Abelian borders and periodicity

(1)

ÉmilieCharlier

(JointworkwithTeroHarju,SvetlanaPuzyninaandLu aZamboni)

(2)

◮

Anite wordis bordered ifithas aproperprexequal to a

sux.

◮

For example,aababbaaab is bordered.

◮

Ifaword isnot bordered,we say itisunbordered or

border-free.

◮

For example,10 n

=

100

· · ·

0areunbordered.

◮

Aperiodi innite word: w

=

uv

ω

=

uvvv

· · ·

.

◮

Apurely periodi innite word: w

=

v

ω

=

vvv

· · ·

.

Theorem(Ehrenfeu ht-Silberger 1979)

An innite wordis purelyperiodi ieverysu ientlylong fa toris

(3)

◮

Finitewords u and v are abelian equivalentif

|

u

|

a

= |

v

|

a for alllettersa .

◮

Anite wordis abelianbordered ifithas a properprex

abelianequivalent toa sux.

◮

For example,abababbaabb is abelian bordered.

◮

Ifaword isnot abelianbordered,we say itisabelian

unbordered orabelian border-free.

◮

For example,abababbaabbb isabelianunbordered.

(4)

◮

An abelian periodi inniteword: w

=

uv 1 v 2 v 3

· · ·

where the v i

'sareallabelianequivalent.

◮

TheThue-Morse word

w

=

0110100110010110

· · ·

abelianperiodi with period2. (Itis the xedpointof the

morphism0

7→

01,1

7→

10.)

Finitely many abelian

unbordered fa tors

←→

?

Abelian

(5)

nitely many abelian unbordered fa tors.

Forexample, theThue-Morseword 0110100110010110

· · ·

is

◮

abelianperiodi with period2

◮

has innitelymany abelian unbordered fa tors: 0p1where p is

apalindrome. E.g. takep

= µ

2n

(

0

)

.

We even have:

Proposition

Ifa uniformlyre urrentaperiodi word ontains innitelymany

(6)

su ient ondition for periodi ity

Proposition

There exists aninnite aperiodi word w and onstants C

,

D su h that everyfa torv of w with

|

v

| ≥

C hasan abelian borderof length atmostD.

Forexample, any aperiodi innite word

w

∈ {

010100110011

,

0101001100110011

}

ω

(7)

Open question1

Let w be aninnite wordwith nitely manyabelianunbordered

fa tors. Doesitfollowthatw is abelian periodi ?

We do not know. But we are ableto answer the questionin aweak

(8)

◮

Thefrequen y

ρ

b

(

w

)

ofaletterb inanite wordw isdened as

ρ

b

(

w

) =

|

w

|

b

|

w

|

.

◮

Aweakly abelian periodi (WAP)inniteword:

w

=

uv 1 v 2 v 3

· · ·

where the v i

'shavethe same letter

frequen ies.

◮

w is alled bounded weakly abelianperiodi ,ifitisWAP with

bounded lengths ofblo ks:

∃

C

∀

i

|

v i

| ≤

C.

On ertainsequen esoflatti epoints[Gerver-Ramsey1979℄

Wordsthatdonot ontain onse utivefa torswithequalfrequen iesofletters

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Aim: w

=

a 1 a 2

· · · ∈ Σ

ω

7→

graph G w

⊆ Z

|Σ|

◮

Inthe binary ase:

0

→

movealongv 0

,

e.g. v 0

= (

1

,

−

1

)

1

→

movealongv 1

,

e.g. v 1

= (

1

,

1

)

Startatthe origin

(

x 0

,

y 0

) = (

0

,

0

)

. At stepn

>

0:

(

x n

,

y n

) = (

x n

−

1

,

y n

−

1

) +

v a n . Then

(

x n

−

1

,

y n

−

1

)

and

(

x n

,

y n

)

are onne ted by a line

segment.

◮

For ak-letter alphabet one an onsiderasimilargraph in

Z

k

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integer pointson alinewith arational slope.

Bounded WAP innite words orrespondtoa graphwith innitely

many integer pointsona linewith arational slopewith bounded

gaps.

Thegraph ofthe Thue-Morse word withv

0

= (

1

,

−

1

)

,v 1

= (

(11)

(12)

00100110001101100010011100110110

· · ·

It an be denedas aToeplitz wordwith pattern0

?

1

?

:

(13)

00100110001101100010011100110110

· · ·

?

1

?

:

(14)

00100110001101100010011100110110

· · ·

?

1

?

:

(15)

00100110001101100010011100110110

· · ·

?

1

?

:

(16)

00100110001101100010011100110110

· · ·

?

1

?

:

(17)

00100110001101100010011100110110

· · ·

The graphof the paperfoldingword with v

0

= (

1

,

−

1

)

, v 1

= (

1

,

1

)

.

It is WAPalong the line y

= −

1 (anda tually alongany line y

=

C,C

= −

1

,

−

2

, . . .

).

(18)

◮

Anite wordw isweaklyabelianbordered ifithas aproper

prex anda suxwith the sameletterfrequen ies.

◮

For example,abaabbaabb isweakly abelian bordered while

abaabbaabbb isn't.

Finitely many weakly

abelian unbordered fa tors

←→

? relation Weak abelian periodi ity

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Ifthe graph ofaword lies betweentwo lines,thenwesay the word

(20)

An innite wordw isK-balan ed ifforea h letteraand two

fa torsu and v of equal length, theinequality

||

u

|

a

− |

v

|

a

| ≤

K

holds. It isbalan ed ifitis K-balan ed forsomeK.

Lemma

(21)

unbordered fa tors implies WAP

Theorem

Let w be aninnite binary word. Ifthereexists a onstantC su h

that everyfa torv of w with

|

v

| ≥

C isweakly abelianbordered, then w isbounded WAP.

Moreover,its graphlies betweentwo rationallinesand has points

(22)

Open question2

Let w be abounded WAPinnite binaryword su hthat itsgraph

lies betweentwo rationallinesand has pointson ea h ofthese two

lineswith bounded gaps.

Doesw ne essarilyhaveonlynitely many weakly abelian

unbordered fa tors? True for

ρ

0

= ρ

1

=

1 2 .

(23)

Let w be aninnite binary word, and i,j beintegers,i

<

j. Ifg w

(

k

) >

g w

(

j

)−

g w

(

i

)

j

−

i k

+

g w

(

i

)

j

−

g w

(

j

)

i j

−

i forea h i

<

k

<

j, then the fa torw

[

i

+

1

..

j

]

is weakly abelianunbordered.

Weakly abelian unborderedfa tor11010.

(24)

Supposew have nitelymany weaklyabelianunbordered fa tors.

Then w is ofbounded width.

We derivea onsequen e inthe abeliansetting:

Proposition

Let w be aninnite wordhaving onlynitely manyabelian

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Lemma

Let whave nitelymany weaklyabelianunbordered fa tors. Then

the letterfrequen iesare rational.

Sket h of the proof

We know thatbalan e impliesthe existen eof letterfrequen ies.

What we havetoshow isthat theyare rational.

Sin e w isof bounded width,i.e.,there exist

α, β

′

_{, β}

′′

su h that

the graphof w lies between two lines:

α

x

+ β

′

_≤

g w

(

x

) ≤ α

x

+ β

′′

.

Here

α

is relatedtothe letterfrequen iesas follows:

ρ

0

=

1

−α

2 and

ρ

1

=

1

+α

2 .

(26)

nitelymanyunborderedfa tors pureperiodi ity Ehrenfeu ht-Silberger

nitelymanyabelianunborderedfa tors abelianperiodi ity

+

C ? Q.1

nitelymanyweaklyabelianunborderedfa tors weakabelianperiodi ity

+

C Maintheorem

? Q.2