n ) is a pr eﬁx of t and the sequence ( µ ˘ x 1 ··· ˘ x n ( a n )) n ≥ 1 is n ot ultimately constan t (since ˘ ∆ is not ultimately constant), th en t = lim n →∞ µ ˘ x 1 ··· ˘ x n ( a n ). Th is fact is a s ligh t generalizat ion of a result of Risley and Zam b oni [105, Prop. I I I.7] on S- adic r epr esentations for standard Arnoux-Rauzy sequences. See also the recen t pap er [23] for S-adic represen tations of Sturmian words. Note that S - adic dynamic al systems w ere in tro duced by F erenczi [50] as minimal dynamic al systems (e.g. , s ee [9 6]) generated b y a ﬁnite n umber of substitutions. In the case of episturmian words, th e n otio n itself is actually a reformulatio n of the well-kno w n R auzy rules , as studied in [98]. In fact, it is w ell known that the subshift of an ap erio dic epistur m ian w ord t (i.e., the top ological closure of th e sh ift orbit of t ) is a minimal d ynamical system, i. e., it consists of all the episturm ian w ords with the same set of factors as t . It is not hard to see that a m orphism is episturm ian (resp. epistand ard) if an d only if it preserve s the set of ap erio dic episturmian (resp. epistand ard ) w ords (see [73]). Eve n more: Theorem 3.9. [73, Theorem 3.13] A morphism ϕ is episturmian (r e sp. epistandar d) if ther e exist strict episturmian (r esp. epistandar d) w or ds m , t such that m = ϕ ( t ) . ⊔ ⊓ Purely morph ic episturmian words (i.e., those generated by morph isms) are d iscussed f urther in Sect ion 4, where we consider the r elat ionsh ip b etw een spins and the shifts that they in d uce. These ideas w ere used in [75] to obtain a complete answe r to the question: if an episturmian w ord is pur ely morphic, whic h shifts of it, if an y , are also p urely morphic? (See Theorem 4.19, to follo w.) Suc h rigidit y issu es are d iscussed in more detail in Sections 4.4 and 8. In [75], Justin and Pirillo also made u se of bi-inﬁ nite words, wh ic h often allo w for more natural form ulations. Indeed, the c h aracte rization (Theorem 3. 3 ) of righ t-inﬁn ite episturmian words b y a sequence ( t ( i ) ) i ≥ 0 extends to bi-inﬁnite episturmian wo r d s, with all the t ( i ) no w bi-inﬁn ite episturmian words. T hat is, as for right-inﬁnite ep istu rmian wo rd s, w e hav e bi-inﬁnite wo rd s of the f orm l ( i ) . r ( i ) where l ( i ) is a left-inﬁnite epistur mian wo rd and r ( i ) is a r igh t-inﬁnite episturmian w ord. Moreo v er, if the bi-inﬁ nite episturm ian wo rd b = l . r is directed by ˘ ∆ with asso ciated b i- inﬁnite episturmian w ords b ( i ) = l ( i ) . r ( i ) , then r is directed b y ˘ ∆ with associated right-inﬁnite episturmian w ords r ( i ) . 10 4 Spins, shifts, and d irectiv e w ord s In this section, w e discuss in more detail the n otion of sp ins, the shifts they in duce, and the concept of blo c k -e quiv alenc e in connection with dir ectiv e words. These notions allo w u s to s tu dy in particular when t wo diﬀerent spinn ed inﬁ nite w ords direct the same epistu rmian w ord. Indeed, as we shall see in Section 4.3, the corresp ondence b et wee n episturm ian wo rd s and spinned directiv e w ords is n ot one-to-one. 4.1 Notation for pure episturmian morphisms F or a ∈ A , let µ a = ψ a and µ ¯ a = ¯ ψ a . Th is op erator µ can b e natur ally extended (as done in [73]) to a pure episturmian morphism: for an y spinn ed ﬁnite wo rd ˘ w = ˘ x 1 · · · ˘ x n o v er A ∪ ¯ A , w e deﬁne µ ˘ w := µ ˘ x 1 · · · µ ˘ x n and set µ ε equal to the identi ty morp hism Id. Viewing w = x 1 x 2 · · · x n as a preﬁx of the directiv e w ord ∆ = x 1 x 2 x 3 · · · ∈ A ω , it is clear from Theorem 3.1 that the words µ x 1 ··· x n − 1 ( x n ) , n ≥ 1 , are p reﬁxes of the epistandard w ord s directed by ∆. Example 4.1. W e observe that any epistandard word s ∈ A ω has the form s = µ w ( s ′ ) for some uniquely determined ﬁnite word w and strict epistandard w ord s ′ . In deed, if ∆ = x 1 x 2 x 3 · · · ∈ A ω is the directiv e word of s and m is the smallest p ositiv e intege r s u c h that Alph( x m +1 x m +2 · · · ) = Alph(∆), then x 1 · · · x m is the shortest p r eﬁx of ∆ that conta ins all the letters n ot app earing inﬁnitely often in ∆. Moreo v er, by Theorem 3.1, s = µ x 1 ··· x m ( s ( m ) ) where s ( m ) is the ep istandard w ord directed by T m (∆) = x m +1 x m +2 · · · . Since Ult(T m (∆)) = Alph(T m (∆)) b y construction, the epistandard w ord s ( m ) is str ict. F or example, with ∆ = c ( ab ) ω , we ha ve s = ψ c ( s (1) ) where s (1) is directed by ( ab ) ω , i.e., s (1) is the we ll-known Fib onac ci wor d o ver { a, b } . F or n ≥ 1, let u n +1 := P al ( x 1 · · · x n ) and set u 1 = ε . Then b y p art iv ) of Theorem 2.3, the epistandard wo rd s directed by ∆ is gi ven by s = lim n →∞ u n . W e ha ve the fol lowing useful form ula from [73 ]: u i +1 = µ x 1 ··· x i − 1 ( x i ) u i for i > 0 . (4.1) F or letters ( x j ) 1 ≤ j ≤ i , f ormula (4.1) inductivel y leads to: u i +1 = µ x 1 ··· x i − 1 ( x i ) · · · µ x 1 ( x 2 ) x 1 = Y 1 ≤ j ≤ i µ x 1 ··· x j − 1 ( x j ) . (4.2) (Note that by con ve ntion, x 1 · · · x 0 = ε in the ab o ve p ro duct.) F or example, with ∆ = abcb · · · , w e compute: u 3 = P al ( abcb ) = µ abc ( b ) µ ab ( c ) µ a ( b ) a = abacab · abac · ab · a. 4.2 Shifts No w let ˘ w = ˘ x 1 ˘ x 2 · · · ˘ x n b e a spinn ed version of w = x 1 x 2 · · · x n (view ed as a preﬁx of a spin ned v ersion ˘ ∆ of ∆). Th en, for any ﬁnite word v , we ha v e µ ˘ w ( v ) = S − 1 ˘ w µ w ( v ) S ˘ w where S ˘ w = Q i = n,..., 1 | ˘ x i = ¯ x i µ x 1 ··· x i − 1 ( x i ). (4.3) Observe that S ˘ w is a preﬁx of P al ( w ); in particular S ¯ w = P al ( w ) by equation (4.2). Note also that µ ˘ w ( v ) = T | S ˘ w | ( µ w ( v )). Th e w ord S ˘ w is called the shifting factor of µ ˘ w and its length | S ˘ w | is called th e shift induc e d by th e preﬁx ˘ w of ˘ ∆ of length n [75 ]. 11 Example 4.2 . If we tak e ˘ w = a ¯ bc ¯ a , then S ˘ w = µ abc ( a ) µ a ( b ) = abacaba · ab. Th u s since µ abca ( ca ) = abacabaab · acabacaba , we ha ve µ a ¯ bc ¯ a ( ca ) = T 9 ( µ abca ( ca )) = acabacaba · abacabaab. Lik ewise, for any inﬁn ite word y ∈ A ω , µ ˘ w ( y ) = S − 1 ˘ w µ w ( y ). F or example, if we tak e ˘ w = ¯ a ¯ b , then S ˘ w = P al ( ab ) = aba , and hen ce µ ¯ a ¯ b ( y ) = ( aba ) − 1 µ ab ( y ) for an y inﬁ nite w ord y . Note. Th e morphisms µ w and µ ˘ w are c onjugate morphisms [99]. 4.3 Blo c k-equiv alence & directiv e w ords By Theorem 2.3 (a n d also Theorem 3.1), an y epistandard w ord s ∈ A ω has a unique directiv e w ord o ve r A , but s also has inﬁn itely many other spinn ed dir ective words (see [73, 75, 63]). F or example, t h e T r ib onacci w ord is directed by ( abc ) ω and also by ( abc ) n ¯ a ¯ b ¯ c ( a ¯ b ¯ c ) ω for eac h n ≥ 0, as well a s inﬁnitely m any other spinned w ords. T h e natural question: “do es any sp inned w ord direct a un ique episturmian word?” was answered in [73]. Prop osition 4.3. [73] 1. Any spinne d inﬁnite wor d ˘ ∆ having inﬁnitely many letters with spin 0 dir e cts a uni q ue episturmian wor d b e gi nning with the left-most letter having spin 0 in ˘ ∆ . 2. Any spinne d inﬁnite wor d ˘ ∆ with al l spins ultimately 1 dir e cts exactly | Ult(∆) | episturmian wor ds. 3. L et ˘ ∆ b e a spinne d inﬁnite wor d having al l its letters with spin 1 and let a ∈ Ult(∆) . Then ˘ ∆ dir e cts e xactly one episturmian wor d st arting with a . ⊔ ⊓ Note. The ab o v e statemen t corrects a small error in Prop osition 3.1 1 of [73] where it em 3 wa s stated in the more general case when ˘ ∆ has all spins ultimat ely 1. In this c ase, ˘ ∆ sti ll directs exactly one epistur mian word f or eac h letter a in Ult(∆), bu t contrary to what is written in [73], nothing can b e said ab out its ﬁ rst letter. Blo ck-e quivalenc e for spinn ed w ords was in tro du ced in [75] as a w a y of studying when ˘ ∆ and ˆ ∆ (t wo spinn ed ve r s ions of a d irectiv e word ∆) direct the same bi-inﬁn ite episturmian wo rd . W e do not recall the fu ll details here, only a f ew notions relating to it. Notation. I f v ∈ A + , th en ¯ v ∈ ¯ A + is v w ith all spins 1. A word of the form xv x , wh ere x ∈ A and v ∈ ( A \ { x } ) ∗ , is called a ( x -based) blo ck . A ( x - based) blo ck- tr ansformation is the replacemen t in a spinned word of an o ccurrence of xv ¯ x (wh er e xv x is a b lock) by ¯ x ¯ v x or vice-v ersa. Two ﬁ nite sp inned words w , w ′ are said to b e blo ck-e quivalent if we can pass from one to the other by a (p ossibly empty) c hain of blo c k-transform ations, in whic h case we w rite w ≡ w ′ . F or example, ¯ b ¯ ab ¯ cb ¯ a ¯ c and babc ¯ b ¯ a ¯ c are blo c k-equiv alen t b ecause ¯ b ¯ ab ¯ cb ¯ a ¯ c → ba ¯ b ¯ cb ¯ a ¯ c → babc ¯ b ¯ a ¯ c and v ice-v ersa. Note that if w ≡ w ′ then w an d w ′ are spinned v ersions of the s ame wo rd o ver A . Blo c k-equiv alence extends to (right- )inﬁ nite words as f ollo ws. Let ∆ 1 , ∆ 2 b e spinned ve rs ions of ∆. W e write ∆ 1 ∆ 2 if there exist inﬁ nitely man y preﬁxes f i of ∆ 1 and g i of ∆ 2 with the g i of str ictly increasing lengths, a n d su c h that, for all i , | g i | ≤ | f i | and f i ≡ g i c i for a su itable spin ned word c i . Inﬁ nite wo rd s ∆ 1 and ∆ 2 are said to b e blo ck-e quivalent (denoted b y ∆ 1 ≡ ∆ 2 ) if ∆ 1 ∆ 2 and ∆ 2 ∆ 1 . 12 Remark 4.4. If x is a letter and v ∈ A ∗ is x -free, then ¯ x ¯ v x and xv ¯ x are b lock-e quiv alen t and they induce the same shift, i.e., µ ¯ x ¯ v x = µ xv ¯ x [75, T h eorem 2.2]. Thus the monoid of pure episturmian morphisms, E p , is isomorphic to the quotient of ( A ∪ ¯ A ) ∗ b y the blo c k-equiv alence generated by { ¯ x ¯ v x ≡ xv ¯ x | x ∈ A , v is x -free } . Note that this h as some relat ion to the study o f conjugacy and episturmian mo r phisms carried out by Ric homme [99]. F rom what we ha ve already learned ab ou t bi-inﬁn ite episturmian w ords (in S ectio n s 2.2 and 3.2), it is clear that Justin and Pirillo ’s results ab out spinned inﬁnite words directing the same bi-inﬁnite episturmian w ord are still v alid f or w ords directing the same (right- in ﬁ nite) episturm ian w ord. Roughly sp eaking, tw o spinn ed inﬁ nite words direct the same episturmian wo rd if and only if th ey are blo c k-equiv alen t. F or ins tance, we hav e the follo wing results for wavy spinned v ersions of ∆ ∈ A ω . A spinned ve rs ion ˘ ∆ of ∆ is said to b e wavy if ˘ ∆ cont ains inﬁ nitely m an y letters of spin 0 and inﬁnitely m any letters of spin 1. Theorem 4.5. [75, Theorem 3.4] Supp ose ˘ ∆ and ˆ ∆ ar e wavy versions of ∆ ∈ A ω with | Ult(∆) | > 1 . Then ˘ ∆ and ˆ ∆ dir e ct the same episturmian wor d if and only if ˘ ∆ ≡ ˆ ∆ . ⊔ ⊓ F or example, ba ( ¯ bc ¯ a ) ω and ¯ b ¯ ab ( c ¯ a ¯ b ) ω direct the same episturmian w ord, namely µ ba ¯ bc ( c r ) (= µ ¯ b ¯ abc ( c r )) where r is the T rib onacci wo rd . Theorem 4.6. [75, Pr op. 3.6] L et ˘ ∆ , ˆ ∆ b e two spinne d version s of ∆ ∈ A ω with | Ult(∆) | > 1 , ˘ ∆ wavy, and ˆ ∆ having al l spins ultimately 0 or 1 . If ˘ ∆ and ˆ ∆ dir e ct the same episturmian wor d, then ˘ ∆ ˆ ∆ . ⊔ ⊓ Similar results also hold when al l spins are ultimately 0 or 1 and in the p erio dic case. See Prop ositions 3.7 and 3.10 in [75]. Remark 4.7. In [75], the study of blo c k-equiv alence for ﬁnite s pinned wo rd s led to numeration systems that resem ble the Ostr owski systems [20] asso ciated with Sturmian wo rd s. A matrix form ula for computing the num b er of repr esentati ons of an in teger in su ch a system was also giv en in [75 , Section 2]. More recen tly , Glen, Lev´ e, and Richomme [63] established t h e follo wing co mp lete c haracter- ization of pairs of sp in ned inﬁn ite words d ir ecting the same unique episturmian w ord . Not on ly do es the follo wing theorem p r o vide th e relativ e forms of t wo spin ned inﬁnite words directing the same episturmian word, but it also fully solves th e p erio dic case, whic h was only partially solv ed in [75]. Theorem 4.8. [63] Given two spinne d inﬁnite wor ds ∆ 1 and ∆ 2 , the fol lowing assertions ar e e qui valent. i) ∆ 1 and ∆ 2 dir e ct the same right-inﬁnite episturmian wor d. ii) 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 ; 13 2. ∆ i = wx 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 a letter, ( v n ) n ≥ 1 is a se quenc e of non-empty x -fr e e wor ds, and ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 ar e se quenc es of non-empty 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 ) . ⊔ ⊓ In items 1 and 2 of Theorem 4.8, the t wo considered directiv e w ords are s pinned versions of the same inﬁnite wo r d . Th is do es n ot hold in item 3, wh ic h concerns only p erio dic episturmian w ords. In p articular, we observ e the follo wing: Remark 4.9. I f an ap e rio dic epistu r mian w ord is directed by t w o diﬀerent spinned inﬁnite w ords ∆ 1 and ∆ 2 , th en ∆ 1 and ∆ 2 are sp inned v ersions of the same word ∆. As an example of item 3, one can consider the p erio dic episturmian word ( bcba ) ω whic h is directed by b oth bca ω and b ¯ ac ω (since µ bc ( a ) = µ b ¯ a ( c )). Note also that ( bcba ) ω is epistandard and has the same set of fac tors as the epistandard word ( babc ) ω directed by bac ω . Actually , in view of Remark 3.5, we observ e the follo win g: Remark 4.10. The s u bshift of any ap erio dic episturmian word cont ains a un ique (ap erio d ic) epistandard word, whereas the sub shift of a p erio dic episturmian word con tains exactly t wo (p e- rio dic) epistandard w ord s, except if this wo rd is a ω with a a letter. W e also observe that x and y can b e equal in item 3 of Theorem 4 .8 ; for example ( ab ) ω is directed by a ¯ bb ω and by ab ω . Example 4.11. [63] F or a, b, c th ree diﬀerent letters in A , the spinned inﬁnite w ords ∆ 1 = a ( bc ¯ a ) ω and ∆ 2 = ¯ a ( ¯ b ¯ c ¯ a ) ω direct the same episturmian w ord that starts with the letter a . Ind eed, these t wo directiv e words fulﬁll item 2 of T heorem 4.8 w ith w = w ′ = ε , x = a , and for all n , v n = bc and ˘ x n = ˆ x n = ¯ a . Moreo v er the fact that ∆ 1 starts with th e letter a sho ws that the word it directs starts with a . Similarly ∆ ′ 1 = ¯ ab ( ca ¯ b ) ω and ∆ ′ 2 = ¯ a ¯ b (¯ c ¯ a ¯ b ) ω direct the same episturmian w ord starting w ith t h e letter b . S in ce ∆ 2 = ∆ ′ 2 , this sh o ws that the relation “direct the same episturmian w ord” o v er spinn ed inﬁnite words is not an equiv alence relation. Items 2 and 3 of Theorem 4.8 sho w that an y episturmian w ord is directed b y a spinned inﬁn ite w ord h a v in g in ﬁ nitely many letters of spin 0, bu t also by a spin n ed wo rd ha vin g both inﬁ nitely man y letters of spin 0 and in ﬁ nitely m an y letters of spin 1 (i.e., a wa vy wo rd ). T o emphasize th e imp ortance of these facts, let us recall from Prop osition 4.3 that if ˘ ∆ is a sp in ned inﬁn ite w ord o v er A ∪ ¯ A with in ﬁnitely man y letters of spin 0, then there exists a u nique epistu rmian wo rd t directed by ˘ ∆. Unicit y comes from the fact that the ﬁ rst letter of t is ﬁxed by the ﬁ rst letter of spin 0 in ˘ ∆. W e also note th at if an epistu r mian w ord t h as t wo directiv e words satisfying items 2 or 3, th en t h as inﬁnitely man y directiv e words (this w as shown in [63]). When studying rep etitions in Sturmian words, Berth ´ e, Holton, and Zam b oni [23] prov ed th at an y Sturmian w ord has a un ique directiv e word o v er { a, b, ¯ a, ¯ b } contai n ing inﬁnitely many letters of spin 0, but n o facto r of the form ¯ a ¯ b n a or ¯ b ¯ a n b with n an int eger. Lev ´ e and Richomme [80] r ecen tly generalized this result to episturmian w ord s b y int r o du cing a wa y to ‘n ormalize ’ the directiv e w ord(s) of an episturmian word s o that any episturm ian word can b e d eﬁned u niquely b y its so-calle d normalize d dir e ctive wor d , deﬁned by some facto r a v oidance, as f ollo ws . This idea has since prov ed useful in the stud y of quasip erio dic epistur mian w ords (see S ection 8); in particular, it p r o vides an eﬀectiv e w a y to decide whether or n ot a giv en episturmian word is quasip erio dic. 14 Theorem 4.12. [63, 80] Any episturmian wor d t ∈ A ω has a spinne d dir e ctive wor d ˘ ∆ c ontaining inﬁnitely many letters of spin 0 , but no factor in S a ∈A ¯ a ¯ A ∗ a . Such a dir e ctive wor d is uniqu e if t is ap erio dic, in which c ase ˘ ∆ is c al le d the normalized directiv e w ord for t . ⊔ ⊓ Note. Unicit y do es n ot n ecessarily hold for p erio dic episturmian wo rd s. F or example, the p erio dic episturmian w ord ( ab ) ω = ψ a ( b ω ) = ¯ ψ b ( a ω ) is directe d by ab ω and b y ¯ ba ω (since ψ a ( b ) = ab = ¯ ψ b ( a )). The follo win g r esu lt tells us precisely whic h epistur mian w ords hav e a uniqu e directiv e word. Theorem 4.13. [63] An episturmian wor d t ∈ A ω has a uniqu e dir e ctive wor d if and only if the (normalize d) dir e ctive wor d of t c ontains 1) inﬁnitely many letters of spin 0 , 2) inﬁnitely many letters of spin 1 , 3) no factor in S a ∈A ¯ a ¯ A ∗ a , and 4) no factor in S a ∈A a A ∗ ¯ a . Such an episturmian wor d is ne c essarily ap erio dic. ⊔ ⊓ F or instance, a particular family of episturmian words ha ving unique d ir ectiv e w ords consists of those dir ected b y r e gular wavy wor ds [58, 64], i.e., spinned in ﬁ nite words ha ving b oth in ﬁnitely man y letters of spin 0 and inﬁn itely many letters of spin 1 suc h that eac h letter o ccurs with the same spin eve ry w here in the directive w ord . More formally , a spinned ve rs ion ˘ w of a ﬁnite or inﬁnite w ord w is said to b e r e g ular if, for eac h letter x ∈ Alph( w ), all o ccurrences of ˘ x in ˘ w hav e the same sp in (0 or 1). F or example, the regular wa vy wo rd ( a ¯ b ¯ c ) ω is th e u nique directiv e word for the epistur m ian word a r = aabacabaabacab · · · wh ere r is the T rib onacci w ord. In the Sturm ian case, we ha ve : Prop osition 4.14. [63] Any Sturmian wor d has either a unique spinne d dir e ctive wor d or in- ﬁnitely many spinne d dir e ctive wor ds. Mor e over, a Sturmian wor d has a uniqu e dir e ctive wor d if and only if its (normalize d) d ir e c tiv e wor d i s r e g ular wavy. ⊔ ⊓ As p oint ed out in [63], Prop osition 4.14 sho ws a great diﬀerence b etw een Stur m ian w ords and episturmian words constructed o ver alphab ets with at least three letters. Indeed, when considerin g w ords ov er a ternary alphab et, one can ﬁn d episturmian wo rd s ha ving exactly m directiv e w ord s for an y m ≥ 1. F or instance, the episturmian word t d irected b y ˘ ∆ = a ( b ¯ a ) m − 1 b ¯ c ( ab ¯ c ) ω has exactly m directiv e words, namely ( ¯ a ¯ b ) i a ( b ¯ a ) j b ¯ c ( ab ¯ c ) ω with i + j = m − 1. Notice that the suﬃx b ¯ c ( ab ¯ c ) ω of ˘ ∆ is regular wa vy , and the other m − 1 spinned v ersions of ∆ that also direct t arise from the m − 1 words that are b lock-e qu iv alen t to th e preﬁx a ( b ¯ a ) m − 1 . 4.4 P erio dic and purely morphic episturmian w ords W e are n o w ready to describ e p erio dic and p urely morphic episturmian words. Recall fr om Remark 2.7 that the p erio dic epistu r mian w ord s corresp ond to | Ult (∆) | = 1. The follo wing theorem giv es the form of such w ords in terms of pu re episturmian morp hisms. Theorem 4.15. [73] An episturmian wor d is p e rio dic i f and only if it is ( µ ˘ w ( x )) ω for some spinne d ﬁnite wor d ˘ w and letter x . ⊔ ⊓ F or example, ( µ a ¯ b ( c )) ω = ( acab ) ω is the p erio dic episturmian w ord directed by a ¯ bc ω (in f act, it is epistandard as it is also d irected by acb ω ). The next t h eorem charact erizes p urely morp hic episturm ian words w ith respect to their di- rectiv e wo rd s. 15 Theorem 4.16. [73, T heorem 3.14 ] A n ap erio dic episturmian wor d is pur ely morphic (i.e., gen- er ate d by a morphism) if and only if it i s dir e cte d by a p erio dic spinne d inﬁnite wo r d ˘ ∆ = ( ˘ f ) ω for some spinne d wor d ˘ f . Mor e over it c an b e gener ate d by µ ˘ f . ⊔ ⊓ W e observ e from Theorem 4 .16 that any p urely morphic episturmian word is s tr ict (i.e., an Arnoux-Rauzy sequ en ce) as Ult (∆) = Alph( f ) = Alph(∆). Th e pro of of th is theorem mak es us e of Pr op osition 4.3 and Theorem 3.9. Example 4.17. T he T rib onacci w ord is generated by µ abc . Notice that µ abc = σ 3 where σ is the T rib onac ci mo rphism deﬁned by σ : ( a, b, c ) 7→ ( ab, ac, a ). Remark 4.18. Purely morph ic stand ard Sturmian w ords were previously c haracterized indep en- den tly in the f ollo wing pap ers: [16, 38, 77]. Y asutomi [118] has since established a c haracterizati on of all pur ely morp hic Stur mian w ord s with resp ect to th eir slop es and in tercepts (when vie wed as cutting sequences). An alternativ e geometric pro of of Y asutomi’s r esult was recen tly give n b y Berth ´ e et al. in [21]. Using the n otio n of b lock-e qu iv alence, Justin and Pirillo [75] explicitly determined wh ic h shifts, if any , of a purely morphic episturm ian w ord are also p urely morphic. Theorem 4.19. [75] If an episturmian wor d t is pur ely morphic, then its shift T i ( t ) is also pur ely morphic if and only if i b elongs to some p articular interval. ⊔ ⊓ See S ectio n 4 of [75] f or sp eciﬁc (and v ery tec hn ical) d etails. Example 4.20. F or the T rib onacci wo rd r , only itself and T − 1 ( r ) are p urely morphic. Note that T − 1 ( r ) corresp ond s to three episturm ian w ord s : a r , b r , c r , d irected b y ( a ¯ b ¯ c ) ω , (¯ ab ¯ c ) ω , (¯ a ¯ bc ) ω , resp ectiv ely . Remark 4.21 . Theorem 4.19 corrects an error in [73, S ectio n 5.1] where it wa s mistak enly s aid that if an epistu r mian word is pu rely morphic then an y shift of it is also pu rely morp h ic. In deed, this is false even in the S turmian case as F agnot [48] has s ho wn that if s is a p urely morphic standard S turmian w ord on { a, b } , then a s , b s , ab s , ba s (wh ic h are pur ely m orphic [17]) are the only pu r ely morphic S turmian w ords related to s by a sh ift. 5 Arnoux-Rauzy sequences W e now br ieﬂy turn our atten tion to Arnoux-Rauzy sequences since their combinatoria l pr op erties are also considered in the sections that follo w. Arno ux- R auzy se quenc es are unif orm ly recurrent inﬁ nite wo rd s ov er a ﬁn ite alphab et A with factor complexit y ( |A| − 1) n + 1 for eac h n ∈ N , and exactly one r ight and one left sp ecial factor of eac h length. They w ere in tro duced by Arnoux a n d Rauzy [9 7, 12], who studied them using R auzy gr aphs , with particular emphasis on th e case |A| = 3. (Note that the foregoing deﬁnition is equiv alen t to the one given in the intro d uction.) As ment ioned previously (in Section 2.3), Arnoux-Rauzy sequences are exactly the strict episturmian w ords ; in particular, an y epistur m ian wo r d has the form ϕ ( t ) with ϕ an episturmian morphism and t an Arnoux -Rauzy sequence. In this sense, episturmian words are only a s light generalizat ion of Ar noux-Rauzy s equ ences. F or example, the family of episturmian w ords on three letters { a, b, c } consists of the Arn oux-Rauzy s equ ences o ve r { a, b, c } , the Sturm ian words 16 o v er { a, b } , { b, c } , { a, c } and their images under epistu r mian morphisms on { a, b, c } , and p erio dic inﬁnite w ords of the form ϕ ( x ) ω where ϕ is an episturmian morphism on { a, b, c } an d x ∈ { a, b, c } . Arnoux-Rauzy sequences h a v e deep prop erties studied in th e framew ork of dynamical systems, with c onn ectio ns to geometrical realizat ions suc h as R auzy fr actals [11] and in terv al exc hanges. When |A| = 3, the condition on th e sp ecial factors d istinguishes Arn oux-Rauzy sequences from other inﬁ nite w ord s of complexit y 2 n + 1, su ch as those obtained by co ding tr a jectories of 3-in terv al exc hange transformations (e.g., see [51]). In [12], it w as sh o wn how Arnoux-Rauzy sequences of complexit y 2 n + 1 (i.e., th e 3-strict episturmian w ords ) ca n b e geometrically r ealiz ed by an exc hange of six in terv als on the un it circle, wh ic h generalizes the repr esen tation of S turmian sequences b y rotations. An alternativ e wa y of introducing and studying Arnoux-Rauzy sequ ences is in the con text of S -adic dyna mic al systems , as done in [105] for instance (see our remarks follo wing Theorem 3.3 in Sect ion 3.2). In [4 0 ], Damanik and Zam b oni giv e a kind of sur v ey on this approac h by con- sidering Ar noux-R auzy subshifts and answ ering v arious combinatorial qu estions concerning linear recurrence, maxima l p o w ers of factors, a nd the n u mb er of palindromes of a g iven length. They also p resen t some applications of their r esults to the sp ectral theory of discrete one-dimens ional Sc hr¨ odinger op erators with p oten tials giv en b y Arnoux-Rauzy sequences. Arnoux-Rauzy sequences also ha ve inte resting arithm etic al p r op erties. F or instance, if one considers the fr e quencies of letters (as discu ssed later in Section 6.4), they are we ll-deﬁn ed, and renormalization by an episturmian morp h ism leads to a generalization of the con tinued fraction algorithm that asso ciates to eac h k -letter Ar noux-Rauzy sequence an inﬁ nite arra y of k × k rational n umb ers. In the sp ecial case k = 2, these f ractio n s are consecutive F arey n umb ers arising from the contin u ed fr acti on expansion of the frequencies of the t wo letters. More g enerally , giv en an Arnoux-Rauzy sequence on k -letters, its dir ecti ve w ord is determined by the ‘multi-dimensional’ con tin u ed fraction expansion of the frequencies of the ﬁrst k − 1 letters. Unfortunately , this generalized algorithm (except for th e ca se k = 2 when it is exactly t h e usual cont inued fraction algorithm) is only deﬁ ned on a set of measure zero in R k − 1 . This redu ces its interest and explains wh y it has n ot b een appr op r iatel y studied sin ce its inception (see Sections 6.2.1 and 6.4 for further details). Nonetheless, a nice arith m etica l c haracterization of 3-letter Ar n oux-Rauzy sequences can b e giv en, as follo ws. W e say that a triple ( a, b, c ) do es not satisfy the triangular inequ alit y if one of the co ord inates is larger than the sum of the other tw o (e.g., a > b + c ). In that case, we can renormalize in a uniqu e wa y to obtain the triple ( a − b − c, b, c ) satisfying th e triangular inequalit y . The set of allo wa b le fr equ encies for 3-letter Arnouxy-Rauzy sequen ces is exactly the set of triples ( a, b, c ) that can b e inﬁ nitely r enormalized, eac h time to a triple that do es not satisfy the tr iangular inequalit y (see [12 ]). The resu lting picture exhibits a kind of Sie rpinski c arp et . F or further details on Arnoux-Rauzy sequ ences, we refer the reader to the inte resting su r v ey [22] in whic h Berth´ e, F erenczi, and Zam b on i discuss connections b et we en Arnoux-Rauzy sequences and rotations of the 2-torus; co ding of t wo -dimens ional actions and t wo -dimen s ional Stur mian w ords; and in terv al e xchanges and sequences of lo w complexit y . See also [35], Section 12 .2.3 in [96], and J. Berstel’s nice survey pap er [15] in whic h he compares some combinato r ial pr op erties of Ar n oux-Rauzy sequences (as we ll as episturm ian w ords) to th ose of Sturmian words. 5.1 Finite Arnoux-Rauzy words A ﬁnite w ord w is said to b e ﬁnite episturmian if w is a factor of some inﬁnite epistur m ian w ord. When c onsid ering f acto rs of (inﬁnite) episturmian w ords, it suﬃces to consider only the strict standard ones (i.e., th e standard Ar n oux-Rauzy sequences). Indeed, for any preﬁ x u of an 17 epistandard w ord, there exists a strict epistandard word also having u as a preﬁx. In particular, the wo rd s µ w ( x ), with w ∈ A ∗ and x ∈ A , a re the standar d ones ( cf. standard words, e.g., [83, Chapter 2]). They can b e obtained b y the R auzy rules [98 ] (see also [43, Theorem 8]), and this has a strong connection with th e set of p erio ds of the p alindromes u n +1 = P al ( x 1 · · · x n ) (giv en in Theorem 2.3) and the Euclidean algorithm. This relation wa s studied by Castelli, Mignosi, and Restiv o [34], who extended the w ell-kno wn Fine and Wilf The or em [82] to w ords h a ving thr ee p erio ds. Justin [70] generalized this r esult ev en fu rther to words having an arbitrary num b er of p erio ds, whic h led to a characte rization of ﬁnite epistur mian w ord s. Finite episturmian words are exactl y the ﬁnite Arno ux- R auzy wor ds . Suc h w ords w ere en u- merated by Mignosi a n d Zam b on i [88], who describ ed a m ulti-dimensional generalization of th e Euler p hi-fu nction that counts the num b er of ﬁn ite Arn ou x -Rauzy w ords of eac h length. Finite episturmian words ha ve also b een c haracterized w ith resp ect to lexicographic ord erings in [62] (see Th eorem 7.5 later). 6 Some prop erties of factors 6.1 F actor complexit y As menti oned previously , an y k -strict episturmian w ord has complexity ( k − 1) n + 1 for all n ∈ N . More generally: Theorem 6.1. [43, Theorem 7] Supp ose t is an episturmian wor d dir e c te d by ˘ ∆ with | Ult(∆) | > 1 . Then, for n lar ge enough, t has c omplexity ( k − 1) n + q for some q ∈ N + , wher e k = | Ult(∆) | . ⊔ ⊓ This theorem can b e easily d educed from the fact that for s u ﬃcien tly large n , an y left sp ecial factor of t of length at least n has exac tly k = | Ult(∆) | diﬀerent left extensions in t (b y Theorem 6 in [43]). 6.2 P alindromic factors The p alindr omic c omplexity of epistur m ian w ords wa s established in [73] by carrying out a similar study to the one f or Sturmian words in [44]. Theorem 6.2. [73, Theorem 4.4] If t is an A -strict episturmian wor d, t hen ther e e xists exactly • one p alindr ome of length n for al l even n , • one p alindr ome of length n and c entr e x for al l o dd n and x ∈ A . ⊔ ⊓ As shown in [44], the ab o ve p rop ert y is c haracteristic in the Sturmian case, bu t not when A con tains more than t wo lette r s b ecause it also holds for bil liar d wor ds , which are not epistur mian (see Borel and Reutenauer [25]). Theorem 6.3. [73, Section 4.2] If t is episturmian, then ther e exist | Ult(∆) | + 1 bi-inﬁnite episturmian wor ds of the form ˜ m . m and ˜ m x m with x ∈ Ult(∆) giving the p alindr omic fa ctors of t . The spinne d versions of ∆ dir e cting these bi-inﬁnite episturmian wor ds c an b e e asily c onstructe d via a simple algorithm. ⊔ ⊓ F or more precise tec hnical details, s ee Section 4.2 in [73]. Example 6.4 . F or th e T rib onacci wo rd , ˜ r . r is directed by ( abc ¯ abca ¯ bcab ¯ c ) ω . 18 6.2.1 Iterated palindromic clos ure In [105], Risley an d Zam b oni gav e an alternativ e construction of the sequence ( u n ) n ≥ 1 of palin- dromic preﬁxes of an epistandard w ord (where u 1 = ε and u i +1 = P al ( x 1 · · · x i ) for all i ≥ 1), using a ‘hat op eration’ as opp osed to palindromic closur e. The so-called hat op eration is d eﬁned as follo ws . W e construct a new alphab et A ′ := A ∪ b A wh ere b A = { b x | x ∈ A} and denote by φ the morphism φ : A ′ → A deﬁned by φ ( x ) = φ ( b x ) = x for all letters x ∈ A . Th e morph ism φ extends t o a morp hism (also denoted by φ ) from wo r d s o ve r A ′ to words o ver A . No w, from a giv en directiv e wo r d ∆ = x 1 x 2 x 3 · · · ∈ A ω , we construct a sequ ence of words ( p i ) i ≥ 1 as follo ws. W e b egin w ith p 1 = ε an d p 2 = b x 1 . Then, for n ≥ 2, p n +1 is obtained fr om p n according to the rule: if b x n do es not o ccur in p n , then p n +1 = p n b x n φ ( p n ); otherwise p n +1 = p n b x n φ ( s n ), wh ere s n is the longest palindr omic suﬃx of p n con taining no o ccurrence of b x n . Example 6.5 . Let ∆ = ( abc ) ω . Th en using th e h at op eration, w e obtain: p 1 = ε p 2 = ˆ a p 3 = ˆ a ˆ ba p 4 = ˆ a ˆ ba ˆ caba p 5 = ˆ a ˆ ba ˆ caba ˆ abacaba p 6 = ˆ a ˆ ba ˆ caba ˆ abacaba ˆ bacabaabacaba . . . No w remo ving all hats (b y applying φ ), we see that the p i ’s are precisely the palindromic preﬁxes of th e T r ib onacci w ord: abacabaabacababacabaabacaba · · · . As demonstrated by th e ab ov e example, the hat op eration is clearly the same as iterated palindromic cl osur e; in fact, the relationship b et w een these t wo constru ctions is eviden t by for- m ula (4.1), whic h we no w rewr ite as: P al ( x 1 · · · x n ) = µ x 1 ··· x n − 1 ( x n ) P al ( x 1 · · · x n − 1 ) for n > 0 . The ab o v e form u la is actually a sp ecial case of form ula (3) fr om [71], whic h also happ ens to be form ula (3) in [73], namely: P al ( v w ) = µ v ( P al ( w )) P al ( v ) for any w ords w , v . (6.1) This form ula is commonly referred t o as Justin ’s F ormula , from whic h w e deduce th e follo wing t wo sp ecial cases: P al ( xw ) = ψ x ( P al ( w )) x and P al ( wx ) = µ w ( x ) P al ( w ) for any wo rd v a n d letter x . (6.2) The ﬁrst formula giv en in (6.2) tells us that P al ( xw ) is obtained from P al ( w ) s imply b y in serting the letter x b efore eac h letter d iﬀerent from x and then app ending x to the resulting word. F or example, P al ( bc ) = bcb and P al ( abc ) = abacaba . T he second formula giv en in (6.2) pro vides another wa y to compute the p alindromic right- closure of wx b y placing the ﬁnite epistandard w ord µ w ( x ) in fr on t of P al ( w ). F or example, to compu te P al ( abcb ) w e need only compute th e w ords µ abc ( b ) = abacab and P al ( abc ) = abacaba , and then w e hav e: P al ( abcb ) = µ abc ( a ) P al ( abc ) = abacab · abacaba. 19 In [71], J ustin established some relations b et ween the w ord s P al ( w ), µ w , P al ( e w ), and µ e w where w is any ﬁ nite word. Moreo v er, he sho wed th at h is r esults can b e explained b y the similarit y of the incidenc e matric es o f µ w and µ e w . O ne curious result is th at | P al ( w ) | = | P al ( ˜ w ) | . F or example, with w = abac , P al ( w ) = abaabacabaaba and P al ( e w ) = cacbcacacacbcac , b oth of length 15. Applying h is results to a 2-letter alphab et, J ustin [71] ga v e a new pro of of a Galois theorem on conti nued fractions, b y considering the epistandard words that are ﬁ x ed p oints of µ w and µ e w for an y ﬁnite word w . F rom this p oint of view, Jus tin’s result h ighligh ts the relev ance of the previously men tioned ‘m ulti-dimensional’ contin ued fraction algorithm, prop osed b y Z am b oni [119, 117] (see also [96, Section 12.2]). Ho w ever, there still remains m uch w ork to b e d one in this direction, esp ecially concerning the gener alize d i nter c ept (coheren t with the Sturmian case) in tro duced in [73, S ectio n 5.4] and the generalized O stro wski numeration systems [20, 75] (r ecall Remark 4.7). Note. Th e aforementio ned Galo is theorem was used in the theory of Stu rmian wo rd s to c harac- terize s o-called Sturm numb ers (see [83, Theorem 2.3.26]). 6.2.2 P alindromic ric hness In [43], Drouba y , Ju stin, and Pirillo obser ved that an y ﬁnite wo r d w con tains at most | w | + 1 distinct palindromes (includin g the emp t y w ord ). Ev en further, they prov ed that a w ord w con tains exactly | w | + 1 d istinct palindromes if and only if the longest palindromic s u ﬃx of an y preﬁx p of w o ccurs exactly once in p (i.e., ev ery preﬁ x of w has Pr op erty J u [43]). Such wo r d s are ‘ric h’ in palindromes in the s en se that they con tain the maxim um num b er of diﬀerent p alindromic factors. Accordingly , w e say that a ﬁnite word w is rich if it conta ins exactly | w | + 1 d istinct palindromes (or equiv alen tly , if ev ery preﬁx of w has Pr op erty J u ). F or example, abac is ric h since it is of length 4 and conta ins th e follo wing ﬁve p alindromes: ε , a , b , c , aba . Naturally , an inﬁ nite w ord is ric h if all of its factors are r ic h. F or example, the p erio dic inﬁnite words a ω = aaa · · · and ( ab ) ω = ababab · · · are clearly r ich, w hereas ( abc ) ω = abcabacabc · · · is not rich since i t co ntains the n on-ric h word abca . Drouba y et al. [43] sho wed that all ﬁ nite and inﬁ nite episturm ian words are r ich. S p eciﬁcally , they pro ved that if an inﬁnite word has prop ert y P i (and hence is epistandard – see Theorem 2.3), then all of its pr eﬁxes ha ve prop erty J u . Consequentl y , an y factor u of an ep istand ard wo rd (and hence, of an episturmian wo rd ) con tains exactly | u | + 1 distinct palindromes, and is therefore ric h (see Corollary 2 in [43]). Another sp ecial class of ric h words the encompasses the ep istu rmian w ord s consists of Fisc hler’s sequences with “abun dan t p alindromic p reﬁxes”. These words were in tro du ced and studied in [54, 55] in the cont ext of Diophan tine app r o ximation. See also pap ers by Adamczewski and Bu gea u d [2, 3] concerning the transcendence of certain real num b ers whose sequ en ces of partial quotien ts con tain arbitrarily long p alindromes. The theory of rich words has r ecen tly b een furth er devel op ed in a series of p ap er s [61, 29, 42, 30]. I n ind ep enden t w ork, Am br o ˇ z, F rougn y , Mas´ ako v´ a, and P elan to v´ a [8] ha v e considered the same class of words whic h they call ful l wor ds , follo win g the earlier w ork of Brlek, Hamel, Niv at, and Reutenauer in [26]. 6.3 F ractional p o w ers & critical exp onen t The stud y of fr actional p owers o ccurring in S turmian words has b een a topic of gro wing interest in recent times. See for instance [14, 23, 39, 72, 86, 111], as well as [73, 105, 59] for similar r esults concerning ep istu rmian wo r d s and Arn oux-Rauzy sequences. 20 The follo wing theorem extends the resu lts in [72] on fractional p ow ers in Sturmian wo rd s. Throughout this section, w e let s denote an epistandard word with directiv e w ord ∆ = x 1 x 2 · · · ∈ A ω (as usual), and for all n ≥ 1, w e d enote by u n +1 the palindromic preﬁx P al ( x 1 · · · x n ) of s giv en in Th eorem 2.3. As in [72], we d enote b y L ( m ) the length of the longest factor v ∈ F ( s ) h a ving p erio d m ∈ N , and write L ( m ) = em + r , e ∈ N + , 0 ≤ r < m . Given a ﬁn ite or inﬁnite word w , w e denote by w ( i ) (resp. w ( i, j )) the letter in p osition i of w (resp. the factor of w b eginning at p osition i and en ding at p osition j ). When L ( m ) ≥ 2 m , all factors of s ha ving p erio d m and length L ( m ) are equal to a p alindrome v , and for 0 ≤ i < e , the word v i := v (1 , im + r ) is a palindromic preﬁx of s by Lemma 4.1 in [73]. Moreo v er, with th e preceding notation, we ha ve : Theorem 6.6. [73, T h eorem 4.2] L et m , n ∈ N b e suc h that | u n | < m ≤ | u n +1 | and s (1 , m ) = w is primitive with s ( m ) = x o c curring in s (1 , m − 1) . Then the fol lowing pr op erties hold . i) L ( m ) ≥ 2 m if and only if w = µ x 1 ··· x n ( x ) and x ∈ Alph( x n +1 x n +2 · · · ) . ii) Supp ose L ( m ) ≥ 2 and deﬁne p = max { i ≤ n | x i = x } and t = min { j ∈ N + | x n + j 6 = x } . Then u n +1 = w t u p is the longest pr eﬁx of s having p erio d m . Mor e over, if x ∈ Alph( x n + t +1 x n + t +2 · · · ) , then e = t + 1 ; that is, v = w t +1 u p , otherwise e = t and v = w t u p . ⊔ ⊓ Remark 6.7. Let us men tion a few n otew orth y facts. • Exp onent s of p o wers in s are b ounded if and only if exp onent s of letters in ∆ are b oun d ed [105, 73]. • An y Stu rmian w ord has sq u are preﬁxes and so do epistand ard wo rd s [5, 105 ]. • An y episturmian word has inﬁ nitely man y p reﬁxes of the form uv 2 with | u | / | v | b ounded ab o v e. The latter fact is r eadily ded u ced from the follo wing r esult of R isley and Zam b oni [105]. Theorem 6.8. [105, P r op. I.3] If t is an Arnoux-R auzy se quenc e, then ther e exists a p ositive numb er ǫ such that t b e gins with inﬁnitely many blo cks of the form U V V V ′ , wher e V ′ is a pr eﬁx of V and min {| V ′ | / | V | , | V | / | U |} > ǫ . ⊔ ⊓ Note. S u c h a result is motiv ated by transcendence issu es; see for ins tance [52]. When s is purely morphic, it is p ossible to giv e a rather explicit form u la for th e critic al exp onent : γ = lim sup n →∞ L ( m ) /m , as follo ws. Notation. Let P b e the fun ction d eﬁned by P ( n ) = sup { i < n | x i = x n } if this intege r exists, undeﬁn ed otherwise. That is, if x n = a , then P ( n ) is the p osition of the righ t-most o ccurrence of the letter a in the pr eﬁ x x 1 x 2 · · · x n − 1 of th e directiv e word ∆ = x 1 x 2 x 3 · · · ∈ A ω . Theorem 6.9. [73, T heorem 5.2] L et s b e an A -strict epistandar d wor d gener ate d by a morphism with dir e ctive wor d ∆ having p erio d q . F urther, let l ∈ N b e maximal such that y l ∈ F (∆) for some letter y , and deﬁne L = { r , 0 ≤ r < q | x r +1 = x r +2 = · · · = x r + l } and d ( r ) = r + q + 1 − P ( r + q − 1) for 0 ≤ r < q . Then the critic al exp onent fo r s is given by γ = l + 2 + su p r ∈ L  lim i →∞ | u r + iq +1 − d ( r ) | / | h r + iq |  . Mor e over, for any letter x in s the limit ab ove c an b e obtaine d as a r ational function with r ational c o eﬃcients of the fr e que ncy α x of this letter. ⊔ ⊓ 21 See also [86, 107, 111] for resu lts on the critical exp onent for the Fib onacci word, T rib onacci w ord, and Sturmian words, resp ectiv ely . Example 6.10. F or the ev er-so p opular Fib onacci word f , directed b y ( ab ) ω , w e ha ve q = 2, l = 1, d (0) = d (1) = 2. Hence, since | u n − 1 | / | h n | has limit 1 /ϕ where ϕ = (1 + √ 5) / 2 is the golden ratio, we obtain the well -known v alue 2 + ϕ for the critical exp onen t, originally pr o v ed by Mignosi and Pir illo [86]. More generally , the k -b onacci word, directed by ( a 1 a 2 · · · a k ) ω , has critical exp onent 2 + 1 / ( ϕ k − 1), where the k -b onac ci c onstant ϕ k is the (un ique) p ositiv e real ro ot o f the k -th degree monic p olynomial x k − x k − 1 − · · · − x − 1. 6.4 F requencies Let w b e a n on-empt y ﬁnite wo rd . F or any v ∈ F ( w ), the fr e quency of v in w is | w | v / | w | where | w | v denotes the num b er of distinct o ccurrences of v in w . T he notion of fr equency can b e extended to inﬁ nite w ords in t wo wa ys, as follo ws. Deﬁnition 6.11. Supp ose v is a non-empty factor of an inﬁnite wor d x . Then: i) the fr e quency of v in x in the we ak sense is lim n →∞ | w (1 , n ) | v /n i f this limit exists; ii) v has fr e quency α v in x in the strong sens e if, for any se quenc e ( w n ) n ≥ 0 of factors of x with incr e asing lengths, we have α v = lim n →∞ | w n | v / | w n | . In a pu rely com binatorial wa y , Ju stin and Pirillo [73 , Section 6] pro ve d that any factor o ccur- ring in an epistu rmian wo r d has frequency in th e strong s en se. W ozn y and Zamb oni [117] also stud ied frequencies (in th e we ak s ense) for Arnoux-Rauzy se- quences. Using a reformulat ion of a vec torial division algo r ith m , originally in tro duced in [105], they compu ted eac h allo wable frequ ency of fact ors of the same length, a s w ell as the n um b er of factors with a giv en frequency . In particular, the authors of [117] ga ve simulta n eous r ational ap- pro ximations of the frequencies by unredu ced fractions ha ving a common denominator. F rom this w ork, one reco v ers the results of Berth ´ e [19] for S turmian w ords in terms of F ar ey appr o ximations arising f rom the cont inued f r actio n exp ansions of the frequ encies of the letters. F or in stance, the frequencies of factors of the same length in a S turmian w ord assume at most three v alues, w h ic h w ere explicitly give n b y Berth ´ e [19], who also disco vered that this r esu lt is in strong connection with the thr e e distanc e the or em in Diophan tine analysis. 6.5 Return w ords Let us n o w recall the notion of a r eturn wor d , whic h wa s int r o du ced indep enden tly b y Durand [45], and Holton and Z am b oni [67] when s tudying primitive substitutive sequences. Deﬁnition 6.12. L et v b e a r e curr ent factor of y ∈ A ω , starting at p ositions n 1 < n 2 < n 3 · · · . Then e ach wor d r i = y n i y n i +1 · · · y n i +1 − 1 is c al le d a return to v in y . Mor e over, y c an b e factorize d in a uniqu e way as y = y 1 · · · y n 1 − 1 r 1 r 2 r 3 · · · wher e r 1 r 2 r 3 · · · , viewe d as a wor d on the r i , i s c al le d the der ived w ord of y with resp ect to v . That is, a r eturn t o v in y is a non-empty factor of y beginnin g at an occurrence of v and ending exactly b efore the next o ccurrence of v in y . Th u s, if r is a return to v in y , then r v is a factor of y that con tains exactly t wo o ccurrences of v , one as a preﬁx and one as a s u ﬃx. W e call r v a c omplete r eturn to v [7 6 ]. 22 Return w ords p la y an imp ortan t role in the study of minimal sub shifts in s y mb olic dyn amics; see for instance [45 , 46, 47, 53 , 106]. In the conte xt of epistur m ian words, su c h w ord s h a v e recen tly pro ven to b e a usefu l to ol in the stud y of quasip erio dicit y (see Section 8 for furth er details). This latest work made use of the follo win g resu lt of Ju s tin and V u illon [76] whic h completely d escrib es the return s to any factor of an epistandard word. In fact, their r esu lt actually characte rizes return words in episturmian words (not ju st epistandard w ords) since, by uniform recurren ce, the returns to any factor v in an ep istand ard wo rd s are the same as the returns to v as a factor of an y episturm ian word t ha ving the same set of factors as s . Theorem 6.13. [76] L et s b e an epistandar d wor d dir e cte d by ∆ = x 1 x 2 x 3 · · · ∈ A ω and c onsider any v ∈ F ( s ) . If u n +1 is the sh ortest p alindr omic pr eﬁx of s c ontaining v with u n +1 = f v g , then the r eturns to v in s ar e the wor ds f − 1 µ x 1 ··· x n ( x ) f wher e x ∈ Alph( x n +1 x n +2 · · · ) . M or e over, the c orr e sp onding c omplete r eturns to v ar e the wor ds f − 1 ( u n +1 x ) (+) g − 1 and the derive d wor d of s with r esp e ct to v is given by s ( n ) = µ − 1 x 1 ··· x n ( s ) . ⊔ ⊓ Note. It follo ws immediately that any factor of an A -strict episturmian w ord has exactly |A| return words. Theorem 6.13 extends earlier work of V uillon on return w ords in Stu rmian words (see [114]). In particular, V uillon pro ved that S turmian wo rd s are c haracterized b y the prop erty that an y non-empt y factor has exactly 2 diﬀerent return wo rd s in the giv en Stur mian word. Ho wev er, con trary to what one migh t exp ect, s uc h a prop erty w ith 2 replaced b y a p ositiv e i ntege r k ≥ 3 do es not c haracterize k -strict episturmian w ords . F or instance, inﬁnite w ords co ding 3-i nterv al exc hange transformations, which constitute a diﬀerent generalization of Sturmian w ords to 3- letter alph ab ets, are kno wn to h a v e the prop erty that every factor has 3 diﬀeren t return words (see the work b y F erenczi, Holton, and Zam b oni in [51]). 7 Balance & lexicographic order 7.1 q -Balance Deﬁnition 7.1. A ﬁnite or inﬁnite wor d is q -balanced if, for any two of i ts factors u , v with | u | = | v | , we have || u | x − | v | x | ≤ q for any letter x, i.e., the numb er of x ’s in e ach of u and v diﬀers b y at most q . Note. A 1-balanced w ord is simply said to b e b alanc e d . The term ‘balanced’ is relativ ely new; it app eared in [16, 17] (also see [83, Ch apter 2]), and the notion itself dates bac k to [91, 37]. In th e p ioneering work of Morse and Hedlund [91], b alanced inﬁnite words ov er a 2-letter alphab et were called ‘Sturmian tra jectories’ and b elong to three classes: aperio dic Sturmian; p erio dic Stu rmian; and inﬁnite w ord s that are ultimately p erio dic (but not perio dic), calle d skew wor ds . That is, the family of balanced inﬁnite words consists of the (recurrent) Sturmian w ords an d the (non-recur ren t) skew inﬁnite w ords , th e factors of which are balanced. Skew w ords a re ultimat ely p eriod ic su ﬃxes of w ord s of the f orm µ ( a p ba ω ), where µ is a pur e standar d Sturmian morphism and p ∈ N . F or example, aba ω and ψ b ( aba ω ) = bab ( ba ) ω are skew. See also [108, 109, 66, 95] for f u rther w ork on ske w wo rd s. Remark 7.2. No wada ys, for m ost authors, only th e ap erio dic Stur mian words are considered to b e ‘Sturmian’. Ho wev er, from now on, w e will use the term ‘Sturmian’ to r efer to b oth ap erio dic 23 and p erio dic Stu rmian wo rd s. In the con text of cutting s equ ences, the ap erio dic (resp. p erio dic) Sturmian words are precisely those with irrational slop e (r esp . rational slop e). It is imp ortan t to note th at a ﬁ n ite w ord is ﬁnite Sturmian (i.e ., a factor of some S turmian w ord) if and only if it is balanced [83, Chapter 2]. Accordingly , the balanced inﬁn ite wo r ds are precisely the inﬁ nite words w hose factors are ﬁnite Sturmian. T his concept w as r ecen tly generalized in [62] by sh o wing that the set of all inﬁnite words whose factors are ﬁnite episturmian consists of the (recurrent) epistu r mian w ord s and the (non-recur ren t) episkew inﬁ nite w ords, as deﬁned in the next section. 7.2 Episk ew words Inspired b y th e skew words of Morse and Hedlun d [91], episkew wor ds we r e recently d eﬁned in [62] as non-recurren t inﬁnite wo rd s, a ll of w h ose factors are (ﬁn ite) episturmian. Th e f ollo win g theorem giv es a num b er o f equiv alen t deﬁn itions of suc h words, similar to those for ( r ecurren t) episturmian w ords. Theorem 7.3. [62] An inﬁnite wor d t with Alph( t ) = A is episkew if e quivalently: i) t i s non-r e curr ent and al l of its factors ar e (ﬁnite) episturmian; ii) ther e exists an inﬁnite se quenc e ( t ( i ) ) i ≥ 0 of non-r e curr e nt inﬁnite wor ds and a dir e ctive wor d x 1 x 2 x 3 · · · ( x i ∈ A ) such that t (0) = t , . . . , t ′ ( i − 1) = ψ x i ( t ( i ) ) , wher e t ′ ( i − 1) = t ( i − 1) if t ( i − 1) b e gins with x i and t ′ ( i − 1) = x i t ( i − 1) otherwise; iii) ther e e xi sts a letter x ∈ A and an e pistandar d wor d s on A \ { x } such that t = v µ ( s ) , wher e µ is a pur e epistandar d morphism on A and v is a non-empty suﬃx of µ ( e s p x ) for some p ∈ N . Mor e over, t is said to b e strict episke w if s is strict on A \ { x } , i.e., if e ach letter in A \ { x } o c curs inﬁnitely often in the dir e ctive wor d x 1 x 2 x 3 · · · . ⊔ ⊓ A simple example of an episkew wo rd on more than t wo letters is the inﬁ nite w ord c f = cabaababa · · · where f is the Fib onacci word and c is a letter (see also Example 3.7). Note that the episkew words on a 2 -letter alphab et are p r ecisely the skew w ords. Certainly , in the Stu r mian case, the w ord e s p x s r educes to a w ord of the form a p ba ω . Remark 7.4. Th anks to Richomme [104], episkew words actually h a v e the follo wing simp ler c haracterization: an inﬁ nite w ord t is episkew if and only if t = ϕ ( x s ) wh ere s is an epistand ard w ord, x is a letter not o ccurr in g in s , and ϕ is a pure epistu rmian morph ism . Episk ew w ords w ere ﬁrst alluded to (but not explicated) i n the recen t pap er [60]. F ollo wing that pap er, these wo r d s sho wed u p again in the stu d y of in equalitie s c haracterizing ﬁ nite and inﬁnite episturmian words w ith resp ect to lexicographic orderings [62]. In fact, as d etail ed in the next section, episturmian words ha v e similar extremal p rop erties to Stur mian w ords. S ee also [74, 69, 93, 94, 95, 60, 62] for other w ork in this direction. 7.3 Extremal prop erties Supp ose the alphab et A is totally ord ered by the r elati on < . Then we can totally order A ∗ b y the lexic o g r aphic or der ≤ deﬁ n ed as follo ws. Given tw o words u , v ∈ A + , we ha ve u ≤ v if and only if either u is a preﬁx of v or u = xau ′ and v = xbv ′ , for some x , u ′ , v ′ ∈ A ∗ and letters a , b 24 with a < b . This is the usual alph ab etic ordering in a dictionary . W e write u < v when u ≤ v and u 6 = v , in w hic h ca se w e sa y that u is (strictly) lexic o gr aphic al ly smal ler than v . The notion of lexicographic order naturally extends to inﬁ n ite w ord s in A ω . W e denote by min ( A ) the smallest letter in A with resp ect to the giv en lexicographic order. Let w b e a ﬁnite or in ﬁnite word o ve r A and let k b e a p ositiv e in teger. W e denote by min( w | k ) (resp. max( w | k )) th e lexicographically smallest (resp . greatest) factor of w of length k for the given order (wh ere | w | ≥ k if w i s ﬁnite). If w is inﬁ nite, then it is clear that min( w | k ) and max( w | k ) are pr eﬁxes of the resp ectiv e words min( w | k + 1) and max( w | k + 1). So w e can deﬁne, by taking limits, the follo wing t wo inﬁnite w ords (see [94]): min( w ) = lim k →∞ min( w | k ) and max( w ) = lim k →∞ max( w | k ) . That is, to any inﬁ n ite word t we can asso ciate t wo inﬁnite w ords min( t ) and max( t ) suc h that an y preﬁ x of min( t ) (resp. max( t )) is the lexicographically smallest (resp. greatest) amongst the factors of t of the same length. F or a ﬁ nite word w o v er A and a giv en order on A , min( w ) d enotes min ( w | k ) wh er e k is maximal suc h that all min( w | j ), j = 1 , 2 , . . . , k , are preﬁxes of min ( w | k ). In the case A = { a, b } , max( w ) is deﬁn ed similarly (see [62]). In 2003 , Pirillo [93] (see also [94]) prov ed th at, for inﬁ nite words s on a 2-let ter alphab et { a, b } with a < b , the inequalit y a s ≤ min( s ) ≤ max( s ) ≤ b s (7.1) c haracterizes standar d St urmian wor ds (ap erio dic and p erio dic). Actually , th is result w as kno wn m uch earlier, dating bac k to the work of P . V eerman [112, 113] in th e mid 80’s. Sin ce that time, these ‘Sturmian inequalities’ ha ve b een redisco ve r ed n u merous times und er diﬀeren t guises, as discussed in the forthcoming survey pap er [6]. Con tinuing his work in relation to inequalit y (7.1), Pirillo [94] pr o v ed further that, in the case of an arbitrary ﬁnite a lph ab et A , an inﬁ nite word s ∈ A ω is epistand ard if and only if, for an y lexicographic ord er, w e hav e a s ≤ min( s ) where a = m in( A ) . (7.2) Moreo v er, s is a strict ep istandard wo rd if and only if (7.2 ) holds with strict equalit y for an y order [74]. In a similar spirit, Glen, J ustin, and Pirillo [62] recentl y established n ew c haracterizations of ﬁnite Sturmian and episturmian words via lexicographic orderin gs. As a consequence, they were able to charac terize by lexicographic ord er all episturmian an d episk ew w ord s. Similarly , they c haracterized b y lexic ographic order all b alanced inﬁnite w ords on a 2-letter alphab et; in o ther w ords, all Sturmian and skew inﬁ nite words, the factors of whic h are ( ﬁ nite) Stu rmian. In the ﬁnite case: Theorem 7.5. [62] A ﬁnite wor d w on A is episturmian if and only if ther e exists a ﬁnite wor d u su c h that, for any lexic o gr aphic or der, au | m |− 1 ≤ m (7.3) wher e m = min( w ) and a = min( A ) for the c onsider e d or der. ⊔ ⊓ 25 Example 7.6. Consider the ﬁ n ite word w = baabacababac . F or th e d iﬀeren t ord ers on { a, b, c } , w e ha ve • a < b < c or a < c < b : min ( w ) = aabacababac , • b < a < c or b < c < a : min ( w ) = babac , • c < a < b or c < b < a : min ( w ) = cababac . It can b e v eriﬁed that a ﬁn ite word u satisfying (7.3) must b egin with aba and one p ossibilit y is u = abacaaaaaa ; thus w is a ﬁn ite episturmian word. Note. In the ab o v e example, an y t w o ord ers with th e same minimum letter giv e the same min( w ), whic h is not tru e in general. A corollary of Theorem 7.5 is the follo wing new c haracterization of ﬁn ite S turmian words (i.e., ﬁnite balanced words). Corollary 7.7. [62] A ﬁnite wor d w on A = { a, b } , a < b , is not Sturmian (in other w or ds, not b alanc e d) if and only if ther e exi sts a ﬁnite wor d u such that aua is a pr eﬁx of min( w ) and bub is a pr eﬁx of max( w ) . ⊔ ⊓ In the inﬁn ite case, the follo wing c haracterization of all inﬁn ite w ords w hose f acto rs are ﬁnite episturmian follo ws almost immediately fr om Theorem 7.5 . Corollary 7.8. [62] An inﬁnite wor d t on A is episturmian or episkew if and only if ther e exists an inﬁnite wor d u such that, for any lexic o gr aphic or der, a u ≤ min( t ) wher e a = min( A ) . ⊔ ⊓ Consequent ly , an in ﬁ nite word s on { a, b } ( a < b ) is balanced (i.e., Sturmian or sk ew) if and only if there exists an in ﬁ nite w ord u s uc h that a u ≤ min( s ) ≤ max( s ) ≤ b u . Corollary 7.8 wa s r ecen tly reﬁned in [58] where it was sh own that, for any ap erio dic epistur m ian w ord t , the inﬁ nite wo rd u ( as giv en in the corolla r y ) is the unique epistandard w ord with the same set of factors as t . As an easy consequence, w e obtain the follo w in g characte rization of strict episturmian words th at are inﬁnite Lyndon wor ds (Th eorem 7.9). Recall that a non-emp t y ﬁnite w ord w o v er A is a Lyndon wor d if it is lexicographically smaller than all of its prop er suﬃxes for the giv en order < on A . Equiv alen tly , w is the lexicographically smallest pr im itive w ord in its conjugacy class; that is, w < v u for all n on -emp t y wo rd s u , v s u c h that w = uv . Th e ﬁrst of these deﬁnitions exte n ds to inﬁn ite words: an inﬁnite word ov er A is an inﬁnite Lyndon wor d if and only if it is (strictly) lexicographically smaller than a ll of its prop er suﬃxes f or the giv en order on A . That is, a ﬁn ite or inﬁ nite word w is a Lyndon wo r d if and only if w < T i ( w ) for all i > 0. Assuming th at |A| > 1 (since there are no Lyn don w ords on a 1-letter alphab et), w e ha ve: Theorem 7.9. [58] An A -strict e pisturmian wor d t is an inﬁnite Lyndon wor d if and only if t = a s wher e a = min( A ) for the given or der on A and s is an (ap erio dic) A -strict epistand ar d wor d. Mor e over, if ∆ = x 1 x 2 · · · ∈ A ω is the dir e ctive wor d of s , then t = a s is the unique episturmian wor d i n the subshift of s dir e cte d by the spinne d version of ∆ having al l spins 1 , exc ept when x i = a . ⊔ ⊓ 26 The ab ov e theorem is actually a generalizatio n of a result on (ap eriod ic) Sturm ian w ords given b y Borel and Laubie [24] (see also [102]). Let A = { a 1 , . . . , a m } b e an alphab et ordered b y a 1 < a 2 < · · · < a m . Then Theorem 7.9 says that an A -strict episturmian wo rd t is a n inﬁn ite Lyn don word if and only if the (normalized) directiv e w ord of t b elongs to { a 1 , ¯ a 2 , . . . , ¯ a m } ω . T his can b e reformula ted as a generalization of Prop osition 6.4 in [81]: Corollary 7.10. [58] An A -strict episturmian wor d t is an inﬁnite Lyndon wor d if and only if it c an b e inﬁnitely de c omp ose d over the set of morphisms { ψ a , ¯ ψ x | x ∈ A \ { a }} wher e a = min( A ) for the given or der on A . ⊔ ⊓ W e observ e that, con trary to the fact that th er e exists |A| ! p ossible orders of a ﬁ nite alphab et A , Theorem 7.9 sho ws that there exist exactly |A| inﬁn ite Lyndon w ords in the subshift of a giv en A -strict epistand ard w ord s (when |A| > 1). That is, for any order with min ( A ) = a , th e subshift of s con tains a unique inﬁn ite Lyndon word b eginning w ith a , namely a s . Example 7.11. With ∆ = ( abc ) ω , the sp inned versions ( a ¯ b ¯ c ) ω , (¯ a b ¯ c ) ω , (¯ a ¯ bc ) ω and their ‘opp osites’ (obtained by exc hange of spins): (¯ a bc ) ω , ( a ¯ bc ) ω , ( ab ¯ c ) ω direct epistu rmian words in th e sub s hift of the T rib onacci word r . Only the ﬁrst three of these spinned inﬁnite words d irect episturmian Lyndon words: a r , b r , c r , resp ectiv ely . The ab o ve results on strict ep isturmian Lyndon wo rd s ha v e ve ry recen tly b een generalized to all epistur m ian w ords by Glen, Lev´ e, and Ric homme [64 ], as follo ws . Theorem 7.12. [64] L et A = { a 1 , . . . , a m } b e an alph ab et or der e d by a 1 < a 2 < · · · < a m and, for 1 ≤ i ≤ m , let B i = { a i , . . . , a m } . An episturmian wor d t is an inﬁnite Lyndon wor d i f and only if ther e exi sts an inte ge r j such that 1 ≤ j < m and the (normalize d) dir e ctive wor d of w b elongs to: ( ¯ B ∗ 2 a 1 ) ∗ · · · ( ¯ B ∗ j a j − 1 ) ∗ ( ¯ B ∗ j +1 a j ) ∗ ( ¯ B + j +1 { a j } + ) ω . ⊔ ⊓ Note. In th e ab o ve theorem, the word normalize d app ears b et wee n brack ets s ince one can easily v erify from Th eorem 4.13 that a s p inned inﬁ nite word of the giv en form is the u n ique dir ectiv e w ord of exactly one episturmian wo rd . Example 7.13. [6 4 ] Let A = { a, b, c, d } . Then the spinn ed inﬁn ite w ord ( ¯ b ¯ ca ) ( ¯ d ¯ cb ) 2 ( ¯ dcc ) ω directs a Lyndon episturmian word, and so do es aa ( ¯ dc ) ω , but ¯ ca ¯ ba ¯ dcd ω do es not (since this spinn ed word directs a p erio dic word). Remark 7.14. Th eorems 4.13 a nd 7.12 sh o w that an y episturmian Lynd on word has a unique spinned d irectiv e word, but th e con verse is not true. F or example, the regular wa vy wo r d ( a ¯ bc ) ω is the uniqu e d irectiv e w ord of the str ict episturmian word: lim n →∞ µ n a ¯ bc ( a ) = acabaabacabacabaabaca · · · whic h is clearly n ot an inﬁnite Lyndon word b y Th eorem 7.12 and also by the fact that acabaaw is not a Lyndon word for an y order on { a, b, c } and for any w ord w . A k ey tool used in the pro of o f Theorem 7.12 w as the follo wing result of R ichomme, whic h c haracterizes ep istu rmian morph isms that preserv e L y n don w ords. A morp hism f is said to preserve ﬁnite (resp. inﬁn ite) Lyn don words if for eac h ﬁnite (resp. inﬁnite) Lyndon w ord w , f ( w ) is a ﬁnite (resp . inﬁn ite) Lyndon wo r d. 27 Theorem 7.15. [100, 103] L et A = { a 1 , . . . , a m } b e an alphab et or der e d by a 1 < a 2 < · · · < a m . Then the fol lowing assertions ar e e qu ivalent for an episturm ian morphism: • f pr eserves ﬁnite Lyndon wor ds; • f pr eserves inﬁnite Lyndon wor ds; • f ∈ ( ¯ Ψ ∗ { a 2 ,...,a m } ψ a 1 ) ∗ { ¯ Ψ a m } ∗ wher e ¯ Ψ A = { ¯ ψ x | x ∈ A} . ⊔ ⊓ 7.4 Im balance W e now return our atten tion to the notion of b alance . Episturmian w ords on thr ee or more lette rs are generally unbala n ced in the sense of 1-balance, except, of cours e, for th ose on a 2-letter alphab et, whic h are precisely the (p erio dic and ap erio dic) Sturmian w ords . In fact, Cassaigne, F erenczi, and Zam b on i [33] ha ve pro ve d , by c onstru ction, that there exists an episturmian wo rd that is not q -balanced for an y q . Note, h o w ev er, that the T rib onacci w ord is 2-bala nced, for exa mp le. More generally , it can b e sh o wn by induction that the k -b onac ci wor d , directed b y ( a 1 a 2 · · · a k ) ω , is ( k − 1)-balanced. Eve n further, o n e can pro ve that an y line arly r e curr ent strict episturmian wo rd (or Arnoux-Rauzy sequence) is q -b alance d for some q . Linearly recurrent Arnoux-Rauzy sequences were completely describ ed in [105, 32]; they are the strict episturmian w ords for whic h eac h letter x o ccurs in ∆ with b ounded gaps. Using th eir main result on retur n wo r d s (Th eorem 6.13), Justin and V u illon [76] p ro v ed that episturmian w ords do in fact satisfy a kin d of b alance prop ert y . Sp eciﬁcally: Theorem 7.16. [76, Th eorem 5.2] L et s ∈ A ω b e an epistandar d wor d and let { d, e } b e a 2 -letter subset of A . Then, for any u , v ∈ F ( s ) ∩ { d, e } ∗ with | u | = | v | , we have || u | d − | v | d | ≤ 1 . ⊔ ⊓ This prop erty of episturmian words reduces to the balance p rop ert y of Stu rmian w ords w hen A is a 2-letter alphab et (in w h ic h case it is c haracteristic); ho wev er, the prop ert y is far from b eing c haracteristic wh en A consists of more than tw o letters. More recen tly , Richomme [101] also pro ved that episturmian w ord s an d Ar noux-Rauzy se- quences can b e c haracterized via a n ice ‘lo cal balance prop erty’. That is: Theorem 7.17. [101] F or a r e curr ent inﬁnite wor d t ∈ A ω , the fol lowing assertions ar e e quiva- lent: i) t i s episturmian; ii) for e ach f actor u of t , ther e exists a letter a such that A u A ∩ F ( t ) ⊆ au A ∪ A ua ; iii) for e ach p alindr omic factor u of t , ther e exists a letter a such that A u A ∩ F ( t ) ⊆ au A ∪ A ua . ⊔ ⊓ Roughly sp eaking, the ab o ve theorem sa ys that for any factor u of a giv en episturmian wo r d t , there exists a uniqu e letter a such that ev ery o ccurrence of u in t is immediately preceded or follo w ed by a in t . When |A| = 2, pr op ert y ii ) of Theorem 7.17 is equiv alen t to the deﬁnition of b alance. Indeed, Co v en and Hedlun d [37] stated that an in ﬁ nite wo r d w o v er { a, b } is not balanced if and only if there exists a palindrome u such that aua and bub are b oth factors of w . As p oint ed out in [101], this prop ert y can b e rephrased as follo ws: an in ﬁnite w ord w is Stur mian if and only if w is ap erio dic and, for any f acto r u of w , the set of factors b elonging to A u A is a subset of au A ∪ A ua or a subset of bu A ∪ A ub . 28 7.5 F raenk el’s conjecture As discussed previously , the r e curr ent balanced in ﬁnite words on t wo letters are exactly the Sturmian w ord s (ap erio dic and p erio dic). A natural question to ask is then: “What are the balanced r ecurren t in ﬁnite wo rd s on more than tw o lette r s?” In this d irectio n , Paquin and V uillon [92] r ecently c haracterized the balanced ep isturmian wo r d s by classifying these words into three families, as follo ws. Theorem 7.18. [92] Any b alanc e d standar d episturmian se qu enc e s on a k -letter alphab et A k = { 1 , 2 , . . . , k } , k ≥ 3 , b e longs to one of the fol lowing thr e e fam ilies (up to letter p ermutation): i) s = p ( k − 1) p ( k p ( k − 1) p ) ω , with p = P al (1 n 2 · · · ( k − 2)) ; ii) s = p ( k − 1) p ( k p ( k − 1) p ) ω , with p = P al (123 · · · ( k − ℓ − 1)1( k − ℓ ) · · · ( k − 2)); iii) s = [ P al (123 · · · k )] ω . ⊔ ⊓ The imp ortance of the ab o ve r esult lies in the fact that it supp orts F r aenkel’s c onje ctur e [56]: a problem that arose in a num b er-theoretic con text and h as remained unsolv ed for o ver thirt y y ears. F raenke l conjectured t h at, for a ﬁ xed k ≥ 3, there is only on e co vering of Z by k Be atty se que nc es of the form ( ⌊ αn + β ⌋ ) n ≥ 1 , where α , β are real n umb ers. A com binatorial inte rp retatio n of this conjecture ma y b e stated as follo w s (tak en from [92]). Over a k -letter alphab et with k ≥ 3, there is only one recurrent balanced inﬁn ite w ord , up to letter p erm utation and shifts, that h as m utually distinct letter fr equencies. This supp osedly un ique in ﬁnite w ord is called F r aenkel’s se que nc e and is giv en by ( F k ) ω where the F r aenkel wor ds ( F i ) i ≥ 1 are deﬁned recursiv ely by F 1 = 1 and F i = F i − 1 iF i − 1 for all i ≥ 2 . (Note that F k = P al (12 · · · k ).) F or further details, see for instance [92, 110] and referen ces therein. Amongst the classes of balanced episturm ian wo rd s giv en in Theorem 7.18, only on e class has mutually distinct letter fr equ encies and, u p to letter p ermutatio n and sh ifts, corresp onds to F raenk el’s s equence. That is: Theorem 7.19 (P aquin-V uillon [92]) . Supp ose t is a b alanc e d episturmian wor d w ith Alph( t ) = { 1 , 2 , . . . , k } , k ≥ 3 . If t has mutual ly distinct letter fr e quencies, then up to letter p ermutation, t is a shift of ( F k ) ω . ⊔ ⊓ More recen tly , it was p r o v ed in [61] that any r ecurren t balanced rich inﬁ nite wo rd is n ecessarily episturmian, and hence such words ob ey F raenk el’s conjecture (recall that ric h wo r d s were deﬁn ed Section 6.2.2). Remark 7.20. An inte resting kno w n fact (e.g., see [68]) is that any balanced recurrent in ﬁnite w ord x o n k ≥ 3 letters having m u tu ally distinct le tter frequencies is necessarily perio dic. Cer - tainly , the image of x u nder an y morp h ism of the form: ( a 7→ a , other x 7→ b ) is a Stur mian word. If, for o n e let ter, the corresp ondin g Stu rmian word is aperio dic (i.e., x h as irrational slop e as a cutting sequen ce), then w e meet imp ossibilit y; thus rather easily x m u st b e p erio dic. 8 Concluding remarks In closing, we men tion a n umb er of very recen t works in vol vin g epistu r mian wo rd s. 29 Rigidit y: Krieger [78] h as s ho wn that an y strict pu r ely morp hic epistandard w ord s is rigid . That is, all of the morphism s that generate s are p o wers of the s ame unique (epistandard ) morphism. Krieger also sh ow ed that a certain class of ‘ultimately str ict’ purely morp hic epistandard words are not rigid, b ut it remains a n open question as to w h ether or not all strict morp hic episturmian w ord s are rigid. Quasip erio dicit y: A ﬁnite or inﬁ nite word w is said to b e quasip erio dic if there exists a wo rd u (w ith u 6 = w for ﬁn ite w ) suc h that the o ccurrences of u in w ent irely co v er w , i.e., every p osition of w falls w ith in some o ccurr en ce of u in w . Su c h a word u is called a quasip e rio d of w . F or example, the w ord w = abaababaabaababaaba has quasip erio ds aba , abaaba , abaababaaba . In the last ﬁfteen y ears, quasip erio dicit y and co v erin gs of ﬁnite wo rd s has b een extensiv ely studied (see [9 ] for a brief su rv ey on quasip erio dicit y in ‘strings’). Qu asip eriod ic ﬁnite w ords were ﬁrst in tro du ced b y Ap ostolico and Eh renfeuc ht in [10]. Th e n otion w as later ex- tended to inﬁnite w ords by Marcus [85] who op ened some questions, particularly concerning quasip erio dicit y of Stu rmian words. After a brief answer to s ome of these questions in [79], the Stur mian case was fu lly studied b y Lev ´ e and Ric homme [81] who prov ed that a Stur mian w ord is non-quasip er io dic if and only if it is an inﬁn ite Lyndon word. The study of quasip eri- o dicit y in Sturmian wo rd s w as very recen tly extended to episturmian w ords b y Glen, Lev´ e, and Ric homme [58, 64, 80], who ha ve completely describ ed all of the quasip erio ds of an episturmian wo r d, yielding a charac terization of quasip erio dic episturmian words in terms of their d irectiv e w ords. They ha ve also c h aracterized episturmian m orphisms that map any w ord onto a quasip erio dic one. T hese results sho w that, un lik e the Sturmian case, there exist non-quasip erio dic episturmian w ord s that are not inﬁn ite Lyn d on words. Key tools used in the study of qu asip eriod icit y in ep isturmian wo rd s w ere epistu rmian morphism s, normalized directiv e words (recall Theorem 4.12 ), and the follo wing equiv alen t d eﬁnition of quasip erio dicit y in te r m s o f return w ords in tro duced b y Gle n in [58]: a ﬁnite w ord v is a quasip erio d of an inﬁnite wo r d w if and only if v is a recurr en t preﬁ x of w suc h that all of the r eturns to v in w ha ve length at most | v | . In [89], Mon teil p ro v ed that an y Sturmian subsh ift con tains a multi- sc ale quasip erio dic wor d , i.e., an in ﬁ nite w ord h a v in g in ﬁ nitely many qu asip er io ds. A shorter pr oof of this fact was pro vided in [81] and this result has also b een p ro v en true for ep istu rmian wo rd s in [64]. F or more recen t wo rk on quasip erio dicit y , see for instance [89, 90]. θ -episturmian words: Recall that an inﬁnite w ord is episturmian if and only if its set of f acto rs is closed under reve r s al and it h as at most one left sp ecial factor of eac h length. With this deﬁnition in mind, Bucci, de L u ca, De Lu ca, and Zamb oni [27, 28] ha v e recen tly int ro du ced and studied a fur ther extension of epistur mian words in wh ic h the rev ersal op erator is replaced by an arbitrary involutory antimorphism (i.e., a map θ : A ∗ → A ∗ suc h that θ 2 = Id and θ ( uv ) = θ ( v ) θ ( u ) for all u , v ∈ A ∗ ). More precisely , an inﬁnite w ord ov er A is said to b e θ -episturmian if it has at most one left sp ecial factor of eac h length and its set of factors is closed un der an inv olutory an timorph ism θ of the free monoid A ∗ . 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