Conjugacy of morphisms and Lyndon decomposition of standard Sturmian words

HAL Id: hal-00013457

https://hal.archives-ouvertes.fr/hal-00013457

Submitted on 8 Nov 2005

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires

Conjugacy of morphisms and Lyndon decomposition of standard Sturmian words

Gwenaël Richomme

To cite this version:

Gwenaël Richomme. Conjugacy of morphisms and Lyndon decomposition of standard Sturmian words.

2005, pp.341-351. �hal-00013457�

Université dePiardie Jules Verne CNRSFRE2733

33, rueSaint Leu,80039Amiensedex 01, Frane

Tel: (+33)[0℄322 82 8877

Fax: (+33)[0℄0322 8254 12

http://www.laria.u-piardie.fr

Conjugay of morphisms

and Lyndon deomposition of

standard Sturmian words

G. Rihomme

a

LaRIARESEARCHREPORT:LRR 2005-09

(Otober2005)

Sturmian words

∗

G. Rihomme

LaRIA, Université de PiardieJules Verne

33, Rue SaintLeu,

F-80039 Amiens edex 1

(gwenael.rihommeu-piardie.fr)

Otober28,2005

Abstrat

Usingthenotionsofonjugayofmorphisms,weansweraquestionofG.Melançononern-

ingthedeompositioninLyndonwordsofstandardSturmianwords. Weshowsomeonnetions

withmorphismspreservingLyndonwords

1 Introdution

Finite (or innite) Lyndon words an be enountered inmany studies (see for instane [8 , 9 ,10 ℄).

Theyarethenonemptywordswhiharesmallerinlexiographiorderthanalltheirpropersuxes.

The Lyndon fatorization theorem [4℄ states that any nite word an be deomposed uniquely in

a produt of noninreasing (in lexiographi order) Lyndon words. This result was extended to

innitewords[19℄ (Insuhaase, thedeomposition anend withan inniteLyndonword). Thus

some works onern the deomposition in Lyndon words of some innite words (see for instane

[3, 5,11,12,18 ℄for suh results).

In [12 ℄, G. Melançon gives a deomposition inLyndon words of standard Sturmian words. He

asksthefollowing question: inwhih ases,thesequeneof noninreasing Lyndonwordsappearing

in the deomposition of a standard Sturmian word an be written

(g ⁿ (ℓ ₀ )) _n≥0

^with

ℓ ₀

^{a Lyndon}

word and

g

^{amorphism. InSetion 5,we answerthisquestion.}

Forthis,weuseresultsabout morphismspreservingLyndonwords[14 ℄ andaboutonjugay of

morphisms[13℄. Inpartiular,weshowthatwhenapositiveanswerexiststo thepreviousquestion,

g

^{preserves Lyndon words and is the onjugate of a morphism}

f

^{that generates the deomposed}

standard Sturmianword.

In Setion 2, we reall notions on Sturmian words and morphisms. Setion 3 realls both

the deomposition in Lyndon words of standard Sturmian words obtained by G. Melançon, and

his question. This setion also reall notions on morphisms preserving Lyndon words. Setion 4

presentsnotionsononjugayofmorphismsandintroduesanewpartiularase,namelythestrong

∗

Thispaperwas presentedat the 5thInternational ConfereneonWords whihheld inMontréalonseptember

2005(PubliationsduLaCIMnuméro36, page341-351(S.Brlek,C.Reutenauereds.).)

that for any standard Sturmian words

w

^over

{a < b}

^,

aw

^{isan innite Lyndon word [3℄. Finally,}

inSetion 5, we answerG. Melançon. Note that at arst step, we expressthe deomposition of a

standard Sturmianword using only morphisms.

2 Sturmian words and morphisms

We reallhere notions onwords (see for instane[8,9 ℄ formore details).

An alphabet

A

^{isa setofsymbolsalledletters. Hereweonsideronly nitealphabets. A word}

over

A

^{is a sequene of letters from}

A

^{. The empty word}

ε

^{isthe emptysequene. Equipped with}

theonatenation operation,theset

A ^∗

^{ofnitewordsover}

A

^{isafreemonoid withneutralelement}

ε

^{andsetofgenerators}

A

^{. Wedenoteby}

A ^ω

^{thesetofinnitewordsover}

A

^{. Asusually,for anite}

word

u

^andaninteger

n

^,the

n ^th

^powerof

u

^,denoted

u ⁿ

^,istheword

ε

^if

n = 0

^andtheword

u ⁿ⁻¹ u

otherwise. If

u

^{isnottheemptyword,}

u ^ω

^{denotes theinniteword obtainedbyinnitelyrepeating}

u

^{. A nite word}

w

^{is said primitive if for any word}

u

^{, the equality}

w = u ⁿ

^(with

n

^{an integer)}

implies

n = 1

^{. Any word isthepowerofa unique primitive word alledtheprimitiveroot of}

w

^.

Given anonemptyword

u = u ₁ . . . u _n

^with

u _i ∈ A

^,thelength

|u|

^of

u

^istheinteger

n

^{. Onehas}

|ε| = 0

^{. Iffor some words}

u, v, p, s

^{(possiblyempty),}

u = pvs

^,then

v

^{is a fator of}

u

^,

p

^{is a prex}

of

u

^and

s

^{is a sux of}

u

^{. When}

p 6= u

^(resp.

s 6= u

^{),we saythat}

p

^{isa proper prex (resp.}

s

^{is a}

proper sux) of

u

^{. By}

|u| _a

^{we denotethenumberof ourrenes of theletter}

a

^intheword

u

^.

Sturmianwordsmaybedenedinmanyequivalentways(see[1℄forinstane). Theyareinnite

binary words. Here we will onsider them as the innitebalaned non ultimately periodi words.

We reallthata (niteor innite)word

w

^over

{a, b}

^{isbalaned ifforanyfators}

u

^and

v

^{of same}

length

||u| _a − |v| _a | ≤ 1

^{,andthataninniteword}

w

^{isultimatelyperiodi if}

w = uv ^ω

^{for somenite}

words

u

^and

v

^.

Many studies of Sturmian words use Sturmian morphisms. Let

A, B

^{be two alphabets. A}

morphism (endomorphism if

A = B

⁾

f

^from

A ^∗

^to

B ^∗

^{isamappingfrom}

A ^∗

^to

B ^∗

^{suh thatfor all}

words

u, v

^over

A

^,

f (uv) = f (u)f (v)

^{. We also saythat}

f

^{is a morphism on}

A

^{or that}

f

^isdened

on

A

^{(withoutanyotherpreisionwhen}

B

^hasnoimportane). Amorphismon

A

^{isentirelyknown}

by theimages oftheletters of

A

^{. A morphism extendsnaturally oninnitewords. We denotejust}

byjuxtapositiontheompositionofmorphisms. Givenanendomorphism

f

^,if

lim n→∞ f ⁿ (a)

^exists,

thenthis limit isdenoted

f ^ω (a)

^{and isa xedpoint of}

f

^{: theword}

f ^ω (a)

^{is saidgenerated by}

f

^.

Sturmian morphisms arethemorphisms in

{E, L _a , L _b , R _a , R _b } ^∗

^where

E, L _a , L _b , R _a , R _b

^arethe

endomorphisms dened on

{a, b}

^by

E(a) = b

^,

E(b) = a

^,

L _a (a) = a

^,

L _a (b) = ab

^,

L _b (a) = ba

^,

L _b (b) = b

^,

R _a (a) = a

^,

R _a (b) = ba

^,

R _b (a) = ab

^,

R _b (b) = b

^{. Many relations existsbetween Sturmian}

words and Sturmian morphisms. For instane, it is known [2 , 6℄ that any Sturmian word an be

dened asan inniteprodutofSturmian morphisms.

A partiular ase of Sturmian words is thestandard (or harateristi) one. For any standard

Sturmian words, there existsa sequene

(d _n ) _n≥0

^{ofintegers, alled the diretive sequene verifying}

d ₁ ≥ 0

^and

d _k ≥ 1

^{for all}

k ≥ 2

^,suhthat

w = lim

n→∞ s _n

where thesequene

(s _n ) _n≥−1

^{ofwordsisdened by:}

s ₋₁ = b

^,

s ₀ = a

^and

s _n = s _n−1 ^{d n} s _n−2

^for

n ≥ 1

^.

Letus observe thatfor every

n ≥ 0

^,

s _2n

^endswith

a

^{. Moreover[1 ℄,}

s _2n = L ^d _a ¹ L ^d _b ² . . . L ^d _a ^{2 n− 1} L ^d _b ²ⁿ (a)

= L ^d _a ¹ L ^d _b ² . . . L _a ^{d 2n− 1} L ^d _b ²ⁿ L ^d _a ²ⁿ⁺¹ (a) s _2n+1 = L ^d _a ¹ L ^d _b ² . . . L _a ^{d 2n− 1} L ^d _b ²ⁿ L ^d _a ²ⁿ⁺¹ (b)

= L ^d _a ¹ L ^d _b ² . . . L _a ^{d 2n− 1} L ^d _b ²ⁿ L ^d _a ²ⁿ⁺¹ L ^d _b ²ⁿ⁺² (b)

3 Lyndon words and morphisms

From now on we onsider ordered alphabets. We denote

{α 1 < . . . < α _n }

^the

n

^{-letter alphabet}

{α ₁ , . . . , α _n }

^{with order}

α ₁ < . . . < α _n

^{. Given an ordered alphabet}

A

^{, we denote by}

^the

lexiographi order whenever used on

A ^∗

^{or on}

A ^ω

^{. Let reall that for two dierent (nite or}

innite) words

u

^and

v

^,

u ≺ v

^{if and only if}

u = xay

^,

v = xbz

^with

a, b ∈ A

^,

a < b

^,

x ∈ A ^∗

^,

y, z ∈ A ^∗ ∪ A ^ω

^{,or if(when}

u

^isnite)

u

^{isaprex of}

v

^.

A nonempty niteword

w

^{isaLyndon word iffor allnonemptywords}

u

^and

v

^,

w = uv

^implies

w ≺ vu

^. Equivalently[4,8℄,anonemptyword

w

^{isaLyndonwordifallitsnonemptypropersuxes}

aregreaterthan itfor thelexiographi order. For instane,on theone-letteralphabet

{a}

^,only

a

is a Lyndon word. On

{a < b}

^{the Lyndonwords oflength at most 5 are}

a

^,

b

^,

ab

^,

aab

^,

abb

^,

aaab

^,

aabb

^,

abbb

^,

aaaab

^,

aaabb

^,

aabab

^,

aabbb

^,

abbbb

^{. Lyndon wordsareprimitive.}

The seonddenitionofLyndonwordsextendsto innitewords: An innitewordisan innite

Lyndonword ifallits propersuxesaregreaterthanitfor thelexiographi order. Ausefulresult

ofG. Melançon[12℄statesthatan inniteword isa Lyndonwordifandonly ifithasaninnityof

prexesthat areLyndonwords. See for instane[7 ℄ for areent exampleof inniteLyndon word.

Any nonempty nite or innite Lyndon words an be deomposed asa noninreasing produt

of Lyndonwords. First,R.C.Lyndon proved (see[8 ℄ for instane):

Anyword

w ∈ A ⁺

^{maybe writtenuniquely asa}noninreasingprodutof Lyndonwords:

w = ℓ ₁ ℓ ₂ . . . ℓ _n

^{where for eah}

i

^,

ℓ _i

^{isa Lyndon word and}

ℓ _n ℓ _n−1 . . . ℓ ₁

^.

This resultwasgeneralized to innitewords [19 ℄:

Anyright inniteword

w

^{maybe uniquely expressed asa} noninreasingprodutof Lyn- don words, nite or innite, in one of the two following forms: either there exists an

innite noninreasingsequene of nite Lyndon words

(ℓ _k ) _k≥0

^suhthat

w = ^Y

n≥0

ℓ _n = ℓ ₀ ℓ ₁ . . .

or there exist nite Lyndon words

ℓ ₀ , . . . , ℓ _m−1

⁽

m ≥ 0

^{) and an innite word}

ℓ _m

^suh

that

ℓ _m ≺ ℓ _m−1 ℓ _m−2 . . . ℓ ₀

^and

w = ℓ ₀ . . . ℓ _m−1 ℓ _m .

some innite words. In[12℄, G. Melançon obtains the deomposition of standard Sturmian words.

We onsider these words here on thealphabet

{a < b}

^{. For any word}

w

^{ending withthe letter}

a

^,

let usdenote

w

^{theword suhthat}

w = wa

^.

Theorem 3.1 [12℄ Let

s

^{be a standard Sturmian word with diretive sequene}

(d n ) n≥1

^{. Let}

ℓ _n = as ^d _2n ^{2n+1 −1} s _2n−1 s _2n

^{. (if}

d ₁ = 0

^then

ℓ ₀ = b

^).

The words

(ℓ n ) n≥0

^{form a stritly dereasing sequene of Lyndon words and the unique fator-}

izationof

s

^{as a} noninreasingprodut of Lyndonwords is

s = ^Y

n≥0

ℓ ^d _n ²ⁿ⁺¹ .

G. Melançon wrote[12, Remark3.7℄ :

Whenisthesequene

(ℓ _n ) _n≥0

^{morphi? Morepreisely,isitpossibletogiveamorphism}

ϕ : {a, b} ^∗ → {a, b} ^∗

^{and a Lyndon word}

ℓ ₀

^{suh that}

ℓ _n+1 = ϕ(ℓ _n )

^{? This question}

has a positive answer inthe ase where the diretivesequene isonstant. For instane,

if

d n = 2

^{for all}

n ≥ 0

^{, then we may set}

ℓ ₀ = aab

^{and use the morphism mapping}

a 7→ aaabaab

^and

b 7→ aab

^.

AharateristiSturmianwordmaybeitselfmorphi. Thatis,maybethelimit

lim _n ϕ ⁿ (a)

of a (nonerasing) morphism(satisfying

ϕ(a) ∈ aA ^∗

^{). Itis knownthat this is}essentially equivalent to the fat that its diretive sequene is periodi. Unfortunately, even when

a harateristi Sturmianword

s

^{has a periodidiretive sequene, it seems that the se-}

quene

(ℓ n ) n≥ 0

^{is notalways morphi, althoughit ispossibleto desribe patterns in the}

fatorization.

Theaimofthispaperisto answerthisquestion. Themainideasofourproofaregeneralizations

of the following remarks: the morphism

a 7→ aaabaab

^and

b 7→ aab

^{is the Sturmian morphism}

L ² _a R ² _b

^{and preserves Lyndonwords. Moreover}

L ² _a L ² _b

^{is a onjugate of}

L ² _a R ² _b

^,

L _a ² L ² _b (a) = (l ₀ ) ² a

^and

L ² _a R ² _b (a) = a(l ₀ ) ²

^{. Let us notethat, in[14℄, similarremarks aremade about the}deomposition of the Fibonai word (the standard Sturmian word of diretive sequene

(1) _n≥0

^{). In Setion 4, we}

reallnotions on onjugay ofmorphisms.

Let us now reall some results on morphisms preserving (nite) Lyndon words. These mor-

phismsare studied in[14℄. Bydenition, a morphism

f

^{preserves Lyndon word iffor eah Lyndon}

word

w

^,

f (w)

^{is a Lyndon word. Eetive} haraterizations of suh morphisms are given in [14 ℄.

Consequently Sturmianwords preservingLyndon wordsareknown:

Proposition 3.2 [14 ℄ A Sturmian morphism on

{a < b}

^{is a Lyndon morphism if and only if it}

belongs to

{L _a , R _b } ^∗

^.

Toend this setion, letus observe thata study ofmorphisms preservinginniteLyndon words

isgiven in[15 ℄.

Inthissetion, wereallthenotionofonjugay(see,e.g.,[9,13℄). Wealsointroduethepartiular

aseof strong onjugay whihwill beuseful to answer G.Melançon.

Let

A

^and

B

^{betwoalphabetsandlet}

f

^and

g

^{betwomorphismsfrom}

A ^∗

^to

B ^∗

^{. Themorphism}

g

^{isa(right)onjugate of}

f

^{ifthereexistsaword}

u

^{suhthatforanyword}

x

^over

A

^,

f(x)u = ug(x)

^.

We will also saythat

f

^and

g

^are

u

-onjugated, and we will denote

f ⊳ _u g

^{. Moreover if}

f (a) = ua

and

g(a) = au

^{for aletter}

a

^,

f

^and

g

^{willbe alledstrongly (on}

a

⁾

u

^-onjugated.

Let us reall that any morphism

f

^{has at least one onjugate: itself (}

f ⊳ _ε f

^{). The Fibonai}

morphism

ϕ = L _a E

^{dened by}

ϕ(a) = ab

^and

ϕ(b) = a

^{has exatlytwo onjugates, itselfand the}

morphism

ϕ ˜ = R a E

⁽

ϕ(a) = ˜ ba

^,

ϕ(b) = ˜ a

^{). A lotofrelations between onjugayof morphismsand}

SturmianmorphismsweregivenbyP.Séébold[17℄andgeneralized to alargerfamily ofmorphisms

in[13 ℄.

Sine

ϕ(a)

^{does not end with the letter}

a

^{, no morphism is strongly onjugate (on}

a

^{) to the}

Fibonai morphism. Nevertheless we an observe that

ϕ ²

⁽

a 7→ aba

^,

b 7→ ab

^{) is strongly}

ab

^-

onjugated to

ϕ ϕ ˜

⁽

a 7→ aab

^,

b 7→ ab

^{). More generally, for all integers}

x

^and

y

⁽

y 6= 0

^{), the mor-}

phism

L ^x _a L ^y _b

^{isstrongly onjugated to themorphism}

L ^x _a R ^y _b

^{. This follows immediatlythe formulas:}

L ^x _a L ^y _b (a) = (a ^x b) ^y a

^,

L ^x _a L ^y _b (b) = a ^x b

^,

L ^x _a R ^y _b (a) = a(a ^x b) ^y

^,

L ^x _a R ^y _b (b) = a ^x b

⁽

L _a ^x L ^y _b ⊳ _(a x b) ^y L ^x _a R ^y _b

^).

Abasipropertyofonjugayis[9 ,13 ℄: formorphisms

f

^,

f ^′

^,

g

^,

g ^′

^,andwords

u

^,

u ^′

^,if

f ⊳ _u g

^and

f ^′ ⊳ _u ′ g ^′

^then

f f ^′ ⊳ _{f (u} ′ )u gg ^′

^(ofourse

f (u ^′ )u = ug(u ^′ )

^{). Thispropertyextendsto strongonjugay:}

Lemma 4.1 Let

f, f ^′ , g, g ^′

^{, (}

a

^{a letter) and}

u, u ^′

^{wordssuh that}

f

^{is strongly(on}

a

⁾

u

^-onjugated

to

g

^and

f ^′

^{isstrongly (on}

a

⁾

u ^′

^{- onjugated to}

g ^′

^{. Then}

f f ^′

^{is strongly (on}

a

⁾

[f (u ^′ )u]

^-onjugated

to

gg ^′

^.

Proof. We already know

f f ^′ ⊳ _f(u ′ )u gg ^′

^{. By} hypothesis,

f (a) = ua

^,

g(a) = au

^,

f ^′ (a) = u ^′ a

^et

g ^′ (a) = au ^′

^{. Thus}

f f ^′ (a) = f (u ^′ a) = f (u ^′ )ua

^and

gg ^′ (a) = g(au ^′ ) = aug(u ^′ ) = af (u ^′ )u

^{. So}

f f ^′

^is

strongly

[f(u ^′ )u]

^{-onjugated to}

gg ^′

^.

We end this setion with a rst use of strong onjugay onerning Sturmian words. One

partiular property of any standard Sturmian word

w

^over

{a < b}

^{is that both}

aw

^and

bw

^are

Sturmian words [16℄. Words

aw

^(with

w

^{standard Sturmian) arealso known as Christoelwords.}

In[3℄, itisshown, thatChristoelwordsareinniteLyndon words:

Proposition 4.2 [3 ℄ For any standard Sturmian word

w

^over

{a < b}

^,

aw

^{is an innite Lyndon}

word.

Proof. Let

w

^{be a standard word with diretive sequene}

(d n ) _n≥1

^{. We have already said that a}

standard word an be viewed as

w = lim n→∞ s n

^{for some words}

s n

^{dened in Setion 2. In fat,}

wean verifythatthen

w = lim _n→∞ s _2n

^{. Let}

n ≥ 1

^{. Weknowthat}

s _2n = L ^d _a ¹ L ^d _b ² . . . L a ^{d 2n− 1} L ^d _b ²ⁿ (a)

^.

Asa onsequene of Lemma 4.1and of thefat thatfor all integers

x

^and

y

^{,the morphism}

L ^x _a L ^y _b

is strongly onjugated to themorphism

L ^x _a R ^y _b

^{,we an verifythat}

L _a ^{d 1} L ^d _b ² . . . L ^d a ^{2n− 1} L ^d _b ²ⁿ

^{is strongly}

onjugated to

L ^d _a ¹ R ^d _b ² . . . L ^d a ^{2n− 1} R ^d _b ²ⁿ

^.

In partiular,

aL ^d _a ¹ L ^d _b ² . . . L a ^{d 2n− 1} L ^d _b ²ⁿ (a) = L _a ^{d 1} R ^d _b ² . . . L ^d a ^{2n− 1} R ^d _b ²ⁿ (a)a

^{. By} Proposition 3.2, the morphism

L ^d _a ¹ R ^d _b ² . . . L a ^{d 2n− 1} R _b ^{d 2n}

^{preservesLyndonwords. Hene}

L _a ^{d 1} R ^d _b ² . . . L a ^{d 2n− 1} R _b ^{d 2n} (a)

^isaLyn-

don word. Consequently the word

w

^{has an innity of Lyndon words as prexes. It is a Lyndon}

word.

we let thereader prove:

Proposition 4.3 Let

A

^{be an alphabet and}

a

^{a letter in}

A

^{. Let}

f, g

^{be two nonerasing endomor-}

phisms on

A

^{and let}

u

^{be a word over}

A

^{suh that}

f

^is

u

^{-strongly onjugateto}

g

^{. Then}

f ^ω (a)

^and

g ^ω (a)

^existand

af ^ω (a) = g ^ω (a)

^.

Thusif

g

^generateson

a

^{aninniteLyndonword(whihistheaseifitpreservesLyndonwords}

or ifitpreservesinniteLyndonwords(see [15℄)),

af ^ω (a)

^{isan inniteLyndon word.}

ThesituationofProposition4.3anbemetfor morphismsthatarenot Sturmian. For instane,

this isthease withthemorphisms:

f :

( a 7→ aba

b 7→ abb g :

( a 7→ aab b 7→ bab

Moreoveroneanseethat

g

^{preservesinniteLyndonwordsandgeneratesaninniteLyndonword.}

5 An answer to G. Melançon

In this setion, we onsider a standard Sturmian word

w

^{over theordered alphabet}

{a < b}

^with

diretive sequene

(d n ) _n≥1

^{(Letreallthat}

d ₁ ≥ 0

^and

d _n ≥ 1

^{for all}

n ≥ 2

^{). Thesequeneofwords}

(s _n ) _n≥0

^and

(ℓ _n ) _n≥0

^{are those dened} respetively at theend of Setion 2 and inTheorem 3.1. In

partiular,

w = lim n→∞ Q

n≥0 ℓ ^d _n ^{2 n}

^isthe deomposition inLyndon words of

w

^{(for eah}

n ≥ 0

^,

ℓ _n

isa Lyndonword and

ℓ _n+1 ℓ _n

^{). Ourresult is:}

Theorem 5.1 Withthehypothesesofthissetion,thereexistsamorphism

g

^{suhthatforall}

n ≥ 0

^,

ℓ _n+1 = g(ℓ _n )

^{ifand onlyif one of the two following ases hold:}

• 1 ≤ d ₁ ≤ d ₃

^{, and for all}

n ≥ 1

^,

d _2n = d ₂

^and

d _2n+1 = d ₃

^{. In this ase,}

ℓ ₀ = a ^{d 1} b

^and

g = L ^d _a ¹ R _b ^{d 2} L ^d _a ^{3 −d 1}

^.

• d ₁ = 0

^,

1 ≤ d ₂ ≤ d ₄

^{, and for all}

n ≥ 1

^,

d _2n+2 = d ₄

^and

d _2n+1 = d ₃

^{. In this ase,}

ℓ ₀ = b

^and

g = R ^d _b ² L _a ^{d 3} R ^d _b ^{4 −d 2}

^.

We observe that in eah ase, the morphism

g

^{is a Sturmian morphism that preserves Lyn-}

don words (see Proposition 3.2). Moreover the word

w

^{is generated by a Sturmian morphism}

(

L ^d _a ¹ L ^d _b ² L ^d _a ^{3 −d 1}

^or

L ^d _b ² L ^d _a ³ L ^d _b ^{4 −d 2}

^).

Inordertoprovetheprevioustheorem,usingthestrongonjugay,werstexpresseahLyndon

word

ℓ _n

^{withmorphisms. For}

n ≥ 0

^{,we denote:}

f _n = (L ^d _a ¹ L ^d _b ² ) . . . (L ^d _a ^{2 n− 1} L ^d _b ²ⁿ )

g _n = (L ^d _a ¹ R ^d _b ² ) . . . (L ^d _a ^{2n− 1} R ^d _b ^{2 n} )

The interestof the morphisms

f _n

^{is immediate sine we have already seenrelations between them}

andthewords

s _n

⁽

s _2n = f _n (a)

^,

s _2n+1 = f _n+1 (b)

^{). Wealsoobservethateah}

g _n

^{isamorphism that}

preservesLyndon words. Asaonsequene ofLemma4.1andof thefatthatforall integers

x

^and

y

^,themorphism

L ^x _a L ^y _b

^{isstrongly onjugated to themorphism}

L ^x _a R ^y _b

^{,we have:}

Lemma 5.2 For all

n ≥ 1

^,

f n

^{isstrongly (on}

a

^{) onjugated to}

g n

^.

Nowwegive anew formula forthewords

(ℓ _n ) _n≥0

^:

Lemma 5.3 For all

n ≥ 0

^,

ℓ n = g n L ^d a ²ⁿ⁺¹ (b)

Proof.

ℓ n a = as ^d _2n ^{2n+1 −1} s _2n−1 s _2n a

= as ^d _2n ^{2n+1 −1} s _2n−1 s _2n

= af _n (a ^{d 2n+1 −1} ba).

If

n = 0

^,

ℓ _n a = a ^{d 1} ba = L ^d _a ¹ (b)a = g ₀ L ^d _a ¹ (b)a

^.

When

n ≥ 1

^,let

u _n

^{be theword suh that}

f ⊳ _{u n} g _n

^{. By Lemma 5.2,}

f _n (a) = u _n a

^,

g _n (a) = au _n

^.

Thus

ℓ _n a = af _n (a ^{d 2 n +1 −1} b)u _n a

= au _n g _n (a ^{d 2n+1 −1} b)a

= g n (a ^{d 2n+1} b)a

= g _n L ^d _a ²ⁿ⁺¹ (b)a

Consequently for all

n ≥ 0

^,

ℓ _n = g _n L ^d a ²ⁿ⁺¹ (b)

^.

Let us observe that Lemma 5.2 allows to give a new proof of the fat that the words

(ℓ n ) n≥0

form a stritly dereasing sequene of Lyndon words. Indeed, by Proposition 3.2, eah morphism

g _n L ^d a ²ⁿ⁺¹

^{isaLyndonmorphism,hene}

g _n L ^d a ²ⁿ⁺¹

^{isaLyndonword. Moreover}

R _b ^{d 2 n} L ^d a ²ⁿ⁺¹ (b)

^foreah

n ≥ 1

^{, then}

R ^d _b ^{2 n} L ^d a ²ⁿ⁺¹ (b) ≺ b

^{whih implies}

ℓ _n = g _n L ^d a ²ⁿ⁺¹ (b) ≺ g _n−1 L ^d a ^{2n− 1} (b) = ℓ _n−1

^{(sine any}

morphism preserving Lyndon words also stritly preserves the lexiographi order on nite words

[14 ℄).

Proof of Theorem 5.1. Note that the if part of the theorem is immediate. Assume the

sequene

(ℓ n ) _n≥0

^{ismorphi. Let}

g

^{bethemorphismsuhthat,forall}

n ≥ 0

^,

g(ℓ n ) = ℓ _n+1

^{. Observe}

thatthemorphism

g

^{annotbeerasingsineotherwisethisontraditsthefatthat}

ℓ ₂

^isaprimitive

word (as aLyndon word).

Werst onsiderthease

d ₁ ≥ 2

^{. Observe}

ℓ ₀ = a ^{d 1} b

^and

g(a ^{d 1} b) = ℓ ₁ = [a(a ^{d 1} b) ^{d 2} ] ^{d 3} a ^{d 1} b.

Assume

g(a) = a

^{, and so}

g(b) = ab(a ^{d 1} b) ^{d 2 −1} [a(a ^{d 1} b) ^{d 2} ] ^{d 3 −1} a ^{d 1} b

^{. The word}

ℓ ₂ = g(ℓ ₁ )

^has

g(a ^{d 1 +1} b)

^{asprex. Thusthewords}

a ^{d 1 +2}

^and

ba ^{d 1} b

^arefatorsof

ℓ ₂

^{. Thisontraditsthefatthat}

ℓ ₂

^{,asafator of aSturmian word,is balaned. Hene}

g(a) 6= a

^.

Sine

d ₁ ≥ 2

^and

g(a ^{d 1} b)

^{starts with}

a ^{d 1 +1} b

^,theword

a ^{d 1 +1} b

^{isa prexof}

g(a)

^{. Morepreisely,}

a(a ^{d 1} b) ^{d 2}

^{must be a prex of}

g(a)

^{. Finally, we an verify that}

g(a) = (a(a ^{d 1} b) ^{d 2} ) ^k

^{for an integer}

k ≥ 1

^{. Itfollows}

g(b) = (a(a ^{d 1} b) ^{d 2} ) ^{d 3 −kd 1} a ^{d 1} b

^{whih implies}

d ₃ ≥ kd ₁

^.

Assume

k ≥ 2

^{. The word}

ℓ ₂ = g(ℓ ₁ )

^ontains

g(ba ^{d 1} b)

^and

g(a ^{d 1 +1} b)

^{as fators. The word}

g(ba ^{d 1} b)

^{ends with}

bub

^where

u = (a ^{d 1} b) ^{d 2} [a(a ^{d 1} b) ^{d 2} ] ^{d 3} a ^{d 1}

^{. Furthermore the word}

g(a ^{d 1 +1} b) =

[a(a ^{d 1} b) ^{d 2} ] ^{d 3 +k} a ^{d 1} b

^startswith

aua

^{. Thisontradits thefatthat}

ℓ ₂

^{is balaned.}

Hene

k = 1

^,

d ₃ ≥ d ₁

^,

g(a) = a(a ^{d 1} b) ^{d 2}

^,

g(b) = [a(a ^{d 1} b) ^{d 2} ] ^{d 3 −d 1} a ^{d 1} b

^{. We observe that}

g = L _a ^{d 1} R ^d _b ² L _a ^{d 3 −d 1}

^{andthatit isan injetive morphism.}

Nowweanprovethat,forall

n ≥ 1

^,

d _2n = d ₂

^and

d _2n+1 = d ₃

^{. Weatbyindutionon}

n

^{. There}

isnothingtodofor

n = 1

^{. Let}

n ≥ 1

^{. Assumethatwehavealreadyproved}

d _2p = d ₂

^and

d _2p+1 = d ₃

for allintegers

p

^with

1 ≤ p ≤ n

^{. Wehave}

ℓ _n+1 = g _n+1 L ^d a ²ⁿ⁺³ (b) = L ^d _a ¹ (R ^d _b ² L _a ^{d 3} ) ⁿ R ^d _b ²ⁿ⁺² L ^d a ²ⁿ⁺³ (b) = (L ^d _a ¹ R _b ^{d 2} L ^d _a ^{3 −d 1} ) ⁿ L ^d _a ¹ R ^d _b ²ⁿ⁺² L ^d _b ²ⁿ⁺³ (b) = g ⁿ (L ^d _a ¹ R ^d _b ²ⁿ⁺² L ^d _b ²ⁿ⁺³ (b))

^{. Moreover}

ℓ _n+1 = g ⁿ (ℓ ₁ )

^{. Sine}

g

^is

injetive,

ℓ ₁ = L ^d _a ¹ R _b ^{d 2 n +2} L _b ^{d 2 n +3} (b)

^{. This implies}

d _2n+2 = d ₂

^and

d _2n+3 = d ₃

^.

Now we onsider the ase

d ₁ = 1

^{. We have}

ℓ ₀ = ab

^and

ℓ ₁ = [a(ab) ^{d 2} ] ^{d 3} ab

^{. As in ase}

d ₁ ≥ 2

^{, we annot have}

g(a) = a

^{. Hene}

g(a)

^{starts with}

aa

^{. We observe that}

g(a)

^{annot ends}

with

a

^{, sine otherwise the balaned word}

ℓ ₂ = g(ℓ ₁ )

^ontains

aaa

^and

bab

^{. We observe also that}

g(a) 6= [a(ab) ^{d 2} ] ⁱ a(ab) ^k

^{for any integer}

k, i

^{suh that}

1 ≤ k < d ₂

^and

i ≥ 0

^{. Indeed otherwise the}

word

ℓ ₂

^{ontaining both}

g(aa)

^and

g(ab)

^{should ontains thefators}

a(ab) ^k aa

^and

b(ab) ^k ab

^(sine

(ab) ^{d 2 +1}

^ends

g(ab)

^{): thisontradits thefatthat}

ℓ ₂

^{isbalaned. Itfollows that}

g(a) = [a(ab) ^{d 2} ] ^k

with

1 ≤ k ≤ d ₃

^and

g(b) = [a(a ^{d 1} b) ^{d 2} ] ^{d 3 −k} a ^{d 1} b

^{. Exatly asinase}

d ₁ ≥ 2

^{,we an thenprove that}

k = 1

^,

g = L _a R _b ^{d 2} L ^d _a ^{3 −1}

^{and for allintegers}

n ≥ 1

^,

d _2n = d ₂

^and

d _2n+1 = d ₃

^.

From now on, we onsider the ase

d ₁ = 0

^{. we have}

ℓ ₀ = b

^{and so}

g(b) = ℓ ₁ = (ab ^{d 2} ) ^{d 3} b

^.

Moreover

ℓ ₂ = R ^d _b ² L ^d _a ³ R ^d _b ⁴ L ^d _a ⁵ (b)

^{,that is}

ℓ ₂ = [ab ^{d 2} [(ab ^{d 2} ) ^{d 3} b] ^{d 4} ] ^{d 5} (ab ^{d 2} ) ^{d 3} b.

Furthermore

ℓ ₂ = g ² (b) = g((ab ^{d 2} ) ^{d 3} b)

^{. It followsthat}

g((ab ^{d 2} ) ^{d 3} ) = [ab ^{d 2} g(b) ^{d 4} ] ^{d 5}

Sine the word

ab ^{d 2} g(b) ^{d 4} = ab ^{d 2} [(ab ^{d 2} ) ^{d 3} b] ^{d 4}

^{is a primitive word,}

g(ab ^{d 2} ) = [ab ^{d 2} g(b) ^{d 4} ] ^x

^and

xd ₃ = d ₅

^{for aninteger}

x ≥ 1

^{. Sine}

ab ^{d 2}

^{is not asuxof}

g(b)

^,

d ₂ ≤ d ₄

^.

Let us prove that

x = 1

^{, that is,}

d ₃ = d ₅

^{. Assume by} ontradition that

x ≥ 2

^{. The word}

ℓ ₂

^has

(ab ^{d 2} ) ^{d 3 +1}

^{as a prex and}

[(ab ^{d 2} ) ^{d 3} b] ²

^{as a sux. Let}

u = ab ^{d 2} g(b) ^{d 4}

^:

g(ab ^{d 2} ) = u ^x

^.

The word

ℓ ₃ = g(ℓ ₂ )

^{ontains the fator}

g((ab ^{d 2} ) ^{d 3 +1} ) = u ^{(d 3 +1)x} = uu ^{d 5} uu ^x−2

^{whih ontains}

the fator

ab ^{d 2} g(b) ^{d 4} u ^{d 5} ab ^{d 2} (ab ^{d 2} ) ^{d 3} b

^{whih starts with}

ab ^{d 2} g(b) ^{d 4} u ^{d 5} (ab ^{d 2} ) ^{d 3} a

^{. Observe now that}

g((ab ^{d 2} ) ^{d 3} b) = [ab ^{d 2} g(b) ^{d 4} ] ^{d 5} g(b)

^endswith

b ^{d 2 +1} g(b) ^{d 4}

^. Consequently theword

ℓ ₃

^{also ontainsthe}

fator

b ^{d 2 +1} g(b) ^{d 4} g(((ab) ^{d 2} ) ^{d 3} b) = bb ^{d 2} g(b) ^{d 4} u ^{d 5} (ab ^{d 2} ) ^{d 3} b

^{. We have a} ontradition with the fat that

ℓ ₃

^{isa balanedword.}

From what preedes,

g(ab ^{d 2} ) = ab ^{d 2} g(b) ^{d 4}

^{and so}

g(a) = ab ^{d 2} g(b) ^{d 4 −d 2} = ab ^{d 2} ((ab ^{d 2} ) ^{d 3} b) ^{d 4 −d 2}

^.

Moreover

g(b) = (ab ^{d 2} ) ^{d 3} b

^{. We observe}

g = R ^d _b ² L _a ^{d 3} R ^d _b ^{4 −d 2}

^{. Asin ase}

d ₁ ≥ 2

^{, we an state that,}

for all integers

n ≥ 2

^,

d _2n = d ₄

^and

d _2n−1 = d ₃

^.

6 Conlusion

This paper shows the interest of onjugay of morphisms and of morphisms preserving Lyndon

words as tools to takle problems onerning Sturmian words and/or Lyndon words. We are now

working to nd other situations where these tools an be useful. Inpartiular, we are looking for

thedeompositioninLyndon wordsof any Sturmianwords.

ManythankstoP.Sééboldfor useful disussionson strong onjugay.

Referenes

[1℄ J. BerstelandP.Séébold. Sturmian words, hapter2 in[9 ℄.

[2℄ V. Berthé,C. Holton,and L.Q.Zamboni. Initial powers ofsturmian sequenes. To appear in

Ata Informatia,seewww.lirmm.fr/

∼

^berthe.

[3℄ J.P.BorelandF.Laubie. Quelquesmots surladroiteprojetiveréelle. JournaldeThéoriedes

Nombres deBordeaux, 5:2351,1993.

[4℄ K.T. Chen, R.H. Fox,and R.C.Lyndon. Freedierential alulusIV thequotient groups of

the lowerentral series. Ann. Math. 68,68:8195, 1958.

[5℄ A. Ido and G. Melançon. Lyndon fatorization of the Thue-Morse word and its relatives.

Disret. Math.andTheoret. Comput.Si., 1:4352,1997.

[6℄ J.JustinandG.Pirillo. Episturmianwordsandepisturmianmorphisms. Theoretial Computer

Siene, 276(1-2):281313, 2002.

[7℄ J. Justin and G. Pirillo. On a harateristi property of Arnoux-Rauzy sequenes. RAIRO

Theoretial Informatisand Appliations, 36:385388, 2002.

[8℄ M. Lothaire. Combinatoris on words, volume 17 of Enylopedia of Mathematis. Addison-

Wesley,1983.Reprintedin1997byCambridgeUniversityPressintheCambridgeMathematial

Library,Cambridge,UK, 1997.

[9℄ M. Lothaire. Algebrai Combinatoris on words, volume 90 of Enylopedia of Mathematis.

Cambridge UniversityPress, Cambridge,UK, 2002.

[10℄ M. Lothaire. Applied Combinatoris on Words. To appear.

(see www-igm.univ-mlv.fr/

∼

^berstel).

[11℄ G.Melançon. Lyndonfatorizationofinnitewords. InSTACS'96,volume1046ofLet.Notes

in Comp. Si.,pages 147154,1996.

[12℄ G. Melançon. Lyndon fatorization of Sturmian words. Disrete Mathematis, 210:137149,

2000.

[13℄ G. Rihomme. Conjugay and episturmian morphisms. Theoretial ComputerSiene, 302:1

34, 2003.

[14℄ G. Rihomme. Lyndon morphisms. Bulletin of the Belgian Mathematial Soiety, 10:761785,

2003.

tomata, Languages and Combinatoris. An extended abstrat appeared in theproeedings of

theonferene"JournéesMontoisesd'InformatiqueThéorique",Liège,Belgium,2004(Prépub-

liation 04.006, Institutde Mathématique, Université de Liège),p325-333.

[16℄ P.Séébold. FibonaimorphismsandSturmianwords. Theoretial ComputerSiene, 88:365

384, 1991.

[17℄ P.Séébold. Ontheonjugation ofstandardmorphisms. Theoretial ComputerSiene,195:91

109, 1998.

[18℄ P.Séébold.LyndonfatorizationoftheProuhetwords. TheoretialComputerSiene,307:179

197, 2003.

[19℄ R.Siromoney,L.Mathew,V.R.Dare,andK. G.Subramanian. InniteLyndonwords. Infor-

mation Proessing Letters,50:101104, 1994.