Words and morphisms with Sturmian erasures

W ORDS AND MORPHISMS W ITH STURMIAN ERASURE S F ABIEN DURAND, ADEL GUERZIZ, AND MICHEL KOSKAS Abstract. W e say x ∈ { 0 , 1 , 2 } N is a w ord with Sturmia n erasures if for any a ∈ { 0 , 1 , 2 } the w ord obtained erasing all a in x is a Sturmian word. A large family of such words is given co ding tra jectories o f balls in the ga me of billiards in the cube. W e prov e that the monoid of morphisms mapping a ll words with Sturmian era sures to words with Sturmian eras ures is not finitely gene r ated. 1. Introduction In this pap er w e are in terested in w ords x defined on the alphab e t A 3 = { 0 , 1 , 2 } ha ving the follo wing prop e rt y: F or an y letter a ∈ A 3 , the w ord obtained erasing all a in x is a Stur mia n w ord. W e sa y x is a wor d wi th Sturmian er asur es . Sturmian words are w ell-kno wn ob jec ts that can b e defined in man y w a ys. F or example, a w ord is Sturmian if and only if f or all n ∈ N the num b er o f distinct finite w ords of length n app earing in x is n + 1 (see [Lo] for complete references ab out Sturmian w ords). Sturmian w ords can also be view ed as tra jectories of balls in the ga me of billiards in the square. W e will see that a large family of w ords with Sturmian erasures is the family o f tra jectories of balls in the game of billiards in the cub e. Here w e are intere sted in the morphisms f : A 3 → A ∗ 3 (the free monoid gener- ated by A 3 ) that send all w ords with Sturmian erasures t o words with Sturmian erasures. W e call suc h f the morph i s ms with Sturmian er a sur es and w e denote b y MSE t he set of all these morphisms. Our main result is the following: Theorem 1. We have: (1) The monoid MSE is not fin i tely g e n er ate d; (2) MSE is the union of MSE ε and the set of p ermutation of A 3 . (3) If f : A 3 → A ∗ 3 is lo c al ly with Sturmian er asur es such that f ( i ) is the empty wor d fo r some i ∈ A 3 then it is a morphism with Sturmian er asur es; Where MSE ε is the set of morphisms with Sturmian erasures having the empt y w ord as an image of a letter and lo cally with Sturmian erasures means tha t there exists a w o rd with Sturmian erasures suc h that f ( x ) is a w ord with Sturmian erasures. W e r ecall tha t F. Mignosi and P . S´ e ´ eb old prov ed in [MS] that the monoid of the morphisms sending all Sturmian words to Sturmian w ords is finitely generated. In the last section we giv e some other informations ab out the words with Stur- mian erasures: sym bolic complexit y , link with the game of billiards in the cub e, balanced prop erty and palyndroms. 1991 Mathematics Subje ct Classific ation. Primar y: 68R15 . Key wor ds and phr ases. W ords with Sturmian erasur es, Sturmian words, Sturmian mor- phisms. 1 2 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS 2. Definitions, not a tions and back ground 2.1. W ords, morphisms and matrices. W e call alp h ab et a finite set of ele- men ts called l e tters . Let A b e an a lphab et and A ∗ b e the free monoid g enerated b y A . The elemen ts o f A ∗ are called wor ds . The neutral elemen t of A ∗ , also called the empty wor d , is denoted by ε . W e set A + = A ∗ \ { ε } . Let u = u 0 u 1 · · · u n − 1 b e a w ord of A ∗ , u i ∈ A , 0 ≤ i ≤ n − 1. Its le n gth is n and is denoted b y | u | . In pa rticular, | ε | = 0. If a ∈ A then | u | a denotes the num b er of o ccurrences of the letter a in the w ord u . W e call in fi nite wor d s the elemen ts of A N and w e set A ∞ = A N ∪ A ∗ . Let x ∈ A ∞ and y ∈ A ∗ . W e say that y is a fac tor o f x if there exist u ∈ A ∗ and v ∈ A ∞ suc h that x = uy v . In particular if u = ε then y is a pr efix of x a nd if v = ε then y is a suffix of x . An infinite w ord x = ( x n ; n ∈ N ) of A N is called eventual ly p erio dic if there exist t w o w ords u ∈ A ∗ and v ∈ A + suc h that x = uv v v . . . . The c omplexi ty function o f an infinite w ord x is the function P x : N → N where P x ( n ) is the n um b er o f factors o f length n of x . Let A , B and C b e three alpha b ets. A mo rphism f is a map from A to B ∗ . It induces b y concatenation a map from A ∗ to B ∗ . If f ( A ) is included in B + , it induces a map from A N to B N . All these maps are also written f . T o a morphism f : A → B ∗ is a sso ciated the matrix M f = ( m i,j ) i ∈ B ,j ∈ A where m i,j is the n um b er of o ccurrences of i in the word f ( j ). If g is a morphism from B to C ∗ then we can c hec k we hav e M g ◦ f = M g M f . 2.2. Sturmian w ords and Sturmian morphisms. Let A b e a finite alphabet. An infinite w ord x ∈ A N is Sturmian if f or all n ∈ N , P x ( n ) = n + 1 . Since P x (1) = 2, w e can supp ose A = { 0 , 1 } (see [Lo] fo r more informations ab out these w ords). A morphism f f rom A t o A ∗ is Sturmian if for all Sturmian word x the w ord f ( x ) is Sturmian. A morphism f is lo c al ly S turmian if there exists at least a Sturmian w ord x suc h that f ( x ) is Sturmian. W e call St the semigroup generated by the morphisms E , ϕ , a nd e ϕ defined by E : A ∗ − → A ∗ ϕ : A ∗ − → A ∗ e ϕ : A ∗ − → A ∗ 0 7− → 1 0 7− → 01 0 7− → 10 1 7− → 0 1 7− → 0 1 7− → 0 Theorem 2. [BS, MS] The fol lowing thr e e c onditions ar e e quivalent (1) f ∈ St ; (2) f lo c al ly Sturmian; (3) f Sturmian. 2.3. W ords with Sturmian erasures. L et A 3 = { 0 , 1 , 2 } and let x b e a infinite w ord of A N 3 . F or i ∈ A 3 w e denote π i : A 3 → A ∗ 3 the morphism defined by π i ( j ) = j if j ∈ A 3 with j 6 = i and π i ( i ) = ε . Definition 3. An infinite wor d x ∈ A N 3 is c al le d w or d with Sturmian erasures if and only if the wor d π i ( x ) is a Sturmian wor d for al l i ∈ A 3 . We say f : A 3 − → A ∗ 3 is a morphism with Sturmian erasures if f ( x ) is a wo r d with Sturmian er asur es for al l wor ds x ∈ A N 3 with Sturmian er asur es. W e call WSE the set of w ords with Sturmian erasures a nd MSE the set of mor- phisms with Sturmian erasures. W e r emark MSE is a monoid for the comp osition WORDS A ND MORPHI SMS WITH STURMIAN ER ASURES 3 la w o f morphism. The image of a Stur mia n w ord b y a morphism with Sturmian erasures is a w ord with Sturmian erasures. Hence WSE is not empty . Example 1. L et g : A 3 → A ∗ 3 b e the morphism defined by : g (0) = 02 , g (1) = 1 0 and g (2) = ε . Let F 0 = 0 and for n ≥ 0 F n +1 = ϕ ( F n ). Let F ∈ { 0 , 1 } N b e the unique fixed p oin t of ϕ in { 0 , 1 } N (see [Qu]). Then for each n ≥ 0 F n is a prefix of F , and we ha v e F = 0 10010100 1001 ... . This word is called the Fib onac ci wor d (remark that | F n +2 | = | F n +1 | + | F n | , n ≥ 0). It is a Sturmian w ord. F rom Theorem 2 we deduce that g ( F ) = 02100202 1 00210020 21002021 002100202100210 . . . is a w ord with Sturmian erasures. Hence WSE is not empt y . Let x ∈ A N 3 b e a w ord with Sturmian erasures . The w ord π 2 ( x ) is a Sturmian w ord and g ◦ π 2 = g . Moreo v er π 2 ◦ g |{ 0 , 1 } is a Sturmian morphism. Hence π 2 ◦ g ( x ) = π 2 ◦ g ◦ π 2 ( x ) = π 2 ◦ g |{ 0 , 1 } ( π 2 ( x )) = g |{ 0 , 1 } ( π 2 ( x )) is a Sturmian w ord. W e can also show that π 0 ◦ g ( x ) and π 1 ◦ g ( x ) are w ords with Sturmian erasures. Hence, g is a morphism with St urmia n erasures and MSE is not empt y . 3. Proo fs of points (2) e t (3 ) of Theorem 1 W e denote by MSE ε the set of morphisms of MSE suc h that there exists l ∈ A 3 with f ( l ) = ε . W e will pro v e that MSE is the union of MSE ε with the set of p erm utations on A 3 . This last set is generated by E 0 : A ∗ 3 − → A ∗ 3 E 1 : A ∗ 3 − → A ∗ 3 E 2 : A ∗ 3 − → A ∗ 3 0 7− → 0 0 7− → 2 0 7− → 1 1 7− → 2 1 7− → 1 1 7− → 0 2 7− → 1 2 7− → 0 2 7− → 2 W e denote b y MSE i , i ∈ A 3 , the set of morphisms with Sturmian erasures suc h that f ( i ) = ε . W e hav e MSE ε = MSE 0 ∪ MSE 1 ∪ MSE 2 . 3.1. Pro of of the p oint (3) of Theorem 1. W e start w ith the f o llo wing prop osition. Prop osition 4. If a morphism f : { 0 , 1 } → A ∗ 3 maps a Sturmian wor d de fi ne d on { 0 , 1 } N into a wor d with Sturmian er asur es then it maps an y Sturmian w o r d into a wor d with Sturmian er asur es. Pr o of. Let x ∈ { 0 , 1 } N b e a Sturmian w ord and f : { 0 , 1 } → A ∗ 3 b e a morphism suc h that f ( x ) is a w ord with Sturmian erasures. Let i b e a letter of A 3 and E : A 3 → { 0 , 1 } ∗ b e a morphism suc h that { 0 , 1 } = { E ( a ); i 6 = a, a ∈ A 3 } . Then E ◦ π i ◦ f ( x ) is Sturmian and E ◦ π i ◦ f : { 0 , 1 } → { 0 , 1 } ∗ is a lo cally Sturmian morphism. Hence, from Theorem 2 it is Sturmian. It follo ws that for ev ery Sturmian w ord y , f ( y ) is a w ord with Sturmian erasures. This ends the pro of.  No w we pro v e the p oin t (3) of Theorem 1. Let f : A 3 → A ∗ 3 , suc h tha t f ( i ) = ε for some i ∈ A 3 , and x ∈ A N 3 b e a word with Stur mia n erasures suc h that f ( x ) is a w ord with Sturmian erasures. W e remark that w e hav e f ◦ π i ( x ) = f ( x ) and that π i ( x ) is a Sturmian word. 4 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS W e can supp ose A 3 \ { i } = { 0 , 1 } . Hence the morphism f ◦ π i |{ 0 , 1 } satisfy the h yp o thesis of Prop osition 4. Consequen tly if y is a w ord with Sturmian erasures then f ( y ) = f ◦ π i ( y ) = f ◦ π i |{ 0 , 1 } ( π i ( y )) is a w ord with Sturmian erasures. ✷ Example 2. W e can r emark that there exist morphisms f : A 3 → A ∗ 3 suc h that for some w ord x ∈ WSE w e ha v e f ( x ) ∈ WSE but f is not a morphism with Sturmian erasures. F or example let F b e the Fib onacci word, f b e define d b y f (0) = 0, f ( 1 ) = 1 and f (2) = 01 2, g : A 3 → A ∗ 3 b e defined by g (0 ) = 0 1 , g (1) = 02 and g ( 2) = ε , and, h : A 3 → A ∗ 3 b e defined by h (0) = 02 , h (1) = 10 and h (2) = ε . As in Example 1 w e can pro v e that g , h and f ◦ g are morphisms with Sturmian erasures and consequen t ly x = g ( F ) and f ( x ) = f ◦ g ( F ) are w ords with Sturmian erasures. But w e remark that f ◦ h ( F ) is not a w ord with Sturmian erasures. Indeed 0012 10 is a prefix of f ( y ) and 00110 is a prefix of w = π 2 f ( y ). Consequen tly P w (2) = 4 and w is not a Sturmian w ord. 3.2. Pro of of the p oin t (2) of Theorem 1. W e need the follow ing lemma that follows f r o m Theorem 2 and the fact that the determinan t of the matrices asso ciated to ϕ , e ϕ and E belong to {− 1 , 1 } . Lemma 5. L et M b e the m atrix asso ciate d to the Sturmian morphi s m f . Then det M = ± 1 . Let us prov e the p oin t (2) of Theorem 1. This pro of is due to D. Bernardi. Let f b e a morphism of MSE. Let i ∈ A 3 and E : A 3 \ { i } → { 0 , 1 } ∗ b e a morphism suc h that { 0 , 1 } = { E ( a ); a 6 = i, a ∈ A 3 } . W e set h = E ◦ π i ◦ f ◦ g |{ 0 , 1 } where g : A 3 → A ∗ 3 is the mo r phism defined by: g (0) = 02, g (1) = 12 and g ( 2) = ε . As the letter i do es not app ear in the images of π i ◦ f , w e consider π i ◦ f as a morphism f rom A 3 to ( A 3 \ { i } ) ∗ . W e set M π i ◦ f = ( u , v , w ) where u , v and w are column v ectors b elonging to R 2 . W e recall g is a morphism with Sturmian erasures (see Example 1 o f the subsection 2.3). Hence the morphism h is Sturmian and w e hav e M h = ( u + w , v + w ). F rom L emma 5 det( u, v ) + det( u, w ) + det( w , v ) = det( u + w , v + w ) = ± 1 . W e do the same with g b eing one of the t w o follo wing morphisms : (0 7− → 01, 1 7− → 12, 2 7→ ε ) and (0 7− → 02, 1 7− → 01 , 2 7→ ε ). W e obtain finally (1) det ( u, v ) + det( u , w ) + det ( v , w ) = ± 1 . (2) det ( u, v ) + det( w , u ) + det( w , v ) = ± 1 . (3) det ( u, v ) + det( u , w ) + det ( w , v ) = ± 1 . The combin ations of the equations (1) and (2), (2) and (3), and, (3) and (1) imply resp ectiv ely that det( u, v ), det( w , u ) and det( w , v ) b elong to {− 1 , 0 , 1 } . F rom (1), o ne o f three determinan t s det( u, v ), det( u , w ) or det( v , w ) is differen t from 0. W e supp o se det ( u, v ) 6 = 0 (the other cases can b e treated in the same wa y). The set { u, v } is a base of R 2 , hence there exist t w o real n um bers a and b suc h that w = au + bv . W e ha v e a = det( w , v ) / det ( u, v ) and b = det( w , u ) / det ( v , u ). Moreo v er from ( 1 ) and (2) we see that det( u , w ) + det( v , w ) and − (det( u, w ) + WORDS A ND MORPHI SMS WITH STURMIAN ER ASURES 5 det( v , w )) = det( w , u ) + det( w , v ) b elong to { det( v , u ) − 1 , det( v , u ) + 1 } whic h is equal to { 0 , 2 } or {− 2 , 0 } . Consequen tly det( u, w ) + det( v , w ) = 0. Hence a = b and w = a ( u + v ) . The v ector w is the column of the matrix of a morphism therefore it has non-negativ e co ordinates whic h implies that a is non-negativ e. Supp ose a > 0. Then det( w , v ) and det( u, v ) are p ositiv e and ha v e t he same sign. Hence one of the equations (2) or (3) is equal to -3 o r 3 whic h is not p ossible. Consequen tly a = 0 and w = 0. Therefore for all i ∈ A 3 the matrix M π i ◦ f = ( m i ( c, d )) c ∈ A 3 \{ i } ,d ∈ A 3 has a column ( m i ( c, d i )) c ∈ A 3 \{ i } with en tries equal to 0. Tw o cases o ccurs. 1- There exists i, j ∈ A 3 ( i 6 = j ) suc h that d i = d j . In this case w e easily c hec k that f ( d i ) = ε . Consequen tly f b elongs to MSE ε . 2- The sets { d 0 , d 1 , d 2 } and A 3 are equal. In this case we can c hec k that f is a p erm utation. ✷ 4. Prime morphisms 4.1. Some t echnical definitions. Let A b e an alphabet and f : A → A ∗ b e a morphism. A letter a is called f -nilp o tent if there exists a n integer n suc h that f n ( a ) = ε (if it is no t am biguous w e will say it is nilp oten t). The set of f -nilp otent letters is denoted by N f . W e call P ′ f the set of letters a suc h tha t there exists an in teger n satisfying π N f ( f n ( a )) = a where π N f ( b ) = ε if b ∈ N f and b otherwise. The set of such letters is denoted b y P ′ f . W e sa y the letter a is f -p ermuting if there exists an integer n suc h that f n ( a ) ∈ ( N f ∪ P ′ f ) ∗ \ N f ∗ . W e denote b y P f the set of suc h letters. W e re- mark that P ′ f is included in P f . A letter a is called f - e xp ans i v e , or expansiv e when the con text is clear, if it is neither nilp oten t nor p ermuting. W e r emark the letter a ∈ A is f -expansiv e if and only if lim n → + ∞ | f n ( a ) | = + ∞ and it is f - p erm uting if and only if the sequence ( | f n ( a ) | ; n ∈ N ) is b o unded and is nev er equal to 0. The morphism f is nilp o tent if f ( A ) is included in N ∗ f , i.e., if there exists an in teger n such that f n ( a ) = ε for all a ∈ A . A morphism f is called exp an s ive if there exists a f -expansiv e letter. A morphism f is a unit if it is neither nilp otent nor expansiv e. In others w ords if f ( A ) is included in ( N f ∪ P f ) ∗ . Let M b e a monoid of morphisms. A morphism f ∈ M is said to b e prime in M if for any morphisms g and h in M suc h that f = g ◦ h , then g or h is a unit of M . W e sa y that f is of de gr e e n in M , n ∈ N , if an y decomp osition of f into a pro duct of prime and unit morphisms of M contains at least n prime morphisms and there exists at least one decompo sition of f into prime and unit morphisms of M con taining exactly n prime morphisms. The set of prime morphisms in St is { f ◦ g ◦ h ; f , h ∈ { I d, E } , g ∈ { ϕ, e ϕ }} . 4.2. Some conditions to b e a prime morphism. In the sequel w e need the follo wing morphisms whic h a r e extensions to the alphab et A 3 of the morphisms ϕ and e ϕ : ϕ 1 : A ∗ 3 − → A ∗ 3 , f ϕ 1 : A ∗ 3 − → A ∗ 3 0 7− → 01 0 7− → 10 1 7− → 0 1 7− → 0 2 7− → ε 2 7− → ε . 6 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS In the next prop osition we need the following lemma. Lemma 6. L et g ∈ MSE 2 . Then, | g ( a ) | ≥ 2 , | g ( a ) | 0 + | g ( a ) | 1 ≥ 1 and | g (01) | a ≥ 1 for al l a ∈ { 0 , 1 } . Pr o of. Supp ose for a ∈ { 0 , 1 } w e ha v e | g ( a ) | = 1, for example g ( a ) = b . Then π b ◦ g ( x ) is p erio dic f o r all x ∈ A N 3 . This con tradicts the fact that g belongs to MSE 2 . If | g ( a ) | = 0 w e hav e the same conclusion. This prov es the first part of the lemma. Supp ose | g ( a ) | 0 + | g ( a ) | 1 = 0. The n π 2 ◦ g ( x ) is p erio dic fo r all x ∈ A N 3 . This pro v es the second inequalit y . Supp ose | g (01) | a = 0, then a do es not app ear in g ( x ). This con tradicts the fact that g b elongs to MSE 2 .  Prop osition 7. L et i ∈ A 3 and f ∈ MSE i . We set A 3 = { i, j, k } . 1) If f ( j ) is ne ither a pr efix nor a suffix of f ( k ) and that f ( k ) is n either a pr efix nor a suffix of f ( j ) , then f is prim e in MSE i . 2) Mor e over, if f is prim e in MSE i and if we have | f (012) | j > | f (012) | k ≥ | f (012) | i , then f ( j ) is neither a pr e fi x nor a suffix of f ( k ) and f ( k ) is neither a pr efix nor a suffix of f ( j ) . Pr o of. W e only mak e the pro o f in the case i = 2. 1) W e supp ose f (0) is neither prefix nor suffix of f (1) and that f (1) is neither prefix nor suffix of f (0). W e pro ceed by con tradiction, i.e., w e supp ose there exist g , h ∈ MSE 2 whic h are not units suc h that f = g ◦ h . Let h 1 = π 2 ◦ h . W e hav e g ◦ h = g ◦ h 1 . W e define h 1 : { 0 , 1 } → { 0 , 1 } ∗ b y h 1 ( i ) = h 1 ( i ) for all i ∈ { 0 , 1 } . W e remark that h 1 is a Sturmian morphism hence it is a pro duct of ϕ, e ϕ and E (Theorem 2). Therefore h 1 is a pro duct of ϕ 1 , e ϕ 1 and π 2 ◦ E 2 . W e consider t w o cases. Supp ose h 1 6∈ { π 2 ◦ E 2 , π 2 ◦ E 2 ◦ E 2 } . Then, for example, h 1 is equal to h 2 ◦ ϕ 1 where h 2 is a pro duct o f ϕ 1 , e ϕ 1 and π 2 ◦ E 2 . The other cases ( h 1 = h 2 ◦ f ϕ 1 or h 1 = h 2 ◦ ϕ 1 ◦ π 2 ◦ E 2 or h 1 = h 2 ◦ f ϕ 1 ◦ π 2 ◦ E 2 ) can b e treated in the same wa y . W e hav e f (0) = g ◦ h 2 ◦ ϕ 1 (0) = g ◦ h 2 (0) g ◦ h 2 (1) and f (1) = g ◦ h 2 (0) . which con tradicts the hypothesis. Supp ose h 1 ∈ { π 2 ◦ E 2 , π 2 ◦ E 2 ◦ E 2 } , then h 1 (0) = 1, h 1 (1) = 0 and h 1 (2) = ε or h 1 (0) = 0, h 1 (1) = 1 a nd h 1 (2) = ε . In b oth case we easily c hec k tha t h is a unit of MSE 2 . This ends the first part of the pro of. 2) W e now suppose f is a prime morphism in MS E 2 suc h that | f (01) | j > | f (01) | k ≥ | f (0 1 ) | 2 , where A 3 = { j, k , 2 } . W e pro ceed b y con tradiction. W e supp o se f (1) is a prefix of f (0). The o ther case can b e treated in t he same w ay . There exist u and v in A ∗ 3 suc h that f (0 ) = uv and f (1) = u . W e define g , h : A 3 → A ∗ 3 b y g (0) = u , g (1 ) = v , g (2) = ε , h (0) = 01, h (1) = 02 and h (2) = ε . W e remark that h is not a unit and f = g ◦ h . T o end the pro o f it suffices to show that g is not a unit of MSE 2 . W e start pro ving g b elongs to MSE 2 . Let x ∈ WSE. As in Example 1 w e can pro ve that h b elongs to MSE 2 . Consequen tly h ( x ) b elongs t o WSE. Moreo v er f ( x ) = g ( h ( x )) b elongs t o WSE. F rom the p oin t (3) of T heorem 1 it comes that g b elongs to MSE 2 . F rom Lemma 6 w e ha v e | f (01) | = | g (010) | ≥ 6. Consequen tly | f (01) | 0 + | f (01) | 1 + WORDS A ND MORPHI SMS WITH STURMIAN ER ASURES 7 | f (01) | 2 ≥ 6. F rom the h yp othesis it comes that | g (0 10) | j = | f (01) | j ≥ 3. Hence 2 | g (01) | j − | g (1 ) | j ≥ 3 a nd | g (01 ) | j ≥ 2. No w w e prov e b y induction that for all n ∈ N we hav e | π 2 ◦ g n (01) | ≥ n + 2 , | π 2 ◦ g n (0) | ≥ 1 , and | π 2 ◦ g n (1) | ≥ 1 . This is true for n = 0. W e supp ose it is true fo r n ∈ N . F rom Lemma 6 we hav e | π 2 ◦ g n +1 (01) | = | π 2 ◦ g n ( g (01)) | ≥ | π 2 ◦ g n ( j j k ) | = | π 2 ◦ g n (01) | + | π 2 ◦ g n ( j ) | ≥ n + 3 . Moreo v er, from Lemma 6 in g ( j ) o ccurs a letter a ∈ { 0 , 1 } . Consequen tly , | π 2 ◦ g n +1 ( j ) | = | π 2 ◦ g n ( g ( j )) | ≥ | π 2 ◦ g n ( a ) | ≥ 1 . W e pro ceed in the same wa y for the letter k . This concludes the induction. Therefore, it is clear g is expansiv e. This concludes the pro of.  5. The monoid MSE is not finitel y ge nera ted 5.1. Some preliminary results. T o pro v e the p oin t (1) o f Theorem 1 w e need the following subset of MSE. Let MSE ′ b e the set of morphisms f ∈ MSE 2 suc h that for some n ∈ N π 2 ◦ f ∈ F n , π 1 ◦ f ∈ G n and π 0 ◦ f ∈ H n where F n = { ϕ 1 , f ϕ 1 } n , G n = E 0 ◦ { ϕ 1 , f ϕ 1 } ◦ E 2 ◦ { ϕ 1 , f ϕ 1 } n − 1 and H n = E 2 ◦ E 0 ◦ { ϕ 1 , f ϕ 1 } n − 1 . With the t w o follow ing lemmata w e prov e that MSE ′ is not empt y . Before w e need a new definition and we mak e some r emarks. Let u ∈ { 0 , 1 } ∗ , v ∈ { 0 , 2 } ∗ and w ∈ { 1 , 2 } ∗ b e three w ords. W e say that u , v and w inter c alate b etwe en them if a nd only if there exists x ∈ A ∗ 3 suc h that π 2 ( x ) = u , π 1 ( x ) = v and π 0 ( x ) = w . Let ( u n ) n ∈ N b e the Fib onacci word : u n +1 = u n + u n − 1 for all n ≥ 1, u 0 = 0 and u 1 = 1. W e can remark that for a ll n ≥ 1 w e hav e M ϕ n 1 = M e ϕ n 1 = M n ϕ 1 =   u n +1 u n 0 u n u n − 1 0 0 0 0   . Lemma 8. L et n ≥ 2 , f ∈ F n , g ∈ G n and h ∈ H n . Then, for al l a ∈ { 0 , 1 } we have | f ( a ) | 0 = | g ( a ) | 0 , | f ( a ) | 1 = | h ( a ) | 1 and | g ( a ) | 2 = | h ( a ) | 2 . Pr o of. It suffices to remark that M f = M ϕ n 1 , M g =   u n +1 u n 0 0 0 0 u n − 1 u n − 2 0   and M h =   0 0 0 u n u n − 1 0 u n − 1 u n − 2 0   .  Lemma 9. L et f , g a nd h b e thr e e morph isms fr om A 3 to A ∗ 3 such that f ( a ) , g ( a ) and h ( a ) ar e r esp e ctively wor ds on the alphab ets { 0 , 1 } , { 0 , 2 } and { 1 , 2 } for al l a ∈ A 3 . Then, f ( a ) , g ( a ) a nd h ( a ) inter c alate b etwe en them for al l a ∈ A 3 if and only if ther e exists a morphism ψ : A 3 → A ∗ 3 such that π 2 ◦ ψ = f , π 1 ◦ ψ = g and π 0 ◦ ψ = h . Pr o of. F or all a ∈ A 3 let ψ ( a ) b e the w ord o bt a ined intercalating f ( a ), g ( a ) and h ( a ). This defines a morphism ψ : A 3 → A ∗ 3 . W e can c hec k it satisfies π 2 ◦ ψ = f , π 1 ◦ ψ = g and π 0 ◦ ψ = h . The reciprocal is left to the r eader.  8 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS Lemma 10. F or al l n ∈ N ∗ , ϕ n 1 (1) is a pr efix but not a suffix of ϕ n 1 (0) . And f o r al l n ∈ N ∗ \{ 1 } if g = E 0 ◦ f ϕ 1 ◦ E 2 ◦ f ϕ 1 n − 1 then g (1 ) is a suffix but not a pr efix of g (0 ) . Pr o of. Let n ∈ N ∗ . W e ha v e ϕ n 1 (0) = ϕ n − 1 1 (01) = ϕ n 1 (1) ϕ n − 1 1 (1). Hence ϕ n 1 (1) is a prefix o f ϕ n 1 (0). W e pro ceed by induction to prov e that ϕ n 1 (1) is not suffix of ϕ n 1 (0). F o r n = 1 is it clear. Supp ose it is true for n ∈ N ∗ . W e prov e it is also true fo r n + 1. W e ha v e ϕ n +1 1 (0) = ϕ n +1 1 (1) ϕ n 1 (1) and ϕ n +1 1 (1) = ϕ n 1 (0). Supp ose ϕ n +1 1 (1) is a suffix of ϕ n +1 1 (0). Looking at M ϕ n 1 w e remark that | ϕ n 1 (1) | < | ϕ n 1 (0) | , therefore ϕ n 1 (1) is a suffix of ϕ n 1 (0) which con tradicts the hypothesis. This concludes the first part of the pro of. The other part can b e achie v ed in the same w a y .  Lemma 11. L et n ∈ N ∗ , f n = ϕ n 1 , g n = E 0 ◦ f ϕ 1 ◦ E 2 ◦ f ϕ 1 n − 1 and h n = E 2 ◦ E 0 ◦ f ϕ 1 n − 1 . T hen ther e exists a morphism ψ n ∈ MSE 2 such that π 2 ◦ ψ n = f n , π 1 ◦ ψ n = g n and π 0 ◦ ψ n = h n . Pr o of. W e easily che c k that if ψ is a morphism suc h that π 2 ◦ ψ = f n , π 1 ◦ ψ = g n and π 0 ◦ ψ = h n , for some n ∈ N , t hen ψ b elongs to MSE 2 . W e pro ceed b y induction on n to pro v e what r emains. F or n = 1, we hav e f 1 : A ∗ 3 − → A ∗ 3 , g 1 : A ∗ 3 − → A ∗ 3 , h 1 : A ∗ 3 − → A ∗ 3 , ψ 1 : A ∗ 3 − → A ∗ 3 0 7− → 01 0 7− → 0 0 7− → 1 0 7− → 01 1 7− → 0 1 7− → 20 1 7− → 2 1 7− → 20 2 7− → ε 2 7− → ε 2 7− → ε 2 7− → ε . The morphism ψ 1 is suc h that π 2 ◦ ψ 1 = f 1 , π 1 ◦ ψ 1 = g 1 and π 0 ◦ ψ 1 = h 1 , and consequen tly ψ 1 b elongs to MSE 2 . F or n = 2, w e ha v e f 2 : A ∗ 3 − → A ∗ 3 , g 2 : A ∗ 3 − → A ∗ 3 , h 2 : A ∗ 3 − → A ∗ 3 , ψ 2 : A ∗ 3 − → A ∗ 3 0 7− → 010 0 7− → 200 0 7− → 21 0 7− → 2010 1 7− → 01 1 7− → 0 1 7− → 1 1 7− → 01 2 7− → ε 2 7− → ε 2 7− → ε 2 7− → ε . The morphism ψ 2 is suc h that π 2 ◦ ψ 2 = f 2 , π 1 ◦ ψ 2 = g 2 and π 0 ◦ ψ 2 = h 2 , and consequen tly ψ 2 b elongs to MSE 2 . Now w e supp ose the result is true for n − 1 and n ≥ 2. W e pro v e it is also true for n + 1. W e ha v e f n +1 (0) = f n − 1 (0) f n − 1 (1) f n − 1 (0) , f n +1 (1) = f n (0) , f n +1 (2) = ε, g n +1 (0) = g n − 1 (0) g n − 1 (1) g n − 1 (0) , g n +1 (1) = g n (0) , g n +1 (2) = ε, h n +1 (0) = h n − 1 (0) h n − 1 (1) h n − 1 (0) , h n +1 (1) = h n (0) a nd h n +1 (2) = ε. F rom the induction hypothesis and Lemma 9 we know f i ( a ), g i ( a ), h i ( a ) in- tercalate b et w een them for all a ∈ A 3 and all i ∈ { n − 1 , n } . Consequen tly , using Lemma 9, there is a mor phism ψ : A 3 → A ∗ 3 suc h t hat π 2 ◦ ψ n +1 = f n +1 , π 1 ◦ ψ n +1 = g n +1 and π 0 ◦ ψ n +1 = h n +1 .  Prop osition 12. F o r al l n ∈ N ∗ , the morp hism ψ n define d in L emma 11 is prime in MSE 2 . Pr o of. W e k eep the notations of L emma 1 1. Let n ∈ N ∗ . F rom Prop osition 7 it suffices to pr ov e that ψ n (1) is neither a prefix nor a suffix of ψ n (0). W e pro ceed b y con tradiction: W e supp ose ψ n (1) is a prefix or a suffix of ψ n (0). WORDS A ND MORPHI SMS WITH STURMIAN ER ASURES 9 Supp ose that ψ n (1) is a prefix of ψ n (0). Then π 1 ◦ ψ n (1) is a prefix o f π 1 ◦ ψ n (0) and consequen tly g n (1) is a prefix of g n (0). This contradicts Lemma 10. Supp ose that ψ n (1) is a suffix of ψ n (0). Then π 2 ◦ ψ n (1) is a suffix of π 2 ◦ ψ n (0) and consequen tly f n (1) is a suffix of f n (0). This contradicts Lemma 10 a nd prov es the lemma.  Corollary 13. The set MSE 2 c ontains infin i tely m any primes. Pr o of. W e left as an exercise to pro v e t hat for all n ∈ N ∗ w e ha v e ψ n 6 = ψ n +1 , where ψ n is defined in Lemma 11. Prop osition 12 ends the pro of.  5.2. Pro of of the p oint (1) of Theorem 1. W e pro ceed b y con tra diction: W e supp ose there exists F = { f 1 , . . . , f l } ⊂ MSE generating MSE , i.e., all g ∈ MSE is a comp osition of elemen ts b elonging to F . Let N = sup a ∈ A 3 , 1 ≤ i ≤ l | f i ( a ) | , ( ψ n ) n ∈ N b e the morphisms defined in Lemma 11 and ( u n ) n ∈ N b e the Fib onacci word defined in the previous section. W e remark lim n → + ∞ max a ∈ A 3 | ψ n ( a ) | ≥ lim n → + ∞ u n +1 = + ∞ . W e fix n ∈ N suc h that max a ∈ A 3 | ψ n ( a ) | > N . By hy p othesis there exist g 1 , . . . , g k in F suc h tha t ψ n = g 1 ◦ · · · ◦ g k . W e set h = g 2 ◦ · · · ◦ g k . The morphism ψ n b elongs to MSE a for some a ∈ A 3 . It implies ψ n (2) = ψ n ( a ) = ε and consequen tly a = 2. There exists b ∈ A 3 suc h t ha t g 1 ∈ MSE b . W e remark ψ n = g 1 ◦ h = g 1 ◦ π b ◦ h . Tw o cases o ccurs. First case: F or a ll a ∈ { 0 , 1 } w e hav e | π b ◦ h ( a ) | = 1. The morphism h b eing a morphism with Sturmian erasures w e cannot ha v e π b ◦ h (0 ) = π b ◦ h (1). Consequen tly ψ n = g 1 ◦ E 2 or ψ n = g 1 . This implies there exists a ∈ A 3 suc h that | g 1 ( a ) | > N whic h is not p ossible. Second case: There exists a ∈ { 0 , 1 } suc h that | π b ◦ h ( a ) | > 1. W e remark π b ◦ h = π b ◦ h ◦ π 2 If b = 2 then π b ◦ h |{ 0 , 1 } : { 0 , 1 } → { 0 , 1 } ∗ ⊂ A ∗ 3 is a Sturmian morphism differen t from E and I d { 0 , 1 } . Hence f rom a remark w e mak e in Subsection 3.2 there exist i and j in { 0 , 1 } , i 6 = j , such that the word π b ◦ h |{ 0 , 1 } ( i ) is a prefix o r a suffix of π b ◦ h |{ 0 , 1 } ( j ). Hence ψ n ( i ) is a prefix or a suffix of ψ n ( j ). Prop o sition 7 implies ψ n is not prime in MSE 2 whic h con tradicts Prop osition 1 2. Let b 6 = 2. W e set { b, c } = { 0 , 1 } . Then, E c ◦ π b ◦ h |{ 0 , 1 } : { 0 , 1 } → { 0 , 1 } ∗ ⊂ A ∗ 3 is a Sturmian morphism differen t from E and I d { 0 , 1 } . Hence f r o m a remark w e mak e in Subsection 3.2 there exist i and j in { 0 , 1 } , i 6 = j , suc h tha t the w ord E c ◦ π b ◦ h |{ 0 , 1 } ( i ) is a prefix or a suffix of E c ◦ π b ◦ h |{ 0 , 1 } ( j ). Hence ψ n ( i ) is a prefix or a suffix of ψ n ( j ). Prop o sition 7 implies ψ n is no t prime in MSE 2 whic h con tradicts Prop osition 12. This concludes t he pro of . 6. Some fur ther f acts about word s with Sturmian erasures 6.1. Geometrical remarks. W e recall that a Sturmian word can b e view ed as a co ding of a straigh t half line in R 2 with direction (1 , α ) where α is a p ositiv e irrational n um ber, or in other terms as a tr a jectory of a ball in the game of billiards in the square with elastic reflexion on the b oundary . W e do not give the details here, w e refer the reader t o [Lo]. 10 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS Let us extend the construction giv en in [Lo] t o obtain what is usually called bil liar d w o r ds in the unit cub e [0 , 1] 3 . Let d = ( d 0 , d 1 , d 2 ) ∈ [0 , + ∞ [ 3 and ρ = ( ρ 0 , ρ 1 , ρ 2 ) ∈ [0 , 1[ 3 . Let D be the half line with direction d and in tercept ρ t ha t is to sa y D = { td + ρ ; t ≥ 0 } . Consider the inte rsections of D with the planes x = a , y = a , z = a , a ∈ Z : W e denote b y I 0 , I 1 , . . . these consecutiv e in tersection p oints . W e say I n crosses the face F i , i ∈ { 0 , 1 , 2 } , if the i + 1-th co ordinate of I n is an in teger and the i + 1- th co or dina t e of I n +1 − I n is not equal to 0. W e set Ω n = { i ∈ { 0 , 1 , 2 } ; I n crosses F i } . Let x = u 0 u 1 . . . b e a w ord suc h that u n =    i if Ω n = { i } , ij if Ω n = { i, j } where i 6 = j, ij k if Ω n = { i, j, k } = { 0 , 1 , 2 } . W e sa y x is a bil l i a r d wor d in the unit cub e [0 , 1] 3 (with direction d and interce pt ρ ). W e can also sa y that x is a co ding of D . Of course a ha lf line can ha v e sev eral co dings. One of t he co dings of a half line is p erio dic if and only if d ∈ γ Z 3 for some γ ∈ R + . When one of the co ordinates of the direction is equal to zero and the tw o others are rationally indep enden t we can easily deduce from [Lo] (Chapter 2) that x is a Sturmian w ord. The recipro cal is also t r ue: All Sturmian words can b e obtained in this wa y (see [Lo]). W e remark that if x is a non-p erio dic cubic billiard w ord then π 0 ( x ) is a Stur- mian w o rd with direction d = (0 , d 1 , d 2 ) and in tercept ρ = (0 , ρ 1 , ρ 2 ) (i.e. the orthogonal pro jection of D onto { 0 } × [0 , + ∞ [ × [0 , + ∞ [). W e hav e the analogous remark for π 1 ( x ) and π 2 ( x ). It is easy to conclude that a cubic billiard w ord is a w ord with Sturmian erasures if and only if it is non p erio dic and d ∈ ]0 , + ∞ [ 3 . There exist w ords with Sturmian era sures that a re not cubic billiard w ords. F or example, tak e the Fib onacci w ord x (Example 1) and the mor phism ψ defined b y ψ (0) = 0 012 and ψ (1) = 01. It is easy to see that y = ψ ( x ) is a w ord with Sturmian erasures. Let us show that if y w as a cubic billiard w ord t hen the w ord 102 should a pp ear in x , whic h is not the case. W e briefly sk etch the pro of. Supp ose y is a cubic billiard word with direction d = (1 , α , β ) and interc ept ρ , then the w ords π 0 ( y ) and π 2 ( y ) a re Sturmian w ords with resp ectiv e directions (0 , α , β ) and (1 , α, 0). It can b e sho wn that α = θ − 1 and β = ( θ − 1) 2 where θ = ( √ 5 + 1) / 2. But with suc h a direction d = (1 , θ − 1 , ( θ − 1) 2 ) easy calculus sho w that the w ord 102 should app ear in y . 6.2. Balanced words. Let us recall a characterization of Sturmian w ords due to Hedlund and Morse [HM2]. Let A b e a finite alphab et. W e sa y a w ord x ∈ A N is balanced if fo r all fa ctor s u and v of x having the same length we ha v e || u | a − | v | a | ≤ 1 f o r all a ∈ A . Supp ose Card A = 2. A word x ∈ A N is Sturmian if and only if x is non ev en tually p erio dic and balanced. P . Hub ert c haracterizes in [Hu] the w ords on a three letters alphab et that are bala nced. This c haracterization sho ws t ha t suc h w ords a re not words with Sturmian erasures. Definition 14. L et A b e a finite al p hab et. We say x ∈ A N is n -b alanc e d if n is the le as t i n te ger such that: F or al l wor ds u and v app e aring in x and having the same length we have || u | a − | v | a |≤ n for al l a ∈ A . Clearly , Sturmian w ords are 1-balanced. WORDS A ND MORPHI SMS WITH STURMIAN ER ASURES 11 Prop osition 15. If x ∈ A N 3 is a wor d with Sturmian er asur es then x is n o n eventual ly p erio dic and 2 -b a lanc e d. Pr o of. It is clear x is non eve n tually p erio dic. F ro m a previous remark w e know that x is not 1-balanced. Supp ose x is n -balanced with n ≥ 3: There exist e ∈ A 3 and t w o w o rds u and v app earing in x and having the same length suc h that || u | e − | v | e | ≥ 3. F or all a ∈ A 3 w e set n ( a ) = | | u | a − | v | a | . The n w e can set A 3 = { a, b, c } where n ( a ) ≥ 3 a nd n ( a ) ≥ n ( b ) ≥ n ( c ). Without loss of g enerality we sup- p ose | u | a − | v | a = n ( a ). As | u | = | v | we ha v e n ( a ) = ( | v | b − | u | b ) + ( | v | c − | u | c ). Consequen tly we necessarily ha v e | v | b − | u | b ≥ 0 and | v | c − | u | c ≥ 0 b e- cause n ( a ) ≥ n ( b ) ≥ n ( c ). Th us n ( b ) = | v | b − | u | b , n ( c ) = | v | c − | u | c and n ( a ) = n ( b ) + n ( c ). W e also see that n ( b ) ≥ 2 a nd n ( c ) ≥ 0. Supp ose there exists a factor u ′ of the w ord u verifying | π c ( u ′ ) | = | π c ( v ) | and | π c ( u ′ ) | a − | π c ( v ) | a ≥ 2 . Then this w ould say tha t π c ( x ) is not balanced and a fortiori not St urmia n whic h w ould end the pro of. Let us find suc h a u ′ . W e hav e | π c ( u ) | ≥ | π c ( v ) | ≥ | v | b ≥ 2 . Hence there exists a non-empt y w ord u ′ satisfying | π c ( u ′ ) | = | π c ( v ) | and ha ving an o ccurrence in u . Moreo v er | π c ( u ′ ) | a + | π c ( u ′ ) | b = | π c ( u ′ ) | = | π c ( v ) | = | v | − | v | c = | u | − | v | c = | u | a + | u | b + | u | c − | v | c = | π c ( u ) | a + | π c ( u ) | b − n ( c ) . Hence | π c ( u ) | a − n ( c ) = | π c ( u ′ ) | a + | π c ( u ′ ) | b − | π c ( u ) | b ≤ | π c ( u ′ ) | a and then 2 ≤ n ( b ) = n ( a ) − n ( c ) = | π c ( u ) | a − | π c ( v ) | a − n ( c ) ≤ | π c ( u ′ ) | a − | π c ( v ) | a , whic h ends the pro of.  6.3. Complexit y. Let x b e a word with Sturmian erasures a nd f b e a mor- phism b elonging to MSE i for some i ∈ A 3 . Then π i ( x ) is a Sturmian w ord and f ( x ) = f ( π i ( x )). Consequen tly from a result o f Co v en a nd Hedlund [CH] w e deduce there exist tw o in tegers n 0 and k suc h that P x ( n ) = n + k fo r all n ≥ n 0 . F or example, let F b e the Fib o na cci w ord and f : { 0 , 1 , 2 } → { 0 , 1 , 2 } ∗ b e the morphism defined b y f (0) = 0102, f (1) = 01 a nd f (2) = ε . It is a morphism with Sturmian erasures and y = f ( F ) is a w ord with Sturmian erasures. In f act it is a cubic billiard word with direction d = (1 , θ − 1 , ( θ − 1) 2 ) and in tercept ρ = (0 , θ − 1 , ( θ − 1 ) 2 ), where θ is the go lden mean ( √ 5 + 1) / 2. This do es not contradict the result in [AMST] say ing that if 1 , α and β are rat io - nally indep enden t then the complexit y of the cubic billiard word with direction (1 , α, β ) and in tercept ρ ∈ ]0 , 1[ 3 is n 2 + n + 1, b ecause − 1 + ( α − 1) + ( α − 1) 2 = 0. 6.4. Conclusion. Many generalizations of the Sturmian w ords were tried (more letters, applications of Z 2 to { 0, 1 } , ...) but none app eared to be entire ly suitable in the sens e tha t it seems imp ossible to extend these prop erties to a more general domain astonishing v arieties of the prop erties characterizing these w ords. The example whic h we c hose for this pa p er, do es no t derogat e f r om this rule. Nev er- theless, the fa ct that MSE is not giv en by a finite generator show s a fundamen tal difference b etw een the Sturmian w ords and an y generalization with more than t w o letters b ecause the definition adopted here w as less “compromising” possible. 12 F ABIEN DURAND , AD EL GUERZIZ, AND MICHEL KOSKAS F urthermore, this definition giv es a words of a complexit y structurally similar t o the one of the Sturmian w o rds. Reference s [AMST] P . Arnoux, C. Ma uduit, I. Shiok a wa and J.-I. T amura, Complexity of se quenc es define d by bil liar ds in the cub e , Bull. So c. Ma th. F rance 1 2 2 (1994), 1–1 2. [BS] J. Bers tel and P . S´ e´ ebold, A char acterization of S turmian morphisms , Lectur e Notes in Comput. Sci. 71 1 (1993), 281- 290. [CH] E. M. Coven and G. A. Hedlund, Se quenc es with Minimal Blo ck Gr owth , Math. Sys- tems Theory 7 (197 3), 138-153 . [He] G. A. Hedlund, St urmian minimal sets , Amer. J. Ma th 66 (1944), 60 5–620 . [Hu] P . Hubert, Suites ´ equili br´ ees , Theoret. Comput. Sci. 24 2 (2000), 91-1 08. [HM1] G. A. Hedlund and M. Mor se, Symb olic dynamics , Amer. J. Math. 60 (1938), 815-866. [HM2] G. A. Hedlund and M. Morse, Symb olic dynamics II. Sturmian t r aje ctories , Amer. J. Math. 62 (1940 ), 1-42. [Lo] M. Lothair e, A lgebr aic Combinatorics on Wor ds , Cambridge University Pres s (2002). [MS] F. Mignosi and P . S ´ e ´ eb old, Morphismes sturmiens et r` egles de R auzy , J. Th´ eor. Nom- bres Bordea ux 5 (1993 ), 2 21–2 33. [Qu] M. Queff´ elec, Substitution Dynamic al Systems-Sp e ct r al Analysi s , Lecture Notes in Math. 1294 (1987 ). Labora toire Ami ´ enois de Ma th ´ ema tiques Fondament ales et A ppliqu ´ ees, CNRS- UMR 6140, Universit ´ e de Picardie Jules Verne, 33 rue Saint L eu, 80039 Amiens Cedex 1 , France. E-mail addr ess : fabien.d urand @u-picardie.fr E-mail addr ess : adel.gue rziz@ u-picardie.fr E-mail addr ess : koskas@l aria. u-picardie.fr