Directive words of episturmian words : equivalences and normalization

DOI:10.1051/ita:2008029 www.rairo-ita.org

DIRECTIVE WORDS OF EPISTURMIAN WORDS:

EQUIVALENCES AND NORMALIZATION

Amy Glen

, Florence Lev´ e

and Gw´ ena¨ el Richomme

Abstract. Episturmian morphisms constitute a powerful tool to study episturmian words. Indeed, any episturmian word can be infinitely de- composed over the set of pure episturmian morphisms. Thus, an epis- turmian word can be defined by one of its morphic decompositions or, equivalently, by a certain directive word. Here we characterize pairs of words directing the same episturmian word. We also propose a way to uniquely define any episturmian word through a normalization of its directive words. As a consequence of these results, we characterize episturmian words having a unique directive word.

Mathematics Subject Classification.68R15.

1. Introduction

Since the seminal works of Morse and Hedlund [21], Sturmian words have been widely studied. Their beautiful properties are related to many fields like number theory, geometry, dynamical systems, and combinatorics on words (see [1,3,20,23]

for recent surveys). These infinite words, which are defined on a binary alphabet, have numerous equivalent definitions and characterizations.

Many recent works have been devoted to generalizations of Sturmian words to arbitrary finite alphabets. Two very interesting generalizations are very close: the

Keywords and phrases.Episturmian word, Sturmian word, Arnoux-Rauzy sequence, epistur- mian morphism, directive word.

1 LaCIM, Universit´e du Qu´ebec `a Montr´eal, C.P. 8888, succursale Centre-ville, Montr´eal, Qu´ebec, H3C 3P8, Canada

2 The Mathematics Institute, Reykjav´ık University, Kringlan 1, IS-103 Reykjav´ık, Iceland;

[email protected]

3Universit´e de Picardie Jules Verne, Laboratoire MIS (Mod´elisation, Information, Syst`emes), 33 rue Saint Leu, 80039 Amiens Cedex 1, France;

[florence.leve;gwenael.richomme]@u-picardie.fr

Article published by EDP Sciences c EDP Sciences 2008

Arnoux-Rauzy sequences(e.g., see [2,14,23,30]) and episturmian words (e.g., see [5,13,15]). The first of these two families is a particular subclass of the second one.

More precisely, the family of episturmian words is composed of the Arnoux-Rauzy sequences, images of the Arnoux-Rauzy sequences byepisturmian morphisms, and certain periodic infinite words. In the binary case, Arnoux-Rauzy sequences are exactly the Sturmian words whereas episturmian words include all recurrentbal- ancedwords, that is, periodic balanced words and Sturmian words (see [10,22,29]

for recent results relating episturmian words to the balanced property). See also [9] for a recent survey on episturmian theory.

Episturmian morphisms play a central role in the study of these words. In- troduced first as a generalization of Sturmian morphisms, Justin and Pirillo [13]

showed that they are exactly the morphisms that preserve the aperiodic epistur- mian words. They also proved that any episturmian word is the image of another episturmian word by some so-calledpure episturmian morphism. Even more, any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms (see Sect. 2.4). This last property allows an episturmian word to be defined by one of its morphic decompositions or, equivalently, by a certaindirective word, which is an infinite sequence of rules for decomposing the given episturmian word by morphisms. In consequence, many properties of episturmian words can be deduced from properties of episturmian morphisms. This approach is used for instance in [4,8,16,28–30] and of course in the papers by Justinet al. In Section2, we recall useful results on episturmian words and explain the vision of morphic decompositions and directive words introduced by Justin and Pirillo in [13].

An episturmian word can have several directive words. The question: “When do two different words direct the same episturmian word?” was considered in [15].

Using a block-equivalence notion for directive words, Justin and Pirillo provided several results to answer this question in most cases (see Sect.3). In Section 4, we state a complete result characterizing the form of words directing the same episturmian word, without using block-equivalence.

In [4], Berth´eet al. showed that any Sturmian word has a unique directive word with some particular properties. In [18], the second and third authors rephrased this result and used it to characterize all quasiperiodic Sturmian words. In Sec- tion 5, we extend this result to all episturmian words by introducing a way to normalize the directive words of an episturmian word so that any episturmian word can be defined uniquely by its normalized directive word, defined by some factor avoidance (Thm.5.2). This result was previously presented at the Sixth In- ternational Conference on Words [17] to characterize all quasiperiodic episturmian words (see also [11]).

As an application of the previous results, we end this paper with a characteri- zation of episturmian words having a unique directive word.

2. Episturmian words and morphisms

We assume the reader is familiar with combinatorics on words and morphisms (e.g., see [19,20]). In this section, we recall some basic definitions and properties relating to episturmian words which are needed later in the paper. For the most part, we follow the notation and terminology of [5,10,13,15].

2.1. Notation and terminology

Let A denote a finite alphabet. A finite word over A is a finite sequence of letters fromA. Theempty wordεis the empty sequence. Under the operation of concatenation, the setA^∗ of all finite words overAis afree monoidwith identity elementεand set of generatorsA. The set ofnon-emptywords overAis thefree semigroupA⁺=A^∗\ {ε}.

Given a finite wordw =x₁x₂. . . x_m ∈ A⁺ with eachx_i ∈ A, thelength of w is |w|=m. The length of the empty word is 0. By|w|a we denote the number of occurrences of the letter a in the word w. If |w|a = 0, then w is said to be a-free. For any integer p ≥ 1, the p-th power of w is the word w^p obtained by concatenatingpoccurrences ofw.

A (right)infinite word x is a sequence indexed by N⁺ with values in A, i.e., x=x₁x₂x₃. . .with eachx_i∈ A. The set of all infinite words overAis denoted by A^ω. Given a non-empty finite wordv, we denote byv^ωthe infinite word obtained by concatenatingvwith itself infinitely many times. Anultimately periodicinfinite word is a word that can be written asuv^ω=uvvv . . ., for someu, v∈ A^∗,v=ε.

If u = ε, then such a word is periodic. An infinite word that is not ultimately periodic is said to be aperiodic. For easier reading, infinite words are hereafter typed in boldface to distinguish them from finite words.

Given a setX of finite words,X^∗(resp.X^ω) is the set of all finite (resp. infinite) words that can be obtained by concatenating words of X. The empty word ε belongs toX^∗.

A finite wordwis afactorof a finite or infinite wordzifz=uwvfor some words u,v (wherevis infinite iffzis infinite). Further,wis called aprefix (resp.suffix) ofz ifu=ε(resp.v=ε). We use the notationp⁻¹w(resp.ws⁻¹) to indicate the removal of a prefixp(resp. suffix s) of the wordw.

Thealphabet of a wordw, denoted by Alph(w) is the set of letters occurring in w, and ifwis infinite, we denote by Ult(w) the set of all letters occurring infinitely often inw.

2.2. Episturmian words

In this paper, our vision of episturmian words will be the characteristic property stated in Theorem2.1. Nevertheless, to give an idea of what an episturmian word is, let us give one of the equivalent definitions of an episturmian word provided in [5]. Before doing so, we recall that a factor u of an infinite wordw ∈ A^ω is right (resp.left) special ifua, ub (resp.au, bu) are factors of w for some letters

a,b∈ A,a=b. We recall also that thereversal wof a finite wordwis its mirror image: ifw=x₁. . . x_m−1x_m, thenw=x_mx_m−1. . . x₁.

An infinite word t ∈ A^ω is episturmian if its set of factors is closed under reversal and t has at most one right (or equivalently left) special factor of each length. Moreover, an episturmian word isstandard if all of its left special factors are prefixes of it.

In the initiating paper [5], episturmian words were defined in two steps. Stan- dard episturmian words were first introduced and studied as a generalization of standard Sturmian words. (Note that in the rest of this paper, we refer to a standard episturmian word as anepistandard word, for simplicity.) Then an epis- turmian word was defined as an infinite word having exactly the same set of factors as some epistandard word.

Moreover, it was proved in [5] that any episturmian word isrecurrent, that is, all of its factors occur infinitely often (actually episturmian words are uniformly recurrent but this will not be needed here). Thus ultimately periodic episturmian words are (purely) periodic.

2.3. Episturmian morphisms

To study episturmian words, Justin and Pirillo [13] introduced episturmian morphisms. In particular they proved that these morphisms (defined below) are precisely the morphisms that preserve the set of aperiodic episturmian words.

Let us recall that given an alphabetA, amorphism f onAis a map fromA^∗ to A^∗ such that f(uv) =f(u)f(v) for any words u, v over A. A morphism on Ais entirely defined by the images of letters in A. All morphisms considered in this paper will be non-erasing: the image of any non-empty word is never empty.

Hence the action of a morphism f on A^∗ can be naturally extended to infinite words; that is, ifx=x₁x₂x₃. . .∈ A^ω, thenf(x) =f(x₁)f(x₂)f(x₃). . .

In what follows, we will denote the composition of morphisms by juxtaposition as for concatenation of words.

Episturmian morphisms are the compositions of the permutation morphisms (the morphismsf such thatf(A) =A) and the morphismsL_a andR_a where, for alla∈ A:

L_a :

a → a

b → ab , R_a :

a → a

b → ba for allb=ain A.

Here we will work only onpure episturmian morphisms, i.e., morphisms obtained by composition of elements of the sets:

L_A={La|a∈ A} and R_A={Ra|a∈ A}.

Note. In [13], the morphismL_a (resp.R_a) is denoted byψ_a (resp. ¯ψ_a). We adopt the current notation to emphasize the action ofL_a (resp.R_a) when applied to a

word, which consists in placing an occurrence of the letteraon the left (resp. right) of each occurrence of any letter different froma.

Epistandard morphismsare the morphisms obtained by concatenation of mor- phisms inL_Aand permutations onA. Likewise, thepure episturmian morphisms (resp.pure epistandard morphisms) are the morphisms obtained by concatenation of morphisms in LA∪ RA (resp. in LA). Note that the episturmian morphisms are exactly theSturmian morphismswhenAis a 2-letter alphabet.

All episturmian morphisms are injective on both finite and infinite words. The monoid of episturmian morphisms is left cancellative (see [26], Lem. 7.2) which means that for any episturmian morphismsf, g, h, if f g=f hthen g =h. Note that this fact, which is a by-product of the injectivity, can also be seen as a consequence of the invertibility of these morphisms (see [7,12,26,32]).

2.4. Morphic decomposition of episturmian words

Justin and Pirillo [13] proved the following insightful characterizations of epi- standard and episturmian words (see Thm. 2.1 below), which show that any episturmian word can be infinitely decomposed over the set of pure episturmian morphisms.

The statement of Theorem2.1needs some extra definitions and notation.

First we define the following new alphabet, ¯A = {x¯ | x∈ A}. A letter ¯x is considered to be x with spin R, whilst x itself has spin L. A finite or infinite word over A ∪A¯ is called a spinned word. To ease the reading, we sometimes call a letter with spinL(resp. spinR) anL-spinned (resp.R-spinned) letter. By extension, anL-spinned (resp.R-spinned) word is a word having only letters with spinL(resp. spin R).

The opposite w¯ of a finite or infinite spinned word w is obtained from w by exchanging all spins inw. For instance, ifv=ab¯a, then ¯v= ¯a¯ba. Whenv∈ A⁺, then its opposite ¯v∈A¯⁺is anR-spinned word and we set ¯ε=ε. Note that, given a finite or infinite wordw=w₁w₂. . .over A, we sometimes denote ˘w= ˘w₁w˘₂. . . any spinned word such that ˘w_i =w_i if ˘w_i has spinLand ˘w_i = ¯w_i if ˘w_i has spin R. Such a word ˘wis called aspinned version ofw. Sometimes we denote by ˆw a spinned version of a wordwthat is different from ˘w.

Note. In Justin and Pirillo’s original papers, spins are 0 and 1 instead ofLandR.

It is convenient here to change this vision of the spins because of the relationship with episturmian morphisms, which we now recall.

For a ∈ A, let μ_a = L_a and μ_¯a = R_a. This operator μ can be naturally extended (as done in [13]) to a morphism from the free monoid (A ∪A¯)^∗ to a pure episturmian morphism: for a spinned finite word ˘w= ˘w₁. . .w˘_n overA ∪A¯, μ_w˘ =μ_w˘1. . . μ_w˘n (μ_ε is the identity morphism). We say that the wordwdirects or is adirective wordof the morphismμ_w. The following result extends the notion of directive words to infinite episturmian words.

Theorem 2.1[13].

(i) An infinite word s ∈ A^ω is epistandard if and only if there exist an in- finite word Δ = x₁x₂x₃. . . over A and an infinite sequence (s⁽ⁿ⁾)_n≥0 of infinite words such that s⁽⁰⁾ = s and for all n ≥ 1, s⁽ⁿ⁻¹⁾ = μ_xn(s⁽ⁿ⁾) (=L_xn(s⁽ⁿ⁾)).

(ii) An infinite wordt∈ A^ω is episturmian if and only if there exist a spinned infinite word Δ = ˘˘ x₁x˘₂x˘₃. . . over A ∪ A¯ and an infinite sequence (t⁽ⁿ⁾)_n≥0 of recurrent infinite words such thatt⁽⁰⁾=tand for all n≥1, t⁽ⁿ⁻¹⁾=μ_x˘n(t⁽ⁿ⁾).

For any epistandard word (resp. episturmian word)tandL-spinned infinite word Δ (resp. spinned infinite word ˘Δ) satisfying the conditions of the above theorem, we say that Δ (resp. ˘Δ) is a(spinned) directive wordfortor thattisdirected by Δ (resp. ˘Δ). Notice that this directive word is exactly the one that arises from the equivalent definition of epistandard words that uses palindromic closure [5,9,13]

and, in the binary case, it is related to the continued fraction of the slope of the straight line represented by a standard word (see [20]). It follows immediately from Theorem2.1that, with the notation of case (ii), eacht⁽ⁿ⁾ is an episturmian word directed by ˘x_n+1˘x_n+2. . .

The natural question: “Does any spinned infinite word direct a unique epistur- mian word?” is answered in [13]:

Proposition 2.2([13], Prop. 3.11).

(1) Any spinned infinite wordΔ˘ having infinitely many L-spinned letters di- rects a unique episturmian word beginning with the left-most letter having spinLin Δ.˘

(2) AnyR-spinned infinite wordΔ˘ directs exactly|Ult(Δ)|episturmian words.

(3) Let Δ˘ be an R-spinned infinite word, and let a be a letter such that

¯

a∈Ult( ˘Δ). ThenΔ˘ directs exactly one episturmian word starting witha.

Note. In [13], item (3) was stated in the more general case where ˘Δ is ultimately R-spinned. In this case, ˘Δ still directs exactly one episturmian word for each letter

¯

a in Ult( ˘Δ), but contrary to what is written in [13], nothing can be said on its first letter.

As a consequence of the previous proposition and part (i) of Theorem 2.1, any L-spinned infinite word directs a unique epistandard word. The following important remark links the two parts of Theorem2.1.

Remark 2.3 [13]. If tis an episturmian word directed by a spinned version ˘Δ of anL-spinned infinite word Δ, thenthas exactly the same set of factors as the (unique) epistandard wordsdirected by Δ.

Moreover (with the same notation as in the previous remark):

Remark 2.4. The episturmian word tis periodic if and only if the epistandard wordsis periodic, and this holds if and only if there is only one letter occurring infinitely often in Δ, that is,|Ult(Δ)|= 1 (see [13], Prop. 2.9). More precisely, a

periodic episturmian word takes the form (μ_w˘(x))^ω for some finite spinned word

˘

wand letterx.

Note. Sturmian words are precisely the aperiodic episturmian words on a 2-letter alphabet.

When an episturmian word is aperiodic, we have the following fundamental link between the words (t⁽ⁿ⁾)_n≥0 and the spinned infinite word ˘Δ occurring in Theorem2.1: ifa_n is the first letter oft⁽ⁿ⁾, thenμ_x˘1...˘xn(a_n) is a prefix oftand the sequence (μ_x˘1...˘xn(a_n))_n≥1is not ultimately constant (since ˘Δ is not ultimately constant), thent= lim_n→∞μ_x˘1...˘xn(a_n). This fact is a slight generalization of a result of Risley and Zamboni [30], Proposition. III.7, onS-adic representationsfor characteristic Arnoux-Rauzy sequences. See also the recent paper [4] for S-adic representations of Sturmian words. Note that S-adic dynamical systems were introduced by Ferenczi [6] asminimal dynamical systems(e.g., see [23]) generated by a finite number of substitutions. In the case of episturmian words, the notion itself is actually a reformulation of the well-knownRauzy rules, as studied in [25].

To anticipate next sections, let us also observe:

Remark 2.5 [13]. If an aperiodic episturmian word is directed by two spinned words Δ₁ and Δ₂, then Δ₁ and Δ₂ are spinned versions of the sameL-spinned word.

This is no longer true for periodic episturmian words; for instanceab^ωand ¯ba^ω direct the same episturmian word (ab)^ω=ababab . . .

3. Known results on directive-equivalent words

We have just seen an example of a periodic episturmian word that is directed by two different spinned infinite words. This situation holds also in the aperiodic case (see [13,15]). For example, theTribonacci word (or Rauzy word [24]) is directed by (abc)^ω and also by (abc)ⁿ¯a¯b¯c(a¯b¯c)^ω for each n≥0, as well as infinitely many other spinned words. More generally, by [13], any epistandard word has a unique L-spinned directive word but also has other directive words (see also [15] and Thm.4.1).

We now consider in detail the following two questions. When do two different finite spinned words direct the same episturmian morphism? When do two differ- ent spinned infinite words direct the same unique episturmian word? We say that two different finite (resp. infinite) spinned words aredirective-equivalent words if they direct the same episturmian morphism (resp. the same episturmian word).

In Section3.1, we recall the characterizations of directive-equivalent finite spinned words. In Section3.2, we recall known results about directive-equivalent infinite words. Section4will present a new characterization of these words.

3.1. Finite directive-equivalent words: presentation versus block-equivalence

Generalizing a study of the monoid of Sturmian morphisms by S´e´ebold [31], the third author [26] answered the question: “When do two different spinned finite words direct the same episturmian morphism?” by giving a presentation of the monoid of episturmian morphisms. This result was reformulated in [27]

using another set of generators and it was independently and differently treated in [15]. As a direct consequence, one can see that the monoid of pure epistandard morphisms is a free monoid and one can obtain the following presentation of the monoid of pure episturmian morphisms:

Theorem 3.1 (direct consequence of [27], Prop. 6.5; reformulation of [15], Thm. 2.2). The monoid of pure episturmian morphisms with{Lα, R_α|α∈ A}as a set of generators has the following presentation:

R_a1R_a2. . . R_akL_a1 =L_a1L_a2. . . L_akR_a1

wherek≥1 is an integer anda₁, . . . , a_k∈ Awith a₁=a_i for alli,2≤i≤k.

This result means that two different compositions of morphisms inLA∪RAyield the same pure episturmian morphism if and only if one composition can be deduced from the other one in a rewriting system, called the block-equivalence in [15].

Although Theorem 3.1 allows us to show that many properties of episturmian words are linked to properties of episturmian morphisms, it will be convenient for us to have in mind the block-equivalence that we now recall.

A word of the form xvx, where x ∈ A and v ∈ (A \ {x})^∗, is called a (x- based) block. A (x-based) block-transformation is the replacement in a spinned word of an occurrence ofxvx¯ (where xvxis a block) by ¯x¯vx or vice-versa. Two finite spinned words w, w are said to be block-equivalent if we can pass from one to the other by a (possibly empty) chain of block-transformations, in which case we write w ≡ w. For example, ¯b¯ab¯cb¯a¯c and babc¯b¯a¯c are block-equivalent because ¯b¯ab¯cb¯a¯c → ba¯b¯cb¯a¯c → babc¯b¯a¯c and vice-versa. The block-equivalence is an equivalence relation over spinned words, and moreover one can observe that if w≡w thenwandw are spinned versions of the same word overA.

Theorem3.1can be reformulated in terms of block-equivalence:

Theorem 3.1. Let w,w be two spinned words over A ∪A¯. Then μ_w =μ_w if and only ifw≡w.

3.2. Infinite directive-equivalent words: previous results

The question: “When do two different spinned infinite words direct the same unique episturmian word?” was tackled by Justin and Pirillo in [15] forbi-infinite episturmian words, that is, episturmian words with letters indexed by Z (and not by N as considered until now). Let us recall relations between right-infinite

episturmian words and bi-infinite episturmian words (see [15], p. 332 and [9] for more details).

First we observe that a right-infinite episturmian wordtcan be prolonged infin- itely to the left with the same set of factors (see [15]). Note also that the definition of episturmian words considered in Section2.2(using reversal and special factors) can be extended to bi-infinite words. Furthermore, the characterization (Thm.2.1) of right-infinite episturmian words by a sequence (t⁽ⁱ⁾)_i≥0 extends to bi-infinite episturmian words, with all thet⁽ⁱ⁾now bi-infinite episturmian words. That is, as for right-infinite episturmian words, we have bi-infinite words of the forml⁽ⁱ⁾.r⁽ⁱ⁾ wherel⁽ⁱ⁾is a left-infinite episturmian word andr⁽ⁱ⁾is a right-infinite episturmian word. Moreover, if the bi-infinite episturmian wordb=l.ris directed by ˘Δ with associated bi-infinite episturmian wordsb⁽ⁱ⁾ = l⁽ⁱ⁾.r⁽ⁱ⁾, then r is directed by ˘Δ with associated right-infinite episturmian wordsr⁽ⁱ⁾.

As a consequence of what precedes, Justin and Pirillo’s results about spinned words directing the same bi-infinite episturmian word are still valid for words directing the same (right-infinite) episturmian word. We summarize now these results, which will be helpful for the proof of our main theorem (Thm. 4.1, to follow).

First of all, Justin and Pirillo characterized pairs of words directing the same episturmian word in the case ofwavydirective words, that is, spinned infinite words containing infinitely manyL-spinned letters and infinitely manyR-spinned letters.

This characterization uses the following extension of the block-equivalence≡for infinite words.

Let Δ₁, Δ₂be spinned versions of Δ. We write Δ₁Δ₂if there exist infinitely many prefixesf_iof Δ₁andg_i of Δ₂with theg_iof strictly increasing lengths, and such that, for alli,|g_i| ≤ |f_i|andf_i≡g_ic_i for a suitable spinned wordc_i. Infinite words Δ₁and Δ₂are said to beblock-equivalent(denoted by Δ₁≡Δ₂) if Δ₁Δ₂ and Δ₂Δ₁.

Theorem 3.2 ([15], Thm. 3.4, Cor. 3.5). Let Δ₁ and Δ₂ be wavy spinned ver- sions of Δ ∈ A^ω with |Ult(Δ)| >1. Then Δ₁ and Δ₂ direct the same (unique) episturmian word if and only ifΔ₁≡Δ₂.

Moreover when Δ₁ and Δ₂ do not have any common prefix modulo ≡, and when there exists a letterxsuch that Δ₁ andΔ₂ begin with xandx¯ respectively, if Δ₁ ≡Δ₂, then Δ₁=x

n≥1v_nx˘_n,Δ₂ = ¯x

n≥1¯v_nxˆ_n for an L-spinned letter x, a sequence (v_n)_n≥1 of x-freeL-spinned words, and sequences of spinned letters (˘x_n)_n≥1,(ˆx_n)_n≥1in{x,x}¯ such that(˘x_n)_n≥1contains infinitely many times theR- spinned letterx, and¯ (ˆx_n)_n≥1contains infinitely many times theL-spinned letterx.

The relation(and hence the block-equivalence≡for infinite words) is rather intricate to understand. So in some way the forms of Δ₁ and Δ₂ at the end of Theorem 3.2 are, although technical, easier to understand. Theorem 4.1, which refines the end of the previous result and proves the converse, describes all possible forms for pairs of directive-equivalent words without any use of notationsand≡.

When one of the two considered directive words is not wavy, Justin and Pirillo established:

Proposition 3.3([15], Prop. 3.6). LetΔ₁andΔ₂ be spinned versions of the same word such that Δ₁ is wavy and letters ofΔ₂ are ultimately of spin L (resp. ulti- mately of spin R). If Δ₁ andΔ₂ are directive-equivalent, then Δ₁Δ₂. More- over there exist spinned wordsw₁,w₂, anL-spinned letterx, andL-spinnedx-free words (v_i)_i≥1 such that μ_w1 =μ_w2,Δ₁ =w₁x¯

i≥1v¯_ix and Δ₂ =w₂x

i≥1v_ix (resp.Δ₁=w₁x

i≥1v_ix¯andΔ₂=w₂x¯

i≥1¯v_ix).¯

With the next two results, they considered the remaining cases of words direct- ing aperiodic episturmian words. In the first one, the spins of the letters in each of the two directive words are ultimately L or ultimately R. The second result shows that if one of the directive words has the spins of its letters ultimately L (resp. ultimatelyR), then the other directive word cannot have the spins of its letters ultimatelyR (resp. ultimatelyL).

Proposition 3.4([15], Prop. 3.7). LetΔ₁andΔ₂ be spinned versions of the same word Δ ∈ A^ω with |Ult(Δ)|> 1. If there exist spinned words w₁, w₂ and an L- spinned infinite wordΔ such that Δ₁=w₁Δ andΔ₂=w₂Δ (resp.Δ₁=w₁Δ¯ andΔ₂=w₂Δ¯), thenΔ₁,Δ₂ are directive-equivalent if and only if μ_w1 =μ_w2. Proposition 3.5([15], Prop. 3.9). Let Δbe anL-spinned infinite word. Then Δ andΔ¯ do not direct the same right-infinite episturmian word.

Actually the previous statement is a corollary of Proposition 3.9 in [15] which considers more generally words directing episturmian words differing only by a shift.

Justin and Pirillo also discussed in [15] the periodic case and proved:

Proposition 3.6 ([15], Prop. 3.10). Suppose that Δ₁ = ˘w˘ya^ω and Δ₂ = ˆwˆy¯a^ω, wherew˘ andwˆ (resp.y˘andy) are spinned versions of the same word andˆ ais an L-spinned letter. ThenΔ₁andΔ₂ are directive-equivalent if and only if there exist sequences of letters(˘a_n)_n≥1 and(ˆa_n)_n≥1 such that w˘˘y

n≥1a˘_n≡wˆˆy

n≥1ˆa_n. We will see in Theorem4.1that other cases can occur for periodic episturmian words.

4. Directive-equivalent words: a characterization

As shown in the previous section, Justin and Pirillo provided quite complete results about directive-equivalent infinite words. Nevertheless they did not system- atically provide the relative forms of two directive-equivalent words. The following characterization does it, moreover without the use of relationsand≡. This re- sult also fully solves the periodic case, which was only partially solved in [15].

Theorem 4.1. Given two spinned infinite wordsΔ₁ andΔ₂, the following asser- tions are equivalent.

(i) Δ₁ andΔ₂ direct the same right-infinite episturmian word;

(ii) Δ₁ andΔ₂ direct the same bi-infinite episturmian word;

(iii) One of the following cases holds for somei, j such that {i, j}={1,2}:

(1) Δ_i =

n≥1v_n, Δ_j =

n≥1z_n where (v_n)_n≥1,(z_n)_n≥1 are spinned words such that μ_vn=μ_zn for alln≥1;

(2) Δ_i =wx

n≥1v_nx˘_n, Δ_j =wx¯

n≥1v¯_nxˆ_n where w,w are spinned words such that μ_w = μ_w, x is an L-spinned letter, (v_n)_n≥1 is a sequence of non-emptyx-freeL-spinned words, and(˘x_n)_n≥1,(ˆx_n)_n≥1 are sequences of non-empty spinned words over {x,x}¯ such that, for all n≥1,|˘x_n|=|ˆx_n|and|˘x_n|x=|ˆx_n|x;

(3) Δ₁=wx andΔ₂=wy wherew,w are spinned words,xandy are letters, and x∈ {x,¯x}^ω,y∈ {y,y}¯ ^ω are spinned infinite words such that μ_w(x) =μ_w(y).

Note. For a, b, c three different letters in A, the spinned infinite words Δ₁ = a(bc¯a)^ω and Δ₂ = ¯a(¯b¯c¯a)^ω direct the same episturmian word that starts with the lettera. Indeed, these two directive words fulfill item (2) of Theorem4.1with w=w=ε,x=a, and for alln,v_n=bcand ˘x_n = ˆx_n= ¯a. Moreover the fact that Δ₁starts with theL-spinned letterashows that the word it directs starts witha.

Similarly Δ₁ = ¯ab(ca¯b)^ω and Δ₂ = ¯a¯b(¯c¯a¯b)^ω direct the same episturmian word starting with the letterb. Since Δ₂= Δ₂, this shows that the relation “direct the same episturmian word” over spinned infinite word is not an equivalence relation.

Items2 and3 of Theorem 4.1 show that any episturmian word is directed by a spinned infinite word having infinitely many L-spinned letters, but also by a spinned word having both infinitely manyL-spinned letters and infinitely many R-spinned letters (i.e., a wavy word). To emphasize the importance of these facts, let us recall from Proposition2.2that if ˘Δ is a spinned infinite word overA ∪A¯ with infinitely manyL-spinned letters, then there exists a unique episturmian word tdirected by ˘Δ. Unicity comes from the fact that the first letter oftis fixed by the firstL-spinned letter in ˘Δ.

Before proving Theorem4.1, let us make two more remarks.

Remark 4.2. In items (1) and (2) of Theorem4.1, the two considered directive words are spinned versions of the same L-spinned word. This does not hold in item (3), which deals only with periodic episturmian words. This is consistent with Remark2.5. As an example of item (3), one can consider the word (ab)^ω = L_a(b^ω) =R_b(a^ω) which, as already said at the end of Section2.4, is directed by ab^ω and by ¯ba^ω (L_a(b) =ab=R_b(a)). Note also that (ab)^ω is directed by (a¯b)^ω, underlining the fact thatxandy can be equal in item (3) of Theorem4.1.

Remark 4.3.If an episturmian wordthas two directive words satisfying items (2) or (3), thenthas infinitely many directive words. Indeed, if item (2) is satisfied

and ¯x occurs in ˘x_p (p ≥ 1), then by Theorem 3.1, x_p−1

k=1v_nx˘_n

v_px˘_px¯ ≡

¯ x_p−1

k=1¯v_nx˘_n

¯

v_px˘_pxwhere ˘x_p is such that ˘x_p≡x˘¯x_p. Thus tis also directed by w¯x_p−1

k=1¯v_nx˘_n

¯ v_p˘x_px

n≥p+1v_nx˘_n. Similarly if item (2) is satisfied andx oc- curs in ˘x_p(p≥1), thentis also directed bywx_p−1

k=1v_nxˆ_n

v_pxˆ_px¯

n≥p+1v¯_nxˆ_n where ˆx_p is such that ˆx_p ≡ xˆx_p. If item (3) is satisfied, then t is periodic and directed bywxwherexis any spinned version ofx^ω.

The rest of this section is dedicated to the proof of Theorem4.1.

Proof of Theorem4.1. We have (i)⇔(ii) by the remarks on bi-infinite words at the beginning of Section3.2.

(iii) ⇒ (i). Assume first that Δ₁ =

n≥1v_n and Δ₂ =

n≥1z_n for spinned words (v_n)_n≥1,(z_n)_n≥1such thatμ_vn =μ_znfor alln≥1. From the latter equality and Theorem3.1, Δ₁ has infinitely manyL-spinned letters if and only if Δ₂ has infinitely manyL-spinned letters.

Let us first consider the case when both Δ₁ and Δ₂ have infinitely many L- spinned letters. Without loss of generality we can assume that for all n ≥ 1, v_n and z_n contain at least one L-spinned letter. Now we need to define some more notations. Lett₁ and t₂ be the episturmian words directed by Δ₁ and Δ₂, respectively (these episturmian words exist and are unique by Prop. 2.2). For n≥0, lett⁽ⁿ⁾₁ andt⁽ⁿ⁾₂ be the episturmian words as in (ii) of Theorem2.1and let a_nandb_nbe their respective first letters. Finally, forn≥1, setp_n=_n

i=1v_iand q_n =_n

i=1z_i. The wordsμ_pn(a_|pn|) (resp. μ_qn(b_|qn|)) are prefixes of t1 (resp. of t₂). The lettera_|pn|(resp.b_|qn|) is the first letter ofμ_vn+1(t^(m)₁ ) (resp.μ_zn+1(t^(m)₂ )) with m =_n+1

i=1 |vi| = _n+1

i=1 |zi|. Since v_n+1 (resp. z_n+1) contains at least one L-spinned letter, a_|pn| (resp.b_|qn|) is the first letter ofμ_vn+1(w) (resp.μ_zn+1(w)) for any wordw. From μ_vn+1 =μ_zn+1, we have a_|pn| =b_|qn| and so μ_pn(a_|pn|) = μ_qn(b_|qn|) for alln≥1. If the sequence (μ_pn(a_|pn|))_n≥1is not ultimately constant, then from t₁ = lim_n→∞μ_pn(a_|pn|) and t₂ = lim_n→∞μ_qn(b_|qn|), we deduce that t₁ =t₂. If (μ_pn(a_|pn|))_n≥1 is ultimately constant, then necessarily there exists a letteraand an integermsuch that for alln > m,v_nandz_nbelong to{a}^∗. Then t₁=μ_v1...vm(a^ω) =μ_z1...zm(a^ω) =t₂.

Now, with the same notations as in the above case, we consider the case when the letters of Δ₁and Δ₂are ultimatelyR-spinned. By Theorem3.1, any equality μ_v = μ_z (for some different spinned wordsv and z) implies that v and z both contain at least one L-spinned letter and one R-spinned letter. Hence, in our current case, there exists an integer m such that v_n =z_n for all n > m. Let t be an episturmian word directed by

n>mv_n =

n>mz_n (such an episturmian word exists by Prop.2.2). Thenμ_pm(t) =μ_qm(t) and Δ1 and Δ₂ are directive- equivalent.

Now consider item (2) of part (iii). We assume that Δ₁ = wx

n≥1v_nx˘_n and Δ₂ =wx¯

n≥1¯v_nxˆ_n where w, w are spinned words such that μ_w = μ_w,

x is an L-spinned letter, (v_n)_n≥1 is a sequence of non-empty x-free L-spinned words, and (˘x_n)_n≥1, (ˆx_n)_n≥1are non-empty spinned words over{x,x}¯ such that, for all n ≥ 1, |˘x_n| = |ˆx_n| and |˘x_n|x = |ˆx_n|x. By injectivity of the morphisms μ_w=μ_w, Δ₁ and Δ₂ are directive-equivalent if and only if w⁻¹Δ₁ andw⁻¹Δ₂ are directive-equivalent. So, from now on, we assume without loss of generality thatw=w =ε.

By Proposition2.2, there exist unique episturmian wordst₁andt₂starting with xdirected by the respective words Δ₁and Δ₂(observe that if ˆx_n∈x¯⁺for alln≥1, then ¯x∈ Ult(Δ₂)). For i ≥1, let Δ⁽ⁱ⁾₁ =x

n≥iv_nx˘_n and Δ⁽ⁱ⁾₂ = ¯x

n≥iv¯_nxˆ_n and lett^[i]₁ and t^[i]₂ be the words beginning with xand directed by the respective words Δ⁽ⁱ⁾₁ and Δ⁽ⁱ⁾₂ . (The episturmian wordst^[i]₁ andt^[i]₂ exist by Prop.2.2.) For i≥1 we also defineα_i:=|˘x_i|x=|ˆx_i|x andβ_i:=|˘x_i|x_¯=|ˆx_i|x_¯.

Assume first that α_i = 0. Then ¯x¯v_ixˆ_i ≡ x¯¯v_ixx^αi−1x¯^βi ≡ xv_i¯xx^αi−1x¯^βi ≡ xv_ix^αi−1x¯^βix¯ and xv_ix˘_i ≡ xv_ix^αi−1x¯^βix. Let p_i = xv_ix^αi−1x¯^βi. From what precedes we deduce that Δ⁽ⁱ⁾₁ and p_iΔ⁽ⁱ⁺¹⁾₁ are directive-equivalent, as Δ⁽ⁱ⁾₂ and p_iΔ⁽ⁱ⁺¹⁾₂ are directive-equivalent. By the choice of wordst^[i]₁ andt^[i]₂ , we deduce that t^[i]₁ = μ_pi(t^[i+1]₁ ) and t^[i]₂ =μ_pi(t^[i+1]₂ ) and each of these words starts with μ_pi(x).

Now let us consider the case when α_i = 0. Then ˘x_i = ˆx_i = ¯x^βi. We have xv_ix˘_i ≡x¯¯v_ix¯^βi−1xand ¯x¯v_ixˆ_i = ¯x¯v_ix¯^βi−1x. Taking¯ p_i = ¯x¯v_ix¯^βi−1, we reach the same conclusion as in the case whenα_i= 0.

It follows from what precedes that t1 and t2 both start with μ_p1...pi(x) for alli ≥ 1. Since v_i = ε, the sequence (μ_p1...pi(x))_i≥1 is not ultimately constant;

whencet₁=t₂= lim_i→∞μ_p1...pi(x).

Lastly, assume that Δ₁ = wx and Δ₂ = wy for some spinned words w, w, some letters xand y, and some spinned infinite words x∈ {x,x}¯ ^ω, y ∈ {y,y}¯ ^ω such thatμ_w(x) =μ_w(y). The word Δ₁ (resp. Δ₂) directs the episturmian word μ_w(x^ω) = (μ_w(x))^ω(resp.μ_w(y^ω) = (μ_w(y))^ω). Hence Δ₁ and Δ₂ are directive- equivalent.

(i)⇒(iii). Suppose Δ₁and Δ₂direct the same (right-infinite) episturmian word t. Let us first assume that tis aperiodic. Then, by Remark 2.5, Δ₁ and Δ₂ are spinned versions of the same infinite word Δ∈ A^ω. We now show that item (1) or item (2) holds using results of Justin and Pirillo in [15].

First consider the case when both Δ₁ and Δ₂ are wavy. Suppose there exist a sequence of prefixes (p_n)_n≥0 of Δ₁ and a sequence of prefixes (p_n)_n≥0 of Δ₂ such that for all n ≥ 0, μ_pn = μ_p

n. Without loss of generality we can assume that p₀ = p₀ =ε and the sequence (|pn|)n≥0 is strictly increasing. For n ≥ 1, let v_n, z_n be such that p_n =p_n−1v_n, p_n = p_n−1z_n; that is Δ₁ =

n≥1v_n and Δ₂ =

n≥1z_n. Let us prove by induction thatμ_vn = μ_zn for all n≥ 1. First μ_v1 = μ_p1 = μ_p

1 = μ_z1. For n ≥ 2, since μ_pn = μ_pn−1μ_vn, μ_p n = μ_p

n−1μ_zn, μ_pn=μ_p

n andμ_pn−1 =μ_p

n−1, we haveμ_pn−1μ_vn =μ_pn−1μ_znand soμ_vn=μ_znby left cancellativity of the monoid of episturmian morphisms. So item (1) is satisfied in this case.