Episturmian morphisms and a Galois theorem on continued fractions

39 DOI: 10.1051/ita:2005012

EPISTURMIAN MORPHISMS AND A GALOIS THEOREM ON CONTINUED FRACTIONS

Jacques Justin

Abstract. We associate with a word w on a finite alphabet A an episturmian (or Arnoux-Rauzy) morphism and a palindrome. We study their relations with the similar ones for the reversal ofw. Then when

|A|= 2 we deduce, using the Sturmian words that are the fixed points of the two morphisms, a proof of a Galois theorem on purely periodic continued fractions whose periods are the reversal of each other.

Mathematics Subject Classification.11A55, 68R15.

Introduction

Sturmian words on a two-letter alphabet have been actively studied, at least since the fundamental paper of Morse and Hedlund [15]. For a survey, see [13].

These words have a deep relation with simple continued fractions. Among the generalizations of Sturmian words to any finite alphabet, the one known as Arnoux- Rauzy sequences [2, 16–18] or, with even slightly more generality, as episturmian words [3, 8, 11, 12] leads to many properties extending those of Sturmian words. In particular the continued fractions point of view has a transposition for episturmian words [20]. In this paper the famous Lagrange theorem relating periodic continued fractions to quadratic numbers was extended to a “multidimensional” continued fraction algorithm (see also [19]). However this one has not the same efficiency as the classical one for approximating irrationals by rationals. Evariste Galois

Keywords and phrases.Episturmian morphism, Arnoux-Rauzy morphism, palindrome, con- tinued fraction, Sturmian word.

1Present address: 19 rue de Bagneux, 92330 Sceaux, France.

2 LIAFA, ERS 586, Universit´e Paris VII, case 7014, 2 place Jussieu, 75251 Paris Cedex 5, France;justin@liafa.jussieu.fr

c EDP Sciences 2005

found, when a student at Lyc´ee Louis le Grand¹, a theorem relating the values of two purely periodic (simple) continued fractions whose periods are reversal of each other [9]. In the same paper Galois also gives a characterization of quadratic numbers with a purely periodic continued fraction expansion. Both results are used in [1] for characterizing the so-called Sturm numbers (see also [13], Th. 2.3.26).

Here we study relations involving some (finite) episturmian words and epis- turmian morphisms. More precisely we associate with a (finite) wordw a palin- drome P al(w) and a morphism µ_w. Such palindromes (which are the bispecial Arnoux-Rauzy words of [14]) play an important role in the theory of (infinite) episturmian words [8], in particular in the Sturmian case where they are called central words [13].

In Section 2 we establish some relations between P al(w), µ_w, P al( ˜w), µ_w˜, where ˜w is the reversal ofw. These relations are explained by the fact that the incidence matrices ofµ_w andµ_w˜ are similar.

In Section 3, applying this to a 2-letter alphabet, we prove the above-mentioned Galois theorem by considering the standard episturmian words which are the fixed points ofµ_wandµ_w˜.

It seems to us that this confirms the interest of the multidimensional continued fraction algorithm in the above sense. However it remains much work to do on the subject, in particular in relation with the generalized intercept introduced in [11]

and the generalized Ostrowski numeration systems [4, 12].

1. Preliminaries

The alphabetA, |A| ≥2, is finite and will be kept throughout the paper, εis the empty word,A^∗ is the set of (finite) words andA^ω the set of (right) infinite words onA.

The length ofv=x₁· · ·x_n,x_i ∈A, is|v|=nand the number of occurences of letterxin v is|v|x. If|v|x= 0, thenvisx-free. Thereversalofv is ˜v=x_n· · ·x₁ andvis apalindromeif ˜v=v.

Ifv∈A^∗the(Parikh) vectorofvis the|A|×1 vector of the|v|x,x∈A, and ifϕ is an (endo)morphism ofA^∗itsincidence matrixis the|A|×|A|matrixM = (m_xy) whose columns are the vectors of the|ϕ(y)|,y∈A,i.e.,m_xy=|ϕ(y)|x.

Foru∈A^∗ its(right) palindromic closure is the shortest palindromeu⁽⁺⁾hav- inguas a prefix. We haveu⁽⁺⁾=uv⁻¹uwithvthe longest palindromic suffix ofu (whenh=f g we sometimes writef =hg⁻¹andg=f⁻¹h).

Now we say some words about (infinite) episturmian words, limiting ourselves to what is really needed here. Let ∆∈A^ω, ∆ =x₁x₂· · ·x_i· · ·,x_i∈A. The infinite word having the sequence of prefixesu₁=ε,u₂=x₁, ...,u_i+1= (u_ix_i)⁽⁺⁾, ... is the standard episturmian word directedby ∆ (also called characteristic Arnoux-Rauzy

1 The author also studied at Lyc´ee Louis le Grand but without discovering any theorem.

sequence if ∆ contains infinitely many occurrences of each of its letters). In par- ticular for|A|= 2 Arnoux-Rauzy sequences are the Sturmian words, while epis- turmian words also include the “periodic Sturmian words”,i.e., the infinite words given by cutting sequences with rational slope.

Now letw=x₁· · ·x_n,x_i∈A, then using the successive letters ofwconstruct as previously the sequenceu₁ =ε, . . . , u_n+1 = (u_nx_n)⁽⁺⁾of all palindromic prefixes ofu_n+1. We writeu_n+1=P al(w) and say thatwdirectsP al(w). Letx∈A. Ifw isx-free thenP al(wx) =P al(w)xP al(w). Ifxoccurs inwwritew=w₁xw₂with w₂ x-free. Then the longest palindromic prefix ofP al(w) followed byxinP al(w) isP al(w₁) whence easily

P al(wx) =P al(w)P al(w₁)⁻¹P al(w). (1) Consider for any lettera the morphismψ_a such thatψ_a(a) =a andψ_a(b) =ab for any letterb=a. Theψ_a generate by composition the monoid ofpure standard episturmian morphisms[11], Section 2.1.

For any w = x₁· · ·x_n as previously we write µ_w = ψ_x1 ◦ψ_x2· · · ◦ψ_xn. In particularµ_εis the identity and, forx∈A,µ_x=ψ_x. We also denote byM_w the incidence matrix ofµ_w.

The relations betweenw,µ_w,M_w are one-to-one and indeed are isomorphisms betweenA^∗, the monoid of pure standard episturmian morphisms and the monoid of theM_w.

For continued fractions see [5] for instance.

2. Words and matrices relations

The first lemma recalls and proves for sake of completeness some relations appearing with different notations in [8, 11]. This allows to prove the curious relation|P al(w)|=|P al( ˜w)|and some other ones. Then we give an interpretation in terms of the matricesM_w andM_w˜.

Lemma 2.1.

1) For any palindromepand letterx,µ_x(p)xis a palindrome.

2) For anyw∈A^∗,x∈Awe have P al(xw) =µ_x

P al(w)

x. (2)

3) For anyv, w∈A^∗,

P al(vw) =µ_v

P al(w)

P al(v). (3)

Proof.

Part 1) follows from the easy fact that, foru∈A^∗,x∈A,µ_x(˜u)x=xµ_x(u).

Part 2) is proved by induction on|w|. If|w|= 0 the assertion is trivial. Other- wise letw=vy,y ∈A. If y occurs in vwrite v=v₁yv₂ withv₂ y-free. Then by

equation (1)P al(w) =P al(v)P al(v₁)⁻¹P al(v) whence µ_x

P al(w)

x=µ_x

P al(v)

xx⁻¹µ_x

P al(v₁)₋₁ µ_x

P al(v) x.

As|v|<|w|and|v₁|<|w|we get by induction hypothesis µ_x

P al(w)

x=P al(xv)P al(xv₁)⁻¹P al(xv).

As xv = xv₁yv₂ and |v₂|y = 0, equation (1) shows that the right member is P al(xw).

If on the contrary|v|y= 0 then P al(w) =P al(v)yP al(v) whence µ_x

P al(w)

x=µ_x P al(v)

xx⁻¹µ_x(y)µ_x P al(v)

x=P al(xv)x⁻¹µ_x(y)P al(xv).

Thus if y = x then |w|y = 0, whence µ_x

P al(w)

x = P al(xv)yP al(xv) = P al(xvy) =P al(xw). Ify =xthen similarlyµ_x

P al(w)

x=P al(xv)P al(xv) = P al(xvx) =P al(xw).

For 3) the proof is by induction on |v|. The assertion is trivial for |v| = 0.

Otherwise letv=yv₁. Using part 2) of the lemma and the induction hypothesis we successively get

P al(vw) =P al(yv₁w) =µ_y

P al(v₁w) y=µ_y

µ_v1

P al(w) P al(v₁)

y

=µ_yv1

P al(w) µ_y

P al(v₁)

y=µ_v

P al(w)

P al(v).

Lemma 2.2. Forw∈A^∗ if x∈A occurs inw writew=w₁xw₂ with w₁ x-free.

Then

|P al(w)|x=|P al(w2)|+ 1. (4)

Proof. By Lemma 2.1P al(w) =µ_w1

P al(xw₂)

P al(w₁) whence asw₁ isx-free

|P al(w)|x=|µ_w1

P al(xw₂)

|x=|P al(xw₂)|x

=|µx

P al(w₂)

x|x=|P al(w₂)|+ 1.

We deduce a rather curious relation between the palindromes directed bywand ˜w.

Theorem 2.3. For any w∈A^∗,P al(w)andP al( ˜w)have the same length.

Proof. The proof is by induction on|w|. This is trivial for |w| = 0. Otherwise set w = vx. If v is x-free then P al(w) = P al(v)xP al(v) whence |P al(w)| = 2|P al(v)|+ 1. Also, by equation (2), P al( ˜w) = P al(x˜v) = µ_x

P al(˜v) x. As P al(˜v) isx-free, |µx

P al(˜v)

| = 2|P al(˜v)|. Thus using the induction hypothesis

|P al( ˜w)|= 2|P al(˜v)|+ 1 = 2|P al(v)|+ 1 =|P al(w)|.

Otherwisexoccurs inv. Writev=v₁xv₂withv₂ x-free. Then by equation (1) P al(w) = P al(vx) = P al(v)P al(v₁)⁻¹P al(v) whence |P al(w)| = 2|P al(v)| −

|P al(v1)|.

Also P al( ˜w) = P al(x˜v) = µ_x

P al(˜v)

x whence |P al( ˜w)| = 2|P al(˜v)|−

|P al(˜v)|x+1. But Lemma 2.2 applied to ˜v= ˜v₂x˜v₁gives|P al(˜v)|x=|P al(˜v₁)|+1.

Thus using induction hypothesis,

|P al( ˜w)|= 2|P al(˜v)| − |P al(˜v₁)|= 2|P al(v)| − |P al(v₁)|=|P al(w)|.

Lemma 2.4. Let w∈A^∗,y∈A.

1) If wisy-free thenµ_w(y) =P al(w)y. Otherwise writew=v₁yv₂,v₂ y-free, thenµ_w(y) =P al(w)P al(v₁)⁻¹.

2)Withx∈A, ifwisx-free ory-free then |µw(y)|x=|P al(w)|x+|y|x. Other- wise writew=v₁yv₂=w₁xw₂ with|v₂|y=|w₁|x= 0. Then

|µw(y)|x=|P al(w)|x− |P al(v₁)|x=|P al(w)|x− |P al( ˜w₂)|y.

Proof. For 1), by equation (3), P al(wy) = µ_w(y)P al(w). If w is y-free then P al(wy) =P al(w)yP al(w) whenceµ_w(y) =P al(w)y. Otherwise, withw=v₁yv₂, v₂ y-free,P al(wy) =P al(w)P al(v₁)⁻¹P al(w) whenceµ_w(y) as claimed.

For 2), if w is y-free then|µw(y)|x = |P al(w)y|x =|P al(w)|x+|y|x. If w is x-free andx=y then|µ_w(y)|x= 0 =|P al(w)|x+|y|x.

Otherwise, writew =v₁yv₂ =w₁xw₂ with |v₂|y =|w₁|x = 0. Thenµ_w(y) = P al(w)P al(v₁)⁻¹ whence|µ_w(y)|x=|P al(w)|x− |P al(v₁)|x.

It remains to show that |P al(v₁)|x = |P al( ˜w₂)|y. If |v₁| ≤ |w₁| then v₁ is x-free, w₂ is y-free, hence|P al(v₁)|x = |P al( ˜w₂)|y = 0. Otherwise, |v₁| >|w₁|.

Writev₁=w₁xu,u∈A^∗. Then using Lemma 2.2, asv₂ isy-free andw₁isx-free,

|P al(v₁)|x=|P al(u)|+1 and similarly|P al( ˜w₂)|y =|P al(˜u)|+1 =|P al(u)|+1.

Corollary 2.5. The traces ofM_w andM_w˜ are equal.

Proof. For x ∈ A, if |w|x = 0 then |µw(x)|x = |µw_˜(x)|x = 1. Otherwise let w = v₁xv₂ = w₁xw₂ with |v₂|x = |w₁|x = 0. Then by Lemma 2.4 |µw(x)|x =

|P al(w)|x−|P al(v1)|x=|P al(w)|x−|P al( ˜w₂)|x. Similarly|µw˜(x)|x=|P al( ˜w)|x−

|P al( ˜w₂)|x. Thus in both cases|µw(x)|x− |µw˜(x)|x=|P al(w)|x− |P al( ˜w)|x. Summing overx∈Awe get tr(M_w)−tr(M_w˜) =|P al(w)| − |P al( ˜w)|= 0.

It is possible from this to show thatM_w and M_w˜ have the same eigenvalues.

Indeed M_wk and M_w˜k have the same traces for any integer k. As the trace is the sum of the eigenvalues and as the eigenvalues ofM_wk and M_w˜k are the k-th powers of those ofM_w andM_w˜ we get that the Newton sums of the eigenvalues ofM_wandM_w˜ are the same and that these matrices have the same characteristic polynomial.

Theorem 2.6. Forx∈AsetL_w(x) =

y∈A|µw(y)|x. Then

L_w(x) = (|A| −1)|P al(w)|x+ 1. (5) Proof. If |w|x = 0 then L_w(x) = |µw(x)|x = 1 = (|A| −1)|P al(w)|x+ 1. Oth- erwise let w = w₁xw₂ with w₁ x-free. Then by Lemma 2.4, 2) |µw(y)|x =

|P al(w)|x−|P al( ˜w₂)|y. Summing overywe getL_w(x) =|A||P al(w)|x−|P al( ˜w₂)|.

As by Lemma 2.2 |P al( ˜w₂)| = |P al(w2)| = |P al(w)|x −1 we get L_w(x) =

(|A| −1)|P al(w)|x+ 1.

Corollary 2.7.

y∈A

|µw(y)|= (|A| −1)|P al(w)|+|A|. (6)

Proof. By summation of equation (5) overx.

Remark 2.1. The |µw(y)| are periods of P al(w) and indeed formula (6) is the same as the one given in [7] for|A|= 3 and [10] for |A| ≥2. More precisely the ordered set of the |µw(y)| is a “good”|A|-uple in the sense of these papers with its GCD equal to 1 andP al(w) is a word of maximal length having these periods

|µw(y)| and not having period 1 (this is a particular case of the Fine and Wilf theorem for|A|periods).

Proposition 2.8. For anyx, y ∈A

L_w(x)−L_w˜(y) = (|A| −1)(|µw(y)|x− |µw˜(x)|y).

Proof. By Theorem 2.6, L_w(x)−L_w˜(y) = (|A| −1)(|P al(w)|x− |P al( ˜w)|y). If

|w|x = 0 or |w|y = 0 then by Lemma 2.4, 2) |µ_w(y)|x = |P al(w)|x+|y|x and

|µ_w˜|y = |P al( ˜w)|y +|x|y whence the result. Otherwise let w = w₁xw₂ with

|w₁|x = 0 . Then by Lemma 2.4, 2) |µ_w(y)|x = |P al(w)|x− |P al( ˜w₂)|y and

|µ_w˜(x)|y=|P al( ˜w)|y− |P al( ˜w₂)|y whence the result.

Example 2.1. WithA={1,2,3},w= 123²2³, ˜w= 2³3²21,

P al(w) = 12131213121213121312121312131212131213121 P al( ˜w) = 22232223222232223222122232223222232223222, both of length 41, and

M_w=



21 5 17 12 3 10

8 2 7



 M_w˜=



1 1 1 16 23 26

4 6 7



.

It is possible to write Proposition 2.8 as a matricial relation, but we will see now a more direct way.

Theorem 2.9. MatricesM_w andM_w˜ are similar and are related by

HM_w= (HM_w˜)^T (7)

withH = (h_xy)given by h_xx= 0,h_xy= 1,x, y∈A,x=y.

Proof. For any letterxit is immediate to verify thatHM_x= (HM_x)^Twhence (7)

by product.

Remark 2.2. LetK= (k_xy) such that,k_xx= 2− |A|,k_xy= 1,x, y ∈A, x=y, thenHK=KH = (|A| −1)I whence

M_w^T_˜ = (|A| −1)⁻¹HM_wK.

It is possible to deduce from Theorem 2.9 some relations given above and even some other ones due to the particular form ofH, for example det(M_w−M_w^T_˜) = 0 for|A|>2 and det(M_w−M_w˜) = 0 for odd|A|.

Corollary 2.10. The wordw is a palindrome if and only if the matrix HM_w is symmetrical.

Proof. By Theorem 2.9HM_w is symmetrical if and only if M_w =M_w˜. Thus, in view of the bijection betweenwandM_w the proof is over.

Remark 2.3. WhenAis a 2-letter alphabet,{1,2}, [6] gives a number theoretical condition on the |µ_w(y)| equivalent to w is a palindrome, namely |µ_w(1)|² ≡ 1 mod (|µ_w(1)|+|µ_w(2)|). It could be asked whether this condition can be extended to any finite alphabets.

3. The case |A| = 2 and a Galois theorem

From now onA ={1,2}. Set M_w = (m_ij), M_w˜ = (m_ij), 1 ≤ i, j ≤ 2. For short we also setp₁=m₁₁+m₂₁,p₂=m₁₂+m₂₂ and similarlyp₁=m₁₁+m₂₁, p₂=m₁₂+m₂₂.

The matrixH of Theorem 2.9 becomes the permutation matrix 0 1

1 0

, thus by this theoremM_w˜=

m₂₂ m₁₂ m₂₁ m₁₁

.

Consider an infinite simple continued fraction, [e₁, e₂, . . .],e₁≥0,e₂, e₃, . . . >0 and the standard Sturmian infinite word s directed by ∆ = 1^e12^e21^e3· · ·. Let α₁, α₂ be the frequencies of 1,2 in s. It is well known [13] that α₂ = [0, e₁+ 1, e₂, e₃, . . .].

Consider in particular an immediately periodic continued fraction, [e₁, e₂, . . ., e_d] (thuse₁ > 0, now) and without loss of generality suppose d is even. Then s is directed by ∆ =w^ωwhere w= 1^e12^e2· · ·2^ed,i.e.,sis the fixed point ofµ_w. Also consider ˜w= 2^ed1^ed−1· · ·1^e1 ands directed by ˜w^ω,i.e., the fixed point ofµ_w˜.

The above-mentioned Galois Theorem is as follows.

Theorem 3.1. With θ = [e₁, e₂, . . ., e_d] andθ = [e_d, e_d−1, . . ., e₁], the algebraic conjugate of the (quadratic by Lagrange’s Theorem) numberθ is−1/θ.

Let us verify that. Set θ^∗ = −1/θ. As said above the frequency of 2 in s is α₂ = [0, e₁+ 1, e₂, e₃, . . ., e_d, e₁] and the frequency of 1 in s is α₁ = [0, e_d+ 1, e_d−1, e_d−2, . . ., e₁, e_d].

It follows

α₂= 1

θ+ 1 α₁= 1

θ+ 1 = θ^∗

θ^∗−1· (8)

Now the vector (α₁, α₂)^Tis an eigenvector ofM_wcorresponding to the dominating eigenvalueξofM_w(i.e.,ξis the greater root ofξ²−(m11+m₂₂)ξ+1). Calculation of this vector gives

α₁= ξ−p₂

p₁−p₂ α₂= p₁−ξ

p₁−p₂ (9)

and similarly for frequenciesα₁, α₂ of 1,2 ins. Then using (8) we get θ=ξ−p₂

p₁−ξ θ^∗= p₂−ξ

p₁−ξ· (10)

In order to verify that θ and θ^∗ are conjugate it suffices to verify that θθ^∗ and θ+θ^∗ are rational and this is easy usingξ²= (m₁₁+m₂₂)ξ−1 and det(M_w) = m₁₁m₂₂−m₁₂m₂₁= 1.

Remark 3.1. This proof is by far less direct than that of Galois but raises the question of possible generalization to finite alphabets and multidimensional con- tinued fractions.

Acknowledgements. I am most grateful to Val´erie Berth´e for fruitful discussions and passing me several related papers.

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