Generalised substitutions and stepped surfaces

(1)

PierreArnoux

1

,ValérieBerthé

2

,andDamien Jamet

2

1

InstitutdeMathématiquedeLuminy,CNRSUMR6206,163avenuedeLuminy,

13288MarseilleCedex9-FRANCE.

arnouximl.univ-mrs.fr

2

LIRMM-UMR5506,UniversitéMontpellierII,161rueAda,34392Montpellier

Cedex5-FRANCE.

{berthe,jamet}lirmm.fr

Abstra t Asubstitutionisanon-erasingmorphismofthefreemonoid.

Thenotionofmultidimensionalsubstitutionofnon- onstantlength

a t-ingonmultidimensionalwordsintrodu edin[AI01,ABS04 ℄isprovedto

bewell-denedontheset oftwo-dimensionalwords related todis rete

approximationsof irrational planes. Su h amultidimensional

substitu-tion anbeasso iatedtoanyusualPisotunimodularsubstitution.The

aim of this paperis to try to extend the domainof denition of su h

multidimensional substitutions. Inparti ular, we study an exampleof

amultidimensionalsubstitution whi ha tsonasteppedsurfa einthe

senseof[Jam04 ,JP04℄.

1 Introdu tion

Sturmianwordsareknowntobe odingsofdigitizationsofanirrationalstraight

line[KR04,LOTH02℄.One ouldexpe tfromageneralizationofSturmianwords

that they orrespond to a digitization of ahyperplane with irrational normal

ve tor. It is thus natural to onsider the following digitization s heme

orre-sponding to thenotionof arithmeti planes introdu edin [Rev91℄: this notion

onsists in approximatingaplanein

R

3

by sele tingpointswith integral

oor-dinatesaboveandwithinaboundeddistan eoftheplane;morepre isely,given

v

∈ R

3

,

µ, ω ∈ R

,thelower(resp.upper) dis retehyperplane

P(v, µ, ω)

isthe

set of points

x

∈ Z

d

satisfying

0 ≤ hx, vi + µ < ω

(resp.

0 < hx, vi + µ ≤ ω

).

Moreover,if

ω =

P |v

i

| = |v|

1

,then

P(v, µ, ω)

issaidtobestandard. Inthislatter ase,oneapproximates aplanewith normalve tor

v

∈ R

3

by

square fa es orientedalong the three oordinatesplanes; for ea h of the three

kinds of fa es, one denes a distinguished vertex; the standard dis rete plane

P(v, µ, |v|

1

)

is then equalto theset of distinguishedverti es;after proje tion

ontheplane

x + y + z = 0

,along

(1, 1, 1)

,oneobtainsatilingoftheplanewith

threekindsofdiamonds,namelytheproje tionsofthethreepossiblefa es.One

an odethisproje tionover

Z

2

byasso iatingtoea hdiamondthenameofthe

(2)

Ageneralizationofthenotionofsteppedplane,theso- alleddis retesurfa es,

is introdu ed in [Jam04℄. Roughly speaking, a dis rete surfa e is a union of

pointed fa es su h that theorthogonal proje tionon the plane

x + y + z = 0

indu es an homeomorphism from the dis rete surfa e to the plane. As done

for stepped planes, one provides any dis rete surfa e with a oding as a

two-dimensional word over a three-letter alphabet. In the present paper, we all

dis rete surfa es stepped surfa es, sin e su h obje ts are not dis rete, in the

sense,thattheyarenotsubsetsof

Z

3

.

Letusre allthatasubstitutionisanon-erasingmorphismofthefreemonoid.

Anotionofmultidimensionalsubstitutionofnon- onstantlengtha tingon

mul-tidimensional words is studied in [AI01,AIS01,ABI02,ABS04℄, inspired by the

geometri alformalismof[IO93,IO94℄.Thesemultidimensionalsubstitutionsare

provedtobewell-denedonmultidimensionalSturmianwords.Su h a

multidi-mensionalsubstitution anbeasso iatedtoanyusualPisotunimodular

substi-tution. The aim of the presentpaperis to explore thedomain of denition of

su h generalizedsubstitutions. Forthe sakeof larity, wehave hosento work

outinfulldetailstheexampleof[ABS04℄.Weprovethattheimageofastepped

surfa e under the a tion of this multidimensional substitution is well-dened.

Our proofswill bebased ona geometri alapproa h.We then usethe

fun tio-nalityandtheproje tionontheplane

x + y + z = 0

along

(1, 1, 1)

tore overthe

orrespondingresultsformultidimensionalwords.

We work here in the three-dimensional spa e for larity issues but all the

resultsandmethodspresentedextendin anaturalwayto

R

n

.

2 Basi notions

2.1 One-dimensionalsubstitutions

Let

A

beanitealphabetandlet

A

⋆

bethesetofnitewordsover

A

.Theempty

wordisdenotedby

ε

.Asubstitution isanendomorphismofthefree-monoid

A

⋆

su hthattheimageofeveryletterof

A

isnon-empty.Su hadenitionnaturally

extendstoinniteorbiinnitewordsin

A

N

and

A

Z

.

Weassume

A = {1, . . . , d}

. Let

σ

bea substitution over

A

. The in iden e

matrix of

σ

,denoted

M

σ

= (m

i,j

)

(i,j)∈{1,...,d}

2

,isdened by:

M

σ

= (|σ(j)|

i

)

_{(i,j)∈{1,...,d}}

2

,

where

|σ(j)|

i

isthenumberof o urren esof

i

in

σ(j)

.

Let

ψ : A

⋆

_{→ N}

d

,

w 7→ (|w|

i

)

i∈{1,··· ,d}

betheParikhmapping,that is,the

homomorphismobtainedbyabelianizationofthefreemonoid.Onehasforevery

w ∈ A

⋆

,

ψ(σ(w)) = M

σ

ψ(w)

.

Example 1. Let

σ : {1, 2, 3} −→ {1, 2, 3}

⋆

bethesubstitutiondenedby

σ : 1 7→

13, 2 7→ 1, 3 7→ 2

.Then,

M

σ

=





1 1 0

0 0 1

1 0 0





.

(3)

Asubstitution

σ

is saidto beaPisot substitution ifthe hara teristi

poly-nomial of its in iden e matrix

M

σ

admits a dominant eigenvalue

λ > 1

su h that all its onjugates

α

satisfy

0 < |α| < 1

. The in iden e matrix of aPisot

substitutionisprimitive[CS01℄,thatis,itadmitsapowerwithpositiveentries.

Finally,asubstitutionissaidtobeunimodular if

det M

σ

= ±1

.

From now on, let

σ

denote a unimodular Pisot substitution over

the three-letteralphabet

A = {1, 2, 3}

.

2.2 Stepped planes

Thereareseveralwaystoapproximateplanesbyintegerpoints[BCK04℄.Usually,

thesemethods onsistinsele tingintegerpointswithinaboundeddistan efrom

the onsideredplane.Su hobje tsare alleddis reteplanes.

Let

{e

1

, e

2

, e

3

}

bethe anoni albasisof

R

3

.We allunit ube anytranslate

ofthefundamental unit ube withintegralverti es,thatis,anyset

x

+ C

where

x

∈ Z

3

and

C

isthefundamentalunit ube:

C =

λ

1

e

1

+ λ

3

e

3

+ λ

3

e

3

| (λ

1

, λ

2

, λ

3

) ∈ [0, 1]

3

.

Let

P : hv, xi + µ = 0

, with

v

∈ R

3 +

and

µ ∈ R

. The stepped plane

P

asso iatedto

P

,also alleddis reteplanein[ABS04℄, isdenedastheunionof

thefa es of theintegral ubesthat onne tthe set

{x ∈ Z

3

_{| 0 ≤ hv, xi + µ <}

kvk

1

=

P v

i

}

.Morepre isely:

Denition1. [IO93,IO94 ℄ We onsider the plane

P : hv, xi + µ = 0

, with

v

∈ R

3 +

and

µ ∈ R

. Let

C

P

be the unionof the unit ubesinterse tingthe open

half-spa eof equation

hv, xi + µ < 0

.The steppedplane

P

asso iatedto

P

is denedby:

P

= C

P

\

◦

C

P

,

where

C

P

(resp.

◦

C

P

)

isthe losure(resp.theinterior) of the set

C

P

in

R

3

, providedwith its usualtopology. The ve tor

v

(resp.

µ

) is

alled the normalve tor(resp. the translation parameter)of the stepped plane

P

.

Itis lear,by onstru tion,thatasteppedplaneis onne tedandisaunion

of fa es ofunit ubes. Infa t,by introdu inga suitabledenition of fa es, we

andes ribethesteppedplaneasapartitionofsu hfa es,asdetailedbelow.

Let

E

1

,

E

2

and

E

3

bethethreefollowingfundamental fa es (seeFigure1):

E

1

=

λe

2

+ µe

3

| (λ, µ) ∈ [0, 1[

2

,

E

2

=

−λe

1

+ µe

3

| (λ, µ) ∈ [0, 1[

2

,

E

3

=

−λe

1

− µe

2

| (λ, µ) ∈ [0, 1[

2

.

For

x

∈ Z

3

and

i ∈ {1, 2, 3}

,thefa e of type

i

pointed on

x

∈ Z

3

isthe set

x

+ E

i

.Letusnoti ethat ea hpointedfa ein ludes exa tlyoneintegerpoint, namely,itsdistinguishedvertex.Asmentionedaboveweobtain:

(4)

PSfragrepla ements

e

1

e

2

e

3

(a)

E

1

. PSfrag repla ements

e

1

e

2

e

3

(b)

E

2

. PSfragrepla ements

e

1

e

2

e

3

( )

E

3

.

Fig.1.Thethreefundamentalfa es.

Finally,aneasywayto hara terizethetypeofapointedfa ein ludedina

steppedplaneisgivenby:

Theorem2. Let

v

= (v

1

, v

2

, v

3

) ∈ R

3 +

and

µ ∈ R

. Let

P

= P(v, µ)

be the

steppedplanewithnormalve tor

v

andtranslationparameter

µ

.Let

I

1

= [0, v

1

[

,

I

2

= [v

1

, v

1

+ v

2

[

and

I

3

= [v

1

+ v

2

, v

1

+ v

2

+ v

3

[

.Then,

∀k ∈ {1, 2, 3}, ∀x ∈ P, x + E

k

⊂ P ⇐⇒ hx, vi + µ ∈ I

k

.

Let

P

0

bethediagonalplaneofequation

x+y +z = 0

andlet

π

betheproje tion on

P

0

along

(1, 1, 1)

.

Theorem3. [ABI02℄Let

P

beasteppedplane.Therestri tion

π

P

of

π

from

P

onto

P

0

isabije tion.Furthermore,thesetofpointsof

P

withinteger oordinates isin one-to-one orrespondan ewith the latti e

Z

π(e

1

) + Zπ(e

2

)

.

This theorem allows us to ode a stepped plane

P

asatwo-dimensional word

u ∈ {ψ, 2, 3}

Z

2

asfollows:forall

(m, n) ∈ Z

2

, for

i = 1, 2, 3

,then

u(m, n) = i ⇐⇒ π

−1

_P

(mπ(e

1

) + nπ(e

2

)) + E

i

⊂ P.

2.3 Stepped surfa es

It is thus naturalto tryto extendthepreviousdenitions and resultstomore

generalobje ts:

Denition2. [Jam04 ℄A onne tedunion

S

ofpointedfa es

x+ E

k

,where

x

∈

Z

3

and

i ∈ {1, 2, 3}

,is alleda steppedsurfa eif the restri tion

π

S

: S −→ P

0

of

π

isabije tion.

Atwo-dimensional word

u ∈ {1, 2, 3}

Z

2

issaid tobea oding ofthe stepped

surfa e

S

iffor all

(m, n) ∈ Z

2

,for

i = 1, 2, 3

,then

u(m, n) = i ⇐⇒ π

−1

_S

(mπ(e

1

) + nπ(e

2

)) + E

i

⊂ S.

(5)

pre-3

3

1

2

Fig.2.Apie eofasteppedsurfa eanditstwo-dimensional oding.

3 Generalized substitutions a ting on fa es of a stepped

plane

Theaimof thisse tionis tore allthenotionofgeneralizedsubstitution a ting

onfa esofasteppedplane[AI01,AIS01,Pyt02℄.

Let

σ

denote aunimodularPisot substitutionoverthethree-letteralphabet

A = {1, 2, 3}

.Let

M

σ

beitsin iden ematrix,andlet

α, λ

1

, λ

2

denoteits

eigen-valueswith

α > 1 > |λ

1

| ≥ |λ

2

| > 0

.Let

P

bethe ontra ting plane of

M

σ

,that is,therealplanegeneratedbytheeigenve torsasso iatedto

λ

1

, λ

2

.

Sin ethein iden ematrixofaPisot substitutionis primitive[CS01℄, then,

a ording to Perron-Frobenius Theorem, the eigenvalue

α

admits a positive

eigenve tor

v

. Let us denote by

P

σ

the stepped plane with normal ve tor

v

andtranslationparameter

µ = 0

.

Example 2. We ontinue Example 1. The hara teristi polynomial of

M

σ

is

x

3

_{− x}

2

_{− 1}

; it admits one eigenvalue

α > 1

(whi h is known as the se ond

smallest Pisot number), and two omplex onjugate eigenvalues of modulus

stri tly smaller than 1. The ontra ting plane of

M

σ

is the plane with equa-tion

α

2

_{x + αy + z0}

.

Denition3. [IO93,IO94 ,ABI02,ABS04 ℄ Let

σ

be a unimodular substitution

overthethree-letteralphabet

A = {1, 2, 3}

.Let

P

σ

bethesteppedplaneasso iated to

σ

.The generalizedsubstitution

Σ

σ

asso iated to

σ

isdenedasfollows:

Σ

σ

(x + E

i

) =

3

[

k=1

[

P

σ

(k)=P iS



M

−1

σ



x

− ψ(P ) −

i

X

j=1

e

j









+

k

X

j=1

e

j

+ E

k

Example 3. Let

σ : 1 7→ 13, 2 7→ 1, 3 7→ 2

.Then,

Σ

σ

:

x

+ E

1

7→ M

−1

σ

x

+ e

1

− e

2

+ E

1

∪ M

σ

−1

x

+ e

1

+ E

2

,

x

+ E

2

7→ M

σ

−1

x

+ e

1

+ E

3

,

(6)

In ombinatorialterms,

Σ

σ

anbe odedas

1 7→

2

1 2 7→ 3

3 7→ 1.

Let

r = r(m, n) = −⌈(α

2

_{m + αn)/(α}

2

_{+ α + 1)⌉ + 1}

.Onehas:

((m, n), 1) 7→ ((1 − n, m − n − r(m, n) − 1), 1) + ((1 − n, m − n − r(m, n)), 2)

((m, n), 2) 7→ ((1 − n, m − n − r(m, n)), 3)

((m, n), 3) 7→ ((1 − n, m − n − r(m, n)), 1).

Theorem4. [AI01 ℄ Let

σ

be a unimodular Pisot substitution over the

three-letteralphabet

A = {1, 2, 3}

,let

P

σ

bethe steppedplane asso iatedto

σ

andlet

Σ

σ

be thegeneralizedsubstitution asso iated to

σ

.

i) Twodistin t fa eshave disjoint imagesunder

Σ

σ

.

ii) Thegeneralizedsubstitution

Σ

σ

maps anypatternof

P

σ

(thatis,any nite unionof fa esof

P

σ

) onapatternof

P

σ

.

iii)

Σ

σ

(P

σ

) ⊆ P

σ

.

Sin e

Σ

σ

iswell-denedon

P

σ

(a ordingtoTheorem4i)),andsin e

P

σ

is invariantunder thea tionof

Σ

σ

,itisnaturaltoinvestigatethea tionof

Σ

σ

on anysteppedplane.Morepre isely,givenasteppedplane

P(v, µ)

, anweextend

the domain of denition of the generalizedsubstitution

Σ

σ

to the patterns of

P(v, µ)

?Infa t:

Theorem5. Let

σ

beaunimodular Pisot substitution,let

M

σ

beits in iden e matrix, andlet

Σ

σ

bethe generalizedsubstitutionasso iatedto

σ

.

For anysteppedplane

P(v, µ)

with

v

∈ R

3 +

,one has:

i) Theimages of twodistin t pointedfa esof

P(v, µ)

by

Σ

σ

aredisjoint. ii) Theimage of

P(v, µ)

isin ludedin thesteppedplane

P(

t

_{M · v, µ)}

:

Σ

σ

(P(v, µ)) ⊆ P(

t

M · v, µ)

Proof (Sket h).TheproofisbasedonthesameideasasintheproofofLemma2

and3in[AI01℄.Itmainlyusesthefollowinggeometri interpretationofTheorem

2:apointedfa e

x

+ E

i

isin ludedin

P(v, µ)

ifandonlythepoint

x

+

P

i−1

k=1

e

k

isabovetheplane

hv, xi + µ = 0

whilethepoint

x+

P

i

k=1

e

k

isbelowthelatter.

4 Generalized substitutions a ting on fa es of a stepped

surfa e

4.1 The general ase

Sin etheimageofasteppedplanebyageneralizedsubstitutionisasubsetofa

stepped plane,itisinterestingto investigatethea tionofgeneralized

(7)

Theorem6. Let

S

beasteppedsurfa e.Let

σ

beaunimodularPisot

substitu-tionoverthe three-letteralphabet

{1, 2, 3}

andlet

Σ

σ

be the asso iated general-izedsubstitution.Then,theimageoftwodistin tpointedfa esof

S

aredisjoint.

Furthermore, the restri tion

π

Σ

σ

(S)

: Σ

σ

(S) −→ P

0

is1-1.

Proof (Sket h).Werstnoti ethatgiventwofa es

x+E

i

and

y+E

j

,thenthere existsasteppedplane

P

withpositivenormalve tor ontainingsimultaneously

x

+ E

i

,

y

+ E

j

and

z

+ E

k

.WethenapplyTheorem5.

Inotherwords,itremainstoprovethat

π

Σ

σ

(S)

isontoandthat

Σ

σ

(S)

isa onne ted unionof fa es to dedu e that

Σ

σ

(S)

is asteppedsurfa e a ording to Denition 2. Let us investigate this problem in the parti ular ase of the

generalized substitution

Σ

σ

asso iated to the substitution

σ : 1 7→ 13, 2 7→

1, 3 7→ 2

.

4.2 The parti ular aseof

σ

: 1 7→ 13, 2 7→ 1, 3 7→ 2

.

In the present se tion,

σ

denotes the substitution

σ : 1 7→ 13, 2 7→ 1, 3 7→ 2

whereas

Σ

σ

isthegeneralizedsubstititionasso iatedto

σ

:

Σ

σ

:

x

+ E

1

7→ M

−1

σ

x

+ e

1

− e

2

+ E

1

∪ M

σ

−1

x

+ e

1

+ E

2

,

x

+ E

2

7→ M

σ

−1

x

+ e

1

+ E

3

,

x

+ E

3

7→ M

σ

−1

x

− e

2

− e

3

+ E

1

.

Letus show that forthis substitution, then theimage ofa stepped surfa e

isstillasteppedsurfa e.First,givenatwo-dimensionalword

u ∈ {1, 2, 3}

Z

2

,we

allhook-word afa torof

u

withthefollowingshape(seeFig.3):

PSfragrepla ements

m

n

Fig.3.Hook-shape.

Theset of hook-wordsof

u

with a hook-shape is alled the hook-language

of

u

.In[Jam04,JP04℄, theauthorsredu edthere ognitionproblem ofthe

two-dimensionalwords odingdis retesurfa estoahookre ognitionproblem.More

pre isely,

Theorem7. [Jam04,JP04℄Let

u ∈ {1, 2, 3}

Z

2

.Then

u

isa odingof adis rete

(8)

1

2

3

2

1

2

1

2

1

2

3

1

3

2

3

1

2

3

1

2

3

1

2

1

3 n

m

x

y

Fig.4. Left: Thepermitted hook-words. Right: The 3-dimensionalrepresentation of

thepermittedhook-words.

We onverselyasso iatetoea hpermittedhook-wordits3-dimensional

rep-resentationas a onne tedunionoffa esasdepi tedin Figure4:the odingof

anyo urren eofthis3-dimensionalrepresentationinasteppedsurfa eisequal

tothe orrespondinghook-word.

Proposition1. The image by

Σ

σ

of all the 3-dimensional representations of the permittedhooks(seeFig.5)are onne tedin

R

3

.

(a) (b) ( ) (d) (e) (f)

(g) (h) (i) (j) (k) (l)

Fig.5.Theimageofthepermittedhooksby

Σ

σ

.

Wethendedu ethat:

Theorem8. The image of a stepped surfa e

S

by

Σ

σ

is onne ted and the restri tionoftheproje tion map

π

tothelatterisinje tive.Furthermore, allthe

hook-wordso urringinthe odingwithrespe ttotheinje tiveproje tion

π

Σ

(9)

Proof (Sket h). A ording to Theorem 6, the image of a stepped surfa e by

Σ

σ

iswell-dened.The onne tednessfollowsfromProposition1.Considernow a union

H

of three fa es whose oding a ording to the inje tive proje tion

π

Σ

σ

(S)

(seeTheorem6)isahook-word

U

H

.Thereexist(atmost)threefa esof whi htheunionoftheimagesby

Σ

σ

ontains

H

. One he ksthat thedistan e (denedas

d(v, w) = |w − v|

1

)betweenthedistinguishedverti esofthosefa es is uniformly bounded. By performing a nite ase study, one he ks that the

hook-word

U

H

ispermitted.

(a)

S

(b)

Σ

σ

(S)

.

( )

Σ

2 σ

(S)

.

Fig.6.Apie eofanon-planarsteppedsurfa e

S

and2iterationsby

Σ

σ

.

Remarks. Given a stepped surfa e

S

ontaining the unit ube

(10)

surfa es

(Σ

n

σ

(S))

n∈N

seemsto onvergetowardsthesteppedplane

P

σ

(seeFig. 6);to bemorepre ise, thelimitpointsofthesequen e

(Σ

n

σ

(S))

n∈N

aresubsets of

P

σ

. We will investigate these onvergen e results and more generally, the possibility of extension of the domain of denition of these multidimensional

substitutions to any stepped surfa e in a subsequent paper. Let us note that

this study an also be applied to obtain an e ient generation methods of

steppedplanesandsurfa es.

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