Substitution invariant sturmian bisequences

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

B

RUNO

P

ARVAIX

Substitution invariant sturmian bisequences

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{1 (1999),}

p. 201-210

<http://www.numdam.org/item?id=JTNB_1999__11_1_201_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Substitution

invariant Sturmian

_bisequences

par BRUNO PARVAIX

RÉSUMÉ. Les suites sturmiennes indexées sur

_Z,

de _pente 03B1 et

d’intercept

p, sont laissées fixes par une substitution non triviale

si et seulement si 03B1 est un nombre de Sturm et _{p appartient} à

Q(03B1).

On remarque aussi que les suites de

Beatty

permettent de définir des _partitions de l’ensemble des entiers relatifs.

ABSTRACT. We _prove that a Sturmian

_bisequence,

with

_slope

03B1

and _{intercept p,} is fixed

_by

some non-trivial substitution if and

only

if 03B1 is a Sturm number and _p

belongs

to

_Q(03B1).

We also detail a

_{complementary}

_system of

_integers

connected with

_Beatty

bisequences.

1. INTRODUCTION

Beatty

sequences

( ( ncY

+ and

_([na

+ have been studied

extensively.

Many

papers deal with the case _p =

0,

see

[1,

_9,

_{10, 14, 15, 28,}

29].

The

_{inhomogeneous}

case is also discussed from several

_points

of view

_[6,

7, 16, 20, 21,

22~.

By

the _way, this Note

_provides

a new contribution about

complementary

systems

of

_integers.

This

_problem

_arose, in various

_forms,

in the works of A. S. Fraenkel

_[13],

R. L. Graham

_[17]

and R.

_Tijdeman

[30,

31].

A natural way to examine

_Beatty

_{sequences is} to consider the class of Sturmian words defined

_by

G. A. Hedlund and M. Morse in the context

of

_{topological dynamics,}

see

[25,

26].

For further

details,

both

[3]

and

[8]

contain extensive lists of references. Here we are

_especially

interested in

substitution invariant Sturmian words. In

_[27]

we elicited

_properties

about

some

_right-sided

infinite Sturmian words the

_intercept

of which is a

particu-lar

_homography

of the

_slope.

We therefore obtained a

_partial

_{generalization}

of

_Crisp

et al.’s main Theorem

_{concerning cutting}

_sequences

_~12~.

The aim

of this Note is the full characterization of Sturmian

bisequences

which are

+

_{b~ ~}

_k

E with

_fl

irrational and 6 real. As usual

_lxJ

is the

integer

part and the

_ceiling

of _any real number x. Let and

r’,j

be

the

_{generating bisequences}

of and

_Z’,6:

we set

for each n E Z. We _say that two subsets of Z are a

complementary

_system

if

_they

form a

_partition

of Z.

Let ,A* be the free monoid

_{generated by}

the two-letter

_alphabet

,A. =

~0,1~.

The set of

_right-sided

infinite words is denoted

_by

AW and is the set of left-sided infinite words. A

_bisequence

is a

doubly

infinite word

and is the set of

_bisequences

over A. We _say that the

_bisequences

... ... and ... ... are

equal

if there exists an

integer k

E Z such

that vi

=

v’ i+k

for each i E Z. In this _event, we note

...

... _cti

... ...

... V-2V-lVOVlV2 --- - ---V-2V- 0 1 2

Let a be irrational and _p be real. Consider the

_bisequences

_{and z’ alp}

defined

_by

’

and

for each n E Z. A

_bisequence

x is said to be Sturmian if x - zo.,p or x -

z’ a,p

for a suitable choice of a and _p. It is clear that

za,P(n)

=

za+i,p(n)

and = for

each n E 7G,

_so without loss of

generality,

_we

may take 0 cx 1.

_Finally,

a

_right-sided

infinite word _y is Sturmian if there

exist a Sturmian

bisequence x

and a left-sided infinite word

y’

such that x -

y’y.

_Noting

that Sturmian words are

_intimately

related to

_straight

lines in the

_plane,

the number a is the

slope

and _p the

_intercept.

A substitution

_f

is a _map from A* into itself such that

f

(uu’)

=

f

(u) f (u’)

for all finite words u and u’. Let w =

wowlw2... be a

_right-sided

infi-nite word. Let Inv be the

_operator

defined

_by

_{Inv(w) =}

... W2Wlwo and

Inv(Inv(w))

= _w. As

usual,

_we set

f (w)

= and

The

_image

of ...

un-der

_f

is ...

f (v_2) f (v_1) f (vo) f (vl) f (v2)

... A one-sided infinite

word y

is

fixed

_by

_f

if

_{f (y)}

=

y, and a

bisequence x

is fixed

by f

if

f (x) -

x.

Moreover a substitution

_f

is Sturmian if

f (w)

is a

right-sided

infinite

Sturmian word whenever w is. F.

_Mignosi

and P. Seebold

_proved

that a

substitutions in _{any order}

and number

_[24].

A substitution

_f

is

_locally

Sturmian if there exists a

right-sided

infinite Sturmian word w such that

_{f (w)}

is Sturmian. J. Berstel and P. S66bold stated that any

locally

Sturmian substitution is

actually

Sturmian

_{[4, 5].}

Furthermore a substitution is non-trivial if it differs from the identical

transformation over ,A. In

[27]

we

_proved

that if a

_right-sided

infinite

Stur-mian word is fixed

_by

some non-trivial substitution then its

slope

_a, with

0 a

_1,

is a Sturrn

_number,

that

_is,

there exists an

_{integer n &#x3E;}

2 such

that:

or

where the

_{partial quotients k2, ... ,}

_belong

to I~1* . Remark that these

numbers were

introduced,

in a

slightly

different _way,

by Crisp

et al.

_[12].

3. RESULTS

As

_usual,

for _any

_quadratic

irrational a, let

Q(a) =

I

be the

_splitting

field of a over

Q.

The main result of this Note is the full

characterization of Sturmian

_bisequences

which are invariant under some

non-trivial substitution:

Theorem 1. Let x be a Sturmian

bisequence

with

slope

0 a 1. The

word x is

_{fixed by}

sorne non-trivial substitution

if

and

only if a is

a Sturm

number

_{and p belongs}

to

_{Q(a) .}

In

_~27],

we

_computed

the

_slope

and the

_intercept

of

f (x)

for _any Sturmian

substitution

_f

and any

right-sided

infinite Sturmian word x. Lemma 2 is a

translation of these formulas for Sturmian

_bisequences:

Lemma 2. Let a be irrational with 0 a 1 and

_{let p}

be real. Then

Moreover

The

_proof

of these

_{properties requires}

a careful

_study

of

_generating

bise-quences of

Beatty

bisequences:

.

As an immediate

_corollary,

we can characterize the occurrences of a letter

in any Sturmian

_bisequence.

More

_precisely

we remark that

for each _~y irrational with 0 ₁ 1 and v real. This result is a

general-ization of earlier work of A. S.

_Fraenkel,

M. Mushkin and U. Tassa

_dealing

with the

_homogeneous

case

[15].

From Lemma 3 we also obtain a

_property

about

_{complementary}

_systems

of

_integers:

Proposition

4. Let

_/3

&#x3E; 1 be irrational and 6 be real. Then and

_a , as well as and

Z 3

_-=L+1’ are

complementary

systems

~_i ~a_i

of

integers.

4. PROOFS

First of

_all,

we examine the

_{generating bisequences}

of

_{Beatty bisequences:}

Proof of

Lemma 3. Let n E Z. If we state that

thus Next comes

I

and

Conversely,

if = 1 there exists _an

integer

k _E Z such that

[k,3

₊

b~

= n. We therefore observe that

It follows that

The truth of

the first statement

is

now

clear,

and we turn to the second

part

of the

_proof.

Let n E Z. If z 1

-8 (n)

=1 then

hence we have

In order to describe the

_{complementary}

system

of

_integers,

connected with

a

Beatty

_bisequence,

we need to introduce the

_following

Lemma:

Lemma 5. Let 0 C cx 1 be irrationnal and _p be real. Then

. - - , . ,

Proof.

We

_only

detail the

_proof

_concerning

the first result. Let n E Z. Since the relation

_[a]

_{= -~-a~}

holds for each real number _a, we

_verify

Proof of

Proposition l~.

Let n E Z. From Lemma

_3,

we remark that

Then Lemma 5

_implies

that

In other

_words,

we

_get

Furthermore,

since

_{3

&#x3E; 1 we can affirm that _any

_integer

occurs at most one

time in

_{.~,~,~ .}

_Clearly

this

_property

also holds for

_{,~’ ~}

5

. In

short,

the

B

sets and

_{.’ p a _ 1}

are a

_{complementary}

_system

of

_integers.

The

0-1, 0-1

part

of

_{proof concerning}

_.’,

, and Z p _{- +1} is similar in all

respects.

D

From now on we

_{study properties}

of substitution invariant Sturmian

bise-quences.

Proof

of

Lemma 2. Assume first that 0 p 1. We

split

the

_bisequence

za, p into the words

Let =

yoyl ... with _yj E A

for j

=

0,1, ...

We observe that

yo = 0 =

Let _{nq+1 be the}

_(q

+

_l)-th

occurrence of the letter 0 in the word

_W(w)

for each

_{q &#x3E;}

1. We

_easily

check:

For each n &#x3E; 1 we state that:

From Lemma

_5,

we _prove that _yn = 0 if and

only

i

short we obtain _cp rIb

_compute

_W(w’),

we remark

that

Indeed,

for each n E N* it is clear that

hence

If we write w’ = ... ar,.t ... a1aO over we

get

because

_cp(0)

= 01 and

cp(1)

= 0. We _can deduce that

Much as

_above,

we

verify

Moreover we observe that =

11-ao

and

cp(d)

=

oll-a

for each _{a E}

{0,1}.

Next comes

_{W (u)}

=

0§3(u)

for _{any u} E

_AW,

and

_consequently

). Bearing

in mind that

_w’w,

and

for

_{arbitrarily fl}

irrational and 5

_real,

we

_directly

claim:

The

_computation

of becomes trivial because we have

_cp(v)

for each v E

_Finally,

the

part

of

_proof

_{concerning z"}

is similar in all

respects.

0

’

For each Sturmian substitution

_f

it is therefore clear that

_{f (x)}

is a Sturmian

bisequence

whenever x is. Now we turn to the

_proof

of Theorem 1. Some

preliminaries

are

_required.

Let x and _y be two Sturmian

_bisequences.

Let

f

be a substitution such that

f (x) ^_~

_y. There exist a word x’ E LV A and a

_right-sided

infinite Sturmian word x" such that z ri x’x". Since we

f (x’ ) f (x" ) ,

the word

f (x")

is a

right-sided

infinite Sturmian

word. Thus

_f

is

_locally

Sturmian and

_{consequently f belongs}

to the monoid

E

Let us recall some basic

_properties

about Sturmian

_bisequences.

_{For any} irrational a we set Z + Za =

{a

₊ ba

I

(a, b) E

7~2~ .

Let A be the set of

_couples

_{(,3, 6)}

with 0

_/3

1 irrational and 6 real. We also set U =

{({3,8)

e A Let e A and

_{(cx’~ P’)}

We

have

_{ZO/,p’ if}

and

_only

if a = a’ and _{p -}

p’

E Z +

_Za,

see

[26].

A

similar result can be stated from the relation

2~ ,.

_Furthermore,

if

_za,,p,

then

_belongs

to U and In

_short,

if two

Sturmian

_bisequences

are

equal

then

they

have the same

slope

in

~0,1 ~.

Bearing

these remarks in

_mind,

we therefore obtain:

Lemma 6. Let x be a Sturmian

bisequence

with

slope

0 a 1.

If

x is

invariant under some non-trivial substitutions then a is a Sturm number.

Proof

(Sketch).

Assume that there exists a non-trivial substitution

f

such

that

_{f (x) ^_J}

x. Then

f belongs

to Let

/3

E

_{]0, 1[}

be the

_slope

of

f (x)

which is obtained

_by

Lemma 2.

_Clearly

this

_computation

can be done

regardless

of

_intercepts,

and there exists an

homography

_h,

with

_integer

coefficients,

such that

₁₃

=

h(a).

Therefore it

only

remains to solve the

equation

a =

h(a).

In this context, we have

_yet

observed that a is a Sturm

number: for a full characterization of the

homographies

connected with

Sturmian

_{substitutions,}

see the

proof

of Theorem 1 in

[27].

D

In order _{to prove} our main

result,

we add here a new _{necessary condition}

of invariance:

Lemma 7. Let x be a Sturmian

bisequence,

with

slope

cx and

_intercept

_p.

Proof.

Assume,

without loss of

_generality,

that 0 a 1. Let

_f

be a

non-trivial substitution such that

_{f (x) -}

x. Lemma 6

_implies

that a is a

Sturm number. Since a is a

quadratic irrational,

the

_image

of a under _any

homography,

with

_integer

_{coefficients,}

_belongs

to

_~(a).

_Using

Lemma

_2,

we

compute

the

_image

of x under

_{f .}

Let

_{3

be the

_slope

and 6 be the

_intercept

we obtain. It is clear that

13

E

_Q(oa)

and 0

_fl

1. We also remark that 6 E

_Q(a)+pQ(a).

Since

_{f (x) -}

x, we must check

_fl

= _a and

6 - p

_E Z + Za.

Next comes _p E

_{Q(a) .}

D

Combining

Lemmas 6 and

_7,

we establish the

_"only

if

_part"

of Theorem 1.

Now we turn to the

_proof

of the "if

_part":

the idea is to use some

_properties

that we

reported

in

[27].

First of

all,

a technical result

_concerning

Sturmian

continuations is

_required

_[26].

Definition 8

_(cf.

_[26]).

Let _y be a

_right-sided

infinite Sturmian word. A

Sturmian continuation

_{of y}

is a left-sided infinite word

y’

such that

y’y

is a

Sturmian

_bisequence.

Lemma 9

_(cf.

_[26]).

Let a be irrational with 0 cx 1 and _p be real. Each

right-sided infinite

Sturmian word _y, with

_slope

a and

_intercept

_p, admits

at least one and at most two Sturmian continuations. In the case

where y

admits

_different

StuTmian continuations there exist two

_{integers kl}

E Z and

k2

E N* such that _p =

kl

₊

k2a.

Definition 10

_(cf.

_(27~).

For each m &#x3E;

_1,

we set

A

_right-sided

infinite Sturmian word _y is said to be

_permitted

if there exist

an irrational a with 0 a

1,

an

_integer

m &#x3E; 1 and a

_couple

of

_integers

(a, b)

E

_C’(m)

such that _y = _or

y =

m _m a

Proposition

11

_(cf.

_[27]).

Let cx be a Sturm number. Each

permitted

word

y, with

slope

ce, is invariant under some non-trivial substitution.

Proof

of

Theorem 1. Let a be a Sturm number and _p E

_Q(()).

Let x be a

Sturmian

_bisequence

such that x -

_Clearly

there exists

_(a,

_b,

_n)

E

Z3

with n &#x3E; - 1 such that _{p =}

_Moreover,

since _{za,j+l - za.a}

n ",

for each

_{real 6,}

we

actually

(inod n)--(b (mod n))a . As

usual,

the

~ n

residue i

_{(mod n)}

is the

_{integer j,}

with 0

_j

n, such that there exists

an

_integer

k E Z

_{satisfying j}

= i ₊ kn. For each real 6 we set

is a

_right-sided

infinite

_permitted

word. From

_Proposition

_11,

it follows

that there exists a non-trivial Sturmian substitution

f

such that

f (y) _

_y.

Noting

that

we have

Hence the word _y admits.

as Sturmian continuations. If the

relation

is not valid then Lemma 9

_implies

that there exists

_{(kl, k2) E}

7Gz

with

_k2

&#x3E; 1 such that

In this

event,

since a is irrational we observe that

_{k2 = 0,}

which leads to a

contradiction. We therefore obtain

_{f (x) ^-,}

x.

If n-~-1 a

(mod n)+b (mod n)

we state that

_{(a (mod n), (b (mod n))-n)}

belongs

to

_{C’ (n) .}

(mod n)-t-((b

_(mod

we

easily verify

that

there exists a non-trivial

substitution g

such that x.

There are no other

_{possibilities}

and the truth of the claim is now clear for

the word _za,p. The

_proof

_{concerning z"}

is similar in all

_respects.

0

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Bruno PARVAIX

LACO, Faculte des _Sciences,

123 avenue Albert Thomas

F-87060 LIMOGES