Qusiperiodic Sturmin words nd morphisms Florence Levé, Gwenël Richomme To cite this version: Florence Levé, Gwenël Richomme. Qusiperiodic Sturmin words nd morphisms. Theoreticl Computer Science, Elsevier, 2007, 372 (1), pp.15-25. <hl-00016679> HAL Id: hl-00016679 https://hl.rchives-ouvertes.fr/hl-00016679 Sumitted on 9 Jn 2006 HAL is multi-disciplinry open ccess rchive for the deposit nd dissemintion of scientific reserch documents, whether they re pulished or not. The documents my come from teching nd reserch institutions in Frnce or rod, or from pulic or privte reserch centers. L rchive ouverte pluridisciplinire HAL, est destinée u dépôt et à l diffusion de documents scientifiques de niveu recherche, puliés ou non, émnnt des étlissements d enseignement et de recherche frnçis ou étrngers, des lortoires pulics ou privés.
LRIA : Lortoire de Recherche en Informtique d Amiens Université de Picrdie Jules Verne CNRS FRE 2733 33, rue Sint Leu, 80039 Amiens cedex 01, Frnce Tel : (+33)[0]3 22 82 88 77 Fx : (+33)[0]03 22 82 54 12 http://www.lri.u-picrdie.fr ccsd-00016679, version 1-9 Jn 2006 Qusiperiodic Sturmin words nd morphisms F. Levé, G. Richomme LRIA RESEARCH REPORT : LRR 2006-01 (Jnury 2006) LRIA, Université de Picrdie Jules Verne, {florence.leve, gwenel.richomme}@u-picrdie.fr
Qusiperiodic Sturmin words nd morphisms F. Levé, G. Richomme LRIA, Université de Picrdie Jules Verne 33, Rue Sint Leu, F-80039 Amiens cedex 1 ({florence.leve,gwenel.richomme}@u-picrdie.fr) Jnury 9, 2006 Astrct We chrcterize ll qusiperiodic Sturmin words: Sturmin word is not qusiperiodic if nd only if it is Lyndon word. Moreover, we study links etween Sturmin morphisms nd qusiperiodicity. Keywords: Sturmin words, qusiperiodicity, Lyndon words, morphisms 1 Introduction The notion of repetition in Strings is centrl in lot of reserches, in prticulr in Comintorics on Words nd in Text Algorithms (see for instnce [9], [10] for recent surveys). In this vein, Apostolico nd Ehrenfeucht introduced the notion of qusiperiodic finite words [2] in the following wy: string w is qusiperiodic if there is second string u w such tht every position of w flls within some occurrence of u in w. The reder cn consult [1] for short survey of studies concerning qusiperiodicity. In [12], Mrcus extends this notion to right infinite words nd he opens six questions. Four of them re nswered in [7]. One of these six questions is: does there exist non-qusiperiodic Sturmin word? In [7], we provide n exmple of such word, ut this positive nswer is not completely stisfying. Since first feeling cn e tht there exists no (or t most very few) such word, one cn sk for complete chrcteriztion of such non-qusiperiodic Sturmin words. After some preliminries in Sections 2, 3 nd 4, we provide two nswers descried elow. Sturmin words hve een widely studied ecuse of their mny eutiful properties nd links with mny fields (see [9, Chpter 2] for recent survey). One spect of these words is tht they cn e infinitely decomposed over the four morphisms L, L, R nd R (see Section 3 for more detils). The first chrcteriztion of non-qusiperiodic Sturmin words proposed in this pper is sed on such decomposition. More precisely, Theorem 5.6 sttes tht Sturmin word is not qusiperiodic if nd only if it cn e decomposed infinitely over {L,R } or infinitely over {L,R }. Our second chrcteriztion (Theorem 6.5) provides more semntic nswer: Sturmin word is not qusiperiodic if nd only if it is n infinite Lyndon word. The proof of our first result uses the fct tht some morphisms otined y compositions of the morphisms L, L, R nd R mp ny infinite words into qusiperiodic one. We cll such morphism strongly qusiperiodic. In Section 7, we chrcterize the Sturmin morphisms which re 1
strongly qusiperiodic. Let us quote tht ny Sturmin morphism f is qusiperiodic, tht is there exists non-qusiperiodic word w whose imge y f is qusiperiodic. 2 Generlities We ssume the reder is fmilir with comintorics on words nd morphisms (see, e.g., [8, 9]). We precise our nottions. Given set X of words (for instnce n lphet A, tht is non-empty finite set of letters), X (resp. X ω ) is the set of ll finite (resp. infinite) words tht cn e otined y conctenting words of X. The empty word ε elongs to X. The length of word w is denoted y w. By w we denote the numer of occurrences of the letter in w. A finite word u is fctor of finite or infinite word w if there exist words p nd s such tht w = pus. If p = ε (resp. s = ε), u is prefix (resp. suffix) of w. A word u is order of word w if u is oth prefix nd suffix of w. A fctor u of word w is sid proper if w u. Given n lphet A, (n endo)morphism f on A is n ppliction from A to A such tht f(uv) = f(u)f(v) for ny words u, v over A. A morphism on A is entirely defined y the imges of letters of A. All morphisms considered in this pper will e non-ersing: the imge of ny nonempty word is never empty. The imge of n infinite word is thus infinite nd nturlly otined s the infinite conctention of the imges of the letters of the word. In wht follows, we will denote the composition of morphisms y juxtposition s for conctention of words. Given set X of morphisms, we will lso note X the set of ll finite compositions of morphisms of X nd X ω the set of ll infinite decompositions of morphisms of X. When word w is equl to lim f 1f 2... f n (), n f i X, we will sy tht w cn e decomposed (infinitely) over X. Given morphism f, powers of f re defined inductively y f 0 = Id (the Identity morphism), f i = ff i 1 for integers i 1. When for letter, f() = x with x ε, the morphism f is sid prolongle on. In this cse, for ll n 0, f n () is prefix of f n+1 (). If moreover, for ll n 0, f n () < f n+1 (), the limit lim n fn () is the infinite word denoted f ω () hving ll the f n () s prefixes. This limit is lso fixed point of f. 3 Sturmin words nd morphisms Sturmin words my e defined in mny equivlent wys (see [9, chpter 2] for instnce). They re infinite inry words. Here we first consider them s the infinite lnced non ultimtely periodic words. We recll tht (finite or infinite) word w over {,} is lnced if for ny fctors u nd v of sme length u v 1, nd tht n infinite word w is ultimtely periodic if w = uv ω for some finite words u nd v. Mny studies of Sturmin words use Sturmin morphisms, tht is morphisms tht mp ny Sturmin word into Sturmin word. Sééold [17] proved tht the set of these morphisms is {E,L,L,R,R } where E,L,L,R,R re the morphisms defined y { { { { { E : L, : L, : R, : R, :. Mny reltions exist etween Sturmin words nd Sturmin morphisms. For instnce, recently the following result ws proved: 2/15
Theorem 3.1 [5] Any Sturmin word w over {,} dmits unique representtion of the form w = lim n Ld 1 c 1 R c 1 Ld 2 c 2 R c 2...Ld 2n 1 c 2n 1 R c 2n 1 L d 2n c 2n R c 2n () where d k c k 0 for ll integer k 1, d k 1 for k 2 nd if c k = d k then c k 1 = 0. Remrk: Let us mention tht this representtion is not expressed s in [5] where it is written w = T c 1 L d 1 T c 2 L d 2 T c 3 L d 3 T c 4 L d 4... where T is the shift mp defined, for ny infinite word ( n ) n 0 with n letter for ny n 0, y T( n ) n 0 = ( n+1 ) n 0. One cn verify tht for integers c, d such tht d c 0 nd for ny infinite word w, T c L d (w) = L d c R(w) c nd T c L d (w) = Ld c R c (w). This explins the links etween the two representtions. The interested reder will lso find reltions etween this representtion nd the notion of S-dic systems defined y Ferenczi [6] s miniml dynmicl systems generted y finite numer of sustitutions. A prticulr well-known fmily of Sturmin words is the set of stndrd (or chrcteristic) Sturmin words. It corresponds to the cse where for ech k 0, c k = 0. Hence ny stndrd Sturmin word dmits unique representtion on the form: where d 1 0 nd d k 1 for ll k 2. w = lim n Ld 1 Ld 2 Ld 3 Ld 4...Ld 2n 1 L d 2n () To end this section, we recll useful reltions etween Sturmin morphisms. Theorem 3.2 [9] (see lso [15] for generliztion) The monoid {L,L,R,R,E} of Sturmin morphisms hs the following presenttion: (1) EE = Id, (2) EL = L E nd ER = R E, (3) L L n R = R R n L, for ny n 0. Note tht from (2) nd (3), we get: L L n R = R R n L for ny n 0. 4 Word qusiperiodicity nd morphisms In this pper, we consider minly infinite qusiperiodic words. However we first recll the notion of finite qusiperiodic words to llow us some comprisons. We consider definitions from [3]. A word u covers nother word w if for every i {1,..., w }, there exists j {1,..., z } such tht there is n occurrence of u strting t position i j + 1 in the word w. When u w, we sy tht u is qusiperiod of w nd tht w is qusiperiodic. A word is superprimitive if it is not qusiperiodic (Mrcus [12] clls miniml such words). One cn oserve tht ny word of length 1 is not qusiperiodic. The word w = hs,, s qusiperiods. Only is superprimitive. More generlly in [3], it is proved tht ny qusiperiodic finite word hs exctly one superprimitive qusiperiod. This is consequence of the fct tht ny qusiperiod of finite word w is proper order of w. 3/15
When defining infinite qusiperiodic words, insted of considering the strting indices of the occurrences of qusiperiod, for convenience, we choose to consider the words preceding the occurrences of qusiperiod. An infinite word w is qusiperiodic if there exist finite word u nd words (p n ) n 0 such tht p 0 = ε nd, for n 0, 0 < p n+1 p n u nd p n u is prefix of w. We sy tht u covers w, or tht w is u-qusiperiodic. The word u is lso clled qusiperiod nd we sy tht the sequence (p n u) n 0 is covering sequence of prefixes of the word w. The reder will find severl exmples of infinite qusiperiodic words in [11, 7]. Let us mention for instnce tht the well-known Fioncci word, the fixed point of the morphism ϕ:, is -qusiperodic. It is interesting to note tht ϕ ω () hs n infinity of superprimitive qusiperiods (see [7] for chrcteriztion of ll qusiperiods of ϕ ω ()). This shows gret difference etween qusiperiodic finite words nd qusiperiodic infinite words. The reder cn lso note tht for ny positive integer n, there exists n infinite word hving exctly n qusiperiods (s for exmple the word () n () ω )), or hving exctly n superprimitive qusiperiods [7]. To end this section, let us oserve tht ny qusiperiod of (finite of infinite) qusiperiodic word w is prefix of w. Hence w hs unique qusiperiod of smllest length tht we cll the smllest qusiperiod of w. When w is finite, the smllest qusiperiod of w is necessrily its superprimitive qusiperiod. When w is infinite, its smllest qusiperiod is lso superprimitive, ut there cn exist other superprimitive qusiperiods (see ove). Moreover: Lemm 4.1 If w is n infinite qusiperiodic word with smllest qusiperiod u, then uu is fctor of w. Proof. If uu is not fctor of w then the prefix v of u of length u 1 is qusiperiod of w. This is not possile if u is the smllest qusiperiod. Let us oserve tht Lemm 4.1 is not true for finite words s shown y the -qusiperiodic word. In the following we will lso use the immedite following fct: Fct 4.2 If w is (finite or infinite) u-qusiperiodic word nd f is non-ersing morphism, then f(w) is f(u)-qusiperiodic. 5 Sturmin non-qusiperiodic words In this section, we prove our min result (Theorem 5.6) which is chrcteriztion of ll nonqusiperiodic Sturmin words. Before this, we prove severl useful results. Let w e Sturmin word. Denoting y n the lest numer of etween two consecutive in w nd y i the initil numer of in w, we cn deduce from the lnce property of w tht w elongs to i { n, n+1 } ω. When 0 < i n, w elongs to { i n i, i n+1 i } ω nd w is i n i+1 -qusiperiodic (nd i n i+1 is the smllest qusiperiod of w). Thus: Fct 5.1 If w is non-qusiperiodic Sturmin word, then there exists n integer n such tht w elongs to n+1 { n, n+1 } ω { n, n+1 } ω. 4/15
Of course some Sturmin words in n+1 { n, n+1 } ω { n, n+1 } ω re qusiperiodic: it is the cse of the imge of ny qusiperiodic Sturmin word strting with y the Sturmin morphism L n R : n+1, n. A consequence of Fct 5.1 is: Lemm 5.2 For ll Sturmin word w nd x {,}, L x R x (w) = R x L x (w) is qusiperiodic. Proof. Without loss of generlity, ssume x =. From Theorem 3.2, L R = R L. Let us recll tht L R () = nd L R () =. From Fct 4.2, if w is qusiperiodic word, then L R (w) is qusiperiodic. Assume now tht w is Sturmin non-qusiperiodic word. By Fct 5.1, w elongs to n+1 { n, n+1 } ω { n, n+1 } ω for n integer n. Hence L R (w) elongs to one of the sets n+1 { n, n+1 } ω or { n, n+1 } ω. So L R (w) is n+2 -qusiperiodic or n+2 -qusiperiodic. Let us oserve tht ω nd L R ( ω ) = ω re not qusiperiodic. This shows tht Lemm 5.2 is not true for ritrry words (even if they re lnced), unlike the next fct which is direct consequence of the definition of L L :,, nd L L :,. Fct 5.3 For ny infinite word w, L L (w) is -qusiperiodic nd L L (w) is -qusiperiodic. Lemm 5.2 nd Fct 5.3 will e useful to prove tht our condition in Theorem 5.6 is necessry. To show it is sufficient, we now consider situtions where the imge of word y Sturmin morphism is not necessrily qusiperiodic. Lemm 5.4 Let x {,} nd let w e lnced word strting with x. The word L x (w) is qusiperiodic if nd only if w is qusiperiodic. Moreover in this cse, the smllest qusiperiod of L x (w) is the word L x (v) where v is the smllest qusiperiod of w. Proof. Without loss of generlity, we consider here tht x =. From Fct 4.2, if w is qusiperiodic then L (w) is qusiperiodic. From now on we ssume tht L (w) is u-qusiperiodic where u is the smllest qusiperiod of L (w). If w hs t most one occurence of, then w = ω or w = n ω for n integer n 0. Since L (w) is qusiperiodic, we hve w = ω nd we verify tht the smllest qusiperiod of w nd L (w) is = L (). From now on we ssume tht w contins t lest two occurrences of the letter. Denoting y n the lest numer of etween two consecutive occurrences of in w nd y i the numer of efore the first, since w is lnced, w i { n, n+1 } ω nd 0 i n + 1. If 0 < i n, then w nd L (w) re qusiperiodic with respective smllest qusiperiod i n i+1 nd i+1 n i+1 = L ( i n i+1 ). By hypothesis, w strts with, so we cnnot hve i = 0. In the cse i = n + 1: w n+1 { n, n+1 } ω nd L (w) n+2 { n+1, n+2 } ω. Since u is qusiperiod of L (w), u is prefix of L (w) nd strts with n+2. By Lemm 4.1, uu is fctor of L (w). It follows tht u ends with nd u = L (v) for word v { n, n+1 }. Now we prove tht v is qusiperiod of w. Let (p k u) k 0 e covering sequence of L (w) (p 0 = ε nd for ll k 0, p k u is prefix of L (w) nd p k+1 p k u ). Since u strts with n+2, for ech k 0, there exists word p k such tht p k = L (p k ). Of course, p 0 = ε. Since v {n, n+1 }, we cn deduce for ech k 0 tht p k v is prefix of w. If for k, p k+1 p k > v, then p k+1 = p kvy for word y nd consequently p k+1 = p k ul (y) which contrdicts the fct tht p k+1 p k u. So 5/15
for ech k 0, p k+1 p k v. We hve shown tht (p k v) k 0 is covering sequence of w, so v is qusiperiod of w. Assume w hs qusiperiod v strictly smller thn v. Both v nd v re prefixes of w, so v = v s for non-empty word s. Then L (v ) = L (v) L (s) < L (v) nd L (v ) is qusiperiod of L (w) strictly smller thn u = L (v). This contrdicts the definition of u, so v is the smllest qusiperiod of w. Lemm 5.5 Let x,y e letters such tht {x,y} = {,} nd let w e word strting with x. The word R y (w) is qusiperiodic if nd only if w is qusiperiodic. Moreover when these words re qusiperiodic, the smllest qusiperiod of R y (w) is the word R y (v) where v is the smllest qusiperiod of w. Proof. Without loss of generlity, we consider here tht x = nd y =. From Fct 4.2, if w is qusiperiodic then R (w) is qusiperiodic. Assume now tht R (w) is qusiperiodic nd let u e its smllest qusiperiod. By hypothesis, w strts with, so does u. Since is not fctor of R (w) wheres y Lemm 4.1 uu is fctor of R (w), we deduce tht u ends with. Thus there exists word v such tht u = R (v). As done in the proof of Lemm 5.4 for the cse w n+1 { n, n+1 } ω, we cn show tht v is qusiperiod of u nd more precisely tht it is its smllest qusiperiod. The reder cn oserve one difference etween the two previous lemms: Lemm 5.4 considers only lnced words when Lemm 5.5 works with ritrry words (strting with x). Note tht Lemm 5.4 ecomes flse if we do not consider lnced words. Indeed the word w = () ω is not qusiperiodic, wheres L (w) = () ω is -qusiperiodic. The two lemms ecome lso flse if we consider Sturmin words strting with y where {x,y} = {,}. Indeed, let us consider the cse x =, y = : it is known [7] tht the word w = (L R ) ω () is not qusiperiodic; this Sturmin word strts with nd the word L (w) (resp. R (w)) is -qusiperiodic (resp. -qusiperiodic). We cn now estlish the nnounced chrcteriztion of non-qusiperiodic Sturmin words. Theorem 5.6 A Sturmin word w is not qusiperiodic if nd only if it cn e infinitely decomposed over {L,R } or over {L,R }. In other words Sturmin word w is not qusiperiodic if nd only if or where d k 1 for ll k 2 nd d 1 0. w = lim n Ld 1 R d 2 Ld 3 R d 4 w = lim n Ld 1 Rd 2 L d 3 Rd 4...L d 2n 1...L d 2n 1 R d 2n () R d 2n () Proof. We first show tht the condition is necessry. Let w e non-qusiperiodic Sturmin word. By Theorem 3.1, w = lim n Ld 1 c 1 R c 1 Ld 2 c 2 R c 2...Ld 2n 1 c 2n 1 R c 2n 1 L d 2n c 2n R c 2n () where d k c k 0 for ll integer k 1, d k 1 for k 2 nd if c k = d k then c k 1 = 0. By Lemm 5.2, for x {,} nd ny Sturmin word, L x R x (w) is qusiperiodic. By Fct 4.2, this implies tht for ll k 1, c k = d k or c k = 0. 6/15
Assume tht c k = 0 nd c k+1 = 0 for n integer k 1. Then w = fl L (w ) or w = fl L (w ) for Sturmin word w nd morphism f. By Fct 5.3, w is qusiperiodic. So for ech k 1, c k = 0 implies c k+1 = d k+1. We know tht for ech k 2, c k = d k implies c k 1 = 0. This is equivlent to sy tht for ech k 1, c k 0 implies c k+1 d k+1. But there for ech k, c k = d k or c k = 0. Thus c k = d k implies c k+1 = 0, the condition is necessry. Let us now show tht ny Sturmin word w tht cn e decomposed infinitely over {L,R } is not qusiperiodic (cse {L,R } is similr). Assume y contrdiction tht it is not the cse. Let S e the set of ll Sturmin words w tht cn e decomposed over {L,R } nd tht re qusiperiodic. Let u e qusiperiod of smllest length mong ll qusiperiods of words in S, nd let w e n element of S hving u s qusiperiod. By definition, w = L (w ) or w = R (w ) for word w in S. Since d 3 0, w strts with the letter. By Lemms 5.4 nd 5.5, u = L (v) or u = R (v) where v is the smllest qusiperiod of w. Since ω nd ω re not Sturmin words (they re lnced ut not ultimtely qusiperiodic), v 0 nd v 0. Consequently v < u. This contrdicts the choice of u. Hence S is empty. Given word w, let us denote X(w) the set of infinite words hving the sme set of fctors thn w: X(w) is invrint y the shift opertor nd is clled the sushift ssocited with w. When w is Sturmin, it is known (see [5]) tht word w elongs to X(w) if nd only if it is Sturmin nd the ssocited sequence (d k ) k 0 in its decomposition of Theorem 3.1 is the sme s the one involved in the decomposition of w. To end this section, we oserve tht ny stndrd Sturmin word (tht is Sturmin word tht cn e decomposed using only L nd L ) is necessrily qusiperiodic. This gives new proof of result y T. Monteil [13, 14]: ny Sturmin sushift contins qusiperiodic word (let us mention tht the resutl of T. Monteil is more precisely: ny Sturmin sushift contins multiscled qusiperiodic word, tht is word hving n infinity of qusiperiods). The interested reder will find mterils in Section 7 to show tht ny stndrd Sturmin word hs n infinity of qusiperiods (see Lemm 7.5). Theorem 5.6 lso shows tht in ny Sturmin sushift, there is non-qusiperiodic word. 6 A connection with Lyndon words The im of this short section is to give nother chrcteriztion of non-qusiperiodic Sturmin words relted to Lyndon words (see Theorem 6.5 elow). Let us recll notions on finite [8] nd infinite [18] Lyndon words. We cll suffix of n infinite word w ny word w such tht w = uw for given word u. When u ε, we sy tht w is proper suffix of w. This definition llows us to dopt the sme definition for finite nd infinite Lyndon word. Let e totl order on A (in wht follows, { } denotes the lphet {,} with ). This order cn e extended into the lexicogrphic order on words over A. A (finite or infinite) word over (A, ) is Lyndon word if nd only if w is strictly smller thn ll its proper suffixes. Any infinite Lyndon word hs infinitely mny prefixes tht re (finite) Lyndon words (nd so n infinite Lyndon word cn e viewed s limit of these prefixes). The following sic property of finite Lyndon words ws pointed out y J.P. Duvl (see Acknowledgments): Fct 6.1 Any finite Lyndon word is unordered, tht is the only orders of Lyndon word w re ε nd w. 7/15
This llows us to stte reltion etween infinite Lyndon words nd non-qusiperiodic infinite words (cf Corollry 6.3). Fct 6.2 If w is n infinite u-qusiperiodic word, then ny prefix of w of length t lest u + 1 is not unordered. Proof. If p is prefix of w of length t lest u + 1, then p hs for suffix prefix s of u (of length t most u ). Since u is prefix of w, u is lso prefix of p, nd so s is order of p. Corollry 6.3 Any Lyndon word is not qusiperiodic. Our min Theorem 6.5 is direct consequence of this corollry nd the following chrcteriztion. Following [16] we sy tht morphism f preserves (finite) Lyndon words if for ny (finite) Lyndon word u, f(u) is lso Lyndon word. We hve: Proposition 6.4 [16] A Sturmin morphism f preserves Lyndon words over { } if nd only if f {L,R }. Theorem 6.5 A Sturmin word w over {, } is non-qusiperiodic if nd only if w is n infinite Lyndon word over { } or over { }. Proof. Let w e Sturmin word. By corollry 6.3, if w is n infinite Lyndon word then w is not qusiperiodic. Assume now tht w is not qusiperiodic. By Theorem 5.6, w = lim n Ld 1 Rd 2... L d 2n 1 R d 2n () or w = lim n Ld 1 Rd 2...Ld 2n 1 R d 2n () for some integers (d k ) k 1 such tht d k 1 for ll k 2 nd d 1 0. Proposition 6.4 implies tht, since is Lyndon word, for ech n 1, L d 1 Rd 2...L d 2n 1 R d 2n () is Lyndon word over nd L d 1 Rd 2...Ld 2n 1 R d 2n () is Lyndon word over. Hence w is n infinite Lyndon word over or over. To end this section we study the converse of Corollry 6.3 nd Fct 6.2. The converse of Corollry 6.3 is not true in generl. For instnce we cn consider ny Sturmin word w over {,} nd the word p =. Then pw is not qusiperiodic since p is not lnced nd so not fctor of w. Moreover, since p strts with the letter, pw cnnot e Lyndon word if. It is neither Lyndon word if since for ny prefix p of w, p w. The converse of Fct 6.2 is lso flse: let w e n infinite word nd p e n integer, if ll prefixes of w of length greter thn p + 1 re unordered, then w is not necessrily qusiperiodic. To prove this, it is sufficient to consider the word w = ω. A more complex ut interesting exmple, pointed out y P. Sééold (see Aknowledgements), is the well-known Thue-Morse word T, fixed point of the morphism µ such tht µ() = nd µ() =. The word T strts with nd ny prefix of length t lest 4 ends with, or. But T is not qusiperiodic: indeed it is well-known tht T is overlp-free ( word is overlp-free if it contins no fctor of the form xuxux where x is letter, or equivlently it contins no fctor tht cn e written oth pv nd vs with p < v ) nd we cn oserve tht: Fct 6.6 An overlp-free infinite word is never qusiperiodic. 8/15
Proof. Let w e u-qusiperiodic infinite word nd let (p n u) n 0 e covering sequence of w. If there exists n 0 such tht p n+1 p n < u, then p n+1 u = p n us for word s such tht s = p n+1 p n < u. Hence there exists word p such tht us = pu, then w is not overlp-free. If for ll n 0 we hve p n+1 p n = u, then w = u ω is lso not overlp-free. Finlly let us mention tht this fct is not vlid for finite words since there exist some overlpfree words tht re squre (see [19], cf. lso [4] for chrcteriztion of such words). 7 Sturmin morphisms nd qusiperiodicity We sy tht morphism f is qusiperiod-free if for ny non-qusiperiodic word w, f(w) is lso nonqusiperiodic. A non-qusiperiod-free morphism will just e clled qusiperiodic. Let us oserve tht ll Sturmin morphisms (except E nd Id) re qusiperiodic. To verify it, it is sufficient to show tht L, L, R nd R re qusiperiodic. For L nd R (cse L nd R re similr) we hve: ω nd ω re non-qusiperiodic lthough L ( ω ) = () ω nd R ( ω ) = () ω re -qusiperiodic. In the previous section, we encounter (Lemm 5.2 nd Fct 5.3) two different kinds of Sturmin morphisms. The morphism L L mps ny word into qusiperiodic one, wheres there exists non-qusiperiodic word w such tht L R (w) is not qusiperiodic. Generlizing these two exmples we oserve tht the set of qusiperiodic morphisms cn e prtitioned using the following notions: 1. A morphism f on A is clled strongly qusiperiodic (resp. on suset X of A ω ) if for ech non-qusiperiodic infinite word w (resp. w X), f(w) is qusiperiodic. 2. A morphism f on A is clled wekly qusiperiodic (resp. on suset X of A ω ) if there exist two non-qusiperiodic infinite words w,w (resp. w,w X) such tht f(w) is qusiperiodic, nd f(w ) is non-qusiperiodic. The im of this section is to nswer the two following questions: Which re the strongly (resp. wekly) qusiperiodic Sturmin morphisms? Which re the strongly (resp. wekly) qusiperiodic Sturmin morphisms on (the set of) Sturmin words? We note tht the two questions hve different nswers. Indeed L R s shown y Lemm 5.2 is strongly qusiperiodic on Sturmin words, ut s lredy sid, L R ( ω ) is not qusiperiodic. Of course, ny strongly qusiperiodic Sturmin morphism is strongly qusiperiodic on Sturmin words, or equivlently (since Sturmin morphism is qusiperiodic), ny wekly qusiperiodic Sturmin morphism on Sturmin words is wekly qusiperiodic. 7.1 A property of strongly qusiperiodic morphisms Before going further, we mention the following immedite result: Lemm 7.1 Let f e morphism. If there exist morphisms f 1, f 2, f 3 such tht f = f 1 f 2 f 3 nd such tht f 2 is strongly qusiperiodic, then f is strongly qusiperiodic. 9/15
We oserve tht (quite nturlly) Lemm 7.1 ecomes flse when replcing strongly qusiperiodic y wekly qusiperiodic. For instnce, tking f 1 = Id, f 2 = L nd f 3 = L, we hve f 2 wekly qusiperiodic nd f 1 f 2 f 3 strongly qusiperiodic. There re cses where we cn hve f 2 wekly qusiperiodic nd f 1 f 2 f 3 qusiperiod-free, ut this is not possile when f 1, f 2 nd f 3 re Sturmin morphisms since ll Sturmin morphisms re qusiperiodic. To give n exmple of such cse, we need the following result: Lemm 7.2 The morphism g defined y g() = nd g() = is qusiperiod-free morphism. Proof. Let w e n infinite word such tht g(w) is qusiperiodic. We show tht w is lso qusiperiodic. Let u e the smllest qusiperiod of g(w). Since u is prefix of g(w), u = g(v)p for proper prefix p of g() = or of g() = : p {ε,,,,, }. First we oserve tht if or does not occur in w, then w is qusiperiodic. From now on we ssume tht oth nd occur in w. Consequently v 0 nd v 0. It follows tht g(v) strts with () 2n for n integer n 0 nd with 4m for n integer m 0: of course m = 0 or n = 0. Moreover g(v) ends with () 2n for n integer n 0 nd with 4m for n integer m 0: once gin m = 0 or n = 0. By Lemm 4.1, uu is fctor of g(w). We then deduce tht p = ε since for ll the other potentil vlues, none of the words in {() 2n, 4m }p{() 2n, 4m } could e fctor of g(w). Let (p l u) l 0 e covering sequence of prefixes of g(w). As done in the proof of Lemm 5.4, we cn find covering sequence (p l v) l 0 of prefixes of w: the word v is qusiperiod of w. Now let us consider the morphisms f 1 = Id, f 2 = L, nd f 3 defined y f 3 () =, f 3 () =. By the previous lemm f 1 f 2 f 3 = g is qusiperiod-free wheres f 2 is wekly qusiperiodic. To end this section, we let the reder verify tht f 3 is qusiperiod-free nd more generlly tht ny morphism h defined y h() = i, h() = j with i 1 nd j 1 is qusiperiod-free. 7.2 Wekly nd strongly qusiperiodic Sturmin morphisms In this section, we chrcterize wekly qusiperiodic Sturmin morphisms. (Equivlently this chrcterizes strongly qusiperiodic Sturmin morphisms since ny Sturmin morphism is wekly or strongly qusiperiodic.) Proposition 7.3 A Sturmin morphism is wekly qusiperiodic if nd only if it elongs to the set {E,Id}{L,R } {L,R } {E,Id}{L,R } {L,R }. The proof, given t the end of the section, is consequence of the next lemms. Lemm 7.4 Let f e morphism in {L,L,R,R } different from the identity. The morphism f elongs to {L,R } {L,R } {L,R } {L,R } if nd only if f cnnot e written f = f 1 f 2 f 3 with f 1,f 3 {L,L,R,R } nd f 2 verifying one of the four following properties: 1. f 2 L {L,L,R,R } L L {L,L,R,R } L, or 2. f 2 = R gl with g {R,L } or f 2 = R gl with g {R,L }, or 3. f 2 R R + R or f 2 R R + R, or 10/15
4. f 2 R + L+ R = L + R+ R or f 2 R + L+ R = L + R+ R. Proof. First we let the reder verify using Theorem 3.2 tht if f elongs to {L,R } {L,R } {L,R } {L,R } then it cnnot e written f = f 1 f 2 f 3 with f 1,f 2,f 3 s in the lemm. From now on ssume tht f cnnot e written f = f 1 f 2 f 3 with f 1,f 2,f 3 s in the lemm. Let g 1,...,g n (n 1 since f is not the identity) in {L,L,R,R } such tht f = g 1...g n. We first consider the cse where g 1 = L. By Impossiility 1 for f 2, for ech i > 1, g i L. If there exists n integer i > 1 such tht g i = R, then g 1...g i = hl R l or g 1...g i = hr R l for morphism h nd n integer l 1. In the first cse y Impossiility 4 for f 2, for ll integer j > i, f j R. In the second cse y Impossiilities 3 nd 4 for f 2, for ll integer j > i, we lso hve f j R. Thus f L {R,L } {L,R }. Assume now the more generl cse (thn g 1 = L ) where there exists n integer i 1 such tht g i = L nd g j L for 1 j < i (the first occurrence of L ppers t the position i). Smely s ove, we show tht g = g i...g n L {R,L } {L,R }. By Impossiility 1 for f 2, for ech integer j, 1 j < i, g j L. Thus g j {R,R } for ech 1 j < i. We hve three cses: If f R g, then y Impossiility 4 for f 2, we hve f L {R,L } {L,R } {R,L }. If f hr + R g for morphism h {R,R }, then y Impossiility 2 for f 2, h R nd so f R+ R g; then y Impossiilities 3 nd 4 for f 2 we hve f {L,R } {L,R }. If f R g, f {L,R } {L,R }. So when there exists n integer i 1 such tht g i = L, f {L,R } {L,R }. The cse where there exists n integer i 1 such tht g i = L leds similrly to f {L,R } {L,R }. Now we hve to consider the cse where for ll i, 1 i n, g i {L,L }. Then y Impossiility 3 for f 2, necessrily, f R R R R. Lemm 7.5 Every morphism f in L {L,L,R,R } L L {L,L,R,R } L is strongly qusiperiodic. Proof. We only prove the result for f in L {L,L,R,R } L (the other cse is similr exchnging the roles of the letters nd ). Let f = L f 1 f 2... f n L with n 0 nd f i in {L,L,R,R } for ll 1 i n. We prove y induction on n tht there exist morphisms g nd h such tht f = gl L h (nd so from Lemm 7.1 nd Fct 5.3, f is strongly qusiperiodic). The property is immedite for n = 0. Assume now n 1. If there exists i etween 1 nd n such tht f i = L or f i = L, we cn pply the induction hypothesis nd Lemm 7.1 to conclude. Now suppose tht for ll i, f i {L,L }. Three cses re possile: if f 1 = R, since L R = R L from Theorem 3.2, f = R L f 1...f n L nd we conclude y the induction hypothesis. If f n = R we cn proceed similrly. Assume now f 1 = R nd f n = R (this implies n 2). Let j e the gretest integer (1 j n) such tht f j = R. Then f = L f 1...f j 1 R R n j L, nd y Theorem 3.2 f = L f 1... f j 1 L L n j R. We conclude y the induction hypothesis. Remrk: we could hve used nother pproch oserving tht L R (w) (L R () =, L R () = ) is -qusiperiodic for every infinite word w strting with, nd deducing tht every morphism of the form L R fl with f {L,R,R } is strongly qusiperiodic. 11/15
Lemm 7.6 Every morphism f = R gl with g {R,L } or f = R gl with g {R,L } is strongly qusiperiodic. Proof. We only prove the first cse, the other one is similr. Let g = g 1...g n (necessrily n 1) such tht g {R,L } nd for ech i etween 1 nd n, g i {L,L,R,R }. If there exists n integer i such tht g i = L then the result is immedite from Lemm 7.5. Consequently we consider tht g ({L,R } R ) + {L,R }. Thus the morphism f cn e decomposed f = f 1 hf 2 with h R L R+ R L. If i,j 0,k 1 re the integers such tht h = L i R R kl R j, Theorem 3.2 shows tht h = L i L L k R R j. Consequently Lemms 7.1 nd 7.5 imply tht h is strongly qusiperiodic. Remrk: here gin we could hve used nother pproch oserving tht R R (w) (R R () =, R R () = ) is -qusiperiodic for every infinite word w strting with, nd deducing tht every morphism of the form R R fl with f {L,R,R } is strongly qusiperiodic. This pproch is used to prove: Lemm 7.7 Any morphism f in R R + R R R + R is strongly qusiperiodic. Proof. Let j 1 e n integer such tht f = R R j R. Let w e word. If w strts with, R R (w) is -qusiperiodic, nd so f(w) is qusiperiodic. If w strts with, R j 1 R (w) lso strts with. Then R R j R () is -qusiperiodic. Lemm 7.8 Every morphism f in R + L + R = L + R + R or in R + L+ R = L + R+ R is strongly qusiperiodic. Proof. Theorem 3.2 implies R + L+ R = L + R+ R nd R + L+ R = L + R+ R. We prove only the first cse, the other one is similr. Let n 1. It is esy to see tht R L n R (w) (R L n R () = n, R L n R () = n ) is n+1 -qusiperiodic if w strts with, nd is n -qusiperiodic if w strts with. By Lemm 7.1, ny morphism in R + L + R is qusiperiodic. Proof of Proposition 7.3. From Theorem 3.2, EL = L E nd ER = R E, so ny Sturmin morphism cn e written fg with f {Id,E} nd g {L,L,R,R }. Thus Proposition 7.3 is consequence of the following one: morphism f in {L,L,R,R } is wekly qusiperiodic if nd only if f elongs to the set X = {L,R } {L,R } {L,R } {L,R }. To prove this, ssume first tht f {L,L,R,R } is wekly qusiperiodic. By Lemm 7.1, this morphism cnnot e written f = f 1 f 2 f 3 with f 2 strongly qusiperiodic morphism. Hence y Lemms 7.4, 7.5, 7.6, 7.7 nd 7.8, f elongs to X. Assume now tht f X. Since f is Sturmin, it is qusiperiodic nd so we just hve to prove the existence of one word such tht f(w) is not qusiperiodic. So we just hve to prove the existence of one word w such tht f(w) is not qusiperiodic. We do it for f {L,R } {L,R } (the other cse is similr). There exist morphisms g {L,R } nd h {L,R } such tht f = gh. We cn verify tht h( ω ) = n ω for n integer n 1, nd so is lnced word. By Lemms 5.4 nd 5.5, we thus deduce tht (g(h( ω )) = f( ω ) is not qusiperiodic. 12/15
7.3 Wekly Sturmin morphisms on Sturmin words Proposition 7.3 nd Lemm 5.2 show tht some morphisms, s for instnce L R, re wekly qusiperiodic wheres they re strongly qusiperiodic on Sturmin words. This section llows us to chrcterize ll these morphisms. Let us recll tht since Sturmin morphism is qusiperiodic, ny Sturmin morphism is wekly or strongly qusiperiodic on Sturmin words. Proposition 7.9 A Sturmin morphism different from E nd Id is wekly qusiperiodic on Sturmin words if nd only if it elongs to {E,Id}{L,R } {E,Id}{L,R }. Proof. Let us mke preliminry remrk: for ny morphism f, f is wekly qusiperiodic on Sturmin words if nd only if Ef is wekly qusiperiodic on Sturmin words (since for ny word w, w is qusiperiodic if nd only if E(w) is qusiperiodic). Assume first f {E,Id}{L,R } {E,Id}{L,R }. Without loss of generlity, we cn ssume f {L,R } {L,R }. If f elongs to {L,R } (resp. to {L,R } ), using Theorem 5.6 we oserve tht f((l R ) ω ) (resp. f((l R ) ω )) is not qusiperiodic. Since ny Sturmin morphism is qusiperiodic, f is wekly qusiperiodic on Sturmin words. Now ssume f is wekly qusiperiodic on Sturmin words. Oserve tht from Theorem 3.2(2), f {E,Id}{L,L,R,R }. Without loss of generlity, from the preliminry remrk, we cn ssume tht f elongs to {L,L,R,R } nd prove tht f {L,R } {L,R }. By Proposition 7.3, f elongs to {L,R } {L,R } {L,R } {L,R }. Assume y contrdiction tht f {L,R } {L,R }. One of the following four cses holds: 1. f = gl R with g {L,R } {L,R } ; 2. f = gr R i with g {L,R }, i 1; 3. f = gl R with g {L,R } {L,R } ; 4. f = gr R i with g {L,R }, i 1. Cse 1: Assume f = gl R nd let w e non-qusiperiodic Sturmin word. By Lemm 5.2, f(w) is qusiperiodic. Cse 2: Assume f = gr R i nd let w e non-qusiperiodic Sturmin word. By Theorem 5.6, w cn e decomposed over {L,R } or over {L,R }. So f(w) = gr R i L (w ) or L (w ) for (non-qusiperiodic) Sturmin word w nd n integer j 0. Thus y Lemm 5.2, Lemm 7.7 nd Lemm 7.6, f(w) is qusiperiodic. Cses 3 nd 4 re respectively similr to cses 1 nd 2. In ll cses, f(w) is qusiperiodic for ny non-qusiperiodic Sturmin word w, nd so for ny Sturmin word (y Fct 4.2). Thus f is strongly qusiperiodic on Sturmin words. This is contrdiction, so f {L,R } {L,R }. f(w) = gr R i R (w ) or f(w) = gr R i+j Acknowledgements Fcts 6.2 nd 6.6 were respectively oserved y J.P. Duvl nd P. Sééold during tlk given y the second uthor t the Premières Journées Mrseille-Rouen en comintoire des mots tking plce in Rouen in June 2005. Mny thnks to them nd to J. Cssigne, C. Muduit nd J.Nérud, the orgnizers of these Journées. 13/15
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