A theorem of Cobham for non-primitive substitutions

A THEOREM OF COBHAM F OR NON-PR IMITIVE SUBSTITUTIONS F ABIEN DURAND 1. Intr oduction. Giv en a subset E of N = { 0 , 1 , 2 , · · · } can w e find an elemen tary a lgorithm (i.e., a finite state a ut o maton) whic h accepts the elemen ts of E and rejects those that do n ot b elong to E ? In 1969 A. Cobham s ho w ed that the ex- istence of suc h an algorithm deeply dep ends on the n umeration base. He stated [Co1]: L e t p and q b e two multiplic atively in dep endent inte gers (i.e. , p k 6 = q l for al l inte gers k , l > 0 ) gr e ater than or e qual to 2 . L et E ⊂ N . Th e set E is b oth p -r e c o gnizable and q -r e c o gnizable if and only if E is a finite union of arithmetic pr o gr essions. What is no w called the theorem o f Cob- ham. W e recall that a set E ⊂ N is p -recognizable f o r some in teger p ≥ 2 if the languag e consisting o f the expansions in ba se p of the elemen ts of E is recognizable b y a finite state automa t o n (see [Ei]). In 1972 Cobham ga v e an other partial answ er to this question sho wing that not all sets ar e p -recognizable. He gav e the follo wing c hara cterization: The set E ⊂ N i s p -r e c o gniz able for som e in te ger p ≥ 2 if and only if the char acteristic se quenc e ( x n ; n ∈ N ) of E ( x n = 1 if n ∈ E and 0 o therw ise) is gener ate d by a substitution of length p , where generated b y a substitution of length p means tha t it is the image by a letter to letter morphism of a fixed p oin t of a substitution of length p . W e remark that E is a finite union of arithmetic progressions if and only if its c haracteristic seque nce is ultimately p erio dic. Consequen tly the theorem of Cobham can b e form ulated as follows (this is an equiv alen t statemen t): L et p and q b e two multiplic a tively indep endent inte gers gr e ater than or e qual to 2 . L et A b e a fin ite alph a b et and x ∈ A N . The se quenc e x is gener ate d by b oth a substitution of length p and a substitution of length q if and onl y if x is ultimately p erio dic. T o a substitution σ is asso ciated an integer square matrix M 6 = 0 whic h has non-negativ e en tries. It is kno wn (see [LM] for instance) t ha t such a matrix has a real eigen v alue α whic h is greater t han or equal to the mo dulus of all others eigen v alues. It is usually called the dominan t eigen v alue o f M . Let S 1991 Mathematics Subje ct Classific ation. Primar y: 11B8 5; Secondary: 6 8R15. Key wor ds and phr ases. substitutions, substitutive sequences, theore m o f Cobham. 1 2 F ABIEN DURAND b e a set of su bstitutions. If x is the image b y a letter to letter morphism of a fixed p oin t of σ whic h belongs to S then w e will sa y that x is α -substitutiv e in S . If S is the set of all substitutions w e will sa y tha t x is α -substitutiv e. An easy computatio n shows that if σ is of length p then α = p . F urt hermore if a sequence is generated by a substitution o f length p then it is p - substitutiv e. Note that the con v erse is not true. This suggests the following conjecture form ulated b y G. Hansel. Conjecture. L et α an d β b e two multiplic atively indep endent Perr on num- b ers. L et A b e a finite alpha b et. L et x b e a s e quenc e of ∈ A N , the fol lowing ar e e q uiva lent: (1) x is b oth α -substitutive and β -substitutive; (2) x is ultimately p erio dic. In this pap er we prov e that 2) implies 1) and, what is the main result of this pap er, that this conjecture holds for a v ery large set of substitutions con ta ining all know n cases, w e call it S goo d . This set con t a ins some non- primitiv e substitutions of non-constant length. More precisely for some sets S of substitutions, we prov e Theorem 1. L et α and β b e two multiplic atively indep e n dent Perr on num- b ers. L et A b e a fi n ite alp h ab et. A se quenc e x ∈ A N is α - substitutive in S and β -substitutive in S if and o n ly if i t is ultimately p erio dic . This result is true for S const , the family o f substitutions with constan t length (this is the theorem of Cobham), and for S prim , the family of primitiv e substitutions [Du2]. In [F a] and [Du3] this result w as pro v ed for families of substitutions related to n umeration systems. These fa milies contain some non-primitiv e substitutions of non- constan t length. Muc h more results ha v e b een prov ed concerning generalizations of Cobham’s theorem to non-standard nume ration systems [BHMV1, BHMV2]. Most of the pro ofs of Cobham’s t yp e results are divided into t w o parts. In the first part it is pro v en that the set E ⊂ N is syndetic (the difference b et w een tw o consecutiv e elemen ts of E is b ounded) whic h corresp onds to the fact that the letters of the c har a cteristic sequence of E a pp ear with b ounded gaps. In the second part the result is pro v en for suc h E . W e will do the same. In Section 2 we recall some results concerning the length o f the w ords σ n ( a ) where σ is a substitution on the alphab et A and a ∈ A . These results hav e a k ey role in this pap er. In Section 3 w e prov e that 2) implies 1) . T o prov e the syndeticit y of E all pro of s use the w ell-kno wn fa ct that, if α and β are multiplicativ ely indep enden t nu m b ers strictly greater than 1 then t he set { α n /β m ; n, m ∈ Z } is dense in R + . H ere we need more. W e need the A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 3 densit y in R + of the set { n d α n /m e β m ; n, m ∈ Z } , where d and e are non- negativ e in tegers. W e prov e this result in Section 4 b ec ause we did not find it in the literature. W e pro ve in Section 5 that the letters with infinitely man y o ccurrences in x ∈ A N app ear with b ounded gaps. This implies the same result for w ords. In t he last section w e restrict ourself to S goo d , w e recall some results obtained in [Du3] and, using return w ords, w e conclude that x is ultimately p erio dic. More precisely we prov e that the conjecture is true for S goo d . W ords and sequences. An alphab et A is a finite set of elemen ts called letters . A wor d on A is an elemen t of the free monoid generated b y A , denoted by A ∗ . Let x = x 0 x 1 · · · x n − 1 (with x i ∈ A , 0 ≤ i ≤ n − 1) b e a w ord, its length is n and is denoted by | x | . The empty wor d is denoted b y ǫ , | ǫ | = 0. The set of non-empty words on A is denoted b y A + . The elemen ts of A N are called se quenc es . If x = x 0 x 1 · · · is a sequence (with x i ∈ A , i ∈ N ), and I = [ k , l ] a n in terv al of N w e set x I = x k x k +1 · · · x l and we sa y that x I is a fa ctor of x . If k = 0, w e sa y tha t x I is a p r efi x of x . The set of fa cto r s of length n of x is written L n ( x ) and the set of factors of x , or la n guage of x , is noted L ( x ). The o c c urr enc e s in x o f a w ord u ar e the in tegers i such that x [ i,i + | u |− 1] = u . When x is a w ord, we use the same terminology with similar definitions. The sequence x is ultimately p erio dic if there exist a w or d u and a non- empt y w ord v suc h that x = uv ω , where v ω = v v v · · · . Otherwise w e sa y that x is non-p eri o dic . It is p e rio dic if u is the empt y w ord. A sequence x is uniformly r e curr ent if fo r eac h factor u the gr eat est difference of tw o success iv e o ccurrences o f u is b ounded. Morphisms and matrices. Let A and B b e t w o alphab ets. A morphism τ is a map fro m A to B ∗ . Such a map induces by concatenation a morphism from A ∗ to B ∗ . If τ ( A ) is included in B + , it induces a map from A N to B N . These t wo maps a re also called τ . T o a morphism τ , fro m A to B ∗ , is naturally asso ciated the matrix M τ = ( 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 τ ( j ). Let M b e a square matrix, w e call dominant eigenvalue of M an eigen v alue r suc h that the mo dulus of all the other eigen v alues do not exceed the mo dulus of r . A square matrix is called primitive if it has a p o w er with p ositiv e co efficien ts. In this case the dominan t eigen v alue is unique, p ositiv e and it is a simple ro ot of the c haracteristic p olynomial. This is P erron’s Theorem. A real num b er is a Perr on numb er if it is a n algebraic in t eger that strictly dominates all its other alb ebraic conjugates. The follo wing result is w ell- kno wn (see [LM] for instance). Theorem 2. L et λ b e a r e al n umb er. T h e n 4 F ABIEN DURAND (1) λ is a Perr on numb er if and only if it is the domina nt eigenvalue of a primitive non-ne ga tive inte gr al matrix. (2) λ is the sp e ctr al r adius of a non-n e gative inte gr al matrix if and only if λ p is a Perr on numb er for some p osi tive inte ger p . Substitutions and substitut ive sequences. In this pap er a substitu- tion is a morphism τ : A → A ∗ suc h that for all letters of A w e ha ve lim n → + ∞ | τ n ( a ) | = + ∞ . Whenev er the matrix asso ciated to τ is primitiv e w e say that τ is a primitive substitution . A fixed p oint of τ is a sequence x = ( x n ; n ∈ N ) such tha t τ ( x ) = x . W e sa y it is a pr op er fixe d p oint if all letters of A hav e an o ccurrence in x . W e remark that all prop er fixed p oints of τ ha v e the same lang uage. Example. The substitution τ defined by τ ( a ) = aaab , τ ( b ) = bc and τ ( c ) = b has t wo fixed p oin ts, one is starting with the letter a and is prop er and t he other one is starting with the letter b and is not prop er. If τ is a primitiv e substitution then all its fixed p oin ts a r e prop er and uni- formly recurren t (for details see [Qu] for example). Let B b e another alphab et, w e sa y that a morphism φ from A to B ∗ is a letter to letter morphism when φ ( A ) is a subset of B . Let S b e a set of substitutions and supp ose that τ b elongs to S . Then the sequence φ ( x ) is called substitutive in S . W e say φ ( x ) is substitutive (resp. primitiv e substitutiv e) if S is the set all substitutions (resp. the set of primitiv e substitutions). If x is a prop er fixed p oin t of τ and θ is the dominant eigen v alue of τ ∈ S (i.e., the do minant eigenv a lue of the matrix asso ciated to τ ) then φ ( x ) is called θ -substitutive in S ; and we say θ -substitutive (resp. primitiv e substitutiv e) if S is the set all substitutions (resp. the set of primitiv e substitutions). W e p oin t out that in the last example the fixed p oin t y of τ starting with the letter b is also the fixed p oin t of the substitution σ defined b y σ ( b ) = bc and σ ( c ) = b . Moreov er the dominan t eigen v alue of τ is 3 and the dominan t eigen v alue of σ is (1 + √ 5) / 2. Hence in the definition of “ θ -substitutiv e” it is v ery imp ortan t fo r x to b e a prop er fixed p oin t, otherwise the conjecture presen ted in the in tro duction w ould not b e t r ue. Clearly , if φ ( x ) is θ -substitutiv e then it is θ p -substitutiv e for all p ∈ N . Consequen tly from Theorem 2 we can alw ay s supp ose θ is a Perron n umber. W e define L ( τ ) =  τ n ( a ) [ i,j ] ; i, j ∈ N , i ≤ j, n ∈ N , a ∈ A  . Let x b e a fixed p oin t of τ . Then L ( τ ) = L ( x ) if a nd only if x is prop er. If τ is primitiv e then for all its fixed p oin ts x hav e the same langua g e L = L ( τ ). A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 5 2. Some p reliminar y lemma t a. This section and the first case o f the pro of of Prop o sition 13 is prompted b y t he ideas in [Ha]. In this section σ will denote a substitution defined on the finite alphab et A , x one of its fixed p oints a nd Θ its dominan t eigen v alue. Lemma 3. Ther e exists a unique p artition A 1 , · · · , A l of A such that for al l 1 ≤ i ≤ l and al l a ∈ A i lim n → + ∞ | σ n ( a ) | c ( a ) n d ( a ) θ ( a ) n = 1 wher e θ ( a ) is the do m inant eigenvalue of M r e s tricte d to A i , d ( a ) its Jor dan or der a n d c ( a ) ∈ R . Pro of. See Theorem I I.10.2 in [SS ]. ✷ F or all a ∈ A w e will call gr owth typ e of a the couple ( d ( a ) , θ ( a )). If ( d , α ) and ( e, β ) are tw o growth types w e sa y that ( d, α ) is less than ( e, β ) (or ( d, α ) < ( e, β )) whenev er α < β or α = β and d < e . Consequen tly if the gro wth t yp e o f a ∈ A is less then the growth ty p e of b ∈ A then lim n → + ∞ | σ n ( a ) | / | σ n ( b ) | = 0. If the growth t yp e of a ∈ A is ( i, θ ) then there exists a letter b with gr owth t yp e ( i, θ ) having an o cc urrence in σ ( a ). W e hav e Θ = max { θ ( a ); a ∈ A } . W e set D = max { d ( a ); θ ( a ) = Θ , a ∈ A } and A max = { a ∈ A ; θ ( a ) = Θ , d ( a ) = D } . W e will sa y t ha t the letters of A max are of maximal gr owth and tha t ( D , Θ) is the gr owth typ e of σ . F or all letters a ∈ A , as lim n → + ∞ | σ n ( a ) | = + ∞ , it comes that θ ( a ) > 1, or θ ( a ) = 1 and d ( a ) > 0 . Hence Lemma 3 implies that there is no letter with growth type (0 , 1). An imp ortan t consequence o f the following lemma is that in fact for all a ∈ A we hav e θ ( a ) > 1 . Lemma 4. I f ( d, θ ) is the gr owth typ e of some letter then for al l i b elonging to { 0 , · · · , d } ther e exists a letter of gr owth typ e ( i, θ ) which app e ars i n finitely often in x . Pro of. See Lemma I II.7.1 0 in [SS]. ✷ W e define λ σ : A ∗ → R u 0 · · · u n − 1 7→ P n − 1 i =0 c ( u i ) 1 A max ( u i ). F rom Lemma 3 we deduce the follo wing lemma. Lemma 5. F or al l u ∈ A ∗ we h ave lim n → + ∞ | σ n ( u ) | /n D Θ n = λ σ ( u ) . W e say that u ∈ A ∗ is of maximal grow th if λ σ ( u ) 6 = 0. 6 F ABIEN DURAND Lemma 6. L et a ∈ A which ha s infinitely many o c c urr enc e s in x . Ther e exist a p os i tive in te ger p , a wor d u ∈ A ∗ of ma ximal gr owth and v , w ∈ A ∗ such that for a l l n ∈ N the w or d σ pn ( u ) σ p ( n − 1) ( v ) σ p ( n − 2) ( v ) · · · σ p ( v ) v w a is a pr efix o f x . Mor e over we have lim n → + ∞ | σ pn ( u ) σ p ( n − 1) ( v ) σ p ( n − 2) ( v ) · · · σ p ( v ) v w a | λ σ ( u )( pn ) D Θ pn + λ σ ( v ) P n − 1 k =0 ( pk ) D Θ pk = 1 . Pro of. Let a ∈ A b e a letter that has infinitely man y o ccurre nces in x . W e set a 0 = a . There exists a 1 ∈ A whic h has infinitely man y o ccurrences in x and suc h that a 0 has an o ccurrence in σ ( a 1 ). In this wa y w e can construct a sequence ( a i ; i ∈ N ) suc h that a 0 = a and a i o ccurs in σ ( a i +1 ), f or all i ∈ N . There exist i, j with i < j suc h that a i = a j = b . It comes that a o ccurs in σ i ( b ) and b o ccurs in σ j − i ( b ). Hence there exist u 1 , u 2 , v 1 , v 2 ∈ A ∗ suc h that σ i ( b ) = u 1 au 2 and σ j − i ( b ) = v 1 bv 2 . W e set p = j − i , v = σ i ( v 1 ) and w = u 1 . There exists u ′ suc h that u ′ b is a prefix of x . W e remark that for all n ∈ N the w ord σ n ( u ′ b ) is a prefix of x to o. W e set u = σ i ( u ′ ). W e hav e σ p ( u ′ b ) = σ p ( u ′ ) v 1 bv 2 . Consequen tly fo r all n ∈ N σ pn ( u ′ ) σ p ( n − 1) ( v 1 ) σ p ( n − 2) ( v 1 ) · · · σ p ( v 1 ) v 1 b is a prefix of σ np ( u ′ b ). Then σ pn ( u ) σ p ( n − 1) ( v ) σ p ( n − 2) ( v ) · · · σ p ( v ) v w a is a prefix of σ np + i ( u ′ b ) and consequen tly of x , for all n ∈ N . The last part of the lemma follow s from Lemma 5. ✷ 3. Asse r tio n 2) implies Ass er tion 1) in the conjecture. In this section w e prov e the following prop osition. It it is the “easy” part of the conjecture, namely Assertion 2) implies Assertion 1). The first part of the pro of is an adaptation of the pro of of Pro p osition 3.1 in [Du1] and the second part is inspired by the substitutions introduced in Section V.4 and Section V.5 of [Qu]. Prop osition 7. L et x b e a se quenc e on a finite alp h ab et a nd α a Perr on numb er. If x is p erio d ic (r esp . ultimately p erio dic) then it is α -substitutive primitive (r esp. α -substitutive). Pro of. Let x b e a p erio dic sequence with p eriod p . Hence w e can supp ose that A = { 1 , · · · , p } and x = (1 · · · p ) ω . Let M b e a primitive matrix whose dominan t eigen v alue is α a nd σ : B → B ∗ a primitiv e substitution whose matrix is M . Let y b e one of its fixed p oin ts. In the sequel w e construct, using σ , a new substitution τ with dominan t eigen v alue α , together with a A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 7 fixed p oin t z = τ ( z ), and a letter to letter morphism φ such that φ ( z ) = x . W e define the alphab et D = { ( b, i ) ; b ∈ B , 1 ≤ i ≤ p } , the morphism ψ : B → D ∗ and t he substitution τ : D → D ∗ b y ψ ( b ) = ( b, 1) · · · ( b, p ) and τ (( b, i )) = ( ψ ( σ ( b ))) [( i − 1) | σ ( b ) | ,i | σ ( b ) |− 1] , for all ( b, i ) ∈ D . The substitution τ is well defined b ecause | ψ ( σ ( b )) | = p | σ ( b ) | . Moreo v er, these morphisms are suc h that τ ◦ ψ = ψ ◦ σ . Hence the substitution τ is primitiv e. The sequence z = ψ ( y ) is a fixed p oint of τ and (using P erron theorem and the fact that M τ M ψ = M ψ M σ ) its dominan t eigen v alue is α . Let φ : D → A b e the letter to letter morphism defined by φ (( b, i )) = i . It is easy to see that φ ( z ) = x . It follow s that x is α -substitutiv e. Supp ose no w that x is ultimately p erio dic. Then there exist t w o non-empt y w ords u and v suc h that x = u v ω . F rom what precedes w e kno w that there exist a substitution τ : D → D ∗ , a fixed p oint z = τ ( z ) a nd a letter to letter morphism φ : D → A suc h t hat φ ( z ) = v ω . Let E ′ = { a 1 , a 2 , · · · , a | u | } b e an a lphab et, with | u | letters, disjoint from D and consider the sequence t = a 1 a 2 · · · a | u | z ∈ ( E ′ ∪ D ) N = F N . It suffices to pro v e tha t t is α -substitutiv e. W e extend τ to F setting τ ( a i ) = a i , 1 ≤ i ≤ | u | . Let G b e the alphab et of the w o r ds of length | u | + 1 o f t , that is to sa y G =  ( t n t n +1 · · · t n + | u | ); n ∈ N  where t = t 0 t 1 · · · . The sequence t = ( t 0 t 1 · · · t | u | )( t 1 t 2 · · · t | u | +1 ) · · · ( t n t n +1 · · · t n + | u | ) · · · ∈ G N is a fixed p oin t of the substitution ζ : G → G ∗ w e define as follows. Let ( l 0 l 1 · · · l | u |− 1 a ) b e an elemen t of G . Let s 0 s 1 · · · s | u |− 1 b e the suffix of length | u | of the w o r d τ ( l 0 l 1 · · · l | u |− 1 ). If | τ ( a ) | ≤ | u | , we set ζ (( l 0 l 1 · · · l | u |− 1 a )) = ( s [0 , | u |− 1] τ ( a ) 0 )( s [1 , | u |− 1] τ ( a ) [0 , 1] ) · · · ( s [ | τ ( a ) |− 1 , | u |− 1] τ ( a ) [0 , | τ ( a ) |− 1] ) , otherwise ζ (( l 0 l 1 · · · l | u |− 1 a )) = ( s [0 , | u |− 1] τ ( a ) 0 ) · · · ( s | u |− 1 τ ( a ) [0 , | u |− 1] )( τ ( a ) [0 , | u | ] ) · · · ( τ ( a ) [ | τ ( a ) |−| u |− 1 , | τ ( a ) |− 1] ) , By induction w e can prov e that fo r a ll n ∈ N w e ha v e ζ n (( t 0 t 1 · · · t | u | )) = ( t 0 t 1 · · · t | u | )( t 1 t 2 · · · t | u | +1 ) · · · ( t | τ n ( t | u | ) |− 1 · · · t | τ n ( t | u | ) | + | u |− 1 ) . Consequen tly t is a fixed p oint of ζ a nd ρ ( t ) = t where ρ : G → F is defined b y ρ (( r 0 r 1 · · · r | u | )) = r 0 . 8 F ABIEN DURAND Moreo ver w e remark that for all n ∈ N w e hav e | ζ n (( r 0 r 1 · · · r | u | )) | = | τ n ( r | u | ) | . F rom this and Lemma 3 it comes that fo r all ( r 0 r 1 · · · r | u | ) ∈ D w e hav e lim n → + ∞ | ζ n +1 (( r 0 r 1 · · · r | u | )) | | ζ n (( r 0 r 1 · · · r | u | )) | = α. Hence α is the dominant eigenv a lue of ζ and t is α -substitutiv e. ✷ Example. Let x = (12) ω and α = (1 + √ 5) / 2. It is t he dominan t eigen v alue of the substitution σ : A = { a, b } → A ∗ giv en by σ ( a ) = ab and σ ( b ) = a . W e ha v e D = { ( a, 1) , ( a, 2) , ( b, 1) , ( b, 2) } and the substitution τ : D → D ∗ defined in the previous pro of is giv en by τ (( a, 1)) = ( a, 1)( a, 2) , τ (( a, 2)) = ( b, 1)( b, 2) , τ (( b, 1)) = ( a, 1) and τ (( b, 2)) = ( a, 2) . Example. Let c b e a letter and x = c (12) ω . W e tak e the notations of the previous example a nd f o r conv enience w e set A = ( a, 1), B = ( a, 2) , C = ( b, 1) a nd D = ( b, 2). The substitution ζ : G → G ∗ , where G = { ( cA ) , ( AB ) , ( B C ) , ( C D ) , ( D A ) , ( B A ) } , defined in the previous pro of is giv en b y ζ (( cA )) = (( cA ))(( AB )), ζ (( AB )) = (( B C ))(( C D )), ζ (( B C )) = (( D A )), ζ (( C D )) = (( AB )), ζ (( DA )) = (( B A ))(( AB )), ζ (( B A )) = (( D A ))(( AB )). Let t b e the fixed p oin t of ζ whose first letter is ( cA ). Let φ : G → { c, 1 , 2 } b e the letter to letter mo r phism giv en b y φ (( cA )) = c, φ (( AB )) = 1 , φ (( B C )) = 2 , φ (( C D )) = 1 , φ (( D A )) = 2 , φ (( B A )) = 2 . W e hav e φ ( t ) = c (1 2 ) ω = x . Using Prop osition 7 w e obtain a sligh t impro vem en t o f the main results of resp ectiv ely [Du2] and [D u3]. More precisely: Theorem 8. L et α and β b e two multiplic atively indep e n dent Perr on num- b ers. L et x b e a se q uen c e on a finite a lphab et. The se q uen c e x is b oth α -substitutive primitive and β -substitutive prim itive if a nd only i f it is p eri- o dic. Theorem 9. L et U and V b e two Bertr and numer ation systems, α an d β b e two multiplic atively i n dep endent β -numb ers such that L ( U ) = L ( α ) and L ( V ) = L ( β ) . L et E b e a subset of N . The set E is b o th U -r e c o gnizable and V -r e c o gnizable if and only if it is a finite union of arithmetic pr o gr essions. (see [D u3] for the terminology) A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 9 4. Mul tiplica tive inde p endence and density. This section is dev o t ed to the pro of of the following pro p osition. Prop osition 10. L et α and γ b e two r ational ly indep ende n t p ositive num- b ers (i.e., α/β 6∈ Q ). L et d and e b e two no n-ne gative in te gers. Then the set { nα + d log n − mβ − e log m ; n, m ∈ N } is de nse in R . The fo llo wing straigh t forw ard corollary will b e essen tial in the next section. Corollary 11. L et α an d β b e two multiplic atively indep end ent p os itive r e a l numb ers. L et d and e b e two non-ne ga tive inte gers. Then the se t  n d α n m e β m ; n, m ∈ N  is de nse in R + . These tw o results are we ll-kno wn f or d = e = 0 (see [HW] for example). W e need t he follo wing lemma to pro v e Prop osition 1 0. Lemma 12. L et β < α b e two r ational ly indep endent n umb ers. Then for al l ǫ > 0 a n d al l N ∈ N ther e exist m, n , with m ≥ n ≥ N , such that 0 < nα − mβ < ǫ . Pro of. The pro of is left to the reader. ✷ Pro of of Prop osition 10. Let l ∈ R and ǫ > 0, we hav e to find N , M ∈ N suc h that | N α + d log N − M β − e log M − l | < ǫ . The pro of is divided into sev eral cases. First case: α > β , e = d a nd l ≥ d log( β α ). F rom Lemma 12 there exist t w o integers 0 < n < m suc h that 0 < nα − mβ < ǫ 2 and d log(1 + ǫ mβ ) ≤ ǫ 2 . Hence w e ha v e (1) d log( β α ) < d log( n ) − e log ( m ) < d log ( β α ) + d log(1 + ǫ mβ ) . Then nα − mβ + d (log n − log m ) < l + ǫ . W e consider f : N → R defined b y f ( k ) = k ( nα − mβ ) − d (log( k m ) − log( k n )) . W e hav e f (1) < l + ǫ , lim k → + ∞ f ( k ) = + ∞ and 0 < f ( k + 1) − f ( k ) = nα − mβ < ǫ . Hence there exists k 0 ∈ N suc h that | f ( k 0 ) − l | < ǫ , that is to sa y | N α + d log N − M β − e log M − l | < ǫ where N = nk 0 and M = mk 0 . 10 F ABIEN DURAND Second case: α > β , e = d and l < d log ( β α ). It suffices to take n, m with 0 < n < m suc h that − ǫ 2 < nα − mβ < 0 and d log(1 + ǫ mβ ) ≤ ǫ 2 , and the same metho d will give the r esult. Third case: α > β and e > d . Let k 0 ∈ N b e suc h that − ǫ < ( d − e ) log(1 + 1 k 0 ) < 0. If tw o in tegers n, m with 0 < n < m a re suc h that 0 < nα − mβ < ǫ then w e ha v e ( d − e ) log( m ) + d log( β α ) < d log( n ) − e log ( m ) < ( d − e ) log( m ) + d lo g( β α ) + d log(1 + ǫ mβ ) , whic h is negative for m large enough. Hence from Lemma 12 it comes that there exist t wo in tegers n, m with 0 < n < m suc h that 0 < nα − mβ < ǫ and (2) d log( n ) − e log( m ) ≤ l − ( k 0 ) ǫ − ( d − e ) lo g( k 0 ) . W e consider f : N → R defined by f ( k ) = k ( nα − mβ ) + d log( k n ) − e log ( k m ) . W e hav e f ( k 0 ) ≤ k 0 ǫ + ( d − e ) log( k 0 ) + d log( n ) − e log ( m ) ≤ l . Moreo ver lim k → + ∞ f ( k ) = + ∞ a nd for all k ≥ k 0 − ǫ < f ( k + 1) − f ( k ) = nα − mβ + ( d − e ) log(1 + 1 k ) < ǫ. Hence there exists an inte ger k 1 ≥ k 0 suc h that | f ( k 1 ) − l | < ǫ , that is to sa y | N α + d log N − M β − e log M − l | < ǫ where N = nk 1 and M = mk 1 . Remaining cases: The same ideas ac hiev e the pro of. ✷ 5. The letters appear with bounded gaps. Let α and β b e tw o m ultiplicativ ely indep en den t Perron n umbers. Let σ and τ b e tw o substitutions on the a lpha b ets A and B , with fixed p oin ts y and z a nd with grow th ty p es ( d , α ) and ( e, β ) resp ective ly . Let φ : A → C and ψ : B → C b e t wo letter to letter morphisms suc h that φ ( y ) = ψ ( z ) = x . This section is dev o t ed to the pro of of the following pro p osition. Prop osition 13. The letters of C which have i n finitely many o c curr enc es app e ar in x with b ounde d gap s in x . A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 11 Pro of: W e prov e this prop osition considering t w o cases. Let c ∈ C which has infinitely many o ccurrences. Let X = { n ∈ N ; x n = c } and A ′ = { a ∈ A ; φ ( a ) = c } . Assume that the letter c do es not app ear with b ounded gaps. Then there exist a ∈ A with infinitely man y o ccurrences in y and a strictly increasing sequence ( p n ; n ∈ N ) o f p ositive integers such tha t the letter c do es not app ear in φ ( σ p n ( a )). Let A ′′ b e the set of suc h letters. W e consider tw o cases. First case: There exists a ∈ A ′′ of maximal growth. Let u ∈ A ∗ suc h that u a is a prefix of y . Of course we can supp ose that u is non-empt y . F or all n ∈ N we call Ω n ⊂ A the set of letters app earing in σ p n ( a ). There exist t wo distinct integers n 1 < n 2 suc h tha t Ω n 1 = Ω n 2 . Let Ω b e the set of letters app earing in σ p n 2 − p n 1 (Ω n 1 ). It is easy to sho w tha t Ω = Ω n 1 = Ω n 2 . Consequen tly the set of letters a pp earing in σ p n 2 − p n 1 (Ω) is equal to Ω and for all k ∈ N t he set of letters app earing in σ p n 1 + k ( p n 2 − p n 1 ) ( A ) is equal to Ω. W e set p = p n 1 and g = p n 2 − p n 1 . W e remark that the letter c do es not app ear in the w ord φ ( σ p + k g ( a )) and that [ | σ p + k g ( u ) | , | σ p + k g ( ua ) | [ ∩ X = ∅ , for all k ∈ N . There exists a letter a ′ of maximal growth ha ving an o ccurrence in σ p ( a ). W e set σ p ( a ) = w a ′ w ′ . F or all k ∈ N w e ha v e | σ p + k g ( ua ) | ≥ | σ k g ( σ p ( u ) w a ′ ) | and (3) [ | σ k g ( v ) | , | σ k g ( v w a ′ ) | [ ∩ X = ∅ where v = σ p ( u ). Because a ′ is of maximal growth we ha v e λ σ ( v ) < λ σ ( v w a ′ ). Consequen tly there exists an ǫ > 0 suc h tha t λ σ ( v )(1 + ǫ ) < λ σ ( v w a ′ )(1 − ǫ ) . F rom Lemma 5 w e o btain that there exists k 0 suc h that fo r all k ≥ k 0 w e ha v e (4) | σ k g ( v ) | ( k g ) d α k g < λ σ ( v )(1 + ǫ ) < λ σ ( v w a ′ )(1 − ǫ ) < | σ k g ( v w a ′ ) | ( k g ) d α k g . F rom Lemma 6 applied to τ w e hav e that there exist s ∈ B ∗ of maximal gro wth, t, t ′ ∈ B ∗ and h ∈ N ∗ suc h that for all n ∈ N ψ  y [ τ hn ( s ) τ h ( n − 1) ( t ) ··· τ h ( t ) tt ′ ]  = c. F rom the second part of Lemma 6 it comes tha t there exists γ ∈ R suc h that lim n → + ∞ | τ hn ( s ) τ h ( n − 1) ( t ) · · · τ h ( t ) tt ′ | ( nh ) e β hn = γ . 12 F ABIEN DURAND F rom Corollar y 11 it comes that there exist tw o strictly increasing seque nces of in tegers, ( m i ; i ∈ N ) a nd ( n i ; i ∈ N ), and l ∈ R suc h that γ ( m i h ) e β m i h ( n i g ) d α n i g − → i → + ∞ l ∈ ] λ σ ( v )(1 + ǫ ) , λ σ ( v w a ′ )(1 − ǫ )[ . Hence from Lemma 5 w e also hav e | τ hm i ( s ) τ h ( m i − 1) ( t ) · · · τ h ( t ) tt ′ | ( n i g ) d α n i g (5) = | τ hm i ( s ) τ h ( m i − 1) ( t ) · · · τ h ( t ) tt ′ | γ ( m i h ) e β m i h γ ( m i h ) e β m i h ( n i g ) d α n i g − → i → + ∞ l. F rom (4) and (5) there exists i ∈ N such tha t | σ n i g ( v ) | < | τ hm i ( s ) τ h ( m i − 1) ( t ) · · · τ h ( t ) tt ′ | < | σ n i g ( v w a ′ ) | , whic h means that | τ hm i ( s ) τ h ( m i − 1) ( t ) · · · τ h ( t ) tt ′ | b elongs to X . This g iv es a con tra diction with (3). Second case: No letter in A ′′ has maximal growth. W e define B ′′ as A ′′ but with resp ect to τ and B . W e can supp ose that no letter of B ′′ has maximal gr owth. There exists a letter a ∈ A ′′ (resp. b ∈ B ′′ ) whic h has infinitely many o ccurrences in y (r esp. z ) and with gro wth ty p e ( d ′ , α ′ ) < ( d, α ) (resp. ( e ′ , β ′ ) < ( e, β )). W e recall that α ′ and β ′ are gr eat er than 1. F urthermore w e can supp ose that ( d ′ , α ′ ) (resp. ( e ′ , β ′ )) is ma ximal with resp ect to A ′′ (resp. B ′′ ). Let w = w 0 · · · w n b e a w ord b elonging to L ( y ) (resp. L ( z )), w e call g ap( w ) the largest in teger k suc h that t here exists 0 ≤ i ≤ n − k + 1 for which t he letter c do es not app ear in φ ( w i · · · w i + k − 1 ) (resp. in ψ ( w i · · · w i + k − 1 )). There exist infinitely man y prefixes of y (resp. z ) o f the type u 1 au 2 a ′ (resp. v 1 bv 2 b ′ ) f ulfilling the conditio ns ı ) and ıı ) b elow: ı ) The gr owth t yp e of u 1 ∈ A ∗ and a ′ ∈ A (resp. v 1 ∈ B ∗ and b ′ ∈ B ) is maximal. ıı ) The w o rds u 2 and v 2 do not con tain a letter of maximal growth. It is easy t o pro v e that there exists a constan t K ′ suc h that gap( τ n ( b ′ )) ≤ K ′ n e ′ β ′ n and gap( σ n ( a ′ )) ≤ K ′ n d ′ α ′ n for all n ∈ N . Due to Lemma 3, lim n → + ∞ | σ n ( a ) | /n d ′ α ′ n and lim n → + ∞ | τ n ( b ) | /n e ′ β ′ n exist and are finite, w e call them µ ( a ) and µ ( b ) resp ectiv ely . A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 13 Let u 1 au 2 a ′ b e a prefix of y fulfilling the conditions ı ) and ıı ), then c ho ose v 1 bv 2 b ′ fulfilling the same conditions and so that (6) K ′ µ ( a )  2 λ σ ( u 1 ) 2 λ τ ( v 1 ) + λ τ ( b ′ )  log α ′ log α  log β log α  e log( β ′ ) log( β ) − e ′ ≤ 1 3 . F rom Corollary 11 there exist four strictly increasing sequenc es of integers ( m i ; i ∈ N ), ( n i ; i ∈ N ), ( p i ; i ∈ N ) a nd ( q i ; i ∈ N ) such tha t (7) lim i → + ∞ n d i α n i m e i β m i = 2 λ τ ( v 1 ) 2 λ σ ( u 1 )+ λ σ ( a ′ ) and lim i → + ∞ p e i β p i q d i α q i = 2 λ σ ( u 1 ) 2 λ τ ( v 1 )+ λ τ ( b ′ ) . As a consequenc e of (7) we hav e (8) lim i → + ∞ n i /m i = log( β ) / log( α ) and lim i → + ∞ p i /q i = log ( α ) / log( β ) , and t here exists i 0 suc h that for all i ≥ i 0 w e ha v e | σ n i ( u 1 au 2 ) | | τ m i ( v 1 ) | ≤ 1 ≤ | σ n i ( u 1 au 2 a ′ ) | | τ m i ( v 1 b ) | and | τ p i ( v 1 bv 2 ) | | σ q i ( u 1 ) | ≤ 1 ≤ | τ p i ( v 1 bv 2 b ′ ) | | σ q i ( u 1 a ) | . It comes that ψ ( τ m i ( b )) (resp. φ ( σ q i ( a ))) has an o ccurrence in φ ( σ n i ( a ′ )) (resp. ψ ( τ p i ( b ′ ))). T o obtain a contradiction it suffices to prov e that there exists j ≥ i 0 suc h that gap( σ n j ( a ′ )) / | τ m j ( b ) | ≤ 1 2 or gap ( τ p j ( b ′ )) / | σ q j ( a ) | ≤ 1 2 . W e will consider sev era l cases. Before we define K to b e the maxim um of the set  K ′ , 2 log β log α , 2 log α log β , 4 λ τ ( v 1 ) 2 λ σ ( u 1 ) + λ σ ( a ′ ) , 4 λ σ ( u 1 ) 2 λ τ ( v 1 ) + λ τ ( b ′ )  . W e remark that K ≥ 2. There exists j 0 suc h that for all i ≥ j 0 the quan tities n i m i , p i q i , n d i α n i m e i β m i , p e i β p i q d i α q i , µ ( a ) q d ′ i α ′ q i | σ q i ( a ) | , µ ( b ) m e ′ i β ′ m i | τ m i ( b ) | and gap( σ n i ( a ′ )) n d ′ i α ′ n i are less than K . Let i ≥ j 0 . T o find j we will consider five cases. 14 F ABIEN DURAND First case: log( α ) log( β ) < log( α ′ ) log( β ′ ) . As β ′ > 1 w e hav e gap( τ p i ( b ′ ) / | σ q i ( a ) | ≤ K p e ′ i β ′ p i µ ( a ) q d ′ i α ′ q i µ ( a ) q d ′ i α ′ q i | σ q i ( a ) | ≤ K 2 µ ( a ) p e ′ i q d ′ i exp  p i q i − log α ′ log β ′  q i log β ′  , whic h tends to 0 when i tends to ∞ (t his comes from (8)). Second case: log( α ′ ) log( β ′ ) < log( α ) log( β ) . As in the first case w e obtain lim i → + ∞ gap( σ n i ( a ′ )) / | τ m i ( b ) | = 0 . Third case: log( α ′ ) log( α ) = log( β ′ ) log( β ) and ( e ′ − d ′ ) lo g β < ( e − d ) log β ′ . W e hav e gap( τ p i ( b ′ )) / | σ q i ( a ) | ≤ K 2 µ ( a ) p e ′ i q d ′ i β ′ p i α ′ q i = K 2 µ ( a ) p e ′ i q d ′ i  β p i α q i  log β ′ log β = K 2 µ ( a ) p e ′ i q d ′ i  q d i p e i  log β ′ log β  p e i β p i q d i α q i  log β ′ log β ≤ K 2 µ ( a )  p i q i  e ′ − e log β ′ log β K log β ′ log β q ( e ′ − d ′ ) − ( e − d ) log β ′ log β i ≤ K 2 µ ( a ) K e ′ +(1 − e ) log β ′ log β q ( e ′ − d ′ ) − ( e − d ) log β ′ log β i , whic h tends to 0 when i tends to ∞ . F ourth case: log( α ′ ) log( α ) = log( β ′ ) log( β ) and ( e ′ − d ′ ) lo g β > ( e − d ) log β ′ . As in the previous case w e obtain lim i → + ∞ gap( σ n i ( a ′ )) / | τ m i ( b ) | = 0 . Fifth case: log( α ′ ) log( α ) = log( β ′ ) log( β ) and ( e ′ − d ′ ) lo g β = ( e − d ) log β ′ . F rom (6), (7) and (8) w e obtain fo r a ll la rge enough i gap( τ p i ( b ′ )) / | σ q i ( a ) | ≤ K ′ µ ( a ) p e ′ i β ′ p i q d ′ i α ′ q i µ ( a ) q d ′ i α ′ q i | σ q i ( a ) | ≤ K ′ µ ( a )  p e i β p i q d i α q i  log α ′ log α  q i p i  e log β ′ log β − e ′ µ ( a ) q d ′ i α ′ q i | σ q i ( a ) | ≤ 1 2 . This ends the pro o f . ✷ A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 15 Corollary 14. T h e wor ds having infin i tely many o c curr enc e s in x app e ar in x with b o unde d g a ps. Pro of. Let u b e a w ord ha ving infinitely man y o ccurrences in x . W e set | u | = n . T o pro v e that u app ear s with b ounded gaps in x it suffices to prov e that the letter 1 app ears with b ounded ga ps in the sequence t ∈ { 0 , 1 } N defined b y t i = 1 if x [ i,i + n − 1] = u and 0 otherwise. In the sequel w e prov e that t is α and β -substitutiv e. The sequence y ( n ) = (( y i · · · y i + n − 1 ); i ∈ N ) is a fixed p oint of the substitution σ n : A n → A ∗ n where A n is the alphab et A n , defined for all ( a 1 · · · a n ) in A n b y σ n (( a 1 · · · a n )) = ( b 1 · · · b n )( b 2 · · · b n +1 ) · · · ( b | σ ( a 1 ) | · · · b | σ ( a 1 ) | + n − 1 ) where σ ( a 1 · · · a n ) = b 1 · · · b k (for more details see Section V.4 in [Qu] for example). Let ρ : A n → A ∗ b e the letter to letter morphism defined b y ρ (( b 1 · · · b n )) = b 1 for all ( b 1 · · · b n ) ∈ A n . W e hav e ρ ◦ σ n = σ ◦ ρ , and then M ρ M σ n = M σ M ρ . Consequen tly the dominant eigenv alue of σ n is α and y ( n ) is α -substitutiv e. Let f : A n → { 0 , 1 } b e the letter to letter morphism defined by f (( b 1 · · · b n )) = 1 if b 1 · · · b n = u and 0 otherwise. It is easy to see that f ( y ( n ) ) = t hence t is α -substitutiv e. In the same wa y we sho w that t is β -substitutiv e and Theorem 13 concludes the pro of. ✷ 6. Proof of Theore m 1. 6.1. Decomp osition of a substitution in to sub-substitutions. The follo wing prop osition is a conseque nce of P aragraph 4.4 and Prop osition 4.5.6 in [LM]. Prop osition 15. L et M = ( m i,j ) i,j ∈ A b e a matrix with non-ne ga tive c o effi- cients and no c olumn e qual to 0. Ther e exis t thr e e p ositive inte gers p 6 = 0 , q , l , wher e q ≤ l − 1 , and a p artition { A i ; 1 ≤ i ≤ l } of A such that the 16 F ABIEN DURAND matrix M p is e q ual to (9)              A 1 A 2 · · · A q A q +1 A q +2 · · · A l A 1 M 1 0 · · · 0 0 0 · · · 0 A 2 M 1 , 2 M 2 · · · 0 0 0 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . A q M 1 ,q M 2 ,q · · · M q 0 0 · · · 0 A q +1 M 1 ,q +1 M 2 ,q +1 · · · M q ,q +1 M q +1 0 · · · 0 A q +2 M 1 ,q +2 M 2 ,q +2 · · · M q ,q +2 0 M q +2 · · · 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . A l M 1 ,l M 2 ,l · · · M q ,l 0 0 · · · M l              , wher e the matric es M i , 1 ≤ i ≤ q ( r es p. q + 1 ≤ i ≤ l ) , ar e primitive or e qual to zer o (r esp. primitive), and such that for al l 1 ≤ i ≤ q ther e ex i s ts i + 1 ≤ j ≤ l such that the m atrix M i,j is differ ent fr om 0. In what follo ws w e k eep the notat ions of Prop osition 1 5. W e will sa y that { A i ; 1 ≤ i ≤ l } is a primi tive c omp onent p artition of A (with r esp e ct to M ) . If i b elongs to { q + 1 , · · · , l } we will say that A i is a princip al pri m itive c omp onent of A (with r e s p e ct to M ) . Let τ : A → A ∗ b e a substitution and M = ( m i,j ) i,j ∈ A its matrix. Let i ∈ { q + 1 , · · · , l } . W e denote τ i the restriction ( τ p ) | A i : A i → A ∗ of τ p to A i . Because τ i ( A i ) is included in A ∗ i w e can consider that τ i is a morphism from A i to A ∗ i whose matrix is M i . Let i ∈ { 1 , · · · , q } suc h tha t M i is not equal to 0. Let ϕ i b e the morphism from A to A ∗ i defined b y ϕ ( b ) = b if b b elongs to A i and t he empt y w ord otherwise. Let us consider t he map τ i : A i → A ∗ defined b y τ i ( b ) = ϕ i ( τ p ( b )) for all a ∈ A i . W e r emark as previously that τ i ( A i ) is included in A ∗ i , consequen tly τ i defines a mo r phism fr o m A i to A ∗ i whose mat r ix is M i . W e will say that the substitution τ : A → A ∗ satisfies Condition (C) if: C1. The matrix M , itself, is of the ty p e (9) (i.e., p = 1) ; C2. The matrices M i are equal to 0 or with p o sitiv e co efficien ts if 1 ≤ i ≤ q and with p ositiv e co effi cien ts o therwise; C3. F or all matrices M i differen t from 0, with i ∈ { 1 , · · · , l } , there exists a i ∈ A i suc h that τ i ( a i ) = a i u i where u i is a no n- empt y w ord of A ∗ if M i is differen t from the 1 × 1 ma t rix [1] and empt y otherwise. F rom Prop osition 15 ev ery substitution τ : A → A ∗ has a p ow er τ k satisfying condition (C) . The definition of substitutions implies that for all q + 1 ≤ i ≤ l w e hav e M i 6 = [1]. Let τ : A → A ∗ b e a substitution satisfying condition (C) (we k eep the previous notations). F or all 1 ≤ i ≤ l suc h that M i is differen t f r om 0 and the 1 × 1 mat rix [1], the map τ i : A i → A ∗ i defines a substitution w e A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 17 will call mai n sub-substitution o f τ if i ∈ { q + 1 , · · · , l } and non-ma in sub- substitution of τ otherwise. Moreo ver t he matrix M i has p ositiv e co efficien ts whic h implies that t he substitution τ i is primitiv e. W e remark that there exists at least one main sub-substitution. In [D u3] the following results w ere obtained and will b e used in the sequel. Lemma 16. L et x b e a pr op er fixe d p oin t of the substitution σ . L et σ : A → A ∗ b e a m ain sub-substitution of σ . T h en for al l n ∈ N and al l a ∈ A the wor d σ n ( a ) app e ars infinitely many times in x . Pro of. The pro of is left to the reader. ✷ In [Du3] the following result is obta ined and will b e used in the sequel. Theorem 17. L et x and y b e r esp e ctively a primitive α -substitutive se quenc e and a prim i tive β -s ubstitutive se quenc e such that L ( x ) = L ( y ) . S upp ose that α and β ar e multiplic atively indep endent, then x and y ar e p erio dic . 6.2. The conjecture for “go o d” substitutions. W e do not succeed y et to prov e the conjecture giv en in the intro duction but we are able to pro ve it for a ve ry large family of substitutions. Un til w e pro ve the whole conjecture w e call them “go od” substitutions. More precisely , let σ : A → A ∗ b e a substitution whose dominan t eigen v alue is α . The substitution σ is said to b e a “ g o o d” substitution if there exists a main sub-substitution whose dominan t eigen v alue is α . F or example primitiv e substitutions and substitutions of constan t length are “go o d” substitutions. Now consider the follo wing substitution σ : { a, 0 , 1 } → { a, 0 , 1 } ∗ a 7→ aa 0 0 7→ 01 1 7→ 0 . Its dominan t eigen v alue is 2 and it has o nly one main sub-substitution (0 7→ 01, 1 7→ 0) whic h dominan t eigenv alue is (1 + √ 5) / 2, hence it is not a “go od” substitution. Theorem 18. Supp ose that w e o nly c onsider “go o d” substitutions. The n the c onj e ctur e is true. Pro of. W e take the notatio ns of the first lines of Section 5. Let σ : A → A ∗ b e a main sub-substitution of σ . The w o r ds of x app ear ing infinitely man y times in x app ear with b ounded g aps (Corollary 14). Hence using Lemma 16 we deduce that for all main sub-substitution σ o f σ and τ of τ w e ha ve φ ( L ( σ )) = ψ ( L ( τ )) = L . F rom Theorem 17 it comes that L is p erio dic, i.e., there exists a w ord u suc h that L = L ( u ω ) where | u | is the least p erio d. There exists an in teger N suc h that all the w ords of length | u | 18 F ABIEN DURAND app ear infinitely man y times in x N x N +1 · · · . W e set t = x N x N +1 · · · a nd we will pro v e that t is p eriodic and consequen tly x will b e ultimately p erio dic. The word u app ears infinitely ma ny times, consequen tly it app ears with b ounded gaps. Let R u b e the set of return w ords to u (a word w is a return w ord to u if w u ∈ L ( x ) , u is a prefix of w u and u has exactly t w o o ccurrences in w u ). It is finite. There exists a n in teger N suc h that all the w ords w ∈ R u ∩ L ( x N x N +1 · · · ) a pp ear infinitely man y times in x . Hence these words app ear with b ounded gaps in x . W e set t = x N x N +1 · · · and w e will pro v e that t is p eriodic and consequen tly x will b e ultimately p erio dic. W e can supp ose that u is a prefix of t . Then t is a concatenatio n of return w ords to u . Let w b e a return w ord to u . It app ears with b ounded gaps hence it app ears in some φ ( σ n ( a )) a nd there exist tw o w or ds, p and q , and an integer i suc h that w u = pu i q . As | u | is t he least p eriod of L it comes that w u = u i . It follows that t = u ω . ✷ The case of fixed p oin ts. This part is dev oted to the pro of of Theorem 1 restricted to fixed p o in ts. More precisely w e prov e: Corollary 19. L et x b e a fixe d p oint of the substitution σ : A → A ∗ whose dominant eige nvalue is α . Supp ose that x is also a fixe d p oint of the substi- tution τ : A → A ∗ whose dominan t eigenvalue is β . Supp ose that α and β ar e m ultiplic a tively indep endent. Then x is ultimately p eri o dic. Pro of. The letters app earing infinitely often in x a pp ear with b ounded gaps (Pro p osition 13). Let σ : A → A b e a main sub-substitution of σ . Let a ∈ A . Supp ose that there exists a letter b , app ear ing infinitely man y times in x , whic h do es not b elong to A . Then the w ord σ n ( a ) do es not con tain b and b could not app ear with b ounded gaps. Consequen tly there exists only one main sub-substitution and the letters whic h app ear with b ounded g aps b elong to A . It comes that σ is a “go o d” substitution. In the same w a y τ is a go o d substitution. Theorem 18 concludes the pro of. ✷ Reference s [BHMV1] V. Bruy` ere, G. Hansel, C. Michaux and R. Villemaire, Lo gic and p - r e c o gnizable sets of int e gers , Bull. Belg. Ma th. So c. Simo n Stevin 1 (199 4), 191-2 38. [BHMV2] V. Bruy` ere, G. Hansel, C. Michaux and R. Villemaire, Corr e ction to: ”L o gic and p -re c o gnizable sets of inte gers” , B ull. B e lg . Ma th. So c. Simon Stevin 1 (1994), 577. [Co1] A. Cobham, On the b ase-dep endenc e of sets of numb ers r e c o gnizable by finite automata , Math. Systems Theo ry 3 (196 9 ), 1 86-19 2. A THEOREM OF COBHAM FOR NON- PRIMITIVE SUBSTITUTION S 19 [Co2] A. Cobham, Uniform tag se quenc es , Math. 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Queff´ elec, Substitut ion Dynamic al S ystems-Sp e ctr al Analysis , Lecture Notes in Mathematics 12 94, Springer-V erlag, Be r lin (1987). [SS] A. Salomaa a nd M. Soittola, Automata-the or etic asp e cts of formal p ower series , T exts a nd Monogra phs in Computer Science , Spr inger-V erlag (1 978). Universit ´ e de P icardie Jules Verne, Labora toire Am i ´ enois de Ma th ´ ema ti- ques Fond ament ales et Appliqu ´ ees, CN RS-FRE 2270, 33 rue Saint Leu, 80039 Amiens Cedex 01, France. E-mail addr ess : fabien .dura nd@u-picardie.fr