A characterization of fine words over a finite alphabet

A c haracterization of fine w ords o v er a fini t e alphab et Am y Glen L aCIM, Universit´ e du Qu´ eb e c ` a Montr ´ eal, C.P. 8888, suc cursale Centr e-vil le, Montr ´ eal, Qu´ eb e c, CANA DA, H3C 3P8 Abstract T o any infinite word t o ver a finite alph ab et A we can asso ciate tw o infi nite words min( t ) and max( t ) suc h that any prefix of min( t ) (resp. max( t )) is the lexic o gr aph- ic al ly smallest (resp. greatest) amongst the factors of t of the same length. W e sa y that an infinite wo rd t ov er A is fine if there exists an infinite word s such that, for any lexicographic order, min( t ) = a s wh ere a = min( A ). In this pap er, we charac- terize fin e words; sp ecifically , we pro ve that an infinite word t is fine if and only if t is either a strict episturmian wor d or a strict “sk ew episturmian w ord” . This c harac- terizatio n generalizes a recent result of G. Pirillo, who p r o ved that a fine word ov er a 2-letter alphab et is either an (aperio dic) Sturmian wor d , or an ultimately per io dic (but not p er io dic) infin ite word, all of whose factors are (fi nite) Stur mian. Key wor ds: com bin atorics on words; lexicographic order; episturmian word; Sturmian word; Ar noux-Rauzy sequence; skew word. MSC (2000): 68R15. 1 In tro duction T o an y infinite w ord t o v er a finite alphab et A w e can asso ciate tw o infinite w ords min( t ) and max ( t ) suc h that an y prefix of min( t ) (resp. max( t )) is the lexic o gr aphic a l ly smallest (resp. greatest) amongst the fa ctors of t of the same length (see Pirillo [14]). In the recen t pap er [15], Pirillo defined fine wor ds ov er t w o letters; sp ecifically , a n infinite w ord t o ve r a 2-letter alphab et { a, b } ( a < b ) is said to b e fi ne if (min ( t ) , max( t )) = ( a s , b s ) for some infinite w ord s . Pirillo [15] characterize d these words, and remarke d that p erhaps his c haracterization can b e generalized to an ar bitr ary finite alphab et; w e do just that in this pap er. F irstly , w e extend the definition of a fine w ord t o more Email addr ess: amy.g len@gmai l.com (Amy Glen). Preprint su bmitted to Theoretical Computer Science Marc h 20, 2007 than tw o letters. That is, w e say tha t an infinite w ord t ov er A is fine if there exists an infinite w ord s suc h that, for an y lexicographic order, min( t ) = a s where a = min ( A ). R o ughly sp eaking, our main result states that an infinite w ord t is fine if a nd only if t is either a strict episturmian wor d or a strict “sk ew episturmian word” (i.e., a particular kind of non- recurren t infinite w ord, all of whose fa ctors are finite episturmian ). 2 Notation and terminology Finite and infinite words Let A denote a finite alphab et. A (finite) wor d ov er A is a n elemen t of the fr e e monoid A ∗ generated b y A , in the sense of concatenation. The iden tity ε o f A ∗ is called the empty wor d , and the fr e e semigr oup , denoted b y A + , is defined b y A + := A ∗ \ { ε } . Giv en w = x 1 x 2 · · · x m ∈ A + with each x i ∈ A , the le n gth o f w is | w | = m (note tha t | ε | = 0). The r eversal e w of w is giv en by e w = x m x m − 1 · · · x 1 , and if w = e w , then w is called a p alindr ome . An infinite wor d (or simply s e quenc e ) x is a sequence indexe d b y N with v alues in A , i.e., x = x 0 x 1 x 2 · · · with eac h x i ∈ A . The set of all infinite w ords ov er A is denoted by A ω , and w e define A ∞ := A ∗ ∪ A ω . An ultimately p erio dic infinite word can b e written as uv ω = uv v v · · · , for some u , v ∈ A ∗ , v 6 = ε . If u = ε , then such a w ord is p erio dic . An infinite w ord that is not ultimately p erio dic is said to b e ap erio dic . A finite w ord w is a factor of z ∈ A ∞ if z = uw v for some u ∈ A ∗ , v ∈ A ∞ . F urther, w is called a pr efix (resp. suffix ) of z if u = ε (resp. v = ε ), and we write w ≺ p z (resp. w ≺ s z ). W e sa y that w is a pr op er factor (resp. prefix, suffix) of z if uv 6 = ε (resp. v 6 = ε , u 6 = ε ). An infinite w ord x ∈ A ω is called a suffix of z ∈ A ω if there exists a w ord w ∈ A ∗ suc h that z = w x . A f actor w of a word z ∈ A ∞ is ri g ht (resp. left ) sp e cial if w a , w b (r esp. aw , bw ) are factors of z for some letters a , b ∈ A , a 6 = b . F or x ∈ A ω , F ( x ) denotes the set of all its factors, and F n ( x ) denotes the set of all factors of x of length n ∈ N , i.e., F n ( x ) := F ( x ) ∩ A n . Moreov er, the alphab et of x is Alph( x ) := F ( x ) ∩ A , and w e denote b y Ult( x ) the set of all letters o ccurring infinitely often in x . An y t w o infinite w ords x , y ∈ A ω are said to b e e quivalent if F ( x ) = F ( y ) , i.e., if x and y hav e the same set of factors. A fa cto r of an infinite word x is r e curr ent in x if it o ccurs infinitely man y times in x , and x it self is said to b e r e curr ent if all of its factors are recurren t in it . 2 Lexicographic order Supp ose the alphab et A is totally ordered b y the relation < . Then w e can totally order A ∗ b y the le xic o g r aphic or der < , defined as follows . Giv en tw o w ords u , v ∈ A + , w e ha ve u < v if and only if either u is a pro p er prefix of v or u = xau ′ and v = xbv ′ , f or some x , u ′ , v ′ ∈ A ∗ and letters a , b with a < b . This is the usual alphab etic ordering in a dictionary , and w e say that u is lexic o gr aphic al ly less than v . This no t io n naturally extends to A ω , as follows . Let u = u 0 u 1 u 2 · · · and v = v 0 v 1 v 2 · · · , where u j , v j ∈ A . W e define u < v if there exists an index i ≥ 0 suc h that u j = v j for all j = 0 , . . . , i − 1 and u i < v i . Naturally , ≤ will mean < or =. Let w ∈ A ∞ and let k be a p ositive in teger. W e denote by min ( w | k ) (resp. max( w | k )) the lexicographically smallest (resp. greatest) factor of w of length k f or the g iv en o rder (where | w | ≥ k for w finite). If w is infinite, then it is clear that min( w | k ) and max( w | k ) are prefixes of the resp ectiv e w ords min( w | k + 1) a nd max ( w | k + 1). So w e can define, b y taking limits, the follo wing tw o infinite w ords (see [14]) min( w ) = lim k →∞ min( w | k ) and ma x( w ) = lim k →∞ max( w | k ) . Morphisms and the free gr oup A morphism on A is a map ψ : A ∗ → A ∗ suc h that ψ ( uv ) = ψ ( u ) ψ ( v ) for all u , v ∈ A ∗ . It is uniquely determined b y its image on the alphab et A . All mor phisms considered in this pap er will b e non-erasing: the image of an y non- empt y w ord is nev er empt y . Hence the action of a mor phism ψ on A ∗ naturally extends to infinite w ords; t ha t is, if x = x 0 x 1 x 2 · · · ∈ A ω , then ψ ( x ) = ψ ( x 0 ) ψ ( x 1 ) ψ ( x 2 ) · · · . The free monoid A ∗ can b e naturally embedded within a free group. W e denote b y F ( A ) the fr ee group ov er A that prop erly con tains A , and is obtained from A by adjoining the in v erse a − 1 of eac h letter a ∈ A . More precisely , w e construct a new a lpha b et A ± that consists of all letters a of A and their ‘in v erses’ a − 1 , i.e., A ± = { a, a − 1 | a ∈ A} . If one defines o n the free mono id ( A ± ) ∗ the in v olution ( a − 1 ) − 1 = a for eac h a ∈ ( A ± ) ∗ , then necessarily , w e ha v e ( uv ) − 1 = v − 1 u − 1 for all u , v ∈ ( A ± ) ∗ . The free group F ( A ) o v er A is the quotien t of ( A ± ) ∗ under t he r elat io n: aa − 1 = a − 1 a = ε for all a ∈ A . In what follows , w e use the notat ion p − 1 w and w s − 1 to indicate the remo v al of a prefix p (resp. suffix s ) from a finite word w . An y morphism ψ on A can be uniquely extended to an endomorphism of F ( A ) b y defining ψ ( a − 1 ) = ( ψ ( a )) − 1 for eac h a ∈ A . 3 3 Episturmian w ords An interes ting generalization of Sturmian wor ds (i.e., a p erio dic infinite w ords of minimal complexit y) to a finite alphab et is the family of A rnoux-R auzy se quenc es , the study of whic h b egan in [2] (also see [10,17] for example). Mor e recen tly , a slightly wider class o f infinite words, aptly called episturmian wor ds , w as in tro duced b y Droubay , Justin, and Pirillo [4] (also see [5,9,11,12] for instance). An infinite w ord t ∈ A ω is episturmian if F ( t ) is closed under rev ersal and t has at most one righ t (or equiv alently left) sp ecial factor of eac h length. Moreov er, an episturmian w ord is standar d if all of it s left sp ecial factors ar e prefixes of it. Sturmian w ords are exactly the ap erio dic episturmian w ords ov er a 2-letter alphab et. Standard episturmian w ords w ere c hara cterized in [4] using the concept o f the p alindr omic rig ht-closur e w (+) of a finite w ord w , whic h is the (unique) shortest palindrome having w as a prefix (see [3]). Sp ecifically , an infinite w ord s ∈ A ω is standard episturmian if and only if there exists an infinite w ord ∆( s ) = x 1 x 2 x 3 . . . ( x i ∈ A ), called the dir e ctive wor d of s , suc h that the infinite sequence of palindro mic prefixes u 1 = ε , u 2 , u 3 , . . . of s (whic h exists b y results in [4]) is giv en b y u n +1 = ( u n x n ) (+) , n ∈ N + . (1) This c haracterization extends to the case of an arbitrary finite alphab et a construction give n in [3] for all standar d Sturmian wor ds . An imp ort a n t p oin t is that a standard episturmian w ord s can b e constructed as a limit o f an infinite sequence of its palindromic prefixes, i.e., s = lim n →∞ u n . Note. Episturmian w ords are (uniformly) recurren t [4 ]. 3.1 R elation with episturmian morphism s Let a ∈ A and denote by Ψ a the morphism on A defined b y Ψ a :      a 7→ a x 7→ ax for all x ∈ A \ { a } . T og ether with t he p ermutations of the alphab et, all of the morphisms Ψ a gen- erate b y comp osition the monoid of epistandar d morphisms ( ‘epistandard’ is an elegant shortcut for ‘standard episturmian’ due to Ric homme [16]). The submonoid generated by the Ψ a only is the monoid of pur e epis tandar d mor- phisms , whic h includes the identity morphism Id A = Id, and consists of all the pur e stand a r d (Sturmian) morphisms when | A| = 2. 4 When view ed as an endomorphism of t he free group F ( A ), the morphism Ψ a is in v ertible; that is, Ψ a is a p ositive automorphism of F ( A ), and its inv erse is giv en by Ψ − 1 a :      a 7→ a x 7→ a − 1 x for all x ∈ A \ { a } . It follo ws that ev ery epistandard morphism is a (p ositiv e) automorphism of F ( A ). See [6,8,16,18] for work inv olving the in v ertibilit y of episturmian mor- phisms. Remark 3.1 If x = Ψ a ( y ) or x = a − 1 Ψ a ( y ) for some y ∈ A ω and a ∈ A , then the letter a is sep ar a ting for x and its factors; that is, any factor of x of length 2 contains the letter a . Another useful c haracterization of standard episturmian w ords is the follow ing (see [9]). An infinite w ord s ∈ A ω is standard episturmian with directiv e word ∆( s ) = x 1 x 2 x 3 · · · ( x i ∈ A ) if and only if there exists an infinite sequence of recurren t infinite w ords s (0) = s , s (1) , s (2) , . . . suc h that s ( i − 1) = Ψ x i ( s ( i ) ) fo r all i ∈ N + . Moreo v er, each s ( i ) is a standard episturmian word with directiv e w ord ∆( s ( i ) ) = x i +1 x i +2 x i +3 · · · , the i -th shift of ∆( s ). T o the prefixes of the directiv e w ord ∆( s ) = x 1 x 2 · · · , w e asso ciate the mor- phisms µ 0 := Id , µ n := Ψ x 1 Ψ x 2 · · · Ψ x n , n ∈ N + , and define the words h n := µ n ( x n +1 ) , n ∈ N , whic h ar e clearly prefixes of s . F o r the palindromic prefixes ( u i ) i ≥ 1 giv en b y (1), w e ha v e t he follo wing useful for m ula [9] u n +1 = h n − 1 u n ; whence, for n > 1 and 0 < p < n , u n = h n − 2 h n − 3 · · · h 1 h 0 = h n − 2 h n − 3 · · · h p − 1 u p . (2) Remark 3.2 Eviden tly , if a standard episturmian w ord s b egins with the letter x ∈ A , then x is separating for s (see [4, Lemma 4]). 3.2 Strict epis turmian wor ds A standard episturmian w ord s ∈ A ω , or an y equiv alent (episturmian) w ord, is said to b e B - s trict (or k - strict if |B | = k , o r strict if B is understo o d) if Alph(∆( s )) = Ult(∆( s )) = B ⊆ A . In particular, a standard episturmian w ord o v er A is A -strict if ev ery letter in A o ccurs infinitely often in its directiv e 5 w ord. The k -strict episturmian w ords ha ve complexit y ( k − 1) n + 1 for eac h n ∈ N ; suc h w ords are exactly the k - letter Arnoux-Rauzy sequence s. Note that the 2- strict episturmian w ords corresp ond to the (ap erio dic) Sturmian w ords. Remark 3.3 Supp ose s ∈ A ω is a standard episturmian w ord. If s is not A -strict, t hen Ult(∆( s )) = B ⊂ A and there exists a B -strict standard episturmian w ord s ′ and a pure epistandard morphism µ on A suc h that s = µ ( s ′ ). More precisely , let ∆( s ) = x 1 x 2 x 3 · · · and let m b e minimal suc h that Alph( x m +1 x m +2 · · · ) = B ⊂ A . That is, x 1 x 2 · · · x m is the shortest prefix of ∆( s ) that con tains all the letters not appear ing infinitely often in ∆( s ), namely the letters in A \ B . Then s = µ m ( s ( m ) ) where s ( m ) is the B -strict standard episturmian word with directiv e word ∆( s ( m ) ) = x m +1 x m +2 · · · . F or example, if ∆( s ) = c ( ab ) ω , then s = Ψ c ( s (1) ) where ∆( s (1) ) = ( ab ) ω , i.e., s (1) is the w ell-known Fib onac ci wor d ov er { a, b } . 4 Fine w ords Recall that a n infinite w ord t o v er A is fi n e if there exists an infinite w ord s suc h that , for an y lexicographic order, min( t ) = a s where a = min( A ). Note. Since there are only t wo lexicographic or ders on words o v er a 2- letter alphab et, a fine w ord t o v er { a, b } ( a < b ) satisfies (min( t ) , max( t )) = ( a s , b s ) for some infinite word s . Recen tly , Pirillo [15] c haracterized fine w ords o v er a 2-letter alphab et. Sp ecif- ically: Prop osition 4.1 [15] S upp ose t i s an infi nite w o r d ove r { a, b } . Then the fol lowing pr op erties ar e e quivalent: i) t is fine; ii) either t is a Sturmian w or d, or t = v µ ( x ) ω wher e µ is a pur e standar d Sturmian morphism on { a, b } , and v is a non-em p ty suffix of µ ( x p y ) f o r some p ∈ N and x , y ∈ { a, b } ( x 6 = y ) . In other w ords, a fine w ord o ve r tw o letters is either a Sturmian w ord o r an ultimately p erio dic (but not p erio dic) infinite w ord, all of whose f a ctors a r e (finite) Sturmian, i.e., a so- called s k ew Sturmian w ord (see [13]). In this paper, w e g eneralize Pirillo’s result to infinite words ov er tw o or more letters. The next tw o prop ositions are needed fo r the pro of of our main result (Theorem 4.6, t o follo w). R ecall that the Arnoux-Rauzy seq uences are precis ely the strict episturmian w ords. 6 Prop osition 4.2 [10] Supp ose s is an infinite wor d over a finite a lphab et A . Then the fol low ing pr op erties ar e e quivalent: i) s is a standar d A rnoux-R auzy se quenc e ; ii) a s = min( s ) for any letter a ∈ A and lexic o gr aphic or der < satisfying a = min( A ) . Prop osition 4.3 [14] Supp ose s is an infinite wor d over a finite a lphab et A . Then the fol low ing pr op erties ar e e quivalent: i) s is standar d e p isturmian; ii) a s ≤ min( s ) for any letter a ∈ A and l e x ic o gr aphic or der < satisfying a = min( A ) . The follo wing k ey lemma is also needed. F rom no w on, it will b e conv enien t to denote b y v p the prefix of length p of a giv en infinite w ord v . Lemma 4.4 L et A b e a finite alphab et an d let a ∈ A . Supp ose t , s ∈ A ω ar e infinite wor ds such that t = Ψ z ( t (1) ) and s = Ψ z ( s (1) ) for so m e z ∈ Alph( t (1) ) . Then min( t (1) ) = a s (1) ⇔ min( t ) = a s . Remark 4.5 Let t , t (1) , s , s (1) ∈ A ω b e suc h that t = Ψ z ( t (1) ) and s = Ψ z ( s (1) ) for some letter z (not necessarily in Alph( t (1) )). Using similar reasoning as in the pro of b elow , it can b e show n that min( t (1) ) = a s (1) ⇔ m in( t ) =    z a s if z < a, a s if z ≥ a. F or example, let A = { a, b, c } with a < b < c and supp ose f is the Fib o nacci w ord ov er { a, b } (i.e., the standard episturmian w ord directed by ( ab ) ω ). Then min( f ) = a f , and hence min(Ψ c ( f )) = a Ψ c ( f ). On the other hand, if f ′ is the Fib onacci w ord ov er { b, c } , then min( f ′ ) = b f ′ and w e ha v e min (Ψ a ( f ′ )) = ab Ψ a ( f ′ ). Lemma 4.4 is a sp ecial case of this result with z ∈ Alph( t (1) ) ⊆ A and is sufficien t for our purp o ses. Pro of of Lemma 4 . 4 ( ⇐ ): W e hav e min( t ) = a s . First observ e that a ∈ Alph( t (1) ). Indeed, if a = z , then a ∈ Alph( t (1) ) since z ∈ Alph( t (1) ) (in f act, z z ∈ F ( t ) since z z is a prefix of a s = z s , and hence a = z ∈ Alph( t (1) )). On the other hand, if a 6 = z , then w e m ust hav e a ∈ Alph( t (1) ), otherwise a is not in the alphab et of t = Ψ z ( t (1) ), whic h is imp ossible since F ( a s ) ⊆ F ( t ). No w we show that F ( a s (1) ) ⊆ F ( t (1) ). Supp ose not, i.e., supp ose F ( a s (1) ) 6⊆ 7 F ( t (1) ). Then there exists a minimal m ∈ N + suc h tha t a s (1) m 6∈ F ( t (1) ). There- fore, if s (1) m = s (1) m − 1 x where x ∈ A , then a s (1) m − 1 x 6∈ F ( t (1) ) . Letting s l = Ψ z ( s (1) m − 1 ), w e ha v e a Ψ z ( s (1) m − 1 x ) = a s l Ψ z ( x ) ∈ F ( a s ) ⊆ F ( t ) , and hence Ψ z ( a ) s l Ψ z ( x ) ∈ F ( t ) since z is separating for t . So, if x 6 = z then a s (1) m − 1 x ∈ F ( t (1) ), whic h is imp ossible; whence x = z . But then, s (1) m +1 = s (1) m − 1 z y ′ for some y ′ ∈ A and w e hav e a Ψ z ( s (1) m +1 ) = a s l z Ψ z ( y ′ ) ∈ F ( a s ) ⊆ F ( t ) . Th us, Ψ z ( a ) s l z z ∈ F ( t ) , and hence a s (1) m − 1 z (= a s (1) m − 1 x ) is a factor of t (1) ; a con tradiction. Therefore, w e conclude tha t F ( a s (1) ) ⊆ F ( t (1) ). No w suppose on the con trary that min( t (1) ) 6 = a s (1) . Then there exists a w ord w (1) ∈ F ( t (1) ) of minimal length | w (1) | = m suc h tha t w (1) < a s (1) m − 1 . Let w (1) = u (1) x ( x ∈ A ) where u (1) is non- empty since a ∈ Alph( t (1) ). Then, b y minimality of m , u (1) ≥ a s (1) m − 2 , and therefore u (1) = a s (1) m − 2 (otherwise u (1) > a s (1) m − 2 implies w (1) > a s (1) m − 1 ). Hence, w (1) = a s (1) m − 2 x with w (1) < a s (1) m − 1 , and therefore s (1) m − 1 = s (1) m − 2 y for some y ∈ A , y > x . No w, letting w = Ψ z ( w (1) ) and s l = Ψ z ( s (1) m − 2 ), w e ha v e w = Ψ z ( a ) s l Ψ z ( x ) ∈ F ( t ) and s l Ψ z ( y ) ≺ p s . No w consider x ′ , y ′ ∈ A suc h that w (1) x ′ ∈ F ( t (1) ) and s (1) m = s (1) m − 2 y y ′ is a prefix of s (1) . Then Ψ z ( w (1) x ′ ) is a factor of t , where Ψ z ( w (1) x ′ ) = w Ψ z ( x ′ ) = Ψ z ( a ) s l Ψ z ( x )Ψ z ( x ′ ) =    Ψ z ( a ) s l z Ψ z ( x ′ ) if x = z , Ψ z ( a ) s l z x Ψ z ( x ′ ) if x 6 = z . 8 Therefore, the w ord v = a s l z x is a factor of t . Moreo v er, Ψ z ( s (1) m ) is a prefix of s , where Ψ z ( s (1) m ) =    s l z Ψ z ( y ′ ) if y = z , s l z y Ψ z ( y ′ ) if y 6 = z , and hence s l +2 = s l z y is a prefix of s . Accordingly , v < a s l +2 since x < y , con tradicting the fact t ha t the prefixes of a s are the lexicographically smallest factors of t . Th us, w e conclude that min( t (1) ) = a s (1) . ( ⇒ ): W e ha v e min( t (1) ) = a s (1) . As ab ov e, it is easily show n t hat F ( a s ) ⊆ F ( t ); whence min( t ) ≤ a s . Let us supp o se min( t ) 6 = a s . Then there exists a w ord w ∈ F ( t ) o f minimal length | w | = l suc h that w < a s l − 1 If w e let w = ux , x ∈ A , then u ≥ a s l − 2 , and hence u = a s l − 2 (otherwise w > a s l − 1 ). Therefore, w = a s l − 2 x < a s l − 1 and hence s l − 1 = s l − 2 y where y ∈ A , y > x . Since the letter z is separating for t , s l − 2 m ust end with z ; otherwise x = y = z , whic h is imp ossible. Th us, w = a s l − 3 z x a nd s l − 1 = s l − 3 z y , y > x. Let s (1) m − 1 = Ψ − 1 z ( s l − 1 ). W e distinguish tw o cases: y = z and y 6 = z . Case 1: y = z . W e ha v e s l − 1 = s l − 3 z z , and th us s (1) m − 1 and s (1) m − 2 b oth end with the letter z . Note that z 6 = a b ecause a ≤ x < y = z . Therefore, since w b egins with a and z is separating f o r t , w e ha v e z w ∈ F ( t ). Now , Ψ − 1 z ( z w ) = Ψ − 1 z ( z a s l − 1 z − 1 x ) = a s (1) m − 1 z − 1 Ψ − 1 z ( x ) = a s (1) m − 1 z − 1 z − 1 x = a s (1) m − 3 x, i.e., z w = Ψ z ( w (1) ) where w (1) = a s (1) m − 3 x ∈ F ( t (1) ). Therefore, as s (1) m − 2 ends with z > x , w e hav e w (1) < a s (1) m − 2 ; a contradiction. Case 2: y 6 = z . In this case, s l = s l − 3 z y z = s l − 1 z , and so s (1) m = s (1) m − 1 y ′ = s (1) m − 2 y y ′ for some y ′ ∈ A . If z 6 = a , then z w = z a s l − 3 z x is a factor of t since w ∈ F ( t ) and z is separating for t . So, letting w ′ = z w if z 6 = a and w ′ = w if 9 z = a , we hav e w ′ ∈ F ( t ) and Ψ − 1 z ( w ′ ) = a Ψ − 1 z ( s l − 3 z x ) = a Ψ − 1 z ( s l − 1 y − 1 x ) = a s (1) m − 1 ( z − 1 y ) − 1 Ψ − 1 z ( x ) = a s (1) m − 1 y − 1 z Ψ − 1 z ( x ) = a s (1) m − 2 z Ψ − 1 z ( x ) . That is, w ′ = Ψ z ( w (1) ) where w (1) ∈ F ( t (1) ) is giv en by w (1) =    a s (1) m − 2 xx if x = z , a s (1) m − 2 x if x 6 = z . Therefore, since s (1) m = s (1) m − 1 y ′ = s (1) m − 2 y y ′ with y > x , w e ha ve w (1) < a s (1) m − 1 ; a con tradiction. Both Cases 1 and 2 lead to a contradiction; whence min ( t ) = a s . ✷ W e no w prov e our main result: a characterization of fine w ords o ve r a finite alphab et. (Recall that v p denotes the prefix of length p of a giv en infinite w ord v , a nd e v p denotes its rev ersal.) Theorem 4.6 Supp ose t is an in finite wor d with Alph( t ) = A . Then, t is fine if and only if one of the fol lowing hol d s: i) t is a strict episturmian wor d; ii) t = v µ ( v ) wher e v is a B -strict standar d episturmian wor d w ith B = A \ { x } , µ is a pur e e p istandar d morphism on A , and v is a non-empty suffix of µ ( e v p x ) for some p ∈ N . PR OOF. In what follo ws, let A denote the alphab et of t . ( ⇒ ): t is fine, so there exists an infinite w ord s suc h that, for an y letter a ∈ A and lexicographic order < satisfying a = min( A ), we hav e min( t ) = a s . F urther, min( t ) ≤ min( s ) since F ( s ) ⊆ F ( a s ) ⊆ F ( t ), and therefore a s ≤ min( s ). Th us, Prop osition 4.3 implies that s is a standard episturmian word o v er A . W e distinguish tw o cases, b elow . Case 1: a s = min( s ) for a ny a ∈ A and lexicographic order suc h that a = min( A ). By Prop osition 4.2, s is an A - strict standard episturmian word. Clearly , F ( s ) ⊆ F ( t ) a nd w e show that F ( s ) = F ( t ) whic h implies t is equiv- alen t to s , and hence t is an A -strict episturmian word. Supp ose, on the 10 con trary , F ( s ) 6 = F ( t ). Then there exists a word u ∈ F ( t ) \ F ( s ), say u = xv with | v | = m minimal a nd x ∈ A . Now , v is not a prefix of s ; otherwise, z v is a factor of s fo r all z ∈ A (since an y prefix of s is left sp ecial and has |A| distinct left extensions in s ), whic h contradicts the fact that xv 6∈ F ( s ). Therefore, fo r some order suc h that x = min( A ), we ha v e u = xv < x s m , con tradicting the fact that min( t ) = x s ; whence F ( t ) = F ( s ). Case 2: x s < min( s ) for some x ∈ A a nd lexicographic order suc h that x = min( A ). In this case, it follows from Prop ositions 4.2 and 4.3 that s ∈ A ω is a standard episturmian w ord that is not A -strict. Therefore, letting ∆( s ) = x 1 x 2 x 3 · · · , there exists a minimal n ∈ N s uch that s = Ψ x 1 Ψ x 2 · · · Ψ x n ( s ( n ) ) = µ n ( s ( n ) ), where s ( n ) is a B -strict standar d episturmian w ord with B = Alph(∆( s ( n ) )) = Alph( x n +1 x n +2 x n +3 · · · ) ⊂ A (see Remark 3.3). Note tha t if s = s (0) is B -strict with B ⊂ A , t hen n = 0 and w e ta ke x n = x 0 to b e a letter in A \ B . Clearly , s b egins with x 1 ∈ A and x 1 is separating for s . Observ e t ha t x 1 m ust also b e separating for t . Indeed, let us supp ose that this is not true. Then, there exist letters z , z ′ ∈ A \ { x 1 } (p ossibly equal) suc h that z z ′ ∈ F ( t ). But, if < is a n order suc h that min( A ) = z ≤ z ′ < x 1 , then z x 1 is a prefix o f z s with z z ′ < z x 1 , con tradicting the fact that min( t ) = z s . Therefore, x 1 m ust b e separating for t . No w, let < b e an order with a = min( A ). Let t ′ = t if t b egins with x 1 . Otherwise, if t b egins with y 6 = x 1 , let t ′ = x 1 t . In the latter case, ax 1 ≺ p a s and x 1 y ≺ p t ′ with ax 1 < x 1 y ; th us min( t ′ ) = min ( t ) = a s . So we may consider t ′ instead of t . Observ e that s = Ψ x 1 ( s (1) ) and, since x 1 is separating for s (and hence fo r t ′ ), we ha ve t ′ = Ψ x 1 ( t (1) ) fo r some t (1) ∈ A ω . Because min( t ′ ) = x s for any letter x ∈ A a nd lexicographic order suc h that x = min( A ), it follo ws that Alph( t (1) ) = A (see argumen ts in the first lines of the pro of of Lemma 4.4); in particular x 1 ∈ Alph( t (1) ). So, by Lemma 4.4, w e ha v e min( t (1) ) = a s (1) . Con tin uing in t he same w ay (and applying L emma 4.4 rep eatedly), w e obtain sequence s ( s ( i ) ), ( t ( i ) ), ( t ′ ( i ) ) for i = 0 , 1 , 2, . . . , n suc h tha t s ( i − 1) = Ψ x i ( s ( i ) ), t ′ ( i − 1) = Ψ x i ( t ( i ) ), where t ′ ( i − 1) = t ( i − 1) if t ( i − 1) b egins with x i , t ′ ( i − 1) = x i t ( i − 1) otherwise, and t (0) = t , t ′ (0) = t ′ . In part icular, w e hav e Alph( t ( n ) ) = A a nd min( t ( n ) ) = a s ( n ) for an y a ∈ A a nd lexicographic order < satisfying a = min( A ). 11 No w w e sho w that B = A \ { x n } , i.e., A = B ∪ { x n } . First observ e that x n ∈ A \ B by minimality of n . Supp ose t ( n ) con tains tw o o ccurrences of the letter x n . Then, since x n +1 is separating for t ( n ) , w e hav e x n w ( n ) x n ∈ F ( t ( n ) ) for some non-empt y word w ( n ) for whic h x n +1 is separating, and t he first and last letter of w ( n ) is x n +1 (that is, w ( n ) x n = Ψ x n +1 ( w ( n +1) x n ), where w ( n +1) = Ψ − 1 x n +1 ( w ( n ) x − 1 n +1 )). Contin uing t he ab ov e pro cedure, we o btain infinite w ords t ( n +1) , t ( n +2) , . . . con taining similar shorter facto r s x n w ( n +1) x n , x n w ( n +2) x n , . . . un til w e reac h t ( q ) , whic h con tains x n x n . But this is imp ossible b ecause x q +1 ∈ B ⊆ A \ { x n } is separating f or t ( q ) . Therefore, t ( n ) con tains only one o ccurrence of x n and w e hav e t ( n ) = ux n v for some u ∈ ( A \ { x n } ) ∗ and v ∈ ( A \ { x n } ) ω . Note that the same reasoning a llo ws to pro v e the unicity of x 0 ∈ A \ B when n = 0. Clearly , for any order suc h that x n = min( A ), w e ha v e min( t ( n ) ) = x n v = x n s ( n ) ; whence v = s ( n ) and so t ( n ) = ux n s ( n ) . Note that if u 6 = ε , then u ends with x n +1 , and in particular x n +1 is separating for ux n since x n +1 is separating for t ( n ) . Let u ′ = x n +1 u if u do es not b egin with x n +1 ; ot herwise let u ′ = u . Then u ′ x n is a prefix of t ′ ( n ) . Moreo ve r, since x n +1 is separating for u ′ x n , w e ha v e u ′ x n = Ψ x n +1 ( u ( n +1) x n ) where u ( n +1) = Ψ − 1 x n +1 ( u ′ x − 1 n +1 ). Hence t ( n +1) = u ( n +1) x n s ( n +1) , where x n +2 is separating fo r u ( n +1) x n (if u ( n +1) 6 = ε ). Contin uing in this w ay , w e a rriv e at the infinite word t ( q ) = x n s ( q ) for some q ≥ n . No w, rev ersing the pro cedure, w e find that t ( n ) = w s ( n ) where w = ux n is a non-empty suffix of Ψ x n +1 · · · Ψ x q ( x n ) . Accordingly , u ∈ B ∗ since x n +1 , . . . , x q ∈ B ; whence A = B ∪ { x n } . Supp ose ( u i ) i ≥ 1 is the sequence of palindromic prefixes of s and the words ( h i ) i ≥ 0 are the prefixes ( µ i ( x i +1 )) i ≥ 0 of s . Then, letting u ( n ) i , h ( n ) i , and µ ( n ) i denote the analog ous elemen ts for s ( n ) , w e hav e µ ( n ) 0 = Id , µ ( n ) i = Ψ x n +1 Ψ x n +2 · · · Ψ x n + i = µ − 1 n µ n + i and h ( n ) 0 = x n +1 , h ( n ) i = µ ( n ) i ( x n +1+ i ) for i = 1, 2 , . . . . 12 No w, if u 6 = ε , then q ≥ n + 1, a nd w e hav e Ψ x n +1 · · · Ψ x q ( x n ) = µ ( n ) q − n ( x n ) = µ ( n ) q − n − 1 Ψ q ( x n ) = µ ( n ) q − n − 1 ( x q x n ) = h ( n ) q − n − 1 µ ( n ) q − n − 1 ( x n ) . . . = h ( n ) q − n − 1 · · · h ( n ) 1 µ ( n ) 0 ( x n +1 x n ) = h ( n ) q − n − 1 · · · h ( n ) 1 h ( n ) 0 x n = u ( n ) q − n +1 x n (b y (2)). Therefore, w = ux n where u is a (p ossibly empt y) suffix of the palindromic prefix u ( n ) q − n +1 of s ( n ) . That is, u is the rev ersal o f some prefix of s ( n ) = v ; in particular u = e v p for some p ∈ N , and hence t ( n ) = e v p x n v . So, passing bac k from t ( n ) to t , w e find that t = v µ n ( v ) = v s where v is a non-empt y suffix o f µ n ( e v p x n ). Cases 1 and 2 giv e prop erties i ) and ii ), resp ectiv ely . ( ⇐ ): F ir stly , if t is an A -strict episturmian word, then Prop osition 4.2 implies that t is fine. No w supp ose t = v µ ( v ) where v is a B -strict standard episturmian w ord with B = A \ { x } , µ is a pure epistandard morphism on A , and v is a non-empt y suffix of µ ( e v p x ) for some p ∈ N . First observ e that if µ = Id, then t = e v q x v for some q ≤ p . Consider a n o rder < suc h that min( A ) = a 6 = x . Then, b y Prop osition 4.2, min( v ) = a v , and it follo ws that min( t ) = min( v ) = a v . Indeed, if t = x v (i.e., q = 0), then it is clear that min( t ) = min( v ). On the other hand, if q ≥ 1, let us supp ose, on the contrary , that min( t ) 6 = min( v ). Then min( t ) is a suffix of e v q x v con taining the letter x , i.e., min( t ) = a e v l x v for some l with 0 ≤ l < q ≤ p (where a e v l = e v l +1 ∈ F ( v )). But, since min( v ) = a v , a v l +1 is a f actor of v (and hence a factor of t ) with a v l +1 = a v l a < a e v l x ; a con tradiction. Th us min( t ) = min( v ) = a v . Moreo v er, it is clear that min( t ) = x v for a n y or der 13 suc h that x = min( A ). So w e hav e sho wn that min( t ) = a v for an y letter a ∈ A and lexicographic order satisfying a = min( A ); whence t is fine. No w consider t he case when µ is not the iden tit y . Let us supp ose that t is not fine and let µ b e minimal with this prop erty . Then, µ = Ψ z η for some z ∈ A and pure epistandard morphism η . Consider t ′ = v ′ µ ( v ), where v ′ = v if v b egins with z and v ′ = z v otherwise. Then v ′ is also a non-empty suffix of µ ( e v p x ) since z is separating for the w ord µ ( e v p x ) (whic h b egins with z ). Letting t (1) = Ψ − 1 z ( t ′ ), w e ha v e t (1) = w η ( v ) where w = Ψ − 1 z ( v ′ ) is a non- empty suffix o f η ( e v p x ). By minimalit y of µ , t (1) is fine, so there exists an infinite w ord s (1) = Ψ − 1 z ( s ) suc h that min( t (1) ) = a s (1) for an y a ∈ A and order < satisfying a = min( A ). But then min( t ) = min( t ′ ) = a s b y L emma 4.4. Th us t is fine. ✷ Example 4.7 Let A = { a, b, c } with a < b < c and supp ose f is the Fib onacci w ord ov er { a, b } . Then, the following infinite words are fine. • f = abaababaabaaba · · · • c f = c abaabab aabaaba · · · • e f 4 c f = aabacabaabab aabaaba · · · • Ψ a ( f ) = aabaaaba abaaabaaaba · · · • Ψ c ( c f ) = c cacbcacacbcacbcacacb cacacbca · · · • Ψ c ( e f 4 c f ) = cacacbcac cacbcacacbcacbcacacb caca · · · Let us note, for example, that Ψ c ( f ) is not fine since it is a no n -strict standard episturmian w ord. That is, Ψ c ( f ) is a standard episturmian w ord with directiv e w ord c ( ab ) ω , so it is not strict, nor do es it take the second form giv en in Theorem 4.6. 5 Concluding remarks It is easy to see that Pro p osition 4.1 is a sp ecial case of Theorem 4 .6 b ecause the 2-strict episturmian w ords are precisely the Sturmian words and the 1- strict standard episturmian w ords are p erio dic infinite w ords of the for m x ω where x is a letter (see [9, Prop osition 2.9 ]). As alluded to in the in tro duction, an infinite word ta king form ii ) in Theo- rem 4.6 is said to b e a strict skew episturmian w ord. Sk ew episturmian w ords 14 (no w called episkew wor ds [1,7]) are explicated in the pap er [7], in whic h w e expand on our work here by c haracterizing via lexicographic order all epistur- mian wor ds in a wide sense , i.e., all infinite w ords whose fa cto r s are (finite) episturmian. Ac knowle dgemen ts The author w ould like to thank Jacques Justin fo r suggesting the problem and giving many helpful commen ts on preliminary ve rsions of this pap er. Thanks also to the r eferees for their careful reading of the pap er and prov iding t hough t- ful suggestions. References [1] J .-P . Allouc he, A. Glen. Extremal prop erties of (epi)sturmian sequences and distribution mo dulo 1, in pr ep aration. [2] P . Arnoux, G. Rauzy , Repr´ esen tation g ´ eom ´ etrique de suites de complexit ´ e 2 n +1, Bul l. So c. Math. F r anc e 119 (1991), 199– 215. [3] A. de Lu ca, S turmian words: structure, combinatorics and their arithmetics, The or et. Comput. Sci. 183 (199 7), 45–82. [4] X. Droubay , J. Justin, G. Pirillo, Epistur mian words and some constructions of de Luca and Rauzy , The or et. Comput. Sci. 255 (2001), 539–553. [5] A. Glen, Po wers in a class of A -str ict standard episturmian words, The or e t. Comput. Sci. 380 (200 7), 330–354. [6] A. Glen, On Sturmian and episturmian wor ds, and r elate d topics , Ph D thesis, The University of A delaide , Aus tr alia, April 2006. ( http:// thesis.library .adelaide .edu.au/public/adt-SUA20060426.164255/index.html ) [7] A. Glen, J. Justin, G. Pirillo, C haracterizatio ns of finite and infinite episturmian words via lexicographic orderings, Eur op e an J. Combin. (in p ress), doi:10.10 16/j.ejc.2007 .01. 002. [8] Ed die Godelle, Automorp hismes de gr oup es libr es, tr esses et morp hismes ´ episturmiens , Preprint (2006), ht tp://hal .archives -ouvertes.fr/hal- 00022410/en/ . [9] J . 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