Quasiperiodic and Lyndon episturmian words

Quasip erio dic and Ly ndon episturmian w ords ∗ Am y Glen † Florence Lev ´ e ‡ Gw ´ ena ¨ el Richomm e ‡ Submitted: May 7, 200 8 ; Revised: Septem b er 14 , 2008 Abstract Recently the second t w o authors c haracterized quasip erio dic Sturmian words, pro ving that a Sturmian w ord is non-quasip erio dic if and only if it is an inﬁnite Lyndon wor d . Here we ex- tend this study to ep isturmian wor ds (a natural generaliza tion of Sturmian w ords) by describing all the quasip erio ds of an episturmian word, which yields a characterization of quas ip er io dic episturmian words in terms of their dir e ctive wor ds . E ven further, w e establish a complete char- acterization of all episturmian words that are Ly ndon words. Our main results show that, unlike the Sturmian case, there is a muc h wider c la ss of epistur mia n words that ar e non-quasip erio dic, bes ides those that are inﬁnite Lyndon words. Our key to ols are mor phisms and directive w ords, in particular normalize d directive words, which we introduce d in a n ear lie r pap er. Also of im- po rtance is the use of r eturn wor ds to characterize qua s ipe rio dic episturmian words, since such a metho d could be us e ful in other co ntexts. Keyw ords : episturmian word; Sturmian w ord; Arno ux-Rauzy sequence; episturmian mor- phism; lex icogra phic order ; inﬁnite Lyndo n word; quasip erio dicity . MSC (20 00): 68R15 . 1 In tro duction Sturmian wor ds are a f ascinating family of in ﬁnite words deﬁned on a 2-letter alphab et wh ic h hav e b een extensive ly stud ied since th e pioneering wo rk of Morse a nd Hedlun d in 1940 (see [36]). Over the y ears, these inﬁnite w ords hav e been shown to hav e numerous equiv alen t deﬁ nitions and c haracterizatio ns, and th eir b eautifu l prop er ties are r elated to m any ﬁelds like Nu mber Theory , Geometry , Dynamical Systems, and Combinatorics on W ord s (see [1, 31, 38, 6] for recent surveys). Man y r ecent works hav e b een d evo ted to generalizat ions of Stur mian words to arb itrary ﬁ nite alphab ets. An especially in teresting generalizati on is the family of episturmian wor ds , in trodu ced by Droubay , J ustin, and Pir illo in 2001 [12] (see also [23, 25] f or example). E pisturmian words include not only the Sturmian w ords, but also the w ell- known Arnoux-R auzy se quenc es (e.g., see [5, 24, 38, 46]). More precisely , the family of epistu r mian words is comp osed of the Arn ou x - Rauzy sequences, images of the Arnoux-Rauzy sequences by episturmian morphisms , and certain p eriod ic inﬁnite words. In the binary case, Arnoux-Rauzy sequences are exactly the Sturmian words ∗ This wo rk combines and exten ds tw o conference pap ers, one by the ﬁrst author [15] and the other by the second tw o authors [28], presen ted at t h e Sixth International Conference on W ords, Marseille, F rance, Septemb er 17 –21, 2007. † Am y Glen LaCIM, Universit ´ e du Qu ´ eb ec ` a Montr´ eal, C.P . 8888 , succur s ale Centre-ville, Montr ´ eal, Q u´ ebec, H3C 3 P8, CANAD A \ The Mathematics Institute, Reykjavik Universit y , Kr inglan 1, IS-103 Reyk javik, ICELAND E-mail: amy. glen@ gmail .com ‡ F. Lev´ e, G. Richomme Univ ersit´ e de Picar die J ules V erne, La b o ratoir e MIS (Mo d´ elisation, Information, Syst` emes), 3 3 Rue Sain t Leu, F-80 039 Amiens c edex 1, FRANCE E-mail: {flo rence .leve , g wenael .rich omme}@u-picardie.fr 1 whereas episturmian words in clude all recurren t b ala nc e d words, that is, perio dic balanced w ords and Sturmian w ords (see [18, 37, 44] f or recen t results relating episturmian words to the balanced prop erty). See also [17] for a recent su rvey on episturmian th eory . Episturmian morph isms pla y a central role in the s tu dy of episturm ian w ords (Section 2.3 recalls the deﬁ n ition of these morphisms). Introduced ﬁ rst as a generalizat ion of Sturmian morphisms, Justin and Pirillo [23] show ed that they are exactly the morph ism s that preserv e the aperio dic episturmian words. They also p rov ed that an y episturmian word is the image of another episturmian wo rd by some so-called pur e episturmian morph ism . Eve n more, an y ep istu rmian word can b e inﬁnitely decomp osed o v er the set of pu re episturmian morp hisms. This last prop erty all o ws an episturmian word to be deﬁned by one of its m orphic decomp ositions or, equiv alen tly , by a certain dir e ctive wor d , which is an in ﬁnite sequence of rules for decomp osing the given episturmian word by morphisms. In consequence, many prop erties of epistur mian words can b e dedu ced from prop erties of ep isturmian morph isms. T his ap p roac h is used for ins tance in [7, 16, 27, 43, 44, 46] and of course in the pap ers of Ju s tin et a l. . Morphic decomp ositions of S turmian words hav e b een u sed in [29] to charact erize quasip eriod ic Sturmian words. Quasip erio dicit y of ﬁnite words was ﬁrst introduced by Ap ostolico and Ehren- feuch t [3] in the follo wing wa y: “a word w is quasip erio dic if there exists a second word u 6 = w such that every p osition of w falls w ithin some o ccurrence of u in w ” . The wo rd u is then called a quasip eriod of w . I n the last ﬁfteen y ears, quasiperio dicity and co v ering of ﬁnite w ords h as been extensiv ely studied (see [2, 21] for s ome surve ys). In [3 3], Ma rcus extended this notion to in ﬁnite wo rds and op ened some questions, particularly concerning quasip erio d icit y of Sturmian words. Af- ter a brief answer to some of these questions in [27], the S turmian case wa s fully studied in [29] where it was prov ed that a S turmian word is non-quasip erio dic if and only if it is an inﬁn ite Lyndon wo rd. Here we extend this stud y to epistur mian words. In Sections 2–3, we r ecall useful r esu lts on ep istu rmian w ords and their directiv e wo rds. A particularly imp ortan t tool is the normalizatio n of words dir ecting the same episturmian word, which we recen tly introduced in [28, 19]. This idea allows an epistur mian word to be deﬁned uniquely by its s o-called normalize d dir e ctive wor d , deﬁned by some f actor av oidance. It can b e seen as a generalization of a previous result by Berth´ e, Holton, and Zamb on i [7], which was u sed in [29] to show that the d irectiv e word of a non-quasip erio dic S turmian word can tak e only tw o p ossible (similar) forms. F or non-binary epistu rmian w ords, even those deﬁned on a ternary alphab et, this simplicit y do es not hold since a combinato rial explosion of the number of cases occurs. In particular there exist ternary non-qu asip erio dic epistu rmian words that hav e inﬁnitely many directive words. As suc h, the metho d we use to c haracterize quasiperio dic epistu r mian wo rds greatly diﬀers from the one in the Stu r mian case. In Section 4, we c haracterize quasiperio dic ep istu rmian w ords. T o prov e it, we in trod u ce a new wa y to tac kle quasip erio dicity by stating an equiv alent deﬁnition th at is related to the n otion of r etu rn wor ds . F rom this, we show that any standar d episturmian wo rd (or epistandar d wor d ) is qu asip eriodic; in particular, suﬃciently long p alindromic p reﬁxes of an epistandard word are quasip erio ds of it. W e then extend this result by describing the qu asip eriod s of any (quasip erio dic) episturmian wo rd (see Theorem 4.19). This yields a charact erization of qu asip er io dic epistu r mian wo rds in terms of their directive wo rds (Theorem 4.28). Note that the set of quasip eriod s of an episturmian word wa s previously describ ed o nly f or the Fib onacci w ord [27]. (In [29] this set was not d escrib ed for qu asip eriodic Sturmian w ords). At th e end of Section 4, using the normalization asp ect, we giv e a second characteriza tion of quasip erio dic episturmian wo rds which itself pr o vides an eﬀectiv e way to decide whether or not a giv en episturmian wo rd is quasip erio dic (see Theorem 4.29). Section 5 is concerned with the study of the action of episturmian morphisms in relat ion to quasip eriod icit y . Th is s tu dy leads to non-trivial extensions of results in [29]. Using this approach, we p rovide a completely diﬀerent pro of of our main characteriza tion of quasip erio dic episturmian wo rds. W e also characterize episturmian morp hisms that map an y wo rd onto a quasip erio dic one (see Section 5.4). T his result naturally allo ws u s to consider quasip eriod icit y of words deﬁned using 2 episturmian m orp hisms. Lastly , in Section 6, we characterize episturmian Lyn don words in terms of their directive w ords. This result sho ws that, unlike the Stu rmian case, there exist non-quasip erio dic episturmian w ords that are not inﬁnite Lynd on words. 2 Episturmian w ords and morphisms W e assume the reader is familiar with combinatorics on words and morph isms (e.g., s ee [30, 31]). In this section, w e recall some basic deﬁnitions and prop erties relating to episturmian words which are needed throughout the pap er. F or the most part, we follow the notat ion and terminology of [12, 23, 25, 18]. 2.1 Notation and terminology Let A denote a ﬁ n ite non-emp ty alphab et . A ﬁ nite wor d ov er A is a ﬁnite sequ en ce of letters from A . Th e empty wor d ε is th e empt y sequence. Under the operation of concatenation, the set A ∗ of all ﬁ nite words ov er A is a fr e e monoid w ith 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 } . Giv en a ﬁnite w ord w = x 1 x 2 · · · x m ∈ A + with each x i ∈ A , the length of w , denoted b y | w | , is equal to m . By conv ention, the empt y word is the u nique word of length 0. W e denote b y | w | a the n umber of o ccurrences of th e le tter a in the word w . If | w | a = 0, then w is said to be a -fr e e . The r eversal of w , denoted by e w , is its mirror image: e w = x m x m − 1 · · · x 1 , and if w = e w , then w is called a p alindr ome . A (righ t) inﬁnite wor d (or simply se quenc e ) x is a sequ en ce indexed by N + with v alues in A , i.e., x = x 1 x 2 x 3 · · · with eac h x i ∈ A . The set of all inﬁnite words ov er A is denoted by A ω . An ultimately p erio dic inﬁn ite word can be written as uv ω = uv v v · · · , for s ome u , v ∈ A ∗ , v 6 = ε . If u = ε , then such a wo rd is (pur ely) p erio dic . An inﬁnite wo rd that is n ot ultimately p erio dic is said to b e ap erio dic . F or easier reading, inﬁnite w ords are hereafter typically t yped in boldface to distinguish th em fr om ﬁ nite words. Giv en a set X of words, X ∗ (resp. X ω ) is the set of all ﬁnite (resp. inﬁnite) words that can b e obtained by concatenating words of X . The empty word ε b elongs to X ∗ . A ﬁnite word w is a f actor of a ﬁnite or inﬁnite wo rd z if z = uw v for some words u , v (where v is inﬁ n ite iﬀ z is in ﬁnite). In the sp ecial case u = ε (resp. v = ε ), we call w a pr eﬁx (resp. suﬃx ) of z . W e use the nota tion p − 1 w (resp. ws − 1 ) to indicate the remo v al of a preﬁx p (resp. su ﬃx s ) of the wo rd w . An inﬁnite w ord x ∈ A ω is called a suﬃx of z ∈ A ω if there exists a w ord w ∈ A ∗ such that z = w x . That is, x is a shift of z , given by x = T | w | ( z ) = w − 1 z , where T denotes the shift m ap : T(( x n ) n ≥ 1 ) = ( x n +1 ) n ≥ 1 . Note th at a preﬁx or suﬃx u of a ﬁn ite or inﬁnite word w is said to b e pr op er if u 6 = w . F or ﬁnite words w ∈ A ∗ , the shift map T acts circularly , i.e., if w = x v where x ∈ A , then T( w ) = v x . The alphab et of a ﬁ nite or inﬁ nite word w , denoted by Alph( w ) is the set of letters occur r ing in w , and if w is inﬁnite, we d enote b y Ult( w ) the set of all letters occurring inﬁn itely often in w . A fac tor o f an inﬁnite w ord x is r e curr ent in x if it occurs inﬁnitely o ften in x , and x itself is said to b e r e curr e nt if all of its factors are r ecur rent in it. F urthermore, x is uniformly r e curr ent if for eac h n th er e exists a p ositive integer K ( n ) such that any factor of x of length at least K ( n ) con tains all factors of x of length n . Equiv alen tly , x is uniformly r ecur rent if any factor of x o ccurs inﬁnitely man y times in x with boun d ed gaps [9]. 2.2 Episturmian words In this pap er, our vision of epistu r mian words will be the characteristic p rop erty s tated in Theo- rem 2.1 (below). How ev er, we ﬁrst giv e one of their equ iv alent d eﬁnitions to aid in u nderstandin g. 3 F or this, we recall that a factor u of a ﬁn ite or in ﬁ nite wo rd w ∈ A ∞ is right (resp . left ) sp e cial if ua , ub (resp. au , bu ) are f actors of w for some letters a , b ∈ A , a 6 = b . An inﬁnite word t ∈ A ω is episturmian if its set of factors is closed und er rev ersal and t h as at most one right (or equiv ale ntly left) special factor of each length. Moreo ve r, an epistur mian word is standar d if all of its left sp ecial factors are preﬁ xes of it. In the initiating pap er [12], episturm ian w ords w ere deﬁned a s an extension of standard epis- turmian words, wh ic h were themselv es ﬁrst introdu ced and studied as a generalization of standard Sturmian words using p alindr omic closur e (see Theorem 4.5 later). Sp eciﬁcally , an inﬁnite word wa s said to be episturmian if it h as exactly the same set o f factors as some standard episturmian wo rd [12]. This deﬁ nition is equiv alen t to the aforementioned one by T heorem 5 in [12]. Moreo ver, it was pro v ed in [12] that episturmian words are uniformly recurrent. Hence u ltimately p eriod ic episturmian words are (purely) p eriodic. Note. Hereafter, we refer to a standard episturmian word as an epistandar d wor d , for simplicity . T o stu dy episturm ian words, Justin and Pirillo [23] introduced episturmian morphisms . In particular they pro ve d that th ese morphisms, which we recall b elow, are precisely the morphisms that preserve the set of aperio dic epistur m ian words. 2.3 Episturmian morphisms Let us recall th at given an alphab et A , a morphism f o n A is a map from A ∗ to A ∗ such that f ( uv ) = f ( u ) f ( v ) for any words u , v ov er A . A m orp hism on A is entirely deﬁn ed by the imag es of letters in A . All morphisms considered in this pap er will b e non-erasing: the image of any non- empty wo rd is neve r emp ty . Hence the action o f a morph ism f on A ∗ can be naturally extended to in ﬁnite 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 ) · · · . In wh at follows, we will denote the comp osition o f morph isms b y juxtap osition as for co ncate- nation of words. Episturmian morphisms are the compositions of the permutation morphism s (i.e., the morph isms f such that f ( A ) = A ) and the morphisms L a and R a where, for all a ∈ A : L a :  a 7→ a b 7→ ab , R a :  a 7→ a b 7→ ba for all b 6 = a in A . Here we will work only o n pur e epistur mian morph isms, i.e., morphisms obtained by comp osition of elements of the sets: L A = { L a | a ∈ A} and R A = { R a | a ∈ A} . Note. In [23], the morphism L a (resp. R a ) is denoted b y ψ a (resp. ¯ ψ a ). W e adop t the cur r ent notation to emph asize the action of L a (resp. R a ) w hen ap p lied to a word, which consists of p lacing an occurrence of the letter a on the left (resp. right) of each occurrence of any letter diﬀerent from a . Epistandar d morphisms (resp. pur e episturmian morph isms , pur e e pistandar d morphisms ) are the morp hisms obtained by concatenatio n of morphisms in L A and p ermutations on A (resp. in L A ∪ R A , in L A ). Note that the episturm ian morph ism s are exactly the Sturmian morphisms when A is a 2-lett er alphab et. 2.4 Morphic decomp osit ion of episturmian w ords Justin and Pirillo [23] pr o ve d the f ollowing insightful characterizati ons of epistandard and epis- turmian words (see Theorem 2.1 below), which show that any episturm ian word can b e inﬁnitely de c omp ose d o ver the set of p ure epistur mian morph isms. The state ment of Theorem 2.1 needs some extra deﬁnitions an d notation. First w e d eﬁne the follo wing new alphabet, ¯ A = { ¯ x | x ∈ A} . A letter ¯ x is considered to b e x w ith spin R , whilst 4 x itself has spin L . A ﬁn ite or in ﬁ nite wo rd o v er A ∪ ¯ A is called a spinne d wo rd. T o ease the reading, we sometimes call a letter with spin L (resp. sp in R ) an L -spinned (resp. R -spin ned) letter. By extension, an L -sp inned (resp. R -spinned) word is a word h a ving only letters with spin L (r esp . sp in R ). The opp osite ¯ w of a ﬁ n ite or inﬁn ite spinned word w is obtained from w by exchanging all spins in w . F or instance, if v = ab ¯ a , then ¯ v = ¯ a ¯ ba . When v ∈ A + , then its opp osite ¯ v ∈ ¯ A + is an R -spinn ed wo rd and we set ¯ ε = ε . Note that, given a ﬁ n ite or inﬁ nite word w = w 1 w 2 · · · ov er A , w e sometimes denote ˘ w = ˘ w 1 ˘ w 2 · · · any spinned word such that ˘ w i = w i if ˘ w i has sp in L and ˘ w i = ¯ w i if ˘ w i has s pin R . S uch a wo rd ˘ w is called a spinne d version of w . Note. In Ju stin and Pirillo’s original p ap ers , sp ins are 0 and 1 instead of L and R . It is conv enien t here to c hange this vision of the spins b ecause of the relationship with episturmian morp hisms, which we now recall. F or a ∈ A , let µ a = L a and µ ¯ a = R a . This op erator µ can b e n aturally extend ed (as done in [23]) to a morphism mapping any word o ve r ( A ∪ ¯ A ) into a pure episturmian m orphism: for a spinned ﬁnite word ˘ w = ˘ w 1 · · · ˘ w n o v er A ∪ ¯ A , µ ˘ w = µ ˘ w 1 · · · µ ˘ w n ( µ ε is the identit y morphism). W e will sa y that the w ord w dir e cts or is a dir e ctive wor d of the morphism µ w . The following r esu lt extends the notion of directive words to inﬁnite epistur m ian words. Theorem 2.1. [23] i ) An i nﬁnite wor d s ∈ A ω is epistandar d if and only if ther e e xi sts an inﬁnite wor d ∆ = x 1 x 2 x 3 · · · over A and an inﬁnite se quenc e ( s ( n ) ) n ≥ 0 of inﬁnite wor ds such that s (0) = s and for al l n ≥ 1 , s ( n − 1) = L x n ( s ( n ) ) . ii ) An inﬁnite wor d t ∈ A ω is episturmian if and only if ther e exists a spinne d inﬁnite wor d ˘ ∆ = ˘ x 1 ˘ x 2 ˘ x 3 · · · over A ∪ ¯ A and an inﬁnite se quenc e ( t ( n ) ) n ≥ 0 of r e c u rr ent i nﬁnite wor ds such that t (0) = t and for al l n ≥ 1 , t ( n − 1) = µ ˘ x n ( t ( n ) ) . F or any epistand ard wo rd (resp. ep isturmian w ord) t and L -spin n ed (resp. sp inned) inﬁ nite w ord ∆ (resp. ˘ ∆) satisfying the conditions of the ab ov e theorem, we say that ∆ (resp. ˘ ∆) is a (spinne d) dir e ctive w or d for t or that t is dir e cte d by ∆ (resp. ˘ ∆). Remark 2.2. It follo w s immediately fr om Theorem 2.1 that if t is an episturm ian w ord directed by a spinned inﬁnite word ˘ ∆, then each t ( n ) (as d eﬁned in part ii )) is an epistur mian word d irected by T n ( ˘ ∆) = ˘ x n +1 ˘ x n +2 ˘ x n +3 · · · . By Theorem 1 in [12] (see also Theorem 4.5 later), any epistandard word has a un ique L -spinn ed directiv e word, but also has inﬁn itely many other directiv e words (see [23, 25, 19]). F or example, the T rib onac ci wor d (or R auzy wor d [39]) is directed by ( abc ) ω and also by ( abc ) n ¯ a ¯ b ¯ c ( a ¯ b ¯ c ) ω for each n ≥ 0, as w ell as inﬁnitely m any other spinn ed wo rds. More generally , by Prop osition 3.11 in [23], any spin n ed inﬁ n ite word ˘ ∆ h a ving inﬁn itely many L -spinned letters directs a uniqu e episturmian wo rd t b eginning with the left-most L -sp inned letter in ˘ ∆. Moreo ve r, by one of the main resu lts in [19] (see Theorem 3.2 later), t h as in ﬁ nitely many other directive words. The follo w in g important fact links th e tw o parts of Th eorem 2.1. F act 2.3. [23] If t is an epistur m ian word d irected by a spin ned version ˘ ∆ of an L -sp in ned in ﬁ nite wo rd ∆ , then t has exact ly the s ame set of factors as th e (un ique) epistand ard word s directed by ∆. Moreo v er, with the same notation as in th e ab o ve r emark, the episturm ian word t is p eriod ic if and only if the epistandard word s is p erio dic, and this h olds if and only if | Ult(∆) | = 1 ( see [23, Prop. 2.9] ). More p r ecisely , a p erio dic epistur mian w ord takes the form ( µ ˘ w ( x )) ω for some ﬁnite spinned word ˘ w and letter x . 5 Note. Sturmian words are precisely th e ap erio dic episturm ian words on a 2-letter alphab et. When an episturmian word is ap erio dic, we hav e the follo wing fu ndamental link b etw een the wo rds ( t ( n ) ) n ≥ 0 and the sp inned inﬁnite word ˘ ∆ occurrin g in Theorem 2.1: if a n is the ﬁ rst let- ter of t ( n ) , then µ ˘ x 1 ··· ˘ x n ( a n ) is a pr eﬁx of t and the sequ ence ( µ ˘ x 1 ··· ˘ x n ( a n )) n ≥ 1 is not ultimately constan t (since ˘ ∆ is not u ltimately constan t), then t = lim n →∞ µ ˘ x 1 ··· ˘ x n ( a n ). Th is fact is a sligh t generalizat ion of a result of Risley and Z amboni [46, Pr op . I I I.7] on S- adic r epr e se ntations for c haracteristic Arnoux-Rauzy sequences. See also the recent p ap er [7] for S-adic representatio ns of Sturmian words. Note that S -adic dynamic al systems were introd u ced by F er en czi [14] as minimal dynamic al systems (e.g., s ee [38]) generated by a ﬁnite number of substitutions. In the case of episturmian words, the notion itsel f is actually a reformulation of the well-kno wn R auzy rules , as studied in [40]. In fact, it is well-kno wn that the subshift of an ap eriod ic episturm ian word t (i.e., the top ological closure of th e shift orbit of t ) is a min imal dyn amical system, i.e., it consists of all the epistu r mian words with the same set of factors as t . 3 Useful results on directiv e w ords Notions concerning dir ectiv e w ords of episturmian words and morphisms hav e b een recalled in the previous section. Two natural questions concerning these words are: When do t wo distinct ﬁnite spinned words direct the same episturm ian morp hism? When do tw o distinct spin ned in ﬁnite wo rds direct the same uniqu e episturmian word? In this section we recall existing answers to these questions. W e also present a wa y to u n iquely d eﬁ ne any episturmian word through a normalization of its directive words. Th is p o we rful to ol was r ecently introduced in our pap ers [19, 28]. 3.1 Presen tation ve rsus blo c k-equiv alence Generalizing a stud y of the monoid of Stu r mian morphisms by S´ e ´ eb old [47], the thir d author [41] answered the question: “When do t wo distinct ﬁn ite sp inned words direct the same epistur mian morphism?” b y giving a presentat ion of th e monoid of episturmian morphisms. Th is result was reformulate d in [42] using another set of generators and it was indepen d ently and diﬀerentl y treated in [25]. As a d irect consequence, one can see that the monoid of pure epistandard morphism s is a f ree monoid and one can obtain the follo wing p resenta tion of the monoid of pure episturmian morphisms: Theorem 3.1. (dir ect consequen ce of [42, Prop. 6.5]; reformulati on of [25, Th. 2.2]) The monoid of pur e episturmian morphisms with { L α , R α | α ∈ A} as a set of gener ators has the fol lowing pr esentation: R a 1 R a 2 · · · R a k L a 1 = L a 1 L a 2 · · · L a k R a 1 wher e k ≥ 1 is an inte ger and a 1 , . . . , a k ∈ A with a 1 6 = a i for al l i , 2 ≤ i ≤ k . This r esu lt means that t w o diﬀerent comp ositions of morphism s in L A ∪ R A yield the same pure episturm ian morphism if and only if one composition can b e d educed fr om the other in a rewriting system, called the blo ck-e quivalenc e in [25]. Although T heorem 3.1 allo ws u s to sh o w that many prop erties of epistur mian words are linked to prop erties of epistur m ian morphisms, it will b e con ve nient for us to hav e in mind the blo ck-e quiv alence that we n ow recall. A word of th e form xv x , where x ∈ A and v ∈ ( A \ { x } ) ∗ , is called a ( x -based) blo c k . A ( x - based) blo c k -tr ansformation is the replacemen t in a spinned w ord of an occurrence of xv ¯ x (where xv x is a b lock) by ¯ x ¯ v x or vice-ve rsa. T wo ﬁnite spinn ed words w , w ′ are said to b e blo ck-e quivalent if we can p ass from one to the other by a (p ossibly empty) c hain of blo c k-transformations, in which case we write w ≡ w ′ . F or examp le, ¯ b ¯ ab ¯ cb ¯ a ¯ c and babc ¯ b ¯ a ¯ c are blo ck-equiv alen t b ecause ¯ b ¯ ab ¯ cb ¯ a ¯ c → ba ¯ b ¯ cb ¯ a ¯ c → babc ¯ b ¯ a ¯ c and vice-versa. The blo c k-equiv alence is an equiv alence relation o v er 6 spinned words, and moreov er one can observe that if w ≡ w ′ then w and w ′ are spinned versions of the same word ov er A . Theorem 3.1 can b e reform ulated in terms of block-equiv alence: Theorem 3.1. L et w , w ′ b e two spinne d wor ds over A ∪ ¯ A . Then µ w = µ w ′ if and only if w ≡ w ′ . 3.2 W ords directing the same episturmian w ord Using the blo ck-equiv alence notion, the q u estion: “When do tw o distinct spinned inﬁnite words direct the same uniqu e episturmian wo rd?” was almost completely solved by Jus tin and Pir illo in [25] for bi-i nﬁnite episturmian wor ds , i.e., episturmian words with let ters ind exed by Z (and not by N as we co nsider here). More recently , in [19], we sho w ed that Justin and Pirillo’s resu lts on directiv e words of bi-inﬁnite episturm ian words are still v alid for w ords directing (right -inﬁnite) episturmian words. W e also established the following complete characteriza tion of p airs of spinned inﬁnite words directing the same u nique epistu rmian word. Not only does our characterizat ion provide the relativ e forms of tw o spinned inﬁnite w ords directing the same episturmian word, b ut it also fully solv es the p eriodic case, which was only partially solv ed in [25]. Theorem 3.2. [19] Given two spinne d inﬁnite wor ds ∆ 1 and ∆ 2 , the fol lowing assertions ar e e quivalent. i) ∆ 1 and ∆ 2 dir e ct the same right-inﬁnite ep isturmian wor d; ii) ∆ 1 and ∆ 2 dir e ct the same bi-inﬁnite e pisturmian wor d; iii) One of the fol lowing c ases hol ds f or some i, j suc h that { i, j } = { 1 , 2 } : 1. ∆ i = Q n ≥ 1 v n , ∆ j = Q n ≥ 1 z n wher e ( v n ) n ≥ 1 , ( z n ) n ≥ 1 ar e spinne d wor ds such that µ v n = µ z n for al l n ≥ 1 ; 2. ∆ i = w x Q n ≥ 1 v n ˘ x n , ∆ j = w ′ ¯ x Q n ≥ 1 ¯ v n ˆ x n wher e w , w ′ ar e spinne d wor ds such that µ w = µ w ′ , x is an L -spinne d letter, ( v n ) n ≥ 1 is a se quenc e of non-empty x -fr e e L -spinne d wor ds, and ( ˘ x n ) n ≥ 1 , ( ˆ x n ) n ≥ 1 ar e se quenc es of non-empty spinne d wor ds over { x, ¯ x } such that, for a l 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 and x ∈ { x, ¯ x } ω , y ∈ { y , ¯ y } ω ar e spinne d inﬁnite wor ds for some letters x , y such that µ w ( x ) = µ w ′ ( y ) . In items 1 and 2 of T heorem 3.2, the two considered directiv e words are sp inned versions of the same L -spinned word. This does not h old in item 3, which concerns only p erio dic episturmian wo rds. In particular, w e make the follo w ing observ ation: F act 3.3. I f an ap erio dic episturmian word is directed by tw o spin ned words ∆ 1 and ∆ 2 , then ∆ 1 and ∆ 2 are spinned versions of the same L -spinned word ∆. As an example of item 3 , one can consider the p eriodic epistur mian word ( bcba ) ω which is directed by b oth bca ω and b ¯ ac ω . Note also that ( bcba ) ω is epistandard and h as the same set of factors as the epistandard word ( babc ) ω directed by bac ω . Actually , in view of F act 2.3, we observe the following: F act 3.4. Th e su b shift of an y ap eriod ic e pisturmian w ord conta ins a unique (ap erio dic) epistan- dard word, wh er eas the su bshift of a p eriodic ep isturmian word con tains exactly t wo (p eriod ic) epistandard words, except if th is word is a ω with a a letter. 7 3.3 Normalized directiv e w ord of an episturmian word Items 2 and 3 of T heorem 3.2 show that an y epistur mian word is directed by a spinned wo rd having inﬁnitely many L -spin ned lette rs, b ut also by a s p inned inﬁnite word having both in ﬁnitely many L -spinned letters and inﬁnitely many R -spinned lette rs. T o emph asize the imp ortance of these facts, let us recall from Pr op osition 3.11 in [23] that if ˘ ∆ is a spinned inﬁn ite word ov er A ∪ ¯ A with inﬁn itely many L -spinned letters, then there exists a unique epistu r mian wo rd t directe d b y ˘ ∆. Un icit y comes from th e fact that the ﬁ rst letter of t is ﬁxed b y the ﬁrst L -spinned letter in ˘ ∆. T o work on S turmian words, Berth ´ e, Holton and Zamboni [7] p rov ed that any Sturmian w ord has a un iqu e directiv e word o v er { a, b, ¯ a, ¯ b } containing inﬁnitely many L -spinned letters b u t no factor of the form ¯ a ¯ b n a or ¯ b ¯ a n b with n an inte ger. Using Theorems 3.1 and 3. 2, w e recently generalized this r esult to episturmian w ords: Theorem 3.5. [19, 28] Any episturmian wo r d t has a spinne d dir e ctive wor d c ontaining inﬁnitely many L -spinne d letters, but no factor in S a ∈A ¯ a ¯ A ∗ a . Such a dir e c tive wor d is uniqu e if t is ap erio dic. Note that unicity does not n ecessarily hold for perio dic episturmian w ords. F or example, the p eriod ic episturm ian word ( ab ) ω = L a ( b ω ) = R b ( a ω ) is d irected b oth b y ab ω and by ¯ ba ω ( L a ( b ) = ab = R b ( a )). A d irectiv e word of an ap erio dic ep istu rmian word t with the ab ov e prop erty is called the normalize d dir e ctive wor d of t . W e extend this deﬁnition to morphisms: a ﬁ nite spinn ed word w is said to b e a normalize d dir e ctive wor d of the morphism µ w if w has no factor in S a ∈A ¯ a ¯ A ∗ a . One can observe from Theorem 3.1 that for any morph ism in L a L ∗ A R a , we can ﬁ nd another decomp osition of the morphism in the set R a R ∗ A L a . Equiv alen tly , for any spinned word in a A ∗ ¯ a , there exists a word w ′ in ¯ a ¯ A ∗ a suc h that µ w = µ w ′ . This w as the main idea used in the pro of of Theorem 3.5. Example 3.6. Let f b e the pure episturmian morphism with directiv e word ¯ a ¯ bc ¯ ba ¯ b ¯ a ¯ c ¯ b ¯ a ¯ ca . By The- orem 3.1, µ ¯ a ¯ c ¯ b ¯ a ¯ ca = µ ¯ a ¯ c ¯ bac ¯ a = µ acb ¯ ac ¯ a and hence f = µ ¯ a ¯ bc ¯ ba ¯ b ¯ a ¯ c ¯ b ¯ a ¯ ca = µ ¯ a ¯ bc ¯ ba ¯ bacb ¯ ac ¯ a and ¯ a ¯ bc ¯ ba ¯ bacb ¯ ac ¯ a is the n ormalized d irectiv e word of f . 3.4 Episturmian words ha ving a unique directiv e word Using our charac terization of pairs of words directing the same episturmian wo rd (Theorem 3.2) together with norm alizatio n (Theorem 3.5), we recen tly charact erized episturmian words h a ving a unique d irectiv e word. Theorem 3.7. [19] An episturmian wor d has a unique dir e ctive wor d if and only if its (normalize d) dir e ctive wor d c ont ains 1) inﬁnitely many L -spinne d letters, 2) inﬁnitely many R - spinne d letters, 3) no factor in S a ∈A ¯ a ¯ A ∗ a , 4) no factor in S a ∈A a A ∗ ¯ a . Such an episturmian wor d is ne c essarily ap erio dic. As an example, a particular family of epistu rmian words having un ique directive words consists of those directed by r e gular wavy wor ds , i.e., spinn ed inﬁn ite wo rds h aving b oth inﬁnitely m any L -spinned letters and inﬁnitely many R -sp inned letters s u ch that each letter occurs with the same spin everywhere in the directiv e word. More formally , a spinn ed ve rsion ˘ w of a ﬁn ite or inﬁnite wo rd w is said to b e r e gular if, for eac h letter x ∈ Alph( w ), all o ccurrences of ˘ x in ˘ w hav e the same spin ( L or R ). F or example, a ¯ baa ¯ c ¯ b and ( a ¯ bc ) ω are regular, whereas a ¯ ba ¯ a ¯ cb an d ( a ¯ b ¯ a ) ω are not regular. In [25], a spinned inﬁnite word is s aid to b e wavy if it con tains inﬁnitely many L -spinned letters and inﬁnitely many R -spinn ed letters. F or example, the t wo preceding inﬁnite words are wa vy . In the Stur m ian case, we hav e: 8 Prop osition 3.8 . [19] A ny Sturmian wor d has a unique spinne d dir e ctive wor d or inﬁnitely many spinne d dir e ctive wor ds. M or e over, a Sturmian wo r d has a u ni q ue dir e ctive wor d if a nd only if its (normalize d) d ir e ctive wo r d is r e gu lar wavy. In the n ext section, we shall see that an y episturm ian word having a uniqu e directive word is necessarily non-quasip erio d ic. This will follo w f rom Theorem 3.7 and our characte rizations of quasip eriod ic episturmian words (Theorems 4.19, 4.28, and 4.29 ). 4 Quasip erio dicit y of episturmian w ords 4.1 Quasiperio dicit y Recall (from [4, 21 , 33]) that a ﬁnite or in ﬁ nite word w is quasip eriodic if it can be constructed by concatenati ons and sup erp ositions of one of its p r op er factors u , wh ic h is called a quasip erio d of w (or the smal lest quasip erio d of w when it is of minimal length). W e also sa y th at u cov ers w or that w is u -quasip erio dic. F or example, the word w = abaababaabaababaaba has aba , abaaba , abaababaaba as quasip erio ds, and the smallest qu asip eriod of w is aba . W ords that are not quasip eriod ic are naturally called non-quasip erio dic words. When d eﬁning inﬁ nite quasip erio dic wo rds, for conv en ience, we co nsider the words preceding the o ccurrences of a quasip er io d: an inﬁnite word w is quasip erio dic if and only if there exist a ﬁnite word u and words ( p n ) n ≥ 0 such that p 0 = ε , | p n | < | p n +1 | ≤ | p n u | , a nd p n u is a preﬁx of w for all n ≥ 0 . Then u is a quasip erio d of w and w e say that the sequence ( p n u ) n ≥ 0 is a c overing se quenc e of pr eﬁxes of the wor d w . Nece ssarily , any quasip erio d of a qu asip eriod ic wo rd must b e a preﬁx of it. Readers w ill ﬁnd several examples of inﬁnite qu asip er io dic words in [27, 32, 33]. Let us mention for in stance that the Fib onac ci w or d , directed by ( ab ) ω , is aba -quasip erio dic (see [27]). Let us now recall a simp le, y et important, fact about quasip erio dic wo rds. F act 4.1. If w is a (ﬁnite or inﬁnite) u -quasip erio d ic w ord and f is a non-erasing morphism, then f ( w ) is f ( u )-quasiperio dic. Note that the conv erse of this fact is not tru e. F or example, let f = R a and w = ab ω , then R a ( w ) = a ( ba ) ω . Th e word R a ( w ) is co v ered by R a ( ab ), but w is not co vered b y ab . 4.2 Return words and quasiperio dicity W e no w use the notion of a ‘return word’ to giv e an equiv alent deﬁnition of quasip eriod icit y ( see Lemma 4.3), which prov es to b e a usefu l tool f or studying quasip eriod icit y in epistur m ian words. Return words w ere introduced indep endently by Durand [13] and by Holton and Zamb on i [2 0] when stu dying pr imitiv e sub stitutive sequences. Such words can b e deﬁned in the following wa y . Deﬁnition 4.2. Let v b e a recurr ent factor of an inﬁ nite word w = w 1 w 2 w 3 · · · , starting at p ositions n 1 < n 2 < n 3 · · · in w . T hen eac h w ord r i = w n i w n i +1 · · · w n i +1 − 1 is called a r eturn to v in w . That is, a return to v in w is a non-empty factor of w beginn in g at an o ccurrence of v and ending exactly b efore the next occur r ence of v in w . Thus, if r is a return to v in w , then r v is a factor of w that conta ins exactly tw o o ccurrences of v , one as a preﬁx and on e as a su ﬃx. As an y episturmian wo rd t is u niformly recurrent [12], each factor of t h as on ly a ﬁn ite number of diﬀerent returns (for more details see Theorem 4.7). Note. A retur n to v in w alw a ys h as v a s a pr eﬁx or is a p reﬁx of v . In particular, we observe that a return to v is not n ecessarily longer than v , in w hich case v has overlapping o ccurrences in w (i.e., v z − 1 v is a f actor of w for some n on-empty w ord z ). W e say that v has adjac ent o ccurren ces in 9 w if vv is a factor of w . In this case, if v is primitive (i.e., n ot an in teger p ow er of a sh orter word), then v is a return to itself; otherwise, the corresp ond ing r eturn to v is the primitive r o ot of v . In terms of return w ords, we ha v e the f ollowing equiv alent deﬁnition of a quasip eriod ic inﬁnite wo rd. Lemma 4.3. A ﬁnite wor d v is a qu asip e rio d of an inﬁnite wor d w if a nd only if v is a r e curr ent pr eﬁx of w such th at any r eturn to v in w has length at most | v | . Pr o of. I f v is a quasip erio d of w , then v is a pr eﬁ x of w and its o ccurrences enti rely cov er w . Th at is, v is recurrent in w and successive o ccurrences of v in w are either adjacent or o v erlap, and hence any retur n to v has length at m ost | v | . Conv ersely , if v is a recur rent pr eﬁx of w s uch that any return to v h as length at m ost | v | , then successive occur rences of v in w are either adj acent or o v erlap, and hence en tirely cov er w . Thus w is v -quasip erio dic. Immediately: Corollary 4.4. An inﬁnite wo r d w is quasip erio dic if an d o nly if ther e exists a r e curr ent p r eﬁx v of w such that any r eturn to v i n w ha s length at most | v | , in which c ase v is a quasip e rio d of w . Mor e over, the shortest such pr eﬁx v is the smal lest quasip e rio d o f w . A n otew orthy f act is th at a quasip eriod ic in ﬁnite word is n ot necessarily recur rent [33], although it m ust hav e a preﬁx that is recurrent in it. 4.3 Return words and palindrom ic closure in episturmian words In this section we recall a characteriza tion of retur n w ords in episturmian words, give n b y Justin and V uillon in [26]. F or this, we ﬁrst recall the construction of epistandard words using p alin- dr omic right-closur e as w ell as some relate d properties from [12, 23] th at will b e us ed throughout Sections 4.4 to 5.4. The p alind r omic right-closur e w (+) of a ﬁnite word w is the (u n ique) sh ortest palindrome having w as a pr eﬁx (see [11]). That is, w (+) = w v − 1 e w wh ere v is the longest palind r omic suﬃx of w . The i ter ate d p alindr omic closur e fu n ction [22], denoted by P al , is deﬁned recursively as follo ws. Set P al ( ε ) = ε and, f or any w ord w and let ter x , d eﬁne P al ( w x ) = ( P al ( w ) x ) (+) . F or instance, P al ( abc ) = ( P al ( ab ) c ) (+) = ( abac ) (+) = abacaba . Generalizing a construction giv en in [11] for stan dar d Sturmian wor ds , Droub a y , Justin and Pirillo established the follo w ing c haracterizat ion of epistand ard words. Theorem 4.5. [12] An inﬁnite wor d s ∈ A ω is epistandar d if and only if ther e exists an inﬁnite wor d ∆ = x 1 x 2 x 3 · · · ( x i ∈ A ) such tha t s = lim n →∞ P al ( x 1 · · · x n ) . Note that the palindromes P al ( x 1 · · · x n ) are very often denoted by u n +1 in the literature. In [23], Justin and Pirillo show ed that the word ∆ is exactly th e dir ectiv e word of s as it occur s in Th eorem 2.1. Moreov er, by construction, ∆ uni q uely determines the ep istand ard word s . Notice also th at by construction, the words ( P al ( x 1 · · · x i )) i ≥ 0 are exact ly the p alindromic preﬁxes of s . There exist man y relations b etw een p alindromes and epistu rmian morphisms . T he following ones will b e useful in the next few sections. First recall from [22, 23] that we hav e P al ( w v ) = µ w ( P al ( v )) P al ( w ) for any words w , v . (4.1) In particular, for an y x a letter, P al ( xv ) = L x ( P al ( v )) x an d P al ( w x ) = µ w ( x ) P al ( w ). F or letters ( x j ) 1 ≤ j ≤ i , form ula (4.1) inductivel y leads to: P al ( x 1 · · · x i ) = 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) 10 Note that by conv en tion, x 1 · · · x 0 = ε in the ab ov e pro duct. No w let ˘ w = ˘ x 1 ˘ x 2 · · · ˘ x n b e a spin n ed version of w = x 1 x 2 · · · x n (view ed as a p reﬁx of a spinn ed ve rsion ˘ ∆ of ∆). Then, f or any ﬁnite w ord v , we hav 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) The word S ˘ w is called th e shifting factor of µ ˘ w [25]. Ob serve that S ˘ w is a preﬁx of P al ( w ); in particular S ¯ w = P al ( w ) by equation (4.2). Not e also th at µ ˘ w ( v ) = T | S ˘ w | ( µ w ( v )). F or example, for ˘ w = a ¯ bc ¯ a , we hav e S ˘ w = µ abc ( a ) µ a ( b ) = abacabaab . Thus since µ abca ( ca ) = abacabaab.acabacaba , µ a ¯ bc ¯ a ( ca ) = T 9 ( µ abca ( ca )) = acabacaba.abacabaab . Like wise, for any inﬁnite w ord y ∈ A ω , µ ˘ w ( y ) = S − 1 ˘ w µ w ( y ) . (4.4) This formula used with ˘ w = ¯ w sho ws that: F act 4.6. An y word of th e form µ w ( y ) with y inﬁnite b egins with P al ( w ). Amongst the n umerous inte rests of th e palindromes ( P al ( x 1 · · · x n )) n ≥ 0 , w e ha v e the following explicit characterizat ion of th e returns to any factor of an epistandard w ord. Theorem 4.7. [26] Supp ose s is an epistanda r d wor d dir e cte d by ∆ = x 1 x 2 x 3 · · · with x i ∈ A , and c onsider any factor v of s . If the wor d u n +1 = P al ( x 1 · · · x n ) i s the shortest p alind r omic pr eﬁx of s c ontaining v with u n +1 = f v g , then the r etu rns to v ar e given by f − 1 µ x 1 ··· x n ( x ) f wher e x ∈ Alph( x n +1 x n +2 · · · ) . Because of the uniform recurrence of ep isturmian words, the follo wing simple bu t imp ortant fact ab ou t return words holds. Lemma 4.8. Supp ose s is an epistanda r d wor d and let t b e any episturmian wor d in th e subshift of s . Then, for any factor v of s , r is a r eturn to v in s if a nd only if r is a r eturn to v in t . That is, the returns to a ny factor v of an epistandard word s are the same as the r eturns to v as a f actor of any epistur m ian word t with the same set of factors as s . Hereafter, we often use the ab ov e result without reference to it. The follo w in g result is particularly us eful in Sections 4.3–4. 6. Prop osition 4.9. Supp ose s is an epistandar d wor d dir e cte d by ∆ = x 1 x 2 x 3 · · · with x i ∈ A , and let t b e an episturmian wor d dir e cte d by a spinne d version of ∆ . Then t b e gins with P al ( x 1 · · · x n ) for some non-ne gative inte ger n if and only if t ha s a dir e ctive wor d of the form ˘ ∆ = x 1 · · · x n ˘ x n +1 ˘ x n +2 · · · wher e the pr eﬁx x 1 · · · x n is L - spinne d. Pr o of. Wh en t is directed by a s pinned version of ∆ of th e form ˘ ∆ = x 1 x 2 · · · x n ˘ x n +1 ˘ x n +2 · · · where the p reﬁx x 1 x 2 · · · x n is L -sp inned, t b egins with P al ( x 1 · · · x n ) by F act 4.6. Conv ersely , supp ose P al ( x 1 · · · x n ) is a preﬁx of t , and s upp ose th at t is directed by ˘ ∆ = ( ˘ x i ) i ≥ 1 (a sp inned version of ∆ = ( x i ) i ≥ 1 ). F rom T heorem 3.2 w e can s upp ose that ˘ ∆ contains inﬁn itely many L -spinned letters. I f n = 0, there is nothin g to prov e. Else let x 1 = a . Assu me ﬁrst that ˘ x 1 = ¯ a . Let k b e th e smallest p ositive integ er su c h that ˘ x k ∈ A . Since t b egins with the letter a (which is the ﬁrst letter o f P al ( x 1 · · · x n )), we hav e x k = a . T hen ˘ ∆ = ¯ a ¯ x 2 · · · ¯ x k − 1 a ˘ x k +1 · · · , and hence by Theorem 3.1 t is also d irected by the spinned inﬁ nite word b eginning with ax 2 · · · x j − 1 ¯ a , with j ≤ k . Therefore w e ma y assume fr om n ow on th at ˘ x 1 = a . Let t ′ b e the episturmian word directed by T( ˘ ∆) = ( ˘ x i ) i ≥ 2 . It is easily s een from the equality P al ( x 1 · · · x n ) = L a ( P al ( x 2 · · · x n )) a that P al ( x 2 · · · x n ) is a preﬁx of t ′ . Hence by in duction t ′ is directed by a sp in ned v ersion of T(∆) with x 2 · · · x n as a pr eﬁx. And so t is directed by a spin ned version of ∆ of the form x 1 x 2 · · · x n ˘ x n ˘ x n +1 · · · where the pr eﬁx x 1 x 2 · · · x n is L -sp inned. 11 4.4 All epistandard words are quasip erio dic A ﬁ r st consequen ce of Theorem 4.7 is that any epistand ard word is quasip erio dic. More pr ecisely: Theorem 4.10. Supp ose s is an epistandar d wor d with dir e ctive wor d ∆ = x 1 x 2 x 3 · · · with x i ∈ A , and let m b e the smal lest p ositive inte ger such that Alph( x 1 x 2 · · · x m ) = Alph( s ) . Then, for al l n ≥ m , P al ( x 1 · · · x n ) is a quasip erio d of s . Pr o of of The or em 4.1 0. W e sup p ose that s is an epistandard word with directive word ∆ = x 1 x 2 x 3 · · · , x i ∈ A . L et m b e the smallest p ositiv e in teger such th at Alp h( x 1 x 2 · · · x m ) = Alph( s ). Clearly , f or n < m , P al ( x 1 · · · x n ) cannot b e a quasip eriod of s since it d oes n ot con tain all of the letters in Alph( s ). No w let n ≥ m . W e kn o w that P al ( x 1 · · · x n ) is a preﬁx of s . Su p p ose that for some k ≥ n , P al ( x 1 · · · x k ) is cov ered by P al ( x 1 · · · x n ). Since by c hoice of m , x k +1 b elongs to { x 1 , · · · , x k } , we hav e | ( P al ( x 1 · · · x k ) x k +1 ) (+) | ≤ 2 | P al ( x 1 · · · x k ) | . Hence since P al ( x 1 · · · x k ) is a palindrome, P al ( x 1 · · · x k +1 ) is cov ered by P al ( x 1 · · · x k ), and so by P al ( x 1 · · · x n ). The result f ollows from Theorem 4.5 by in duction. Example 4.11. Recall the T rib onacci word: r = abacabaabacababacabaabacabacabaabaca · · · , which is the epistandard word directed by ( abc ) ω . Ob serve that P al ( abc ) = abacaba is the shortest palindromic preﬁx of r such that Alph( P al ( abc )) = Alph( r ) = { a, b, c } . By T h eorem 4.7, the returns to P al ( abc ) in r are: µ abc ( a ) = abacaba , µ abc ( b ) = abacab , µ abc ( c ) = abac , none of which are longer than P al ( abc ). Hence r is abacaba -quasip erio dic; in fact P al ( abc ) = µ abc ( a ) = abacaba is the smallest quasip eriod of r sin ce its preﬁxes abac , abaca , abacab h a ve returns longer than themselves. This latter fact is also evid ent from our description of quasip erio ds of a (qu asip eriodic) ep isturmian wo rd (Th eorem 4.19, to follo w ). More generally , the k -b onac ci wo r d , which is directed by ( a 1 a 2 · · · a k ) ω , is quasip erio dic with smallest quasip erio d P al ( x 1 · · · x k ). This fact was also obser ved in [27] by noting that the k - b onacci word is generated b y the morph ism ϕ k on { a 1 , a 2 , . . . , a k } deﬁn ed by ϕ k ( a i ) = a 1 a i +1 for all i 6 = k , and ϕ k ( a k ) = a 1 . Remark 4.12. F rom F act 4.1 and Th eorem 4.10, we immediately deduce that ϕ ( s ) is a quasiperi- od ic inﬁnite word for any epistandard word s . Moreov er , if ϕ is a pu re episturmian morphism, then ϕ ( s ) is a quasip erio dic episturmian wor d . More pr ecisely , µ ˘ w ( s ) is a quasip erio dic epistu rmian wo rd for any e pistandard wo rd s and spin ned w ord ˘ w . Suc h an episturmian w ord is directed by a spinned inﬁ nite w ord of the form ˘ w ∆ where ∆ is the L -sp in ned directiv e word of s . Hence, if an episturmian w ord t is dir ected by a spin ned inﬁ nite w ord with all sp ins ultimately L , then t is quasip eriod ic. More generall y , we hav e the follo wing consequ ence of Theorem 4.10 (a conv erse of this result is stated in Theorem 4.1 9, and a generalization is pro vided by Th eorem 5.1 later). Corollary 4.13. If an episturmian wor d t is dir e cte d by ˘ ∆ = ˘ w v ˘ y for some spinne d wor ds ˘ w , ˘ y and L -spinne d wo r d v such that Alph( v ) = Alph( v y ) , then t is quasip erio dic. Mor e over, any wor d of the form µ ˘ w ( P al ( v )) p with p a pr e ﬁx of S − 1 ˘ w P al ( w ) is a quasip erio d of t . Pr o of. L et ( t ( n ) ) n ≥ 0 b e the in ﬁnite sequence of episturm ian words associated to t and ˘ ∆ in T heo- rem 2.1. Then by Prop osition 4.9, the episturm ian word t ( | w | ) , which is d ir ected by v ˘ y , b egins with the palindromic preﬁx P al ( v ) of the epistandard word s ( | w | ) directed b y the L -spinned version o f v ˘ y . Moreov er, since Alph( v ) = Alph( v y ), P al ( v ) is a quasip erio d of s ( | w | ) by Theorem 4.10. By F act 2.3, s ( | w | ) is in the subshift of t ( | w | ) . It follo ws f r om Lemma 4.8 th at P al ( v ) has th e same 12 returns in t ( | w | ) as it does in s ( | w | ) , and since it is a preﬁx of t ( | w | ) , it is also a quasip erio d of t ( | w | ) . Therefore b y F act 4.1 t = µ ˘ w ( t ( | w | ) ) is quasip erio dic and µ ˘ w ( P al ( v )) is a quasip eriod of t . No w by formula (4.4) and by F act 4.6, eac h o ccurren ce of µ ˘ w ( P al ( v )) is follow ed by S − 1 ˘ w P al ( w ) in t , and so is follo w ed by p for an y p reﬁx p of S − 1 ˘ w P al ( w ). Th is sh o ws that µ ˘ w ( P al ( v )) p is a quasip eriod of t . 4.5 Ultimate quasiperio ds Theorem 4.10 shows that long enough palind r omic p r eﬁxes of an epistandard word s are quasip erio ds of s . O ur n ext goal is to extend Th eorem 4.10 b y describin g all the qu asip eriod s of any (quasip eri- od ic) episturm ian wo rd. Several lemmas are required, us ing the notion of ultimate quasip eriod s that w e now deﬁne. A f actor v of an inﬁnite word w is said to b e an ultimate quasip erio d of w if it is a qu asip eriod of a suﬃx of w . In particular, when w is a uniformly recurrent word (whic h is the case for episturm ian wo rds), a recurrent factor v in w is an u ltimate quasip erio d of w if any r eturn to v in w has length at m ost | v | . Clearly , if an u ltimate quasip erio d is a p reﬁx of w , th en w is quasip erio dic by Lemma 4.3. Also note th at the set of quasip erio ds of a (quasip erio dic) inﬁnite word w consists of all its ultimate quasip eriod s that are preﬁxes of it. Remark that we hav e Prop osition 4.14. A l l epist urmian wor ds ar e ultimately qu asip erio dic. Pr o of. F rom Theorem 4.10, all epistand ard words are quasip erio dic. Let s b e an epistand ard word and let q b e a quasip eriod of s . Then any episturmian w ord t in the subshift of s has a su ﬃx t ′ b eginning w ith q . S ince t has the same set of factors than s , w e can ﬁnd inﬁnitely man y preﬁxes of t ′ which are co v ered by q , i.e., t ′ is qu asip eriodic. Hence t is ultimately quasiper io dic. No w w e recall some insight about palindromic closure that will be useful later. As previously , let s denote an epistand ard word with dir ective word ∆ = x 1 x 2 x 3 · · · with x i ∈ A . F or n ≥ 0 let u n +1 = P al ( x 1 · · · x n ). Not e in particular that u 1 = ε and by Theorem 4.5, s = lim n →∞ u n . As in [26, 23], let us deﬁ n e P ( i ) = su p { j < i | x j = x i } if th is num b er exists, un deﬁned otherwise. That is, if x i = a , then P ( i ) is the position of the right-most o ccurrence of the letter a in the preﬁx x 1 x 2 · · · x i − 1 of the directive word ∆. F or in s tance, if ∆ = ( abc ) ω , P ( i ) = i − 3 f or any i ≥ 4 and P (1), P (2), and P (3) are un d eﬁned. F rom the deﬁn itions of palindromic closure and the palindromes ( u i ) i ≥ 1 , it follo ws that, for all i ≥ 1, u i +1 = ( u i x i u i if x i / ∈ Alph( u i ), u i u − 1 P ( i ) u i otherwise . (4.5) Therefore, using Theorem 4.7 with f = g = ε , we d ed uce that for n ≥ 0, the length of the longest r eturn r n +1 to u n +1 in s satisﬁes | r n +1 | = ( | u n +1 | + 1 if some x ∈ Alph( s ) do es not occur in u n +1 , | u n +1 | − | u p n | otherwise , where p n = inf { P ( i ) | i ≥ n + 1 } (see also [26, Lem. 5.6]). In other words, p n = sup { i ≤ n | Alph( x i · · · x n ) = Alph( x i · · · x n · · · ) } . F or instance, if ∆ = ( abc ) ω , then p 4 = 1. The next lemma gives the set of all ultimate quasip erio ds of any epistur m ian word t . It simp ly amounts to determining all of the factors of t that hav e no r eturns longer than themselv es. 13 Notation. Hereafter, we denote b y F ( w ) the set of factors of a ﬁnite or inﬁnite w ord w . Lemma 4.15. Supp ose s is an epistandar d wor d dir e cte d by ∆ = x 1 x 2 x 3 · · · , x i ∈ A . L et m b e the smal lest p ositive i nte ger such that Alph( x 1 x 2 · · · x m ) = Alp h( s ) and let u n +1 = P al ( x 1 · · · x n ) for al l n ≥ 0 . Then the set of al l ultimate quasip erio ds of any episturmian wor d t in th e subshift of s is given by Q = [ n ≥ m Q n with Q n = { q ∈ F ( u n +1 ) | | q | ≥ | u n +1 | − | u p n |} , wher e p n = su p { i ≤ n | Alph( x i · · · x n ) = Alph( x i · · · x n · · · ) } . Pr o of. Firs t observe that the number p n exists for all n ≥ m . Indeed, the set { i ≤ n | Alph( x i · · · x n ) = Alph( x i · · · x n · · · ) } is not empty , as by the d eﬁnition of m it contains i = 1. Clearly , if n < m , then no factor of u n +1 can be an ultimate quasip eriod of t since u n +1 do es not cont ain all of the letters in Alph ( t ) = Al ph(∆). So let us now ﬁx n ≥ m . F or any q ∈ Q n , | q | ≥ | u n +1 | − | u p n | > | u n | . Ind eed from f ormula (4.5), if n = m then u n +1 = u n x n u n , and if n ≥ m + 1, then u n +1 = u n u − 1 P ( n ) u n where P ( n ) ≥ p n . Hence q ∈ F ( u n +1 ) \ F ( u n ), i.e., u n +1 is the shortest palindromic preﬁx of s con taining q . T h erefore, by Th eorem 4.7, the returns to q ∈ Q n are a certain circular shift of the returns to u n +1 and the longest of these return words h as length | u n +1 | − | u p n | . Thus any retur n to q has length at most | q | ; whence q is an ultimat e quasip erio d of t . It remains to sho w that any other f actor w ∈ F ( u n +1 ) \ F ( u n ) with | w | < | u n +1 | − | u p n | is not an ultimate q u asip eriod . This is clearly true s ince the longest return to an y such w h as length | u n +1 | − | u p n | > | w | . That is, at lea st one of the retur ns to w is longer than it, wh ic h im p lies that w is not an ultimate quasip eriod of t . Example 4.16. Let us consider the Fib onacci case. As observe d in [10], | u n | = F n +1 − 2 for all n ≥ 1 wh ere F k is the k -th Fib on acci number ( F 1 = 1, F 2 = 2, F k = F k − 1 + F k − 2 for k ≥ 2). Since for n ≥ 2 , p n = n − 1 a nd | u n +1 | − | u p n | = F n +1 , the ultimate quasip erio ds of the Fib on acci w ord are the factors of u k of length b et wee n F k and F k +1 − 2 f or all k ≥ 3. Th e ﬁ r st few ultimate quasip eriod s of th e Fibonacci word (in order of increasing le ngth) are: aba ( u 3 ), abaab , baaba , abaaba ( u 4 ), abaababa , baababaa , aababaab , ababaaba , abaababaa , baababaab , aababaaba , abaababaab , baababaaba , abaababaaba ( u 5 ), . . . Lemma 4.15 yields the following trivial charac terization of quasip erio dic episturmian words. Corollary 4.17. Supp ose s is an epistand ar d wor d with set of ultimate quasip erio ds Q . Then an episturmian wor d t in the subshift o f s is quasip erio dic if and only if som e v ∈ Q i s a pr eﬁx of t . Moreo v er, L emma 4.15 can b e reform ulated (more n icely) using episturmian morph isms, together with the iterated palindromic closure function. Lemma 4.18. Supp ose s is an e pistandar d wor d dir e cte d by ∆ ∈ A ω . Then the set of ultimat e quasip erio ds of any episturmian wor d t in the subshift of s is the set of al l wor ds q ∈ F ( P al ( w v )) , with | q | ≥ | µ w ( P al ( v )) | , wher e w , v ar e wor ds such that ∆ = wv y with Alph( v ) = Alph ( v y ) . Pr o of. L et ∆ = x 1 x 2 x 3 · · · and let m b e th e smallest p ositive in teger such that Alph( x 1 x 2 · · · x m ) = Alph( s ). T hen, by Lemma 4.15, the set of all ultimate quasiperio ds of s (and hence of t ) is given by Q = S n ≥ m Q n with Q n = { q ∈ F ( u n +1 ) | | q | ≥ | u n +1 | − | u p n |} , where p n = su p { i ≤ n | Alph( x i · · · x n ) = Alph( x i · · · x n · · · ) } . So, for ﬁxed n ≥ m , ∆ can b e written as ∆ = w v y where w = x 1 · · · x p n − 1 , v = x p n · · · x n , y = x n +1 x n +2 · · · , an d Alph( v ) = Alph( v y ) (by the deﬁnition of p n ). Then u n +1 = P al ( x 1 · · · x n ) = P al ( w v ) and u p n = P al ( x 1 · · · x p n ) = P al ( w ). 14 So n o w, u sing formula (4.1), we hav e P al ( w v ) = µ w ( P al ( v )) P al ( w ); in p articular, | u n +1 | − | u p n | = | P al ( w v ) | − | P al ( w ) | = | µ w ( P al ( v )) | . Thus, Lemma 4.15 tells u s that for q ∈ Q n , there exist words w , v suc h that q ∈ F ( P al ( w v )) and | q | ≥ | µ w ( P al ( v )) | . Conv ersely , assu me ∆ = w v y with Alph( v ) = Alph( v y ). Let v 1 , v 2 b e such that v = v 1 v 2 with v 2 the smallest su ﬃx of v such that Alph( v 2 ) = Alph( v 2 y ). Let n = | wv | . It f ollo ws from the c hoice of v 2 that p n = | w v 1 | +1. By Theorem 4.1 0, P al ( v 2 ) is a quasip erio d of th e epistandard wo rd directed by v 2 y . By F act 4.1 µ w v 1 ( P al ( v 2 )) is a quasip eriod of s . S ince any occurrence of µ w v 1 ( P al ( v 2 )) in s is follo w ed b y P al ( w v 1 ), we deduce that any fac tor of µ w v 1 ( P al ( v 2 )) P al ( w v 1 ) of length greater than | µ w v 1 ( P al ( v 2 )) | is an ultimate quasiperio d of t . Usin g formula (4.1), we see that P al ( w v 1 ) = µ w ( P al ( v 1 )) P al ( w ) so that µ w v 1 ( P al ( v 2 )) = µ w ( µ v 1 ( P al ( v 2 )) P al ( v 1 )) = µ w ( P al ( v 1 v 2 )) = µ w ( P al ( v )). Hence any f actor of µ w ( P al ( v )) P al ( w ) = P al ( w v ) of length greater than | µ w ( P al ( v )) | ( ≥ | µ w v 1 ( P al ( v 2 )) | ) is an ultimate qu asip er io d of t . 4.6 Quasiperio ds of episturmian words W e are now r eady to s tate the main theorem of this section, which describ es all of the qu asip eriods of an episturmian wo rd. Theorem 4.19 . The set of quasip erio ds of an episturmian wor d t is the set of al l wo r ds µ ˘ w ( P al ( v )) p, with p a pr eﬁx of S − 1 ˘ w P al ( w ) , (4.6) wher e w , v ar e L -spinne d wor ds suc h that t is dir e cte d by ˘ w v ˘ y for some spinne d version ˘ w of w and some spinne d version ˘ y of an L -spinne d inﬁnite wor d y with Alph( v ) = Alph( v y ) . Mor e over, the smal lest quasip erio d of t is the wor d µ ˘ w ( P al ( v )) wher e w v is of minimal length for the pr op erty Alph( w v ) = Alph( wv y ) , and amongst al l de c omp ositions of wv into w and v , the wor d v is the shortest su ﬃx of wv such that Alph( v ) = Alph( v y ) . The ab ov e theorem shows that if there do n ot exist words ˘ w , y , ˘ y , and v (as deﬁned ab ov e) such that t is directed by ˘ w v ˘ y with Alph( v ) = Alph( v y ), then t does not hav e any quasip eriods , and h ence t is non-quasip erio dic. F or in stance, any regular wa vy word ˘ ∆ (recall the deﬁnition from Section 3.4) clearly d irects an episturmian word with no quasip erio ds since ˘ ∆ is the only directive wo rd for t and it d o es n ot contain an L -spinned factor v containing all lette rs that follow it in ˘ ∆. F or example, ( a ¯ bc ) ω directs a non-quasip erio dic episturmian word in th e su b shift of the T ribonacci wo rd. Let u s n o w illus tr ate the last p art of the theorem. F or the epistandard w ord directed by ∆ = ca ( ab ) ω , i.e., the image of the Fib onacci w ord by the morp hism L c L a , the shortest w ord wv such that ∆ = wv y for some in ﬁnite word y with Alph( wv ) = Alph( wv y ) is the word caab . There are three wa ys to deco mp ose this w ord caab in to w v : 1) w = ε and v = caab ; 2) w = c and v = aab ; 3) w = ca and v = ab . The corresp onding quasip eriod s of the form µ w ( P al ( v )) are resp ectively: 1) P al ( caab ) = cacacbcacac ; 2) µ c ( P al ( aab )) = cacacbcaca ; 3) µ ca ( P al ( ab )) = cacacbca . Pr o of of The or em 4.1 9. Corollary 4.1 3 pro ve s that an y word of the form (4.6) is a quasiperio d of t . No w supp ose that q is a quasip eriod of t . W e show that t has at least one directive w ord of the form ˘ w v ˘ y where ˘ w , ˘ y are sp inned versions of some L -spinned words w , y , and where v is a n L -spinned word such that Alph( v ) = Alph( v y ). Moreo ver, we sh ow that q = µ ˘ w ( P al ( v )) p for some preﬁx p of S − 1 ˘ w P al ( w ). Let s be an epistandard word with L -spinn ed directive w ord ∆ such that t is directed by a spinned version of ∆. T h e quasip erio d q is an ultimate quasip erio d of t that o ccurs a s a preﬁx of t . S o, by Lemma 4.18, q ∈ F ( P al ( w v )) , w ith | q | ≥ | µ w ( P al ( v )) | , 15 for some L -spinned w ords w , v , and y suc h that ∆ = w v y with Alph ( v ) = Alph( v y ). In particular, we ha v e P al ( w v ) = f q g f or some words f , g with | f | + | g | ≤ | P al ( w ) | , where P al ( w v ) = µ w ( P al ( v )) P al ( w ). By deﬁnition of the P al fu nction, P al ( w ) is a p reﬁx of P al ( w v ). Consequently f is a preﬁ x of P al ( w ). Note that q = f − 1 µ w ( P al ( v )) P al ( w ) g − 1 . (4.7) Thus q = ( f − 1 µ w ( P al ( v )) f ) p w ith p := f − 1 P al ( w ) g − 1 a preﬁx of f − 1 P al ( w ). The follo w in g result shows that f = S ˘ w for s ome spinned version ˘ w of w . Lemma 4.20. Given a wor d w and a pr eﬁx f of P al ( w ) , ther e exists a spinne d v ersion ˘ w of w such that f = S ˘ w . Pr o of. T he pr o of pro ceeds b y indu ction on | w | . The lemma is clearly true for | w | = 0 since in this case f is a pr eﬁx of P al ( w ) = P al ( ε ) = ε , and hence f = ε = S ε . No w supp ose | w | ≥ 1 and let u s w rite w = xw ′ where x is a letter. Since f is a preﬁx of P al ( w ) = P al ( xw ′ ) = µ x ( P al ( w ′ )) x (see formula (4.1)), w e hav e f = µ x ( f ′ ) or f = µ x ( f ′ ) x for some preﬁx f ′ of P al ( w ′ ). Moreov er, by the induction h yp othesis, f ′ = S ˘ w ′ for some sp inned v ersion ˘ w ′ of w ′ . Hence, u sing f ormula (4.3), w e hav e f = µ x ( S ˘ w ′ ) = S x ˘ w ′ or f = µ x ( S ˘ w ′ ) x = S ¯ x ˘ w ′ . That is, f = S ˘ w for s ome spin ned version ˘ w of w = xw ′ . No w we h av e to prov e that t is directed by a spinned inﬁnite word b eginnin g with ˘ w v . F or this we n eed some further in termediate results, as follows. Lemma 4.21. F or any wo r d u c ontaining at le ast two diﬀer ent letters and for any o ther wor d w , ther e exists a wor d u w c ontaining at le ast two diﬀer ent letters such that µ w ( u ) = P al ( w ) u w . Pr o of. T he pr o of proceeds by in duction on | w | . The lemma is trivially true for | w | = 0. Now supp ose | w | ≥ 1 and let us write w = xw ′ where x is a letter. Then µ w ( u ) = µ xw ′ ( u ) = µ x ( µ w ′ ( u )) where, by the induction hypothesis, µ w ′ ( u ) = P al ( w ′ ) u w ′ for some word u w ′ con taining at least t wo diﬀerent letters. Hence, µ w ( u ) = µ x ( P al ( w ′ ) u w ′ ) = µ x ( P al ( w ′ )) µ x ( u w ′ ), and therefore by formula (4.1) we hav e µ w ( u ) = P al ( xw ′ ) x − 1 µ x ( u w ′ ) where the word x − 1 µ x ( u w ′ ) c onta ins at least t wo d iﬀerent letters. This completes the p roof of the lemma. Corollary 4.22. F or any letter x and for any wo r ds v , w such that v c onta ins at le ast one letter diﬀer ent f r om x , | µ w ( P al ( v x )) | ≥ | P al ( w v ) | + 2 . (4.8) Pr o of. W e d istinguish t w o cases: x 6∈ Alph( v ), x ∈ Alph ( v ). If x 6∈ Alph( v ), then P al ( v x ) = P al ( v ) xP al ( v ) and the word u = xP al ( v ) contains at least t wo diﬀerent letters. O n the other hand, if x ∈ Alph( v ), then v = v 1 xv 2 where v 2 is x -free, in w hich case P al ( v x ) = P al ( v ) P al ( v 1 ) − 1 P al ( v ) and the word u = P al ( v 1 ) − 1 P al ( v ) contains at least tw o diﬀeren t lett ers. Hence, in either case, P al ( v x ) = P al ( v ) u w here u is a word conta ining at least t wo diﬀeren t letters. Thus, it follows fr om Lemma 4. 21 that µ w ( P al ( v x )) = µ w ( P al ( v )) P al ( w ) u w for some word u w con taining at least t w o diﬀerent letters. S ince P al ( wv ) = µ w ( P al ( v )) P al ( w ) and | u w | ≥ 2, the p ro of is thus complete. Note. Inequ ality (4.8) is n ot true in general. F or instance, when v = ε and w = xx , we hav e µ w ( P al ( v x )) = x and P al ( w v ) = xx . Let us come bac k to our p ro of of Theorem 4.19. W riting v = v ′ x , we hav e | q | ≥ | µ w ( P al ( v )) | > | P al ( w v ′ ) | . Hence as an immediate consequence of Corollary 4.22, when v contains at least t wo diﬀerent letters, P al ( w v ) is the smallest palindromic preﬁx of s of w hich q is a factor. Therefore by Theorem 4.7 and Lemma 4.8, the return s to q in t are the words f − 1 µ w v ( α ) f where α ∈ Alph( y ). Consequently , each o ccurrence of q is preceded by the word f . Thus, the set of factors of f t is exactly the s ame as the set of factors of t ; wh ence, the inﬁnite word f t (wh ic h is clearly recurrent) 16 is epistur mian. Moreo v er, returns to f q in f t are of the form µ w v ( α ) f for letters α ∈ Alph( y ). Hence we dedu ce that there exists an inﬁ nite word t ′ such that f t = µ w v ( t ′ ). Moreov er , we deduce from the follo wing lemma that t ′ is epistu rmian. Lemma 4.23. F or any letter α , an inﬁnite wor d w i s episturmian if and only if µ α ( w ) is epistur- mian. Pr o of. ( ⇒ ): I m mediately follo ws fr om T h eorem 2.1 (see also Corollary 3.12 in [23] wh ic h shows more generally th at if w is episturmian and ϕ is an epistur mian morph ism, then ϕ ( w ) is episturmian). ( ⇐ ): Conve rsely , supp ose z := µ α ( w ) is an episturmian word. Then z b egins with the letter α . Hence, by Prop osition 4.9, z is directed by a spinn ed inﬁn ite w ord ˘ ∆ b eginnin g with α , say ˘ ∆ = α ˘ y for some spinn ed inﬁn ite word ˘ y . So b y Theorem 2.1 and Remark 2.2 , z = µ α ( z ′ ) where z ′ is an episturmian word dir ected by ˘ y . By the injectivit y of µ α , z ′ = w ; whence w is episturmian. No w, since f = S ˘ w , w e hav e t = µ ˘ wv ( t ′ ) by formula (4.4), an d so t is directed by a spin n ed inﬁnite word b eginning with ˘ w v . It r emains to co nsider th e case when v is a p ow er of a letter x . In this case, the condition Alph( v ) = Alph ( v y ) imp lies that y = x ω ; thus s (and h ence t ) is p eriodic. More pr ecisely , s = ( µ w ( x )) ω and q = f − 1 µ w ( x ) f . Since f = S ˘ w , it follows from formula (4.3) that q = µ ˘ w ( x ) and t = q ω showing that t is directed by ˘ w x ω . The pro of of Theorem 4.19 is thus complete, except for the last part concerning th e smallest quasip eriod which we prov e b elow. As previously , let s b e an epistandard word su ch that t b elongs to the sub shift of s and let ∆ b e the L -spinned directive wo rd of s . First of all, we observe that the s m allest quasip eriod of t is of th e form q := µ ˘ w ( P al ( v )), w here ˘ w is a s p inned version of a word w such that ∆ = w v y and Alph( v ) = Alph( v y ). Assume ﬁrst that v has a p rop er suﬃx s suc h th at Alph( s ) = Alph( s y ), so that writing v = ps for a n on-empty word p , µ ˘ wp ( P al ( s )) is also a qu asip eriod of t . By formula (4.1), µ p ( P al ( s )) is a proper p reﬁx of P al ( ps ) = P al ( v ). Thus µ ˘ wp ( P al ( s )) is a p rop er preﬁx of µ ˘ w ( P al ( v )) showing that the second of these wo rds is not the smallest quasip eriod of t . F rom now on, we consider L -spinn ed wo rds w 1 , w 2 , v 1 , v 2 , s pinned versions ˘ w 1 and ˘ w 2 of w 1 and w 2 resp ectiv ely , spinned wo rds y 1 and y 2 such that µ ˘ w 1 ( P al ( v 1 )) and µ ˘ w 2 ( P al ( v 2 )) are quasip eriods of t , ∆ = w 1 v 1 y 1 = w 2 v 2 y 2 , Alph( v 1 ) = Alph( v 1 y 1 ), an d Alph( v 2 ) = Alph( v 2 y 2 ). Moreo v er we assume that v 1 and v 2 ve rify the follo wing hyp othesis: (H) f or i = 1 , 2, v i has n o p r op er s uﬃx s i with Alph( s i ) = Alph( s i y i ) Note that | µ ˘ w 1 ( P al ( v 1 )) | = | µ w 1 ( P al ( v 1 )) | and | µ ˘ w 2 ( P al ( v 2 )) | = | µ w 2 ( P al ( v 2 )) | , so that to deter- mine whether µ ˘ w 1 ( P al ( v 1 )) or µ ˘ w 2 ( P al ( v 2 )) is th e smallest quasiperio d, we just ha v e to determine which of the tw o words µ w 1 ( P al ( v 1 )) and µ w 2 ( P al ( v 2 )) is the s hortest word. But then we can use the fact that w 1 v 1 and w 2 v 2 are b oth preﬁxes of ∆ . When | w 1 | = | w 2 | , Hyp othesis (H) implies that w 1 = w 2 and v 1 = v 2 . Before considering the case wh er e | w 1 v 1 | < | w 2 v 2 | , let us recall that if α is a letter and if x is an α -free w ord, then µ x ( α ) = P al ( x ) α (see formula ( 4.1)). Moreo ver, we ob s erve that for any wo rd x and distinct letters α and β , P al ( xα ) is a preﬁx of µ x ( P al ( αβ )). In d eed, µ x ( P al ( αβ )) = µ x ( µ α ( β ) α ) = µ xα ( β α ) and this wo rd contains P al ( xα ) as a preﬁx b y Lemma 4.21. Assume | w 1 v 1 | < | w 2 v 2 | (the case | w 1 v 1 | > | w 2 v 2 | is symmetric) and let us show th at | µ ˘ w 1 ( P al ( v 1 )) | is less than (or equal to) | µ ˘ w 2 ( P al ( v 2 )) | . In this case w 1 v 1 is a prop er preﬁx of w 2 v 2 . Let x b e the non-empty word such that w 1 v 1 x = w 2 v 2 . Hyp othesis (H) implies th at v 1 cannot b e a factor of v 2 except as a preﬁx. Thus there exists a p ossibly empt y word y s u ch that v 1 x = y v 2 . Assume x has length at least 2 and is a su ﬃx of v 2 . Then Hyp othesis (H) implies that the t w o last letters of v 2 and so of x are diﬀerent . Let α and β b e the t wo last lett ers of x , and let x ′ b e the wo rd su c h that x = x ′ αβ . Then P al ( v 1 ) is a preﬁx of P al ( v 1 x ′ ) wh ich is a preﬁx of 17 µ v 1 x ′ ( αβ ) b y Lemma 4.21. Finally we can see that µ v 1 x ′ ( αβ ) is a preﬁx of µ y ( P al ( v 2 )). T his shows that P al ( v 1 ) is a prop er preﬁx of µ y ( P al ( v 2 )) and so µ ˘ w 1 ( P al ( v 1 )) is a shorter quasip eriod than µ ˘ w 2 ( P al ( v 2 )). No w consider | x | = 1 and | v 2 | ≥ 2. Let α be th e last letter of v 1 and let v ′ 1 and v ′ 2 b e the wo rds such that v 1 = v ′ 1 α , and v 2 = v ′ 2 αx . Then P al ( v 1 ) = P al ( v ′ 1 α ) is a prop er pr eﬁ x of µ v ′ 1 ( P al ( αx )) = µ y v ′ 2 ( P al ( αx )) (see the paragraph before last, taking αβ = αx ), wh ic h is a preﬁx of µ y ( P al ( v 2 )). S o again we ﬁ nd th at µ ˘ w 1 ( P al ( v 1 )) is a shorter qu asip eriod than µ ˘ w 2 ( P al ( v 2 )). No w we come to the case w hen v 2 = α for a letter α . If x contains a letter diﬀerent from α , then P al ( v 1 ) is a proper preﬁx of µ v 1 ( µ x ( α )) = µ y ( α ) = µ y ( P al ( v 2 )). W e still conclude as pr eviously . Lastly , assu me th at x is a p ow er of α . Hyp othesis (H) imp lies that α is the ﬁ rst letter of v 1 and more precisely v 1 = αv ′ 1 with v ′ 1 an α -fr ee wo rd. Thus P al ( v 1 ) = µ α ( P al ( v ′ 1 )) α = µ α ( P al ( v ′ 1 ) α ) = µ α ( µ v ′ 1 ( α )) = µ αv ′ 1 ( α ) = µ v 1 ( x ) = µ y ( v 2 ). So in th is particular case µ ˘ w 1 ( P al ( v 1 )) = µ ˘ w 2 ( P al ( v 2 )), and once again µ ˘ w 1 ( P al ( v 1 )) is the smallest qu asip eriod . Remark 4.24. Th eorem 4. 19 shows in particular that if an epistur mian w ord t is directed by a spinned word ˘ ∆ with all spins ultimately L , th en t has inﬁ n itely many quasip erio ds since th ere are inﬁnitely m any facto rizations of ˘ ∆ into the give n f orm (i.e., an y p ositive shift of an epistandard wo rd is quasip erio dic). Let us demonstrate Theorem 4.19 with s ome examples. First we pr o vide an example of an episturmian word ha ving inﬁnitely many qu asip eriod s, bu t which is not epistandard and all of its directiv e w ords are wavy (recall that a wa vy word is a spinned inﬁnite word co ntai ning inﬁnitely many L -spin ned letters and inﬁnitely many R -sp inned letters). Example 4.25. Consid er the episturm ian word t with normalized directiv e w ord ˘ ∆ = abc ( ¯ ba ) ω . F rom Theorem 3.2, we observe that t has inﬁ nitely many directive words: ˘ ∆ = abc ( ¯ ba ) ω and ˘ ∆ i = a ¯ b ¯ c ( ¯ b ¯ a ) i ba ( ¯ ba ) ω for each i ≥ 0 , all of which are wa vy . Hence, by T heorem 4.19, the set of quasip eriods of t consists of P al ( abc ) = abacaba and, for eac h i ≥ 0, the words: µ ˘ w ( bab ) and µ ˘ w ( bab ) a where ˘ w = a ¯ b ¯ c ( ¯ b ¯ a ) i . Note that P al ( ba ) = bab and S − 1 ˘ w P al ( w ) = ( P al ( w ) a − 1 ) − 1 P al ( w ) = a . Next we give some examples of quasip erio dic episturmian words having only ﬁ nitely many quasip eriod s. Example 4.26. C onsider the quasip erio dic epistu r mian word µ adbc ¯ d ( a r ) where r is the T r ib onacci wo rd; it has only t wo directive w ords: adbc ¯ d ( a ¯ b ¯ c ) ω and a ¯ d ¯ b ¯ cd ( a ¯ b ¯ c ) ω (the ﬁrst one being its normalized d irectiv e word). So we see th at t has only one quasip erio d, namely P al ( adbc ) = adabadacadabada . Similarly , the ep istu rmian word µ cbaa ( c r ), which is directed by exactly three dif- ferent spin ned inﬁnite w ords ( cbaa (¯ a ¯ bc ) ω , cba ¯ aa ( ¯ bc ¯ a ) ω , and cb ¯ a aa ( ¯ bc ¯ a ) ω ), has only t wo qu asip eriods : P al ( cba ) = cbcacbc and P al ( cbaa ) = cbcacbcacbc . W e can also construct episturmian words having exactly k quasiperio ds for any ﬁxed integer k ≥ 1, as s hown by the follo wing example. Example 4.27. Cons ider the episturmian word with normalized dir ectiv e word ˘ ∆ = abc ¯ a ( b ¯ c ¯ a ) ω . By Th eorem 3.2 , t h as exactly two directive words: ˘ ∆ and ¯ a ¯ b ¯ ca ( b ¯ c ¯ a ) ω . Hence, by Theorem 4.19, t has only one quasip erio d, namely P al ( abc ). No w, to construct an episturm ian wo rd having exactly k quasip eriod s for a ﬁxed integ er k ≥ 1, w e consider ˘ w ˘ ∆ = ˘ w abc ¯ a ( b ¯ c ¯ a ) ω where ˘ w is a spinned ve rsion of any L -spin ned word w with Alph( w ) ∩ { a, b, c } = ∅ and | P al ( w ) | = k − 1. Th en, by 18 Theorem 4.19, the set of quasip eriod s of the episturmian word directed by ˘ w ˘ ∆ consists of the k wo rds: µ ˘ w ( P al ( abc )) p where p is a preﬁx of P al ( w ). F or example, d k − 1 ˘ ∆ = d k − 1 abc ¯ a ( b ¯ c ¯ a ) ω directs an episturmian wo rd with exactly k quasip erio ds: µ k − 1 d ( P al ( abc )) p = d k − 1 ad k − 1 bd k − 1 ad k − 1 cd k − 1 ad k − 1 bd k − 1 ap w here p is a p reﬁx of d k − 1 . 4.7 Characterizations of quasip erio dic episturmian w ords F rom T h eorem 4.19, we im m ediately obtain the follo wing charac terizatio n of q u asip eriod ic epistur - mian words. Theorem 4.28. An e pisturmian wor d is quasip erio dic if and only if ther e exists a spinne d wor d ˘ w , an L - spinne d wor d v and a spinne d version ˘ y of an L -spinne d wor d y such that t is dir e cte d by ˘ w v ˘ y with Alph( v ) = Alph( v y ) . The abov e c haracterizatio n is cle arly not useful when one wa nts to decide w h ether or not a giv en epistur mian word is quasip erio dic. In th is regard, our normalized directive word pla ys an imp ortant r ole as it pro vides a more eﬀective wa y to decide. Theorem 4.29. A n episturmian wor d t is quasip erio dic i f and only if the (unique) normalize d dir e ctive wor d of t takes the form ˘ w av 1 ¯ av 2 ¯ a · · · v k ¯ av ˘ y for some L -spinne d letter a , spinne d wor d ˘ w , a -fr e e L - spinne d wor ds v , v 1 , . . . , v k ( k ≥ 0 ), and a spinne d version ˘ y of an L - spinne d wor d y such that Alph( av ) = Alph( av y ) . Pr o of. Firs t assume that an episturmian word t is directed by ˘ w av 1 ¯ av 2 ¯ a · · · v k ¯ av ˘ y (as in the hy- p otheses). Th en by Theorem 3.1, t is also dir ected by ˘ w ¯ a ¯ v 1 ¯ a ¯ v 2 ¯ a · · · ¯ v k av ˘ y . By Theorem 4.28, t is quasip eriod ic. No w assume that t is quasiperio dic. By Theorem 4.28, one of its directiv e words is ˘ w 1 v ′ ˘ y for spinned words ˘ w 1 , ˘ y and an L -spinned wo rd v ′ with Alph( v ′ ) = Alph( v ′ y ). If v ′ con tains only one le tter a , ˘ y = a ω and t he normalized directive word of t ends with aa ω , then the condition is ve riﬁed. No w assume that v ′ con tains at least t w o (diﬀeren t) letters: let u s write v ′ = av . Without loss of generality , we can assume that ˘ y and ˘ w 1 are normalized. If ˘ w 1 has n o su ﬃx in ¯ a ¯ A ∗ , then ˘ w 1 av y is normalized and of the requir ed form (with k = 0). O therwise ˘ w 1 = ˘ w ¯ a ¯ v 1 ¯ a ¯ v 2 ¯ a · · · ¯ v k for some spinned w ords ˘ w , ˘ y and some a -free L -spinned wo rds v 1 , . . . , v k ( k ≥ 0) such that ˘ w h as no suﬃx in ¯ a ¯ A ∗ . Then the normalized d irectiv e word of t is ˘ w av 1 ¯ av 2 ¯ a · · · v k ¯ av ˘ y . Example 4.30. The epistur m ian w ords with directive w ords ( a ¯ b ¯ c ) ω , ( a ¯ bc ¯ ab ¯ c ) ω or ( a ¯ b ¯ aca ¯ a ¯ bcbc ¯ b ) ω are not quasip erio dic wh ereas the one w ith normalized directive word ab ¯ abc ¯ c (¯ abc ) ω is quasip erio dic (it is also directed by ¯ a ¯ babc ¯ c (¯ abc ) ω ). Remark 4.31. It f ollows from Th eorem 4.28 that any qu asip eriod ic epistur m ian wo rd has at least t wo directiv e words. Indeed, if t is a qu asip eriod ic episturmian word, then t h as a directive word of the form ˘ ∆ = ˘ w v ˘ y wh ere the words ˘ w , ˘ y are spin n ed versions of some L -spinned words w , y and v is an L -spinned word such that Alph( v ) = Alph( v y ). If ˘ y (and h ence ˘ ∆) has all spins ultimately L , then item 2 in part iii ) of Theorem 3.2 s hows that t also has a wa v y directiv e wo rd. Now sup p ose ˘ y do es not hav e all spins ultimately L . Then ˘ y must b e w a vy , and h ence con tains in ﬁnitely many R -spinn ed letters. C ho ose x to b e the left-most R -spinned letter in ˘ y . Then ˘ y b egins with u ¯ x for some L -spinned wo rd u (possibly empt y). Hence, since x o ccurs in v , we see that ˘ ∆ = ˘ w v ˘ y con tains the factor v u ¯ x = v ′ xv ′′ u ¯ x . Thus ˘ ∆ does not satisfy the conditions of Theorem 3.7 as it con tains a factor in x A ∗ ¯ x , and therefore t do es not hav e a un iqu e directiv e w ord. Moreo v er, we easily d ed uce from Th eorems 4.28 and 3.7 that any episturmian word ha ving a uniqu e dir ective wo rd is necessarily non-quasip eriod ic. 19 In view of the abov e remark, one might susp ect th at an epistur mian wo rd is non-quasiperio dic if and only if it has a unique directiv e word. But this is not true. F or example, b oth ba ( ¯ bc ¯ a ) ω and ¯ b ¯ ab ( c ¯ a ¯ b ) ω direct the same n on-quasip erio d ic epistur mian word t by Theorems 3.2 and 4.28. T hese t wo spinned inﬁn ite words are the only directive words for t , wh ic h might lead one to guess th at an episturmian word is non-quasip erio dic if and only if it has ﬁnitely m any directive w ords. But again, this is not true. F or example, as stated in E xample 4.26, adbc ¯ d ( a ¯ b ¯ c ) ω and a ¯ d ¯ b ¯ cd ( a ¯ b ¯ c ) ω are the only t wo d irectiv e w ords of the quasip erio dic epistur mian word µ adbc ¯ d ( a r ) where r is the T rib onacci w ord. Moreo v er there exist non-quasiperio dic epistur mian w ords, such as the one directed by ( ab ¯ b ¯ c ) ω , that hav e inﬁnitely many directive words (the w ords ( ab ¯ b ¯ c ) n a ¯ b ¯ c ( ab ¯ b ¯ c ) ω are pairwise diﬀeren t directiv e wo rds). Nev ertheless, in the Sturmian case we h a ve : Prop osition 4.32. A Sturmian wor d is non-quasip erio dic if and only if it h as a unique dir e ctive wor d. Pr o of. Firs t le t us su p p ose b y wa y of con tradiction that t is a non-qu asip er io dic Sturmian w ord, but has more than one directiv e w ord. Then, by Proposition 3.8, t has inﬁn itely many directiv e wo rds. Moreo ver, as t is ap erio dic, all of the directiv e words of t are spinned versions of the same ∆ ∈ { a, b } ω \ ( A ∗ a ω ∪ A ∗ b ω ) by F act 3.3 and none of th ese directive words are regular wa vy . Hence the normalized directive word of s con tains ab or ba . But then t is qu asip eriodic b y Th eorem 4.28 , a con tradictio n. Conv ersely , by Prop osition 3.8, if a Stur m ian word t has a uniqu e directiv e word, then its (normalized) directiv e word is r egular wa vy . Suc h a word clearly d oes not fulﬁll Theorem 4.28 and so t is not quasip erio dic. Let us recall th at a Sturm ian word is n on-quasip eriod ic if and only if it is a L yndon wo rd [27 ]. The generaliz ation of this asp ect will b e discussed in Section 6. 5 Quasip erio dicit y and episturmian morphisms In this section w e dr a w connections b etw een the results o f the pr evious section and str ongly quasip erio dic morphisms. As in [29], a morp hism f on A is called str ongly quasip erio dic (on A ) if for any (p ossibly non-quasip erio dic) inﬁnite word w , f ( w ) is quasip erio dic. 5.1 Strongly quasip erio dic epistanda rd morphisms Quasip erio dicit y of epistandard w ords can also b e explained by the strong quasip erio dicit y of epi- standard m orp hisms. Theorem 5.1. L et v b e a wor d over an alphab et A c onta ining at le ast two letters. The epistandar d morphism µ v is str ongly quasip erio dic if and only if Alph( v ) = A . M or e over P al ( v ) is a quasip erio d of µ v ( w ) for any inﬁnite wor d w . T o pro v e this result, w hich is a direct consequence of Lemma 5.2 (b elo w), w e need to consider inﬁnite words co v ered by seve ral words. W e say that a set X of words cov ers an inﬁ nite word w if and only if there exist tw o sequences of w ords ( p n ) n ≥ 0 and ( z n ) n ≥ 0 such that, for all n ≥ 0, p n z n is a preﬁx of w , z n ∈ X , p 0 = ε and | p n | < | p n +1 | ≤ | p n z n | . The last inequalitie s mean that p n is a preﬁx of p n +1 which is a preﬁx of p n z n , itself a preﬁx of w . Once again the sequence ( p n z n ) n ≥ 0 is called a cov ering sequence of pr eﬁ xes of the word w . O bserve that a w ord is co v ered by X if and only if it is cov ered by X ∪ { ε } . Lemma 5.2. F or w an inﬁnite wor d over A , y a letter, X a subset of A and u a wor d, if w is c over e d by { u } ∪ { ux | x ∈ X ∪ { y }} then L y ( w ) is c over e d by { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X \ { y }} . 20 Pr o of. I f y ∈ X , then L y ( w ) is cov ered by { L y ( u ) } ∪ { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X \ { y }} , an d eac h o ccurrence of L y ( u ) is follow ed by the lett er y . So L y ( w ) is co v ered by { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X \ { y }} . If y 6∈ X , thus X = X \ { y } and then L y ( w ) is cov er ed by { L y ( u ) } ∪ { L y ( u ) y x | x ∈ X } , an d similarly eac h o ccurren ce of L y ( u ) is follow ed by the letter y . So L y ( w ) is cov ered by { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X } . Pr o of of The or em 5.1. Let v = v 1 · · · v | w | b e a word (with eac h v i a lette r) suc h that Alph( v ) = A . Let u | v | = ε and, for any i from | v | to 1, let u i − 1 = L v i ( u i ) v i : observe that u i − 1 = P al ( v i · · · v | v | ). It is immediate that any inﬁ nite word w is co ve red by the set A , wh ic h can b e expressed as A = { ε } ∪ { εx | x ∈ A } = { u | v | } ∪ { u | v | x | x ∈ A} . By induction u sing Lemma 5.2, we can state that, for 1 ≤ i ≤ | v | , µ v i ··· v | v | ( w ) is cov ered by { u i − 1 } ∪ { u i − 1 x | x ∈ A \ { v i , . . . , v | v | }} . S o, since u 0 = P al ( v ), µ v ( w ) is co v ered b y { P al ( v ) } ∪ { P al ( v ) x | x ∈ A \ { v 1 , . . . , v | v | }} . Therefore, since Alph( v ) = A , µ v ( w ) is co ve red by P al ( v ), that is, w is P al ( v )-quasip eriod ic. Hence µ v is strongly quasip erio dic. No w let us consid er th e case where Alph( v ) 6 = A . First supp ose that v = ε . Th en, since A con tains at least t wo letters (and so there exists at le ast one non-quasiperio dic w ord ov er A ), th e morphism µ v , whic h is the id ent it y , is not s trongly quasip erio dic. Assume now that v 6 = ε . Let a b e a letter in A \ Alph( v ) and let b b e a letter in Alph( v ). The word µ v ( ab ω ) cont ains only one occurr ence of the letter a , and so it is n on -qu asip eriodic. Thus the m orphism µ v is not strongly quasip eriod ic. Remark 5.3. Note th at any inﬁnite word ∆ can b e wr itten ∆ = v y with Alph( v ) = Alp h ( v y ). So if t (resp. t ′ ) is the epistandard word d ir ected by ∆ (resp. y ), then t = µ v ( t ′ ) and t is qu asip erio dic by Theorem 5.1. The same approac h allows u s to sh o w that if an episturmian wo rd is directed by a spinned inﬁ nite word with all spins ultimately L , th en it is quasiperio dic, as deduced previously (see Remark 4.12). Not e also th at C orollary 4.1 3 is a direct consequence of Theorem 5.1. 5.2 Useful lemmas W e now introduce useful material for a second pro of of Theorem 4.19 using morphisms (see S ec- tion 5.3). Th e next results (that maybe some readers will r ead when necessary to und erstand the pro of of Theorem 4.19) s how how to obtain some quasip eriod s (or a co v ering set) of a word of the form L x ( w ) or R x ( w ) for an in ﬁnite word w (not necessarily episturmian). Lemma 5.4. (conv erse of Lemma 5.2) F or an inﬁnite wor d w over A , y a letter, X a sub se t of A and u a wor d, if L y ( w ) is c over e d by { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X \ { y }} , then w is c over e d by { u } ∪ { ux | x ∈ X ∪ { y }} . Pr o of. By h yp othesis, there exist ( p n ) n ≥ 0 , ( u n ) n ≥ 0 such that p 0 = ε , u n ∈ { L y ( u ) y } ∪ { L y ( u ) y x | x ∈ X \ { y }} and for all n ≥ 0, both | p n | < | p n +1 | ≤ | p n u n | and p n u n is a preﬁx of w . F or all n ≥ 0, p n L y ( u ) y is a preﬁx of L y ( w ) so there exists p ′ n such that p n = L y ( p ′ n ). W e consider the following t wo complementary cases: 1. Case u n = L y ( u ) y . The word p n L y ( u ) y = L y ( p ′ n u ) y is a preﬁx of L y ( w ) so p ′ n is a pr eﬁx of w . F or the preﬁxes p n +1 and p n u n of w , we h a ve | p n +1 | ≤ | p n u n | , thus p n +1 is a preﬁx of p n u n . If | p n +1 | < | p n u n | , let u ′ n = u : p n +1 = L y ( p ′ n +1 ) is a p reﬁx of p n L y ( u ) = L y ( p ′ n u ). Moreo v er p n +1 y and p n L y ( u ) y are preﬁxes of L y ( w ). So p ′ n +1 is a preﬁx of p ′ n u which itself is a p reﬁx of w . If | p n +1 | = | p n u n | , then p n +1 = p n u n and from p n +1 u n +1 preﬁx of w , we kn o w th at p n +1 y is a preﬁx of w , hence p ′ n uy = p ′ n +1 is a preﬁx of w . I n this case let u ′ n = uy . 21 2. Case u n = L y ( u ) y x for some x ∈ X \ { y } . In this case (as previously), we can see that with u ′ n = ux , p ′ n +1 is a preﬁx of p ′ n u ′ n itself a pr eﬁx of w . W e h a v e pr o ve d that the sequ ence ( p ′ n u ′ n ) n ≥ 0 is a co v ering sequence of p reﬁxes of w . Thus the wo rd w is co v ered b y { u } ∪ { ux | x ∈ X ∪ { y }} . Lemma 5.5. (generalization of [29, Lem. 5.5]) L et w b e an inﬁnite wor d starting with a letter x . F or any letter y 6 = x , the wor d R y ( w ) is quasip erio dic if and only if w is qu asip e rio dic. Mor e over if q is a q uasip erio d of R y ( w ) th en two c ases ar e p ossible: 1. q = R y ( q ′ ) with q ′ a quasip erio d of w ; 2. q y = R y ( q ′ z ) with z 6 = a a letter such tha t b oth q ′ and q ′ z ar e quasip erio ds of w . Pr o of. Ass ume ﬁrst that w is quasiperio dic. By F act 4.1, R y ( w ) is qu asip eriod ic. More precisely , if v is a quasip erio d of w , then R y ( v ) is a qu asip erio d of R y ( w ). Assume no w that R y ( w ) is quasip eriod ic and le t q b e one of its quasiperio ds. By hypothesis, we kn o w that q sta rts w ith x (as w ). A ﬁrst case is that q = R y ( v ) (for a word v ) w hich is equiv alen t to the fact that q ends w ith th e letter y . Let ( p n ) n ≥ 0 b e a sequence of words such that ( p n q ) n ≥ 0 is a co ve ring sequence of pr eﬁxes of R y ( w ). Let n ≥ 0. F rom the f act that q starts with x , we d educe that there exists a wo rd p ′ n such that p n = R y ( p ′ n ). Th en we can see that p ′ n v is a preﬁx of w . Moreov er the inequality | p n | < | p n +1 | ≤ | p n q | implies | p ′ n | < | p ′ n +1 | ≤ | p ′ n v | . So ( p ′ n v ) n ≥ 0 is a co v ering sequence o f p reﬁxes of w : v is a quasiperio d of w . No w assume that q ends with a lette r z diﬀeren t from y . Th en q = R y ( v ) z for a word v and eac h occur r ence of q is follo wed b y the letter y . Let ( p n ) n ≥ 0 b e a sequence of words suc h that ( p n q ) n ≥ 0 is a co ve ring sequence of pr eﬁxes of R y ( w ). Let n ≥ 0. F rom the fact that q starts with x , we deduce that there exists a word p ′ n such that p n = R y ( p ′ n ). Then w e can see th at p ′ n v is a preﬁ x of w . No w since q does not end with y , the inequality | p n | < | p n +1 | ≤ | p n q | is actually | p n | < | p n +1 | ≤ | p n R y ( v ) | . And so ( p ′ n v ) n ≥ 0 is a cov erin g sequence of preﬁxes of w ; thus v is a quasip eriod of w . Since each o ccurrence of R y ( v ) is follo wed by y , ( p ′ n v z ) n ≥ 0 is also a cov ering sequence of preﬁ x es of w ; whence v z is also a qu asip eriod of w . Lemma 5.6. L et x, y b e two diﬀer ent letters, let w b e an inﬁnite wor d, and let u b e a ﬁnite wo r d such that L x ( w ) is uy -quasip erio dic. Then ther e e xists a wor d v such that w is v y -qu asip e rio dic, uy = L x ( v y ) , and | uy | > | v y | . Pr o of. L et x, y , u, w b e as in th e lemma. Since uy is a preﬁx of L x ( w ) and since x 6 = y , there exists a w ord v suc h that uy = L x ( v y ). W e hav e | uy | > | v y | . Let ( p n ) n ≥ 0 b e suc h that ( p n uy ) n ≥ 0 is a co v ering sequ ence of preﬁxes of L x ( w ). Since uy is a preﬁx of L x ( w ), x is the ﬁrst letter of u , and so from p n x preﬁx of L x ( w ), there exists for any n ≥ 0 a word p ′ n such that p n = L x ( p ′ n ). F rom the preﬁx L x ( p ′ n v y ) of L x ( w ), we deduce that p ′ n v y is a preﬁx of w . Th e inequality | p n | < | p n +1 | ≤ | p n uy | , that is | L x ( p ′ n ) | < | L x ( p ′ n +1 ) | ≤ | L x ( p ′ n v y ) | , implies th at | p ′ n | < | p ′ n +1 | ≤ | p ′ n v y | . Hence v y co vers w . Remark 5.7 . Lemmas 5.5 and 5.6 are not true wh en x = y . F or instance ab ω is not quasip erio dic whereas R a ( ab ω ) = a ( ba ) ω is quasip erio dic. Mo reo ve r the word L a ( b ω ) = ( ab ) ω is aba -quasip erio dic whereas b ω has n o quasiper io d ending with a . Lemma 5.8. Su pp ose t is an episturmian wor d starting with a letter z . L et x b e a letter diﬀer ent fr om z and let u b e a non-empty wor d. Then ther e e xists a set B of letters suc h that the wor d R x ( t ) is c over e d by { u } ∪ { u }B if and only if this wo r d is c over e d by { u, ux } . 22 Pr o of. T he if part is imm ed iate. Assume that R x ( t ) is co v ered by { u } ∪ { u }B . There is n othing to prov e if B ⊆ { x } ; h en ce we assu me that B contains a letter y diﬀerent from x . Since t starts with z and z 6 = x , the word R x ( t ) and its p reﬁx u start with z . If the wo rd uy is a factor of R x ( t ), the wo rd u ends with the letter x . Let ( p n ) n ≥ 0 and ( u n ) n ≥ 0 such that ( p n u n ) n ≥ 0 is a co v ering sequ ence of pr eﬁxes of R x ( t ) and for all n ≥ 0, u n ∈ { u } ∪ { u }B . Let n be a ny in teger such th at u n = uy . Since y 6 = x and z 6 = x , th e word y z is not a factor of R x ( t ). S ince | p n +1 | ≤ | p n u n | , since p n +1 and p n u n are preﬁxes of R x ( t ), and since u n +1 starts with z , necessarily | p n +1 | < | p n u n | = | p n uy | . W e deduce f rom w h at pr ecedes that R x ( t ) is cov ered by { x } ∪ { u } ( B \ { y } ). Hence considering successiv ely all letters of B diﬀeren t from x , it follo ws that R x ( t ) is cov er ed by { u, ux } . Lemma 5.9. L et t b e an episturmian wor d start ing with a letter z . L et x b e a letter diﬀer ent fr om z and let q b e a wor d. If the wor d R x ( t ) is c over e d by { q , q x } th en L x ( t ) is c over e d by xq . Pr o of. I f q is empty , the r esult is immediate since in this ca se t = x ω . If q ends w ith a letter diﬀerent from x , then each o ccurrence of q is follo we d by an x and so R x ( t ) is cov ered by q x . Hence we can assum e without loss of generality that q ends w ith x . Then q = R x ( v ) and q x = R x ( v x ) for some word v . By the same technique used in the p ro of of Lemma 5.5, we can see that t is cov ered by { v , v x } . Th is implies th at L x ( t ) is cov ered by { L x ( v ) , L x ( v ) x } with all the occur rences of L x ( v ) follo w ed by an x . S o L x ( t ) is cov ered by L x ( v ) x . It is w ell-kno wn that L x ( v ) x = xR x ( v ) = xq . Lemma 5.10. Supp ose t is an inﬁnite wor d (not ne c essarily episturmian) c over e d by the set { q , q a } and by the set { q , q c } for some wor d q and two diﬀer ent letters a and c . Then t is q - quasip erio dic. Pr o of. L et ( p n ) n ≥ 0 b e preﬁxes of t and ( q n ) n ≥ 0 b e w ords b elonging to { q , q c } suc h that ( p n q n ) n ≥ 0 is a co v ering sequence o f t . Moreov er let us assum e that there is no wo rd p except the elemen ts of ( p n ) n ≥ 0 such that pq is a preﬁx of t . Let n b e an y in teger such that q n = q c . Since t is also co v ered by { q , q a } and since all th e w ords p such that pq is a preﬁx of t b elong to { p n | n ≥ 0 } , w e necessarily hav e | p n +1 | ≤ | pq | (that is | p n +1 | 6 = | p n q c | ). Th e sequence ( p n q ) is a co vering sequence of t ; hence t is q -quasip erio dic. 5.3 A second pro of of Theorem 4.19 W e now giv e a second pro of of T heorem 4.19 wh ic h pr ovides further insight into the connection b et wee n quasiperio dicity and morphisms. F or any s pinned w ord ˘ w , let us denote b y q ˘ w the word S − 1 ˘ w P al ( w ) app earing in Theorem 4.19. Then, for any spinned word ˘ v and letter a , w e hav e: • q ε = ε ; • q a ˘ v = L a ( q ˘ v ) a ; • q ¯ a ˘ v = R a ( q ˘ v ). Indeed by form ula (4.3 ), we hav e S a ˘ v = L a ( S ˘ v ) and S ¯ a ˘ v = L a ( S ˘ v ) a = aR a ( S ˘ v ). Hence q a ˘ v = S − 1 a ˘ v L a ( P al ( v )) a = ( L a ( S ˘ v )) − 1 L a ( P al ( v )) a = L a ( S − 1 ˘ v P al ( v )) a = L a ( q ˘ v ) a and, sin ce L a ( P al ( v )) a = aR a ( P al ( v )), q ¯ a ˘ v = S − 1 ¯ a ˘ v ( aR a ( P al ( v ))) = ( aR a ( S ˘ v )) − 1 ( aR a ( P al ( ˘ v ))) = R a ( S − 1 ˘ v ( P al ( ˘ v )). These form ulae, which deﬁne r ecursively th e word q ˘ w , will b e helpful in the follo wing p roof. First assu me that t is an episturmian word d irected by ˘ w v ˘ y where ˘ w is a spinned word, ˘ y is a sp inned version of an L -spin n ed inﬁn ite word y and v is an L -sp inned word suc h that Alph( v ) = Alph( v y ). Then, by Theorem 5.1, P al ( v ) is a quasip erio d of µ v ( t ′ ) where t ′ is the epistu r mian word directed by ˘ y . No w we p rov e by indu ction on | ˘ w | that µ ˘ w ( P al ( v )) p is a quasip eriod of t for an y preﬁx p of q ˘ w . More precisely we p rov e th at µ ˘ w ( P al ( v )) is a quasip eriod of t = µ ˘ w ( µ v ( t ′ )) (this is also a consequence of F act 4.1) and eac h occurren ce of µ ˘ w ( P al ( v )) in t is follo we d by the word q ˘ w (which is a direct consequen ce of the ab ov e indu ctiv e formula e). 23 Observe that the pr evious fact ob viously holds if ˘ w = ε . When it holds, let ( p n ) n ≥ 0 b e a se- quence of preﬁx es such that | p n | ≤ | p n +1 | ≤ | p n µ ˘ w ( P al ( v )) | and p n µ ˘ w ( P al ( v )) q ˘ w is a preﬁx of t . Then L a ( p n µ ˘ w ( P al ( v )) q ˘ w ) = L a ( p n ) µ a ˘ w ( P al ( v )) L a ( q ˘ w ) a nd is fol lo we d as a preﬁx of L a ( t ) b y a . Note that the word L a ( p n ) µ a ˘ w ( P al ( v )) L a ( q ˘ w ) a = L a ( p n ) µ a ˘ w ( P al ( v )) q a ˘ w is a p reﬁx of L a ( t ). Moreo v er we ca n v erify that | L a ( p n ) | ≤ | L a ( p n +1 ) | ≤ | L a ( p n ) µ a ˘ w ( P al ( v )) | since p n is a p r eﬁx of p n +1 itself a pr eﬁx of p n µ ˘ w ( P al ( v )). Similarly R a ( p n µ ˘ w ( P al ( v )) q ˘ w ) = R a ( p n ) µ ¯ a ˘ w ( P al ( v )) R a ( q ˘ w ) = R a ( p n ) µ ¯ a ˘ w ( P al ( v )) q ¯ a ˘ w is a preﬁx of R a ( t ) and | R a ( p n ) | ≤ | R a ( p n +1 ) | ≤ | R a ( p n ) µ ¯ a ˘ w ( P al ( v )) | . T o end the p ro of of Theorem 4.19, w e pro ve by induction on | q | that if q is the quasip eriod of an episturm ian word t then (at least) one directive word of t can b e wr itten ∆ = ˘ w v ˘ y where ˘ w is a spinned word, ˘ y is a spinn ed version of an L -spin ned inﬁ nite word y and v is an L -spin n ed word s uch that Alph( v ) = Alph( v y ). Moreo v er we hav e q = µ ˘ w ( P al ( v )) p with p a pr eﬁx of q ˘ w = S − 1 ˘ w P al ( w ). W e observe that | q | ≥ 1 since q cov ers the inﬁnite word t . When | q | = 1, for a le tter a , q = a a nd the word t = a ω is directed by a ω which is o f the form ˘ w v a ω with ˘ w = ε and v = a . Then q ˘ w = ε , and taking p = ε , q = µ ˘ w ( P al ( v )) p . Assume from no w on that | q | ≥ 2. Without lo ss of generalit y , we assume t hat ∆ = ˘ x 1 ˘ x 2 · · · is the n ormalized d irectiv e word of t , and let ( t ( n ) ) n ≥ 0 b e the inﬁn ite sequence of episturm ian words associated to t and ∆ in Th eorem 2.1. Let us ﬁr st consider: Case t = L a ( t (1) ) (that is, ˘ x 1 = a f or a letter a ). Assume that q do es n ot end with the letter a . By Lemma 5.6, q = L a ( q 1 ) for a quasip erio d q 1 of t (1) such that | q 1 | < | q | . By the ind uction hyp othesis, t (1) has a directive word of the form ˘ w v ˘ y with ˘ w , v , ˘ y as in the theorem, and q 1 = µ ˘ w ( P al ( v )) p for a pr eﬁx p of q ˘ w . W e hav e q = L a ( µ ˘ w ( P al ( v )) p ) = µ a ˘ w ( P al ( v )) L a ( p ). Observing that t is dir ected by ( a ˘ w ) v ˘ y and that L a ( p ) is a preﬁx of L a ( q ˘ w ) a = q a ˘ w , we get the result for q and t . No w assume that q ends with the letter a , that is q = L a ( q 1 ) a for a word q 1 . By Lemma 5.4, t (1) is cov ered by the set { q 1 , q 1 a } . Po ssibly q 1 a is a qu asip eriod of t (1) . In th is case, q = L a ( q 1 a ). Ob serve that | q | = | L a ( q 1 a ) | ≥ | q 1 a | . The equality holds only wh en a is the only lette r occurring in q 1 so in q . I n this case, taking ˘ w = a | q |− 1 and v = a , the word t is directed by ˘ wv a ω and w e hav e q = aa | q |− 1 = P al ( a ) q a | q |− 1 = µ a | q |− 1 ( P al ( a )) q ˘ w = µ ˘ w ( P al ( v )) q ˘ w : th e induction result holds here. When | q | = | L a ( q 1 a ) | > | q 1 a | , the proof ends, using the in duction hyp othesis, as in th e previous case where q did not end b y a (the only diﬀerence is that q en d s with the w ord L a ( p ) a but L a ( p ) a is still a p reﬁx of q a ˘ w ). It is also p ossible that q 1 is a qu asip erio d of t (1) , but in this case we c an once again conclude using the indu ction hypothesis. No w we assume that neither q 1 nor q 1 a is a quasip eriod of t (1) . T his implies in p articular that t (1) has a factor q 1 b for a letter b diﬀerent from a . So the follo wing prop erty holds for n = 1 (and a 1 = a ): Pr op(n) : t = L a 1 L a 2 · · · L a n ( t ( n ) ) (that is, ˘ x i = a i , for 1 ≤ i ≤ n ), t ( n ) is co ve red by the set { q n } ∪ { q n a i | i = 1 , . . . , n } with q 0 = q , and for i = 1 , . . . , n , q i − 1 = L a i ( q i ) a i . Moreo v er q n a and q n b are b oth factors of t ( n ) . Let u s assume that Prop erty Pr op(n) holds for some n ≥ 1 with q n 6 = ε and t ( n ) = L a n +1 ( t ( n +1) ) (that is ˘ x n +1 = a n +1 ). F r om a 6 = b (so a 6 = a n +1 or b 6 = a n +1 ), we ded uce that q n c is a factor of t ( n ) for a letter c 6 = a n +1 and so q n must end with the letter a n +1 , that is, q n = L a n +1 ( q n +1 ) a n +1 for some word q n +1 . By Lemma 5.4 , q n +1 is cov ered by the set { q n +1 } ∪ { q n +1 a i | i = 1 , . . . , n, n + 1 } . Moreo v er it is quite immediate that q n +1 a and q n +1 b are b oth facto rs of t ( n +1) . Hence P r op(n+1) holds. Observing that for any i ≥ 1 we hav e | q i +1 | < | q i | , we deduce that one of the two follo wing cases holds: 24 1. Ther e exists an inte ger n ≥ 1 such that Pr op erty Pr op(n) h olds with q n = ε . In this case, we ve rify that q = P al ( a 1 · · · a n ). Let ˘ w = ε , v = a 1 · · · a n and let ˘ y b e an y dir ective word of the episturmian wo rd t ( n ) . Then t is directed by ˘ w v ˘ y and q = µ ˘ w ( P al ( v )). Since t ( n ) is co v ered by the letters of v , we dedu ce that Alph( v ) = Alph( v y ). Hence the indu ction result holds. 2. Ther e exists an inte ger n ≥ 1 and a letter c such that Pr op erty Pr op(n) holds with q n 6 = ε and t ( n ) = R c ( t ( n +1) ) (t hat is ˘ x n +1 = ¯ c ). By Theorem 3.5, since ∆ is normalized, the word t ( n +1) do es not start w ith the letter c (otherwise the inﬁnite word ˘ x n +1 ˘ x n +2 · · · starts with a facto r in ¯ c ¯ A c ). Hence the ﬁrst letter of q n is not c whereas its la st lette r is c sin ce a 6 = b and both q n a and q n b are fact ors of R c ( t ( n +1) ). By Lemma 5 .8, we deduce that R c ( t ( n +1) ) is co v ered by { q n , q n c } . Sin ce for i = 1 , . . . , n , q i − 1 = L a i ( q i ) a i , w e can ind u ctiv ely verify that the words t ( i ) ( i = n, . . . , 1) are cov ered b y { q i , q i c } . In p articular t (1) is co ve red by { q 1 , q 1 c } . By Lemma 5.10 , since we hav e assumed that t (1) is not q 1 -quasip eriod ic, w e hav e c = a . So R a ( t ( n +1) ) is cov ered by { q n , q n a } . By Lemma 5.9, L a ( t ( n +1) ) is aq n -quasip eriod ic. Observe that t = L a 1 L a 2 · · · L a n R a 1 ( t ( n +1) ) = R a 1 R a 2 · · · R a n L a ( t ( n +1) ), an d that | q n | < | q | . By induction hyp othesis, L a ( t ( n +1) ) is d ir ected by a word ˘ w ′ v ˘ y for a ﬁnite spin ned word ˘ w ′ , a ﬁn ite L -spinn ed word v and a spinned v ersion ˘ y of an inﬁnite L -spinned word y such that Alp h( v ) = Alph( v y ), and aq n = µ ˘ w ′ ( P al ( v )) p ′ with p ′ a p reﬁx of q ˘ w ′ . Thus we ha ve prov ed that the w ord t is directed by ˘ w v ˘ y wh ere ˘ w = ¯ a 1 ¯ a 2 · · · ¯ a n ˘ w ′ , and µ ¯ a 1 ¯ a 2 ··· ¯ a n ( q a n ) = µ ˘ w ( P al ( v )) p ′ . T o end the pro of of th e current case we hav e to prov e that q = µ ˘ w ( P al ( v )) p ′ . First assume that a i = a for s ome i b etw een 2 and n . W e kno w that the w ord t ( i ) is cov ered by { q i , q i a } and t ( i − 1) = L a i ( t ( i ) ) = L a ( t ( i ) ). Hence by Lemma 5.2, t ( i − 1) is co v ered by L a ( q i ) a = q i − 1 . Using F act 4.1, w e can deduce th at t (1) is q 1 a -quasip eriod ic, a con tradictio n. Hence for i = 2 , . . . , n , a i 6 = a . By in duction we can prov e t hat µ ¯ a i ¯ a i +1 ··· ¯ a n ( aq n ) = aq i − 1 for i = n + 1 , . . . , 2. Ind eed if µ ¯ a i +1 ··· ¯ a n ( aq n ) = aq i then µ ¯ a i ¯ a i +1 ··· ¯ a n ( aq n ) = µ ¯ a i ( aq i ) = R a i ( aq i ) = aa i R a i ( q i ) = aL a i ( q i ) a i = aq i − 1 . Now µ ¯ a 1 ¯ a 2 ··· ¯ a n ( aq n ) = R a 1 ( aq 1 ) = aR a ( q 1 ) = L a ( q 1 ) a = q . No w we come to: Case t = R a ( t (1) ) . By Theorem 3.5, sin ce ∆ is the normalized d irectiv e word of t , the word t d o es not start with th e letter a . By Lemma 5.5, the word t (1) is qu asip eriodic and more precisely there exists a quasip erio d q 1 of t (1) such that q = R a ( q 1 ) or q = R a ( q 1 ) z with z 6 = a : in this last case, q 1 z is also a quasip eriod of t (1) . Since q does not start with a , | q 1 | < | q | . By induction hypothesis, t (1) has a directed w ord of the form ˘ w v ˘ y with ˘ w , v , ˘ y as in the theorem, and q 1 = µ ˘ w ( P al ( v )) p for a preﬁ x p of q ˘ w . When q = R a ( q 1 ), since R a ( µ ˘ w ) = µ ¯ a ˘ w and R a ( p ) is a preﬁx of R a ( q ˘ w ) = q ¯ a ˘ w , q = µ ¯ a ˘ w ( P al ( v )) R a ( p ) and t (which is directed b y ¯ a ˘ w v ˘ y ) ve rify the ind uction r esu lt. When q = R a ( q 1 ) z with z 6 = a , sin ce q 1 z is also a quasip erio d of t (1) , we dedu ce that pz is a preﬁx of q ˘ w and so R a ( p ) z is a pr eﬁx of R a ( p ) z a a p reﬁx of q ¯ a ˘ w , whic h allo ws u s to conclude once aga in that q = µ ¯ a ˘ w ( P al ( v )) R a ( p ) z and t (which is directed by ¯ a ˘ w v ˘ y ) verify the in duction result. This ends the second proof of Th eorem 4.19. 5.4 Strongly quasip erio dic episturmian morphisms The aim of this section is to characterize all the epistur mian morphism s that are strongly quasip eri- od ic, i.e., the epistu rmian morph isms that map a ny inﬁnite word on to a quasip erio dic w ord. O ur c haracterizatio n (Theorem 5.14) generalizes Theorem 5.1 to all epistur mian morph isms. Looking at Theorem 5.1 and Th eorem 4.19, one might guess th at an episturmian morp hism µ ˘ u is strongly qu asip eriod ic if and only if ˘ u has a n L -spinned factor v . W e will see that it is not the case. In p articular, there exist R -spinned words ˘ u suc h that µ ˘ u is strongly quasip erio dic, as shown by the follo wing result. Lemma 5.11. L et ˘ u b e a ﬁnite spinne d wor d over A ∪ ¯ A and let a b e a letter in A . If ther e exist L -spinne d wor ds v , y and a spinne d wor d ˘ w such that ˘ u = ˘ w ¯ a ¯ v ¯ y and Alph( v ) = A \ { a } , then for any inﬁnite wor d t , µ ˘ u ( a t ) is quasip erio dic. 25 Let us m ent ion that for instance the word ¯ a ¯ b ¯ c ¯ a ¯ b veriﬁes the condition of the p revious lemma for eac h of the lette rs a , b and c , so that µ ¯ a ¯ b ¯ c ¯ a ¯ b is str on gly quasip eriod ic o v er { a, b, c } . Lemma 5.11 is a consequence of the following one. Lemma 5.12. Supp ose a is an L - spinne d letter, x is an L -spinne d wor d, v is an a -fr e e L -spinne d wor d, and w is an i nﬁnite wor d over A . Then the wor ds µ av ( xa ) , µ ¯ a ¯ v ( ax ) , and µ ¯ a ¯ v ( a w ) ar e P al ( av ) -quasip erio dic. Pr o of. (1) By Theorem 5 .1, the word µ av ( xa ω ) is P al ( av )-quasip erio dic where P al ( av ) = µ av ( a ) since P al ( av a ) = µ av ( a ) P al ( av ) = P al ( av ) P al ( av ) by form ulae (4.1) and (4.5). Consequently , since µ av ( xa ) ends with µ av ( a ), it is P al ( av )-quasip erio dic. (2) By formulae (4.3) and (4.2), µ ¯ a ¯ v ( xa ) = S − 1 ¯ a ¯ v µ av ( xa ) S ¯ a ¯ v and S ¯ a ¯ v = P al ( av ). So P al ( av ) µ ¯ a ¯ v ( ax ) = µ av ( ax ) µ av ( a ) = µ av ( axa ) = µ av ( a ) µ av ( xa ) = P al ( av ) µ av ( xa ), that is µ ¯ a ¯ v ( ax ) = µ av ( xa ). It follows from (1) that µ ¯ a ¯ v ( ax ) is P al ( av )-quasip erio dic. (3) F r om (2), we d educe that µ ¯ a ¯ v ( a w ) has inﬁnitely many quasip eriod ic preﬁxes with quasip erio d P al ( av ). Hence µ ¯ a ¯ v ( a w ) is P al ( av )-quasip erio dic. Pr o of of L emma 5.11. Let ˘ u , a , ˘ w , v , y b e as in the hypotheses of Lemma 5.11. Let t b e an inﬁn ite wo rd ov er A . The word µ ¯ y ( a t ) starts with a . By Lemma 5.12, µ ¯ a ¯ v ( µ ¯ y ( t )) is quasiperio dic and so µ ˘ u ( a t ) = µ ˘ w ¯ a ¯ v ¯ y ( a t ) is quasip erio dic. The following remark will be u seful several times (see the p roof of Theorem 4.29 for more details): Remark 5.13. An inﬁnite wo rd ov er A ∪ ¯ A h as a deco mp osition ˘ w v ˘ y with Alph( v ) = Alph( v ˘ y ) if and only if its norm alized decomp osition can b e written in the form ˘ w ′ av 1 ¯ av 2 · · · ¯ av k ˘ y ′ with Alph( av i ) = Alph( av i av i +1 · · · av k ˘ y ′ ) for some 1 ≤ i ≤ k . No w we state our characterizati on of strongly qu asip erio dic episturmian morphisms. Theorem 5.14. L et A b e an alphab et c ontaining at le ast th r e e letters. An episturmian morphism is str ongly quasip erio dic on A if and only if its normalize d dir e ctive wor d ˘ u v eriﬁes one of the fol lowing thr e e c onditions: i) ˘ u = ˘ w av 1 ¯ av 2 ¯ a · · · v k ¯ av ˘ y wher e a is a letter of A (with spin L ), ˘ w , ˘ y ar e spinne d wor ds, v 1 , . . . , v k ( k ≥ 0 ) ar e a -fr e e L -spinne d w or ds, a nd v is an L -spinne d wor d such tha t Alph ( av ) = Alph( av ˘ y ) . ii) F or any letter a in A , ˘ u = ˘ w ¯ a ¯ v ¯ y for some spinne d wo r d ˘ w and L -spinne d wor ds v , y such that Alph( v ) = A \ { a } . iii) ˘ u veriﬁes c ase ii ) for al l letters in A exc ept for one letter a ∈ A such that ˘ u = ˘ w v ¯ a ¯ y wher e v and y ar e L -spinne d wor ds verifying Alph( v ) = A \ { a } . Before p rovi ng this theorem, let us observe that we do not include in this result the Stu r mian case. Indeed, T heorem 5.14 is n o longer v ali d wh en n ≤ 2. F or instance the morp hism µ ¯ a ¯ b ( a 7→ aba , b 7→ ba ) is strongly quasip eriodic b ut its normalize d directiv e word fulﬁlls n one of the ab ov e conditions. A complete description of strongly quasip erio dic Sturmian morphisms is p r o vided in [29]. 26 Pr o of. L et ˘ u b e th e norm alized d irectiv e word (of the morph ism µ ˘ u ). If ˘ u veriﬁes i ), then by Theorem 3.1, av 1 ¯ av 2 ¯ a · · · v k ¯ av ≡ ¯ a ¯ v 1 ¯ a ¯ v 2 ¯ a · · · ¯ v k av . Moreo ver fr om Theo- rem 5.1, the epistandard morphism µ av is strongly quasip erio dic. T hus µ ˘ u is strongly qu asip eriod ic on A . If ˘ u veriﬁes ii ), then by Lemma 5.11, µ ˘ u is str on gly quasip eriod ic o v er A . If ˘ u v eriﬁes iii ), then Lemma 5.11 , µ ˘ u ( t ) is quasip erio dic for an y word t that do es not start with a . Let us d ecompose y = y 0 ay 1 · · · ay k where k is the number of o ccurrences of a in y . In the pro of of Lemma 5.12, we hav e seen that µ ¯ a ¯ y k ( ax ) = µ ay k ( xa ) for an y ﬁnite word x . This formula naturally extends to any inﬁnite word t , µ ¯ a ¯ y k ( a t ) = µ ay k ( t ). Hence µ ˘ u ( a t ) = µ ˘ wv ¯ a ¯ y 0 ¯ a ¯ y 1 ··· ¯ y k − 1 ay k ( t ). By The- orem 3.1, µ ¯ a ¯ y i a = µ ay i ¯ a for eac h i , so that µ ˘ u ( a t ) = µ ˘ wv ay 0 ¯ a ¯ y 1 ¯ a ··· y k − 1 ¯ ay k ( t ). F rom Theorem 5.1, the epistandard morphism µ va is strongly quasip erio dic. Thus µ ˘ u ( a t ) is quasip erio dic. Consequ ent ly µ ˘ u is str on gly quasip erio dic on A . T o end , we pro ve that if ˘ u veriﬁes none of the conditions i )– iii ), then th ere exists (at lea st) one word t suc h that µ ˘ u ( t ) is n ot quasip erio dic (and so µ ˘ u is not str on gly quasiperio dic). Th is is immediate if ˘ u = ε . • Let us ﬁ rst consider the case wh ere ˘ u ends with an L -sp inned letter a , that is ˘ u = ˘ w a for some spinned word ˘ w . Since |A| ≥ 2, there exist m ≥ 2 pairwise diﬀerent letters a 1 = a, a 2 , . . . , a m such that A = { a 1 , a 2 , . . . , a m } . Let t b e th e episturmian word with normalized directiv e word (¯ a 2 · · · ¯ a m a ) ω . Since ˘ w a is n ormalized, ˘ w a (¯ a 2 · · · ¯ a m a ) ω is also n ormalized. Moreo v er since ˘ u do es not v erify condition i ), the word ˘ w a ( ¯ a 2 · · · ¯ a m a ) ω cannot b e deco mp osed in to the form ˘ w ′ bv 1 ¯ bv 2 · · · ¯ bv k ˘ y ′ · · · w here b is an L -spinned letter, ˘ w ′ is a spinned word, ˘ y ′ is a spinned ve rsion of an L -spinned word y ′ , and v 1 , . . . , v k ( k ≥ 0) are b -free L -spinn ed wo rds such that Alph( bv ′ ) = Alph( bv ′ y ′ ). By Remark 5.13 and Th eorem 4.29, the w ord µ ˘ u ( t ) is not quasip eriod ic. • Now we consider the case when ˘ u ends with an R -spinned le tter, that is ˘ u = ˘ w ¯ v for a non- empty L -sp in ned wo rd v and a spinn ed word ˘ w su c h that ˘ w = ε or ˘ w ends with an L -spinned letter. Tw o cases can hold: Case 1: A 6 = Alph( v ). Let a b e a letter in A \ Alph ( v ), and let b b e an y other lett er (remem ber |A| ≥ 2). Let t b e the episturmian word with normalized d irectiv e word ( a ¯ b ) ω and let t ′ = µ ˘ u ( t ). Then t ′ is directed by ˘ u ′ = ˘ w ¯ v ( a ¯ b ) ω and, since ˘ w ¯ v is norm alized and ¯ a 6∈ Alph( ¯ v ), ˘ u ′ is normalize d. Moreo v er sin ce ˘ w ¯ v does not ve rify i ) , ˘ u ′ cannot b e decomposed into the form ˘ w ′ cv 1 ¯ cv 2 · · · ¯ cv k ˘ y ′ · · · with an L -spinned letter c , a spin ned word ˘ w ′ , a spinned ve rsion ˘ y ′ of a n L -spinned word y ′ and so me c -free L -sp in ned words v 1 , . . . , v k ( k ≥ 0) such that Alph( cv k ) = Alph( cv k y ′ ). By Remark 5.13 and T heorem 4.29, the word µ ˘ u ( t ) is n ot q u asip eriod ic. Case 2: A = Alph( v ). Since ˘ u do es not verify ii ), there exists a letter a and a -free L -spinn ed words v 0 , . . . , v k ∈ A ∗ such that ˘ u = ˘ w ¯ v 0 ¯ a ¯ v 1 · · · ¯ a ¯ v k and for all i , 1 ≤ i ≤ k , Alph( v i ) 6 = A \ { a } . Moreov er since ˘ u does not ve rify iii ) , th en either v 0 6 = ε , or ˘ w cannot be written in the form ˘ w = ˘ w ′ v ′ for a spinned word ˘ w and an L -spinned word v ′ such that Alp h( v ′ ) = A \ { a } . Since A contai ns at least thr ee letters, there exist m ≥ 3 pairwise diﬀerent lette rs a 1 = a , a 2 , . . . , a m such that A = { a 1 , a 2 , . . . , a m } . Let t b e the epistu rmian word with normalized d irectiv e word ( aa 2 ¯ a 3 · · · ¯ a m ) ω and let t ′ = µ ˘ u ( t ). Then t ′ is directed by ˘ w ¯ v 0 ¯ a ¯ v 1 · · · ¯ a ¯ v k aa 2 ¯ a 3 · · · ¯ a m ( aa 2 · · · ¯ a m ) ω ≡ 27 ˘ w ¯ v 0 av 1 ¯ a · · · ¯ av k ¯ aa 2 ¯ a 3 · · · ¯ a m ( aa 2 ¯ a 3 · · · ¯ a m ) ω . Since ˘ w ends with an L -spinned letter and eac h v i (0 ≤ i ≤ k ) is a -free, this word is normalized. Since ˘ w ¯ v do es not ve rify i ) and f rom the previous observ atio ns, the wo rd ˘ w ¯ v 0 av 1 ¯ a · · · ¯ av k ¯ aa 2 ¯ a 3 · · · ¯ a m ( aa 2 · · · ¯ a m ) ω cannot b e decomp osed into the form ˘ w ′ bv ′ 1 ¯ bv ′ 2 · · · ¯ bv ′ k ˘ y ′ · · · w ith an L -spinn ed letter b , a sp inned word ˘ w ′ , a spinned ve rsion ˘ y ′ of an L -sp inned word y ′ and some b -free L -spinned words v ′ 1 , . . . , v ′ k ( k ≥ 0) such that with Alph( bv ′ k ) = Alph( bv ′ k ˘ y ′ ). By Remark 5.13 and T h eorem 4.29 , the word µ ˘ u ( t ) is not quasiper io dic. 6 Episturmian Lyndon w ords Theorem 4.28 p rovides a c h aracterizat ion of quasip eriod ic episturm ian words. In the b inary case, it was pro v ed in [29] that a Sturm ian w ord is quasip erio dic if and only if it is not an inﬁn ite Lyndon w ord. A n atural question to ask is then : “does this result still hold for episturmian w ords on a larger alphabet?” By a resu lt in [29], one can see that an y inﬁnite Lyn don w ord is non- quasip eriod ic. In this section, we sho w that there is a muc h wider class of epistu r mian w ords that are non-qu asip eriodic, besid es those that are inﬁnite Lyndon words. This follo ws f r om our c haracterizatio n of episturmian Lynd on words (Theorem 6.1, to follo w). Let us ﬁrst recall the notion of lexicographic order and the deﬁnition of Lyn don words (see [30] for in stance). Supp ose the alphab et A is totally ordered by the relation < . Then we can totally order A ∗ by the lexic o gr aph ic or der ≤ deﬁn ed as follo ws. Given t w o w ords u , v ∈ A + , we hav e 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 with a < b . This is the usual a lphab etic ordering in a dictionary . W e w rite u < v wh en u ≤ v and u 6 = v , in which case we sa y that u is (strictly) lexic o gr aphic al ly smal ler than v . The notion of lexicographic order naturally extends to inﬁnite words in A ω . W e den ote by min( A ) the smallest letter w ith resp ect to the lexicographic order. A non-empty ﬁnite word w o ve r A is a Lyndo n wor d if it is le xicographically smaller than all of its prop er suﬃxes for th e giv en order < on A . Equiv alently , w is the lexicog raphically small est primitive wo rd in its conjugacy class; that is, w < v u for all non-empt y words u , v su ch that w = uv . The ﬁ rst of these deﬁnitions extend s to inﬁnite words: an in ﬁ nite word ov er A is an inﬁnite Lyndon wor d if and only if it is (strictly) lexicographically sm aller than all of its prop er suﬃxes for the giv en order on A . Th at is, a ﬁnite or inﬁ nite word w is a Lynd on w ord if and only if w < T i ( w ) for all i > 0. In this section, we assum e that |A | > 1 since on a 1-letter alphab et there are n o inﬁ nite Lyndon wo rds. Also note that an inﬁnite Lyndon word cannot b e p erio dic. Th erefore we consider only ap er io dic episturmian words (i.e., those with | Ult(∆) | > 1). 6.1 A c omplete characterization In this section, generalizing previous results in [29] (S turmian case) and [15] (Arn ou x -Rauzy se- quences or strict episturmian words), we prov e: Theorem 6.1. L et A = { a 1 , . . . , a m } b e an alphab 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 w is an inﬁnite Lyndon wor d if and only if ther e exists an inte ger j such that 1 ≤ j < m and the (normalize d) d ir 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 . 28 Note. In the abov e theorem, we ha v e p ut the wo rd norm alize d b et we en brac k ets since one can easil y ve rify from T heorem 3.7 that a spin ned inﬁnite word of the given form is the uniqu e directiv e word of exactly one episturmian word. Example 6.2. Let A = { a, b, c, d } . Then the word ( ¯ b ¯ ca )( ¯ d ¯ cb ) 2 ( ¯ dcc ) ω directs a Lyn don ep isturmian wo rd, so does aa ( ¯ dc ) ω , but ¯ ca ¯ ba ¯ dcd ω do es not (this spinned word directs a p erio dic word). Remark 6.3. The “if and only if ” condition can b e reformulated as follo ws: The (normalize d) dir e ctive wor d of w takes the form v 1 · · · v j y wh er e: • for 1 ≤ k ≤ j , v k is a spinne d wor d in ( ¯ B ∗ k +1 a k ) ∗ ; • y is a spinne d inﬁnite wor d b elonging to ( ¯ B + j +1 { a j } + ) ω . Remark 6.4. Theorems 3.7 and 6.1 show th at any episturm ian Lyndon word has a uniqu e spinned directiv e word, but the conv erse is not true. Certainly , there exist episturmian words with a unique directiv e word which are not inﬁ nite Lyndon w ords. F or example, the regular wa vy word ( a ¯ bc ) ω is the u nique dir ective word of the strict episturmian word: lim n →∞ µ n a ¯ bc ( a ) = acabaabacabacabaabaca · · · which is clearly not an inﬁn ite Lynd on word by T heorem 6.1 and also b y the fact th at acabaaw is not a Lyndon w ord for any order on { a, b, c } and for any word w . In order to prov e the Theorem 6.1, we recall tw o useful results and state a n ew one (Lemma 6.7). Lemma 6.5. [34] An inﬁnite wor d is a Lyndon wor d if and only if it has inﬁnitely many d iﬀer ent Lyndon wor ds as pr eﬁxes. A morp hism f is said to preserve ﬁ nite (resp. in ﬁ nite) Lyndon words if for eac h ﬁn ite (resp. in- ﬁnite) Lyndon word w , f ( w ) is a ﬁnite (resp. inﬁnite) Lyn don w ord. F or episturmian morphisms, we h a ve : Prop osition 6.6. [42, 45] L et A = { a 1 , . . . , a m } b e an alph ab et or der e d by a 1 < a 2 < · · · < a m . Then the fol lowing assertio ns ar e e quivalent for an episturmian morphism : • f pr eserves ﬁnite Lynd on wor ds; • f pr eserves i nﬁnite Lynd on wor ds; • f ∈ ( R ∗ { a 2 ,...,a m } L a 1 ) ∗ { R a m } ∗ . No w we prov e a lemma concerning the acti on of morphisms in L A ∪ R A on in ﬁnite Lyndon wo rds. Lemma 6.7. Supp ose w is an inﬁnite wor d over an or der e d alphab et A and let a , b ∈ A with a < b . Then, the fol lowing pr op e rties hold. i) L b f L a ( w ) is not an inﬁnite Lynd on wor d for any non-er asing morp hism f . ii) R a f L b ( w ) is not an inﬁnite Lynd on wor d for any morphism f in R ∗ A . iii) If w is r e curr ent, then R x f L x ( w ) is not an inﬁnite Lynd on wor d for any letter x a nd mor- phism f in R ∗ A . 29 Pr o of. i ) Th e inﬁn ite word L b f L a ( w ) starts with b an d cont ains an occurr ence of the letter a ; thus, since a < b , it cannot be an inﬁnite Lyn don word. ii ) As f ∈ R ∗ A , the inﬁ nite word R a f L b ( w ) starts w ith b and co nta ins an o ccurrence of the letter a ; thus, since a < b , it cannot b e an inﬁ nite Lynd on word. iii ) T o b e an inﬁnite Lyn don word, R x f L x ( w ) m ust b e aperiod ic, in which case a lett er diﬀeren t from x occurs in it; in particular this letter occur s in f L x ( w ). Moreov er, as f ∈ R ∗ A , the inﬁn ite wo rd f L x ( w ) b egins with x and hence with a p r eﬁx x n y for s ome integer n ≥ 1 and a letter y 6 = x . The recurrence of the in ﬁnite word w implies th e recurrence of f L x ( w ), and so f L x ( w ) con tains a factor z x n + r y for some letter z 6 = x and integer r ≥ 0. Now R x f L x ( w ) b egins with x n y and cont ains z x n + r +1 y , and so con tains x n +1 . T o be an inﬁ nite Lyndon w ord, it needs x = min(Alph( R x f L x ( w ))), bu t then x n +1 < x n y . T hus R x f L x ( w ) is n ot an inﬁnite Lyndon wo rd. Lastly , we need an imp ortant easy fact: F act 6.8. [45] Any morphism f in L A ∪ R A preserves the lexicog raphic order for in ﬁnite words. More precisely , for any in ﬁnite words w and w ′ , w < w ′ if and only if f ( w ) < f ( w ′ ). A consequence of the abov e fact is that for any wo rd w and for an y morp hism f in L A ∪ R A , if f ( w ) is a Lyndon w ord then necessarily w is also a Lynd on word. Pr o of of The or em 6.1. Assume that ∆ is the n ormalized directiv e w ord of a Lynd on episturmian wo rd. T hen ∆ contains no factor of the f orm ¯ x ¯ v x for any letter x and v ∈ A ∗ . By Lemma 6.7 , it do es not con tain any factor of the form bv a or ¯ a ¯ v b with v ∈ A ∗ and a < b . Thus ∆ tak es the f orm giv en in the statement of th e theorem. I n deed b y item i ) of Lemma 6.7 and by F act 6.8, only one letter (namely a j ) c an ha ve all s p ins ultimat ely L . Sin ce a Lyndon word is not perio dic, at least one other letter in A should o ccur inﬁn itely ofte n. By items ii )- iii ) of Lemma 6.7, su c h a lett er should b elong to ¯ B j +1 . Moreov er, the sequence of letters with spin L must b e order-increasing and items ii )– iii ) of Lemma 6.7 determine the conditions on lette rs with spin R . Conv ersely , sup p ose that the (normalized) directiv e word ∆ of the episturmian wo rd w tak es the form giv en in the statemen t of the theorem. W rite ∆ = v 1 · · · v j y where, for 1 ≤ k ≤ j , v k is a spinned w ord in ( ¯ B ∗ k +1 a k ) ∗ and y is a spinned inﬁnite w ord b elonging to ( ¯ B + j +1 { a j } + ) ω . Then, b ecause of the recurrence of the letter a j and of at least one other letter in B j +1 in y , there exists a sequence of sp inned words ( v n ) n ≥ j , with eac h v n in ¯ B + j +1 { a j } + , such that y = Q n ≥ j v n . Now, for eac h k ≥ 1, µ v k is a Lyndon morphism on B k by Prop osition 6.6. Hence, for eac h k ≥ 1, the word µ v 1 ··· v k ( a j ) is a Lyndon wo rd. F rom w = lim k →∞ µ v 1 ··· v k ( a j ), we deduce from Lemma 6.5 that the episturmian word w is an inﬁnite Lynd on word. 6.2 Strict episturmian Lyndon w ords Let u s recall fr om [23] that an epistandard w ord s , or an y ep isturmian w ord in the subshift of s , is said to b e A -strict if its L -sp inned directive word ∆ veriﬁes Ult(∆) = A . F or these words, also called Arn oux-Rauzy sequ ences [5], Theorem 6.1 give s: Corollary 6.9. L et A = { a 1 , . . . , a m } b e an alphab et or der e d by a 1 < a 2 < · · · < a m . An A -strict episturmian wor d w is an inﬁnite Lyndon wor d if and only if the (normalize d) dir e ctive wor d of w b elongs to { a 1 , ¯ a 2 , . . . , ¯ a m } ω . This can b e reformulated as a generalization of Prop osition 6.4 in [29]: Corollary 6.10. [15] An A -strict ep isturmian wor d t is a n inﬁnite Lyndon wo r d if a nd only if it c an b e inﬁnitely de c omp ose d over the set of morp hisms { L a , R x | x ∈ A \ { a }} wher e a = min( A ) for the g i ven or der on A . 30 The ab ov e r esult also follo ws from the following generali zation of a result on Stur mian w ords giv en by Borel and L au b ie [8] (see also [43]). Theorem 6.11. An A -strict episturmian wor d t i s an inﬁnite Lyndo n wor d if and onl y 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 epistanda r d wor d. Mor e over, if ∆ is the L -spinne d dir e ctive wor d of s , th en t = a s is th e unique episturmian wor d in the subshift of s dir e cte d by the spinne d version of ∆ ha ving al l spins R , exc ept wh en x i = a . The pro of of th e ab o ve t heorem requires the following result that is essential ly Theorem 3.17 from [23], apart from the fact that a s is in the subshift of s , wh ic h follows fr om F act 2.3. Theorem 6.12. Supp ose s is an epistandar d wor d dir e c te d by ∆ = x 1 x 2 x 3 · · · and let a b e a letter. Then a s is an episturmian wor d if and only if a ∈ Ult(∆) , in which c ase a s is the unique episturmian wor d in the sub shift of s dir e cte d by the spinne d version of ∆ having al l spins R , exc ept when x i = a . Pr o of of The or em 6.1 1. Let A = { a 1 , a 2 , . . . , a m } w ith a 1 < a 2 < · · · < a m . By Corollary 6.9, an A -strict episturmian word t is an inﬁnite Lyndon w ord if and only if the (normalized) directive word ˘ ∆ of t belongs to { a 1 , ¯ a 2 , . . . , ¯ a m } ω , i.e., if and only if t = a 1 s where s is the uniqu e epistandard wo rd d irected by the L -spinned version of ˘ ∆, by Theorem 6.12. Note. If s is an epistandard w ord ov er A , then a s is an inﬁnite Lyndon w ord for any order s uch that a = m in( A ). Let u s p oint out th at completely diﬀerent pro ofs of Corollary 6.9 and Th eorem 6.11, usin g a c haracterizatio n of episturmian words via lexicographic orderin gs, were given in [15] by the ﬁ rst author. A reﬁnement of one of th e main results in [18] is also give n in [15]. T o end, let us observe that, con trary to the f act that there exists |A| ! p ossible ord ers of a ﬁnite alphab et A , Th eorem 6.11 sh o ws that there exist exactly |A| inﬁ nite Ly n don words in the su b shift of a giv en A -strict epistandard w ord s , when |A| > 1 (since there are no Ly n don words when |A| = 1). That is, for any order with min( A ) = a , the subs hift of s cont ains a unique inﬁnite Lyndon w ord b eginning with a , namely a s . Example 6.13. With ∆ = ( abcd ) ω , the spinn ed versions ( a ¯ b ¯ c ¯ d ) ω , (¯ ab ¯ c ¯ d ) ω , (¯ a ¯ bc ¯ d ) ω , (¯ a ¯ b ¯ cd ) ω , (¯ a ¯ bcd ) ω , (¯ ab ¯ cd ) ω , (¯ abc ¯ d ) ω and their opposites direct n on-quasip eriod ic epistur mian words in the subshif t of the 4-b onacci word z . Only the ﬁ rst four of these words dir ect Lyndon episturmian wo rds: a z , b z , c z , d z , resp ectiv ely . 7 Concluding remarks In [35 ], Mont eil p rov ed that an y Stu rmian sub shift contains a multi-sc ale quasip erio dic wor d , i.e., an inﬁnite word having inﬁn itely many quasip erio ds. A sh orter pro of of this fact wa s provided in [29]. This can be easily extended to epistur m ian w ords. Certainly , by F act 3.4, an y episturmian subshif t contains at most tw o epistandard words (one in the ap erio dic case and t wo in the p erio dic case) and any epistandard w ord has inﬁnitely many quasip erio ds (by Theorem 4.10). Actually the c haracterizat ion of qu asip eriodic Sturmian words in [29] sho ws that in an y Sturm ian subshif t there are only tw o n on-quasip eriod ic Sturmian words and all other (S turmian) w ords in the subs h ift h a ve inﬁn itely m any q u asip eriod s. It is easy to see that th e same result does not hold for epistu rmian w ords deﬁn ed ov er an alphab et conta ining more than t wo lette rs. F or instance, any epistu r mian wo rd having a spinned directive w ord in { ab ¯ c, a ¯ b ¯ c } ω is non-quasip erio dic: all of these non-quasip erio dic episturmian words b elong to the subsh ift of the T r ib onacci word r , directed by ( abc ) ω . Moreo ve r, one can v erify (us in g Theorem 3.2) that the quasip erio dic episturmian wo rd 31 t directed by ( abc ) n ( ab ¯ c )( a ¯ b ¯ c ) ω for some n ≥ 1 (which is in the subsh ift of r ) has exactly n + 1 directiv e words: ( abc ) i ab ¯ c (¯ a ¯ bc ) n − i ( a ¯ b ¯ c ) ω , 0 ≤ i ≤ n. Hence it is cl ear from Th eorem 4.19 that t has only ﬁn itely many quasip erio ds. (S ee also Exam- ples 4.25–4.27.) Ac kno wledgments: T h e authors wo uld li ke to th ank the t w o anonymous referees for their sug- gestions to improv e the pap er. In particular, one referee indicated the simp le p ro of of Theorem 4.10 and t he other noticed the interest of F act 4.6. Man y thanks also to D. Krieger who suggested to the third author th e notion of ultimate qu asip er io dicity and observed the fact that all Sturmian wo rds are ultimatel y quasiperio dic. The in terest of this notion b ecomes eviden t when one referee underlined th at the term quasi-factor in trodu ced in [15] was n ot completely satisfactory . References [1] J.-P . Allouc he and J. Shallit. Au tomatic se quenc es: The ory, Applic ations, Gener ali zations . Cambridge Universit y Press, 2003. [2] A. Ap ostolico and M. Crochemore. String pattern matc hing for a deluge surv iv al kit. In J. A b ello, P .M. Pardalos , and M.G.C. Resende, editors, Handb o ok of Massive Data Sets, Massive Comput. , volume 4. Kluw er Academic Publishers, 2001. [3] A. A p ostolico and A. Ehrenfeuc h t. Eﬃcient detection of quasip eriodicities in strings. The or et. Comput. Sci . , 119:247 –265, 1993. [4] A. A p ostolico, M. F arach, and C. S. Iliopoulos. O ptimal sup erp rimitivity t esting for strings. Inf orm. Pr o c ess. L ett. , 39(1):17–20, 1991. [5] P . A rnoux and G. Rauzy . R epr´ ese ntation g´ eom ´ etriqu e de suites de complexit ´ es 2 n + 1. Bul l. So c. Math. F r anc e , 119:199 –215, 1991. [6] J. Berstel. St u rmian and episturmian words (a survey of some rec ent results). In Pr o c e e dings of CAI 2007 , vol ume 4728 of L e ctur e N otes in Com puter Scienc e , p ages 23–47. Sp rin ger-V erla g, 2007. [7] V. Berth´ e, C. Holton, and L. Q. Zam boni. Initial pow ers of Sturmian sequences. Ac ta Arith. , 122:315–34 7, 2006. [8] J.P . B orel and F. Laubie. Quelques mots sur la d roite pro jective r´ eelle. Journal de Th ´ eorie de s Nombr es de Bor de aux , 5:23–51, 1993. [9] E. M. Cov en and G.A. H edlund. Sequen ces with minimal block grow th. Math. Systems The ory , 7:138– 153, 1973. [10] A . de Luca. A combinatorial prop erty of the Fib onacci wo rds. Inform. Pr o c ess. L ett. , 12(4):193–195, 1981. [11] A . de Luca. Sturmian wor ds: structure, com binatorics and their arithmetics. The or et. Comput. Sci. , 183:45–82 , 1997. [12] X . Drou b a y , J. Justin, and G. Pirillo. Episturmian words an d some constru ctions of de Luca and R auzy . The or et. Comput. Sci. , 255(1-2):539– 553, 2001. [13] F. Durand. A characterizatio n of substitutive sequences using return wo rds. Discr ete Math. , 179:89–101, 1998. [14] S F erenczi. Co mplexity of sequences and dyn amical systems. Discr ete Math. , 206:145 –154, 1999. [15] A . Glen. Order and quasip erio dicit y in episturmian w ords. I n Pr o c e e di ngs of the 6 th International Confer enc e on Wor ds, Marseil le, F r anc e, Septemb er 17-21 , pages 144–158 , 2007. [16] A . Glen. A characterization of ﬁne words o ver a ﬁnite alphab et. The or et. Comput. Sci. , 391:51–6 0, 2008. [17] A . Glen and J. Justin. Episturmian words: a survey , preprin t, 2007. [18] A . Glen, J. Justin, and G. Pirillo. Characterizations of ﬁnite and inﬁnite episturmian w ords v ia lexicographic orderings. Eur op e an J. Combi n. , 29:45–5 8, 2008. [19] A . Glen, F. Lev ´ e, and G. Richomme. D irective w ords of ep isturmian w ords: equiv alence and normalization, preprint, 200 8. [20] C. Holton and L.Q. Zamboni. Descendan ts of primitive substitutions. The ory Comput. Syst. , 32:133–15 7, 1999. [21] C.S. Iliopoulos and L. Mouc hard. Quasiperiod icit y and string cov ering. The or et. Comput. Sci. , 218(1):205–216, 1999. [22] J. Justin. Episturmian morphisms an d a Galois theorem on con tin ued fractions. The or et. I nform. Appl. , 39(1):207– 215, 2005. 32 [23] J. Justin and G. Pirillo. Episturmian wo rds and episturmian morphisms. The or et. Comput. Sci . , 276(1-2):281– 313, 2002. [24] J. Justin and G. Pirillo. On a c haracteris tic prop erty of Arnoux-Rauzy sequences. The or et. Inf orm. Appl. , 36(4):385– 388, 2003. [25] J. Justin and G. Pirillo. Episturmian w ords: shifts, morphisms and n umeration systems. I nternat. J. F ound. Comput. Sci. , 15(2):329–3 48, 2004. [26] J. Justin and L. V uillon. R eturn w ords in Sturmian and episturmian w ords. The or et. Inf orm. Appl. , 34:343 –356, 2000. [27] F. Lev´ e and G. Richomme. Quasip erio dic inﬁn ite w ords: some answers. Bul l. Eur. Asso c. The or. Comput. Sci. , 84:128– 138, 2004. [28] F. Lev´ e and G. Ric homme. Quasip eriod ic episturmian wor ds. In Pr o c e e dings of the 6 th International Confer enc e on Wor ds, Marseil le, F r anc e, Septemb er 17-21 , pages 201–211 , 2007. [29] F. Lev´ e and G. Richomme. Quasip erio dic Sturmian wo rds and morphisms. The or et. Comput. Sci. , 372:15–2 5, 2007. [30] M. Lothaire. Combinatorics on Wor ds , v olume 17 of Encyclop e dia of Mathematics and i ts Appli c ations . Addison- W esley , 1983. [31] M. Lothaire. Algebr aic Combinatorics on Wor ds , v olume 90 of Encyclop e dia of Mathematics and i ts Applic ations . Cam bridge U niversi ty Press, 2002. [32] S . Marcus. Bridging tw o hierarchies of inﬁnite w ords. J. UCS , 8:292–296, 2002. [33] S . Marcus. Quasiperio dic inﬁnite w ords. Bul l. Eur. Asso c. The or. Com put. Sci. , 82:170 –174, 2004. [34] G. Melan¸ con. L y ndon factorization of S turmian words. Discr ete Math. , 210:137–149 , 2000. [35] T. Monteil. Il lumination dans les bil lar ds p olygonaux et dynamique symb oli que . PhD thesis, Universit ´ e de la M ´ editerran ´ ee, F acult´ e des Sciences de Luminy , December 2005. [36] M. Morse and G.A. Hedlun d. Sy mbolic Dyn amics I I. Sturmian tra jectories. Amer. J. Math. , 61:1–42, 1940. [37] G. P aquin and L. V uillon. A c haracterization of b alanced episturmian sequences. Ele ctr on. J. Combin. , 14, 2007. #R33, p p. 12. [38] N . Py theas F ogg. Substitutions in Dynamics, Arithmetics and Combinatorics , volume 1794 of L e ctur e Notes in Mathematics . S pringer, 2002. [39] G. Rauzy . Nom bres alg ´ ebriques et substitu tions. Bul l . So c. M ath. F r anc e , 110:147–1 78, 1982. [40] G. Rauzy . Mots inﬁnis en arithm ´ etique. In M. N iv at and D. Perri n, editors, A utomata on Inﬁnite wor ds , volume 192 of L e ctur e Notes i n Computer Scienc e , pages 165–171. Springer-V erlag, Berlin, 1985. [41] G. Richomme. Conjugacy and episturmian morphisms. The or et. Comput. Sci. , 302(1-3):1–34 , 2003. [42] G. Richomme. Lyn don morphisms. Bul l. Belg. Math. So c. Simon Stevin , 10:761–785 , 2003. [43] G. R ic homme. Conjugacy of morphisms and Lyndon decomp osition of standard Sturmian w ords. The or et. Comput. Sci. , 380(3):393– 400, 2007. [44] G. Richomme. A lo cal b alance prop erty of episturmian w ords. In Pr o c. DL T ’07 , volume 4588 of L e ctur e Notes in Computer Scienc e , pages 371–381. Springer, Berlin, 2007. [45] G. Richomme. On morphisms preserving inﬁnite Lyndon w ords. Discr ete Math. The or. C omput. Sci. , 9:89–108 , 2007. [46] R .N. Risley and L.Q Zamboni. A generalization of Stu rmian sequences: com binatorial structure and transcen- dence. A cta Arith. , 95:167–184 , 2000. [47] P . S´ e ´ eb old. Fibonacci morp h isms and Sturmian words. The or et. C om put. Sci. , 88(2):365– 384, 1991. 33

Quasiperiodic and Lyndon episturmian words

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