Chaos and Turing Machines on Bidimensional Models at Zero Temperature

(1)

HAL Id: tel-03213208

https://tel.archives-ouvertes.fr/tel-03213208

Submitted on 30 Apr 2021

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

abroad, or from public or private research centers.

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

publics ou privés.

Zero Temperature

Gregorio Dalle Vedove Nosaki

To cite this version:

Gregorio Dalle Vedove Nosaki. Chaos and Turing Machines on Bidimensional Models at Zero

Temper-ature. Mathematical Physics [math-ph]. Université de Bordeaux; Universidade de São Paulo (Brésil),

2020. English. �NNT : 2020BORD0309�. �tel-03213208�

(2)

IME-USP

THÈSE EN COTUTELLE PRÉSENTÉE

POUR OBTENIR LA GRADE DE

DOCTEUR DE

L'UNIVERSITÉ DE BORDEAUX

ET DE L'UNIVERSITÉ DE SO PAULO

ÉCOLE DOCTORALE MATHÉMATIQUES ET INFORMATIQUE

INSTITUTO DE MATEMÁTICA E ESTATÍSTICA

SPÉCIALITÉ: Mathématiques Appliquéeset Cal ulS ientique

Par Gregório DALLE VEDOVE NOSAKI

Chaos and Turing Ma hines on Bidimensional

Models at Zero Temperature

Sous la dire tionde PhilippeTHIEULLEN

et de Rodrigo BISSACOT

Soutenu le15 dé embre de 2020

Membres du jury :

M.EduardoGARIBALDI Professeurasso ié UniversidadedeCampinas Présidentdujury M.SamuelPETITE Maîtrede onféren es UniversitédePi ardieJulesVerne Examinateur M.Mathieu SABLIK Professeur UniversitédeToulouseIIIPaulSabatier Rapporteur M.Aernout VANENTER Professeurémérite UniversityofGroningen Rapporteur M.Pierre PICCO Dire teurdere her he Institut deMath. deMarseille Examinateur M.PhilippeTHIEULLEN Professeur UniversitédeBodeaux CoDire teur M.RodrigoBISSACOT Professeurasso ié UniversidadedeSãoPaulo CoDire teur Mme. NathalieAUBRUN Chagédere her he UniversitéParis-Sa lay Examinatri e M.ArturLOPES Professeurasso ié UniversidadeFederaldoRioGrandedoSul Invité

(3)

Titre: Ma hine de Turing et Chaos pour des Modèles Bidimensionnels à Température

Zéro

Résumé: En mé anique statistique d'équilibre ouformalisme thermodynamique un des

obje tifsestdedé rirele omportementdesfamillesdemesuresd'équilibrepourun

poten-tielparamétré par latempératureinverse

β

. Nous onsidérons i iune mesure d'équilibre omme une mesure shiftinvariantequi maximise lapression. Il existe d'autres

onstru -tions qui prouvent le omportement haotique de es mesures lorsque le système se ge,

'est-à-dire lorsque

β

Ñ 8. Un des exemples lesplus importantsa été donné par Cha-zottes et Ho hman [11℄ oùils prouvent la non- onvergen e des mesures d'équilibre pour

un potentiello alement onstantlorsqueladimension est supérieureà3. Dans e travail,

nous présentons une onstru tionetun exemplepotentiello alement onstant telqu'il

e-xisteunesuitep

β

k

q

k

¥

0

oùlanon- onvergen eestassuréepourtoute hoixsuitedemesures

d'équilibre à l'inverse de la température

β

k

lorsque

β

k

Ñ 8. Pour ela nous utilisons la onstru tion dé ritepar Aubrunet Sablik[2℄quiaméliorelerésultat de Ho hman[19℄

utilisé dans la onstru tion de Chazottes et Ho hman[11℄.

Mots lés: formalismethermodynamique,measure d'équilibre,dé alage.

Title: Chaos and Turing Ma hine onBidimensionalModels atZero Temperature

Abstra t: In equilibriumstatisti alme hani s orthermodynami s formalismone of the

mainobje tivesistodes ribethebehavioroffamiliesofequilibriummeasuresfora

poten-tial parametrized by the inverse temperature

β

. Here we onsider equilibrium measures as the shift invariant measures that maximizes the pressure. Other onstru tions

al-readyprovethe haoti behaviorofthesemeasureswhenthesystemfreezes, thatis,when

β

Ñ 8. Oneofthe mostimportantexampleswasgivenbyChazottesandHo hman[11℄

where they prove the non- onvergen e of the equilibriummeasures for a lo ally onstant

potentialwhenthe dimensionis biggerthanorequalto3. In this workwepresent a

on-stru tionofabidimensionalexampledes ribedbyanitealphabetandalo ally onstant

potential in whi h there exists a subsequen e p

β

k

q

k

¥

0

where the non- onvergen e o urs

for any sequen e of equilibrium measures at inverse temperatures

β

k

when

β

k

Ñ 8. In order to des ribe su h an example, we use the onstru tion des ribed by Aubrun and

Sablik[2℄whi himprovestheresultofHo hman[19℄usedinthe onstru tionofChazottes

and Ho hman [11℄.

(4)

Título: Caos e Máquinas de Turing emModelos Bidimensionaisà TemperaturaZero

Resumo: Em me âni a estatísti a de equilíbrio ou formalismo termodinâmi o um dos

prin ipais objetivos é des rever o omportamento das famílias de medidas de equilíbrio

paraum dado poten ial parametrizadopeloinverso datemperatura

β

. Entendemos aqui pormedidasdeequilíbrioasmedidasshiftinvariantesquemazimizamapressão. Diversas

onstruções já demonstraram um omportamento aóti o destas medidas quando o

sis-tema ongela,ouseja,

β

Ñ 8. Umdosprin ipaisexemploséo onstruídoporChazottes eHo hman [11℄ onde eles onseguem provar anão onvergên ia de umafamília de

medi-dasde equilíbrioparaumdadopotentiallo almente onstantenos asosonde adimensão

é maior ou igual a 3. Neste trabalho apresentaremos a onstrução de um exemplo no

aso bidimensionalsobre um alfabeto nito eum poten ial lo almente onstante talque

existe uma sequen iap

β

k

q

k

¥

0

onde não o orre a onvergên ia para qualquer sequên iade

medidasdeequilíbrioaoinversodatemperatura

β

k

quando

β

k

Ñ 8. Paratal,usaremos a onstrução des rita por Aubrun e Sablik em [2℄ que melhora o resultado de Ho hman

[19℄ usado na onstrução de Chazottes e Ho hman[11℄.

Palavras- have: formalismo termodinâmi o,medidade equilíbrio,subshift.

This study was finan ed in part by the

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior

(5)

(6)

L'undesproblèmeslesplusimportantsdanslamé aniquestatistiqueàl'équilibre onsiste

à dé rire des famillesde mesures de Gibbs pour un potentiel donné ou pour une famille

d'intera tions. Nous travaillons ave des systèmes lassiques, e qui signie que notre

espa ede ongurationsera

Σ

d

p

A

q

:

A

Z

d

où

A

est un alphabet ni et

d

P

N

est la dimension du réseau. Nous introduisons la fon tion

ϕ : Σ

d

p

A

qÑ

R

qu'il s'appelle potentiel par site et peut être physiquement interprétée omme la

ontri-butionénergétique de l'originedu réseau pour haque onguration

x

P

Σ

d

p

A

q.

À partir de es éléments, nous désignons pour haque

β

¡

0

l'ensemble

G

p

βϕ

q qui est l'ensembledes mesures de Gibbs asso iées à

βϕ

à la températureinverse

β

. Il existe plusieursdénitionsquenouspouvons onsidérer ommeunemesuredeGibbs,enutilisant

des mesures onformes, des équations DLR, des limites thermodynamiques, et . Voir

Georgii [17℄, le livre lassique sur les mesures de Gibbs et [25℄ pour les équivalen es de

plusieurs des es dénitions. Par ompa ité, nous savons que et ensemble a au moins

unemesurede Gibbsinvariantepourtranslation. Dans ettethèse,nous nousintéressons

au omportementdel'ensembledesmesures de Gibbsquisontdesmesures de probabilité

invariantes, appelées mesures d'équilibre, lorsque la température tend vers zéro,

'est-à-dire lorsque

β

Ñ 8.

Unemesuredeprobabilité

µ

β

sur

Σ

d

p

A

qestunemesured'équilibre(ouétatd'équilibre) à la température inverse

β

¡

0

pour un potentiel

βϕ

si 'est une mesure invariantepar dé alage (ou mesure invariantepar translation) qui maximisela pression, 'est-à-dire si

P

p

βϕ

q

:

sup

µ

P

M

σ

p

Σ

d

p

A

qq "

h

p

µ

q »

βϕdµ

*

h

p

µ

β

q »

βϕdµ

β

.

Nous onsidérerons par la suite l'ensemble uniquement es mesures d'équilibre

µ

β

, elles qui maximisent la pression

P

p

βϕ

q i-dessus sur toutes les mesures de probabilité invariantes pour translation dénies sur

Σ

d

p

A

q. La fon tion

h

p

nu

q dans l'expression de

(7)

Dansle asunidimensionnel,siun potentiel

ϕ

est Hölder ontinu,nousavonstoujours unemesuredeGibbsunique quiest aussilaseulemesured'équilibre. Pour unedimension

d

¡

1

la situation est radi alement diérente et nous pouvons avoir plusieurs mesures

de Gibbs même pour un potentielà ourte portée, l'exemplele plus onnu est le modèle

d'Ising.

Les états d'équilibreà températurezéro (les états fondamentaux)sont lesmesures de

probabilité invariantes qui minimisent

»

ϕdν

sur toutes les mesures de probabilité invariantes

ν

. En autres termes, étant donné un potentiel, nous avons que les points d'a umulation pour la topologie faible* des états

d'équilibrequand

β

Ñ 8sontné essairementlesmesuresminimisantespourlepotentiel

ϕ

. Uneétudeplus détailléesur leslimitespossibles lorsquelesystème sege et omment

ellesont liées aux ongurations ave une énergieminimalepeut être trouvée dans [36℄.

Chazottes et Ho hman [11℄ ont montré dans le as unidimensionnel un exemple de

potentielLips hitz

ϕ

(mais àlongueportée)oùlasuite

µ

βϕ

ne onverge pas lorsque

β

Ñ 8. I i,

µ

βϕ

est l'uniquemesurede Gibbs invariantepar translation (oul'unique mesure de Gibbs)àlatempératureinverse

β

¡

0

(quiest égalementl'uniquemesured'équilibre). En revan he, [8, 10, 16, 27℄ ont montré qu'une intera tion de ourte portée dans le as

unidimensionnel sur un alphabet ni implique la onvergen e de

µ

βϕ

. Le as où

A

est un ensemble dénombrable a également été étudié dans [23℄. La onstru tion d'exemples

de non- onvergen e a été donnée par van Enter et W. Ruszel [37℄, où un exemple de

potentiel de ourte portée sur un espa e d'états ontinu et un omportement haotique

ontété onstruits. Ré emment,l'argumentde vanEnter etRuszelaétéimplémentépour

le as où

A

est un ensemble ni dans [7, 3, 12℄.

ChazottesetHo hman[11℄ontégalementmontréquelemêmetypedenon- onvergen e

peut être observé lorsque ladimension est

d

¥

3

mêmepour un potentiello alement on-stant(à ourteportée). La onstru tionde leurexemplen'estpossiblequepour

d

¥

3

ar ilss'appuientfortementsurlathéoriedessous-shiftsmultidimensionnelsdetypenietdes

Ma hines de Turing, développée par Ho hman [19℄ qui fournit une méthode pour

trans-férerune onstru tionunidimensionnelleà un sous-shifts de type ni, mais de dimension

supérieure. Grâ e au théorème de Ho hman, Chazottes et Ho hman ont pu onstruire

un exemple pour

d

3

ave un potentiel

ϕ

lo alement onstant sur un espa e d'états ni. Leur onstru tionpeut être fa ilementétendue à n'importe quelle dimension

d

¥

3

. Ces résultats nous amènent à roire que l'énon é est également vrai pour

d

2

. Notre résultat prin ipal est double: nous étendons le théorème du omportement haotique de

(8)

larionslerle de la re onstru tion etla omplexitérelative fon tion de l'extensionpar

un sous-shift de type ni qui manquedans les argumentsde Chazottes-Ho hman.

Le résultat prin ipal d'Aubrun et Sablik [2℄, appelé théorème de simulation, arme

quetoutsous-shift

d

-dimensionneldéniparunensembledemotifsinterditsénuméréspar unema hinedeTuringestunesous-a tiond'unsous-shiftdetypenip

d 1

q-dimensionnel. Il existe d'autres travaux dans lesquels les résultats de simulationobtenus jusqu'i i dans

ette théorie ont été améliorés [14, 15℄. Dans es travaux les auteurs améliorent les

résultatsobtenusjusqu'àprésenten diminuantladimension dusous-shift de typeni qui

génère le sous-shift ee tif ; ependant les preuves sont basées sur le théorème du point

xede Kleeneet n'utilisentpas d'arguments géométriques.

La onstru tion d'Aubrun etSablik[2℄améliorela méthode de Ho hman [19℄ en

aug-mentant uniquement de 1 la dimension du SFT, en parti ulier, elle permet d'obtenir la

onstru tionde Chazottes etHo hman [11℄ en dimension 2.

Dans le deuxième hapitre, nous présentons les prin ipales dénitions du formalisme

thermodynamique, lesrésultats lassiques et lesnotations standards. Nous ommençons

par ladénition des sous-shift etdénissons une lasse spé ialede sous-shiftbasée sur la

on aténationde blo sde mêmetaillean de former haque ongurationpossible. Dans

ladeuxièmese tionde e hapitre,nous présentons unebrèverevue de l'entropietraitant

des partitions, de l'entropie d'une partition,de l'entropie métrique et topologique etdes

on epts de pression, de mesure d'équilibre et de mesure de Gibbs. Dans la troisième

se tion nous donnons une idée générale des opérations transformant un sous-shift en un

autre basé sur [1℄ an d'appréhender la notion de simulation d'un sous-dé alage par un

autre. Enn, nous présentons une dénition formelle d'unema hine de Turing, omment

représenter letravail d'unema hine de Turing dans un diagramme espa e-temps et aussi

une idée de la onstru tiond'Aubrun etde Sablik[2℄.

Letroisième hapitre est dédiéàla onstru tionde notreexemple en s'inspirantde la

onstru tionprésentée dans lestravaux de Chazottes et Ho hman [11℄. Nous dénissons

d'abord un sous-shiftunidimensionnelbasé sur un pro essus d'itérationquinous donneà

haque étape des blo sde même longueur qui sont on aténés pour former un sous-shift

telquedéniaudeuxième hapitre. Nousmontronsquele ontrlequenousavons obtenu

sur l'ensemble des mots interdits de e sous-shift, implique qu'il existe une ma hine de

Turing qui liste tous les mots interdits, 'est-à-dire que notre sous-shift est un sous-shift

ee tivement fermé. De là, nous pouvons utiliser le théorème de simulation

d'Aubrun-Sablik[2℄ etobtenir un sous-shift bidimensionnelde type ni qui simule notre sous-shift

ee tivement fermé unidimensionnel pré édent. Toujours dans la deuxièmese tion de e

hapitre,nousprouvonsquelquesrésultatsimportantsquiexpliquent omment

dé onstru-ireune ongurationdanslesous-shiftbidimensionnelentantquemotifs on aténésdans

(9)

dénis-Ho hman[11℄, qui onsisteàdupliquerun symboledistin tif,an de transférerl'entropie

du sous-shift initialvers le sous-shift de typeni obtenu par lethéorème de simulation.

Après toutes es onstru tions, onseretrouveave un sous-shiftde typeni

bidimen-sionnel

X

déni sur un alphabet ni

A

, un entier

D

¥

1

et un ensemble ni de motifs interdits

F

A

J1,DK

2

. On dénitensuite lepotentiello alement onstantpar site suivant

ϕ : A

Z

2 Σ

2

p

A

q Ñ

R

x

ÞÑ

ϕ

p

x

q

1 F

p

x

q

où

F

est l'ensemble lopen égal à l'uniondes ylindres générés par haque motifdans

F

. Ledernier hapitreestdédiéàladémonstrationdurésultatprin ipalquiestlesuivant.

Theorem 1. Il existe un potentiel lo alement onstant

ϕ : Σ

2

p

A

q Ñ

R

, il existe une sous-suite p

β

k

q

k

¥

0

qui tendvers l'innietdeux ensembles ompa ts etinvariantsquisont

disjoints et non vides

X

A

, X

B

de

Σ

2

p

A

q, tels que si

µ

β

k

est une mesure d'équilibre la

températureinverse

β

k

asso iée au potentiel

β

k

ϕ

, le support de n'importe quelle mesure d'a umulation pour la topologie faible* de la suite p

µ

β

2 k

q

k

¥

0

est in lus dans

X

B

, et le supportden'importequellemesured'a umulationpourlatopologiefaible*dep

µ

β

2 k

1

q

k

¥

0

est in lusdans

X

A

.

Le théorème pré édent arme qu'ilexiste une sous-suite p

β

k

q

k

P

N

ave

β

k

Ñ 8 telle que tout hoixde mesure d'équilibreasso iéaupotentiel

β

k

ϕ

alterneentre deux mesures de probabilité supportées par des ensembles ompa ts et disjoints. C'est-à-dire qu'il

ex-iste un potentiel lo alement onstant par site qui présente une onvergen e haotique à

températurezéro.

Nous al ulons en annexe une borne supérieure de la omplexité relative et de la

fon tion de re onstru tion du sous-shift de type ni donnée dans [2℄; nous remer ions

(10)

1 Introdu tion 11

2 Subshifts 15

2.1 Forbidden words . . . 15

2.2 Entropy and variationalprin iple . . . 18

2.3 Potential . . . 25

2.4 Turing Ma hines and the SimulationTheorem . . . 27

2.5 The Aubrun-Sablik simulation theorem . . . 35

3 Main Constru tion 39 3.1 One-dimensional ee tively losed subshift . . . 39

3.2 BidimensionalSFT . . . 48

3.3 The new oloring . . . 55

4 Analysis of the zero-temperature limit 61

(11)

(12)

Introdu tion

One of the most importantproblems inequilibriumstatisti alme hani s onsists in

des- ribing families of Gibbs states for a given potential or an intera tion family. We work

with lassi allatti e systems, whi h means that our ongurationspa e willbe

Σ

d

p

A

q

:

A

Z

d

where

A

is a nite set and

d

P

N

is the dimension of our latti e. Let us introdu e the fun tion

ϕ : Σ

d

p

A

qÑ

R

whi h is alled per site potential and an be physi ally interpreted as the energy

on-tributionof the origin of the latti e for ea h onguration

x

P

Σ

d

p

A

q, sin e we are only onsideringonly translation invariantmeasures.

Given theseelementswedenoteforevery

β

¡

0

theset

G

p

βϕ

qwhi histheset ofGibbs measures asso iated to

βϕ

at the inverse temperature

β

. The are several denitions we ould onsider as a Gibbs measure, using onformal measures, DLR equations,

thermo-dynami limits et . See Georgii [17℄, the lassi al book about Gibbs measures and [25℄

forthe equivalen e of several of these denitions. By ompa tness we know that this set

has at least one shift translation invariant Gibbs measure. In the present thesis we are

interested on the behaviorof the set of Gibbs measures whi hare translational-invariant

probability measures, alled equilibrium measures, when the temperature goes to zero,

that is,when

β

Ñ 8.

A probabilitymeasure

µ

β

over

Σ

d

p

A

q isanequilibrium measure(orequilibrium state) at inverse temperature

β

¡

0

for a potential

βϕ

if it is a shift invariant (or translation invariant)measure whi h maximizesthe pressure, that isif

P

p

βϕ

q

:

sup

µ

P

M

σ

p

Σ

d

p

A

qq "

h

p

µ

q »

βϕdµ

*

h

p

µ

β

q »

βϕdµ

β

.

(13)

We will onsider later the whole set of equilibrium measures

µ

β

whi h maximize the pressure

P

p

βϕ

qaboveoverallshiftinvariantprobabilitymeasureson

Σ

d

p

A

q. Thefun tion

h

p

ν

qin the expression of

P

p

βϕ

qis the Kolmogorov-Sinaientropy of

ν

.

In the one-dimensional ase if a potential

ϕ

is Hölder ontinuous we always have a uniqueGibbs measure whi his alsothe onlyequilibriummeasure. Foradimension

d

¡

1

the situation is dramati ally dierent and we an have multiple Gibbs states even for a

potentialwith nite range, the most famous example isthe Isingmodel.

The zero-temperature equilibrium states (ground states) are the shift invariant

prob-ability measures whi h minimize

»

ϕdν

overallshift-invariantprobabilitymeasures

ν

. In otherwords, given apotential,we have that the weak* a umulation points of equilibrium states as

β

Ñ 8 are ne essarily minimizing measures for the potential

ϕ

. A more detailed study on the limit when the system freezes and how it is related with the ongurations with minimal energy an be

found in [36℄.

Chazottes and Ho hman [11℄ showed in the one-dimensional ase an example of a

Lips hitz potential

ϕ

(but long-range) where the sequen e

µ

βϕ

does not onverge when

β

Ñ 8. Here

µ

βϕ

is the unique shift-invariant Gibbs measure (or the unique Gibbs

measure)attheinversetemperature

β

¡

0

(whi hisalsotheuniqueequilibriummeasure). On the other hand, [8, 10, 16, 27℄ showed that an intera tion of nite-range in the

one-dimensional ase over a nite alphabet implies the onvergen e of

µ

βϕ

. The ase when

A

is a ountable set was also studied in [23℄. The breakthrough for the onstru tion of

examples of the non- onvergen e was given by van Enter and W. Ruszel [37℄, where an

example of nite range potential on a ontinuous state spa e and haoti behavior was

onstru ted. Re ently the argument of van Enter and Ruszel was implemented for the

ase where

A

is anite set in[7, 3,12℄.

Chazottes and Ho hman[11℄ alsoshowed thatthe same kindofnon- onvergen e may

o urwhenthe dimensionis

d

¥

3

evenforalo ally onstantpotential. The onstru tion of their example is possible only for

d

¥

3

be ause they rely heavily on the theory of multidimensionalsubshiftsofnitetypeandTuringMa hines,developedbyHo hman[19℄

thatprovidesamethodtotransferaone-dimensional onstru tiontoahigher-dimensional

subshift of nite type. Thanks to Ho hman's theorem, Chazottes and Ho hman ould

onstru t anexamplefor

d

3

with apotential

ϕ

lo ally onstantonanitestatespa e. Their onstru tion an be easily extended to any dimension

d

¥

3

. These results led us to believe that the statement is also true for

d

2

. Our main result is two-fold: we extendChazottes-Ho hman'stheoremof haoti behaviortodimension2usingadierent

(14)

Aubrun-extensionby a subshiftof nite typethat is missinginChazottes-Ho hman'sarguments.

The main result of Aubrun and Sablik [2℄, alled simulation theorem, asserts that

any

d

-dimensional subshift dened by a set of forbidden patterns that is enumerated by a Turing ma hine is a suba tion of a p

d

1

q-dimensional subshift of nite type. There are other works in whi h the simulation results obtained so far in this theory have been

improved [14, 15℄. In these works they improve the results obtained sofar by de reasing

thedimensionofthe subshiftofnitetypewhi hgeneratesthe ee tivesubshift,butthey

are based onKleene's xed point theorem and they donot uses geometri arguments.

The onstru tion of Aubrun and Sablik [2℄ improves the method of Ho hman [19℄,

be ausethey in rease the dimensionby1 andthis leads ustoimprovethe Chazottes and

Ho hman [11℄ onstru tion for the dimension 2.

In the se ond hapter we present the main denitions of thermodynami formalism

and omputability, lassi al resultsand standard notations. Webeginwith the denition

ofsubshifts anddene aspe ial lassof subshiftsbased onthe on atenationof blo ks of

the same size in order to form ea h possible onguration. In the se ond se tion of this

hapterweprovideabriefreviewofentropydealingwithpartitions,entropyofapartition,

metri and topologi al entropy and the on epts of pressure, equilibrium measure and

Gibbs measure. In the third se tion we give a general idea of operations transforming a

subshift intoanother one based on [1℄inorder to omprehendthe notion of simulatinga

subshiftby another one. Finally,we presentaformaldenitionofaTuringma hine,how

to represent the work of a Turing ma hine in a spa e-time diagram and also an idea of

the onstru tionof Aubrun and Sablik [2℄.

Thethird hapterisdedi atedtodeneand onstru t ourexamplethat isinspiredby

the onstru tion presented inthe workof Chazottesand Ho hman[11℄. Firstwe denea

one-dimensionalsubshiftbasedonaniterationpro ess thatgivesusatea hstepblo ksof

thesamelengththatare on atenatedtoformasubshiftasdenedinChapter2. Weprove

thatthe ontrolwehaveobtained overthe setof forbiddenwords ofthis subshift,implies

there exists a Turing ma hine that lists all of the forbidden words, that is, our subshift

isanee tively losed subshift. From there weare able to use the simulation theorem of

Aubrun-Sablik [2℄ and obtain a bidimensional subshift of nite type that simulates our

previous one-dimensional ee tively losed subshift. Also in the se ond se tion of this

hapter,weprovesomeimportantresultsthatexplainhowtode onstru ta onguration

inthe

2

-dimensionalsubshift as on atenated patterns inagiven di tionary. In the third and lastpart of this hapter, we dene a new oloring for the bidimensional subshift, as

in Chazottes and Ho hman [11℄, that onsists in dupli ating a distinguished symbol, in

order to transfer the entropy of the initialee tive subshift to the simulated subshift of

nitetype.

After all these onstru tions, we end up with a bidimensional SFT

X

dened over a nite alphabet

A

, an integer

D

¥

1

and a nite set of forbidden patterns

F

A

J1,DK

2

(15)

Wethen dene the followinglo ally onstant persite potential

ϕ : A

Z

2 Σ

2

p

A

q Ñ

R

x

ÞÑ

ϕ

p

x

q

1 F

p

x

q

where

F

is the lopen set equal to the union of ylinders generated by every pattern in

F

.

The last hapter isdedi ated to prove the main result whi his the following.

Theorem 2. There exists a lo ally onstant potential

ϕ : Σ

2

p

A

q Ñ

R

, there exists a subsequen e p

β

k

q

k

¥

0

goingto innity and two disjoint non-empty ompa t invariant sets

X

A

, X

B

of

Σ

2

p

A

q, su h that if

µ

β

k

is an equilibrium measure at inverse temperature

β

k

asso iatedtothe potential

β

k

ϕ

,thesupport ofanyweak

a umulationpointofp

µ

β

2k

q

k

¥

0

isin luded in

X

B

,the supportof anyweak

a umulationpointof p

µ

β

2k

1

q

k

¥

0

isin luded in

X

A

.

The previous theorem asserts that there exists asubsequen e p

β

k

q

k

P

N

with

β

k

Ñ 8 su hthat any hoi eof equilibriummeasure asso iated withthe potential

β

k

ϕ

alternates between two disjoint ompa t sets ofprobabilitymeasures. That isthere exists a lo ally

onstant persite potentialthat exhibits azero-temperature haoti onvergen e.

We ompute in the appendix an upper bound of the relative omplexity and

re on-stru tion fun tions of the SFT given in [2℄; we thank S.B. for many dis ussions on this

(16)

Subshifts

2.1 Forbidden words

Inthis hapterweestablishthebasi denitions,notationsandmainresultsoftheobje ts

that we use in this work. We begin by two denitions of asubshift: one topologi al and

one ombinatorial. These twodenitions oin ide.

Wewillalways workwith aniteset oflettersthatwe allalphabetandwewilldenote

it with a ursive letter

A

. With this alphabet we onstru t the set of ongurations dened over

Z

d

where

d

¥

1

isthe dimension.

Denition 1. Let

A

be a nite alphabet, and

d

¥

1

. Let

S

Z

d

be a subset. A

pattern with support

S

is an element of

p

of

A

S

. We write

S

supp p

p

q for the support of the pattern

p

. If

S

1

S

, the pattern

p

1

p

|

S

1

denotes the restri tion of

p

to

S

1

. A

ongurationis a patternwith full support

S

Z

d

.

When

d

1

aone-dimensional nite patternis alled aword.

The set of all possible

Z

d

- ongurations dened over an alphabet

A

is denoted by

Σ

d

p

A

q

:

A

Z

d

. On this set we dene the shifta tion as follows.

Denition2. Theshifta tionona ongurationspa e

Σ

d

p

A

qisa olle tion

σ

p

σ

u

q

u

P

Z

d

su h that

σ

u

_{: Σ}

d

p

A

q Ñ

Σ

d

p

A

q

x

ÞÑ

σ

u

p

x

q

y,

where

v

P

Z

d

_{, y}

v

x

u v

.

Wewilluse thesamenotationfortheshifta tingonanitepattern,thatis,if

S

Z

d

isanite set and

p

P

A

S

isa pattern, then we an write forall

u

P

Z

d

theshift a tingon

the pattern

p

as

σ

u

p

q

w

P

A

S

u

where

w

v

u

v u

,

v

P

S

u

(17)

p

q

if and onlyif

q

σ

u

p

q, for some

u

P

Z

d

. In that sense, the shape of the support

of the pattern isxed, but the form an be lo atedin any translateof this support.

Let

S, T

Z

d

are two subsets, and

p, q

be two patterns with support

S

and

T

, respe tively. We say that

p

is a sub-pattern of

q

, if

S

T

and

p

q

|

S

. Similarly we say that

p

is a sub-pattern of a onguration

x

P

A

Z

d

, if

p

x

|

S

. We an alsosay that a pattern

p

P

A

S

appears in another pattern

q

P

A

T

(respe tively, in a onguration

x

P

A

Z

d

)if thereexists ave tor

u

P

Z

d

su hthat

σ

u

p

qisasub-patternof

q

(respe tively,

σ

u

p

q is asub-pattern of

x

). Inthat ase wewrite

p

q

(respe tively,

p

x

).

Denition3. If

p

P

A

S

isapatternwithsupport

S

,the ylindergeneratedby

p

,denoted by r

p

s, is the subset of ongurations dened by

r

p

s

:

t

x

P

Σ

d

p

A

q

: x

|

S

p

u

.

For

a

P

A

and

i

P

Z

d

wedenote the ylinder

r

a

s

i

t

x

P

Σ

d

p

A

q

: x

i

a

u

.

Denition 4. Let

P

A

S

beasubset of patterns of support

S

. The ylindergenerated by

P

is the subset,

r

P

s

:

¤

p

P

r

p

s

.

The following is the topologi al denition of one of the most importantobje ts that

we work with.

Denition 5. A subshift

X

is a losed subset of

Σ

d

p

A

q whi h is invariant under

σ

u

_:

Σ

d

p

A

qÑ

Σ

d

p

A

qfor all

u

P

Z

d

, that is,

σ

u

p

X

q

X

.

As said before, there is a ombinatorialdenition of a subshift, whi h is given by the

set offorbidden patterns as presented below.

Denition 6. Let

X

be a subset of

Σ

d

p

A

q. We say that

X

is a subshift generated by a set

F

of forbidden patternsif

F

R

¥

1 A

J1,RK

d

isasubsetofpatterns withnitesupport

and

X

Σ

d

p

A, F

q

:

t

x

P

Σ

d

p

A

q

:

p

P

F , p

x

u

.

Thefollowingpropositionassuresthateverysubshiftisgeneratedbyasetofforbidden

patterns.

Proposition 1. The twodenitions of subshift (Denition 5 and Denition6) oin ide.

The entire onguration spa e

Σ

d

p

A

q

A

Z

d

is a subshift, and we all it the full

shift. We willdenote by p

Σ

d

(18)

generatedby the ylindersets in

Σ

d

p

A

q. Wewilldes ribea lassi ationforthe subshifts based onthe set of forbidden patterns. For the full shift the set of forbiddenpatterns is

empty. If the set of forbidden patterns is nite we willsay that subshift is a subshift of

nite type or SFT. When the set of forbidden patterns an be enumerated by a Turing

ma hine, then we say that the subshift is an ee tively losed subshift (we explain what

weare onsideringas a set enumerated by aTuring ma hine inSe tion 2.4).

Another way of des ribing a subshift is by itslanguage, that we dene next.

Denition 7. Let

A

be a nite alphabet, and

d

¥

1

. Let

X

be a subshift of

A

Z

d

. The

language of

X

, denoted

L

p

X

q, is the set of square patterns that appear in

X

, or more formally,

L

p

X

q

:

§

ℓ

¥

1

!

p

P

A

J1,ℓK

d

:

D

x

P

X,

s.t.

p

x

)

.

(2.1)

We willdenote the set of square patterns of a xed length

ℓ

as

L

p

X, ℓ

q

:

!

p

P

A

J1,ℓK

d

:

D

x

P

X,

s.t.

p

x

|

J1,ℓK

2

)

.

(2.2)

A di tionary

L

of size

ℓ

and dimension

d

over the alphabet

A

is a subset of

A

J1,ℓK

d

.

A di tionaryis aspe ializedsubset of patterns. Wesay that adi tionary

L

of size

ℓ

is a sub-di tionary of

L

1

of size

ℓ

1

(where both have the same dimension

d

), if every pattern of

L

is a sub-pattern of a pattern of

L

1

. Given a di tionary we an dene the set of all

ongurations obtained by the innite on atenation of patterns of this di tionary. In

fa t,this subset is asubshift asdes ribed below.

Denition 8. The on atenated subshift of a di tionary

L

of size

ℓ

and dimension

d

is the subshift of the form

x

L

y ¤

u

P

J1,ℓK

d

£

v

P

Z

d

σ

p

u vℓ

q r

L

s

,

!

x

P

Σ

d

p

A

q

:

D

u

P

J1, ℓK

d

_,

v

P

Z

d

_,

p

σ

u ℓv

p

x

qq|

J1,ℓK

d

P

L

)

.

Another important on ept on erns the admissibility of a pattern. Given a set of

forbiddenpatterns, we dene lo aland global admissibility.

Denition 9. Let

F

A

J1,DK

d

for a xed

D

¥

2

. We say that a pattern

w

P

A

J1,RK

d

where

R

¥

D

is lo ally

F

-admissibleif

σ

u

p

x

q|

J1,DK

d

R

F ,

u

P

J0, R

D

K

d

_,

(19)

w

P

A

J1,RK

d

is globally

F

-admissibleif there exists

x

P

Σ

d

p

A, F

qsu h that

x

|

J1,RK

d

w.

It is lear that if a pattern is globally admissible, then it is lo ally admissible, but

the reverse itnot always true. The next propositionassures that forevery

d

-dimensional subshift, every really large pattern that is lo ally admissible has a entral blo k that is

globally admissible.

Proposition 2. Let

X

Σ

d

p

A, F

q bea subshiftgiven by a set of forbiddenpatterns

F

. There exists a fun tion

R : N

Ñ

N

so that if

q

P

A

J

R

p

n

q

,R

p

n

q

K

d

is lo ally admissible, then

p

q

|

J

n,nK

d

, the restri tion of

q

to

A

J

n,nK

d

, is globally admissible.

Proof. The proof follows froma standard ompa tnessargument asdes ribed inLemma

4.3of [5℄ ina more generalsetting.

Suppose su h a fun tion does not exist, then there exists

n

P

N

su h that for every

m

¥

n

there exists alo allyadmissiblepattern

q

m

of size

m

su h that

p

m

q

m

|

J

n,nK

d

is

notgloballyadmissible. Let

x

m

P

Σ

d

p

A

qbea ongurationsu hthat

x

m

|

J

m,mK

d

q

m

. By ompa tness of

Σ

d

p

A

q, we may extra t a onverging subsequen e

x

m

p

k

q

whi h onverges

to some

x

¯

P

A

Z

d

.

We laim

x

¯

P

X

. Indeed, if not, there is aforbidden patternwhi ho urssomewhere in

x

¯

. In parti ular, there is

k

P

N

su h that the pattern is ompletely ontained in

J

m

p

k

q

, m

p

k

q

K

d

. It follows by onvergen e of the sequen e t

x

m

p

k

q u

k

P

N

that eventually every pattern

q

m

p

k

q

ontains the forbiddenpattern. This isa ontradi tion be ause

q

m

is lo ally admissible. Hen e

x

¯

P

X

.

As

x

¯

P

X

, then

x

¯

|

J

n,nK

d

isglobally admissible,but this isequalto

p

m

forsome

m

P

N

and thus not globally admissible. This is again a ontradi tion. Therefore the fun tion

R

must exist. It is non-de reasing as subpatterns of globally admissible patterns are

themselves globally admissible.

2.2 Entropy and variational prin iple

We establish here some of the most important results about entropy of subshifts. The

resultshere were developed by several authorsindierent approa hes andthey were able

togeneralizetheseresultsevenforamenablegroupa tionsandnon- ompa t onguration

spa es. Here we fo us onthe

Z

d

-a tionovera ompa t ongurationspa e

Σ

d

p

A

q

A

Z

d

. We always onsider

Σ

d

p

A

q

A

Z

d

and

σ

p

σ

u

q

u

P

Z

d

the shift a tion. We will denote

by

M

1

p

Σ

d

p

A

qq the set of allprobability measures dened on

Σ

d

p

A

q and by

M

σ

p

Σ

d

p

A

qq the set of shift-invariant probability measures. Here we always onsider p

Σ

d

p

A

q

, B, µ

q as a probability spa e where

B

is the sigma algebra generated by the ylinder sets and

µ

P

M

σ

p

Σ

d

(20)

Denition10. A olle tion

P

t

P

1 , P

2 , ..., P

n

uofmeasurablesets isanite partitionof

Σ

d

p

A

qif

• P

i

X

P

j

∅

for

i

j

;and

• i

P

i

Σ

d

p

A

q. Foraprobabilityspa ep

Σ

d

p

A

q

, B, µ

qwe alla olle tionofmeasurablesets

P

t

P

1 , P

2 , ..., P

n

u a

µ

-partition if

• µ

p

P

i

q¡

0

,

i

;

• µ

p

P

i

X

P

j

q

0

, for

i

j

; and

• µ

Σ

d

p

A

qz

n

¤

i

1 P

i

0

.

One of the most important on epts in thermodynami s is the entropy of a system.

Herewepresentthe denitionofShannon entropy andsomeuseful propertiesthatweuse

inthis text. The denitions and results an be found in Keller[22℄ and Kerr-Li [24℄.

Denition 11. The information of a

µ

-partition

P

t

P

1 , P

2 , ..., P

n

u is the fun tion

I

P

: Σ

d

p

A

qÑ

R

dened as

I

P

p

x

q

:

¸

P

log

p

µ

p

P

qq

1 P

p

x

q

.

The entropy of a partitionwith respe t a measure

µ

isgiven by

H

p

P, µ

q

:

»

I

P

p

x

q

dµ

n

¸

i

1 µ

p

P

i

q

log

p

µ

p

P

i

qq

We will use the notation

H

p

P

q

H

p

P, µ

q when there is no onfusion over whi h measure we are onsidering inorder to not overload the notation.

Given two

µ

-partitions

P

t

P

1 , P

2 , ..., P

n

u and

Q

t

Q

1 , ..., Q

m

u of a onguration spa e

Σ

d

p

A

q, we an dene the onditional information of

P

given

Q

as the fun tion

I

P

|

Q

: Σ

d

p

A

qÑ

R

dened as

I

P

|

Q

p

x

q

:

n

¸

i

1 m

¸

j

1 log

µ

p

P

i

X

Q

j

q

µ

p

Q

j

q

1 P

i

X

Q

j

p

x

q

.

Inthe same fashion we an denethe onditional entropy of

P

given

Q

withrespe t toa measure

µ

as the value

H

p

P

|

Q, µ

q

:

»

I

P

|

Q

dµ

»

H

p

P, µ

Q

x

q

dµ

p

x

q (2.3)

(21)

where p

µ

Q

x

q

x

P

Σ

d

p

A

q

is the family of onditional probabilities with respe t to

Q

. We an alsoexpress the onditionalentropy as the sum

H

p

P

|

Q, µ

q

n

¸

i

1 m

¸

j

1 µ

p

P

i

X

Q

j

q

log

µ

p

P

i

X

Q

j

q

µ

p

Q

j

q

.

As before we will use the notation

H

p

P

|

Q

q

H

p

P

|

Q, µ

q when there is no onfusion over whi h measure we are onsidering in orderto not overload the notation.

We say that a partition

P

1

is a renement of another partition

P

if every element of

P

1

is ontained in anelement of

P

. We denoteas

P

1

©

P

.

Wedenotethe ommonrenementoftwopartitionsdenoted by

P

_

Q

asthepartition generated by

P

_

Q :

t

P

i

X

Q

j

: P

i

P

P, Q

j

P

Q

u

.

Fora subset

S

Z

d

wedenote by

P

S

:

ª

u

P

S

σ

u

P

the ommon renement of the partitions

σ

u

P

where

u

P

S

. A partition

P

is a

µ

-generated partition of p

Σ

d

p

A

q

, B, µ

q if the sigma algebragenerated by

P

S

for every nite

subset

S

Z

d

is equal to

B mod µ

.

The next lemma gives usthe Jensen inequality that willbe used many times.

Lemma 1 (Jensen's Inequality). Consider

I

R

an open interval and

ψ : I

Ñ

R

a on ave fun tion. If

f : Σ

d

p

A

qÑ

I

a

µ

-integrable fun tion, then the integral of

ψ

f

is well dened and

ψ

»

f dµ

¥ »

ψ

f dµ.

If we onsider

ψ :

r

0, 1

sÑ

R

dened as

ψ

p

x

q #

x log

p

x

q

, 0

x

¤

1 0,

x

0,

(2.4)

then

ψ

is astri tly on ave fun tion and therefore we obtain

ψ

n

¸

i

1 λ

i

x

i

¥

n

¸

i

1 λ

i

ψ

p

x

i

q

,

(2.5)

where

x

i

P r

0, 1

s and

λ

i

¡

0

for ea h

i

P

J1, nK

with °

n

i

1 λ

i

1

. We will use this

inequality for the proof of the next lemma whi h presents some important properties of

(22)

Lemma 2. Consider

P

t

P

1 , ..., P

n

u and

Q

t

Q

1 , Q

2 , ..., Q

m

u two

µ

-partitions of

Σ

d

p

A

q. Then p

i

q

0

¤

H

p

P

|

Q

q¤

H

p

P

q¤

log

|

P

|; p

ii

q

H

p

P

_

Q

q

H

p

P

q

H

p

Q

|

P

q; p

iii

q

H

p

P

q¤

H

p

Q

q

H

p

P

|

Q

q; p

iv

q if

Q

©

P

,then

H

p

P

|

Q

q

0

. p

v

q if

Q

©

P

, then

H

p

P

_

Q

q

H

p

Q

q¥

H

p

P

q;

Proof. p

i

q The inequality

0

¤

H

p

P

|

Q

q follows from the denition of the entropy of a partition. Nowwewillprovethatif

R

t

C

1 , ..., C

l

uisapartitionsu hthat

Q

©

R

wehave that

H

p

P

|

Q

q¤

H

p

P

|

R

q

.

(2.6) Denote

λ

k,j

:

µ

p

B

j

X

C

k

q

µ

p

C

k

q and

x

j,i

µ

p

A

i

X

B

j

q

µ

p

B

j

q

.

As we are onsidering

Q

©

R

,

µ

p

B

j

X

C

k

q is equal to

µ

p

B

j

q or

0

, be ause either

B

j

C

k

or

B

j

X

C

k

∅

. Thus for axed

i

and

k

m

¸

j

1 λ

k,j

x

j,i

¸

j

P

J1,mK

B

j

C

k

µ

p

A

i

X

B

j

q

µ

p

C

k

q

µ

p

A

i

X

C

k

q

µ

p

C

k

q

.

H

p

P

|

Q

q

n

¸

i

1 m

¸

j

1 µ

p

P

i

X

Q

j

q

log

µ

p

P

i

X

Q

j

q

µ

p

Q

j

q

n

¸

i

1 m

¸

j

1 µ

p

Q

j

q

ψ

p

x

j,i

q

n

¸

i

1 m

¸

j

1 l

¸

k

1 µ

p

C

k

q

λ

k,j

ψ

p

x

j,i

q

n

¸

i

1 l

¸

k

1 µ

p

C

k

q

m

¸

j

1 λ

k,j

ψ

p

x

j,i

q ¤

n

¸

i

1 l

¸

k

1 µ

p

C

k

q

ψ

m

¸

j

1 λ

k,j

x

j,i

H

p

P

|

R

q

.

Ifwe take

R

t

Σ

d

(23)

In (2.5) if we onsider

x

i

µ

p

P

i

q and

λ

i

1

{

n

we obtain that

1 n

log

1 n

ψ

1 n

ψ

1 n

n

¸

i

1 µ

p

P

i

q ¥

1 n

n

¸

i

1 ψ

p

µ

p

P

i

qq

1 n

H

p

P

q

,

and therefore

H

p

P

q¤

log

p

n

q

log

|

P

|.

p

ii

q Ea h element of the partition

P

_

Q

is of the form

P

X

Q

where

P

and

Q

P

Q

. Then

I

P

_

Q

p

x

q ¸

P

¸

Q

P

Q

log

p

µ

p

P

X

Q

qq

1 P

X

Q

p

x

q ¸

P

¸

Q

P

Q

log

µ

p

P

X

Q

q

µ

p

P

q

µ

p

P

q

1 P

X

Q

p

x

q ¸

P

¸

Q

P

Q

log

µ

p

P

X

Q

q

µ

p

P

q

1 P

X

Q

p

x

q ¸

P

¸

Q

P

Q

log

p

µ

p

P

qq

1 P

X

Q

p

x

q ¸

P

¸

Q

P

Q

log

µ

p

P

X

Q

q

µ

p

P

q

1 P

X

Q

p

x

q ¸

P

log

p

µ

p

P

qq

1 P

p

x

q

I

P

|

Q

p

x

q

I

P

p

x

q

.

Byintegrating with respe t toa measure

µ

weobtain that

H

p

P

_

Q

q

H

p

P

q

H

p

Q

|

P

q

.

p

iii

q By the previous items we obtainthat

H

p

P

q

H

p

P

_

Q

q

H

p

Q

|

P

q

¤

H

p

P

_

Q

q

H

p

Q

q

H

p

P

|

Q

q

.

p

iv

q Forany twopartitions

P

and

Q

, we have

H

p

P

|

Q

q ¸

P

¸

Q

P

Q

µ

p

P

X

Q

q

log

µ

p

X

Q

q

µ

p

Q

q ¸

P

¸

Q

P

Q

µ

p

Q

q

ψ

µ

p

P

X

Q

q

µ

p

Q

q

.

(24)

Hen e ea h term of the sum above is equal to zero be ause either

µ

p

P

X

Q

q

µ

p

Q

q

0

or

µ

p

P

X

Q

q

µ

p

Q

q

1

,and inboth ases wehave that

H

p

P

|

Q

q ¸

P

¸

Q

P

Q

µ

p

Q

q

ψ

µ

p

P

X

Q

q

µ

p

Q

q

0.

p

v

q It follows fromthe items p

iii

q and p

iv

q.

Lemma 3. Consider p

Σ

d

p

A

q

, B, µ

q ashift-invariantprobabilityspa e and

P

anite par-titionof

Σ

d

p

A

q. The dynami al entropy relative to the partition

P

isgiven by

h

p

P, µ

q

:

inf

n

¥

0

1

|

Λ

n

|

H

p

P

Λ

_n

q

lim

n

Ñ 8

1

|

Λ

n

|

H

p

P

Λ

_n

q

whi h iswelldened, where

Λ

n

:

J

n, n

K

d

for

n

¥

1

.

Proof. Forea h

n

¥

1

wewill onsider

Λ

n

:

J

n, n

K

d

Z

d

. Foraxed

m

¥

1

wedenote

Λ

m

J

m, m

K

d

and

l

m

2m

1

. Consider the set

V

n

:

p

Pp

l

m

Z

q

2 :

p

Λ

m

qX

Λ

n

∅

( Then

Λ

n

˜

Λ

n

:

¤

u

P

V

n

p

Λ

m

u

q

.

Notethat |

˜

Λ

n

||

V

n

||

Λ

m

|¤|

Λ

n m

|. We obtainthat

H

p

P

Λ

n

q ¤

H

p

P

˜

Λ

n

q ¤ ¸

u

P

V

n

H

p

σ

u

P

Λ

m

q |

V

n

|

H

p

P

Λ

_m

q ¤ |

Λ

n m

| |

Λ

m

|

H

p

P

Λ

m

q

,

and therefore

lim sup

n

Ñ 8

1

|

Λ

n

|

H

p

P

Λ

n

q¤

lim sup

n

Ñ 8 |

Λ

n m

| |

Λ

m

|

1

|

Λ

m

|

H

p

P

Λ

m

q

1

|

Λ

m

|

H

p

P

Λ

m

q

.

The lastestimate holdsfor every xed

m

, thus we on lude that

lim sup

n

Ñ 8

1

|

Λ

n

|

H

p

P

Λ

_n

q¤

inf

m

¡

0

1

|

Λ

m

|

H

p

P

Λ

_m

q¤

lim inf

m

Ñ 8

1

|

Λ

m

|

H

p

P

Λ

_m

q

.

(25)

Theorem 3 (Shannon-M Millan-Breiman). Letp

Σ

d

p

A

q

, B, µ

qa shift-invariant probabil-ity spa e and

P

a nite partitionof

Σ

d

p

A

q. Then

lim

n

Ñ 8

1

|

Λ

n

|

log

p

µ

p

P

Λ

n

qq

h

p

P, µ

q

pointwise a.e. and in

L

1

.

The previoustheorem has already been proved for alarger lass ofgroup a tionsonly

with the assumptions that the group is amenable [29, 24, 35℄. The proof for Theorem 3

as statedhere an befound inKrengel [26℄.

Nowwe denethe Kolmogorov-Sinai entropy also alled dynami alentropy of a

mea-sure.

Denition 12. The entropy of the spa e p

Σ

d

p

A

q

, B, µ

q, also known as the dynami al entropy of

µ

is given by

h

p

µ

q

sup

P

t

h

p

P, µ

q

: P

is anite partitionu

.

Denition 13. The topologi alentropy of asubshift

X

Σ

d

p

A

q is given by

h

top

p

Σ

d

p

A

qq

lim

n

Ñ 8

1

|

Λ

n

|

log

p|

L

p

X, 2n

1

q|q

.

In Chazottes-Meyerovit h [20℄ they establish important results about the

hara teri-zationofthe entropy formultidimensionalSFT.Next wepresentthe variationalprin iple

for the entropy.

Theorem 4 (VariationalPrin iple). Let

X

Σ

d

p

A

q be a subshift,then

h

top

p

X

q

sup

µ

h

p

µ

q

wherethesupremumistakenoverthesetofshift-invariantprobabilitymeasures

M

σ

p

Σ

d

p

A

qq.

The VariationalPrin ipleasstatedabovehasalreadybeen proved foramenablegroup

a tionsin[24℄. Oneimportantresultforthe hara terization ofthe dynami alentropy of

a measure is given by the followingtheorem.

Theorem 5 (Kolmogorov-Sinai). If

P

is

µ

-generated partition for p

Σ

d

p

A

q

, B, µ

q and

H

p

P

q 8, then

h

p

µ

q

h

p

P, µ

q

.

Proof. For any nitesubset we have that

h

p

P

Λ

(26)

Indeed, onsider a xed

N

¡

0

su h that

Λ

N

, thenwehave that

h

p

P

Λ

, µ

q

lim

n

Ñ 8

1

|

Λ

n

|

H

p

P

Λ

q

Λ

n

¤

lim

n

Ñ 8

1

|

Λ

n

|

H P

Λ

n

N

¤

lim

n

Ñ 8 |

Λ

n N

| |

Λ

n

|

1

|

Λ

n N

|

H P

Λ

n

N

h

p

P, µ

q ¤

h

p

P

Λ

, µ

q sin e

P

Λ

©

P

.

Now onsider

P

a nite

µ

-generated partitionwith nite entropy and

Q

a nite par-tition. From 2.7and Lemma2 we obtainthat

h

p

Q, µ

q ¤

h

p

P

Λ

_n

, µ

q

H

p

Q

|

P

Λ

_n

q

h

p

P, µ

q

H

p

Q

|

P

Λ

_n

q

.

As

lim

n

Ñ 8

H

p

Q

|

P

Λ

_n

q

H

p

Q

|

B

q

0

,itfollowsthatforanarbitrarypartition

Q

,istrue that

h

p

Q, µ

q¤

h

p

P, µ

q, and therefore the result follows.

2.3 Potential

A fun tion

f : Σ

d

p

A

qÑ

R

is upper semi- ontinuous if the set t

x

P

Σ

d

p

A

q

: f

p

x

q

c

u is anopen set for every

c

P

R

.

Denition 14. A potential

ϕ : Σ

d

p

A

qÑ

R

isregular if 8 ¸

n

1 n

d

1 δ

n

p

ϕ

q 8

,

where

δ

n

p

ϕ

q

:

sup

t|

ϕ

p

w

q

ϕ

p

v

q|

: w, v

P

Σ

d

p

A

q

, w

|

Λ

n

v

|

Λ

n

u.

We say that a potential

ψ

has nite range if there exists

n

0

P

N

su h that

δ

n

p

ψ

q

0

, forall

n

¥

n

0

. If apotentialhas nite range,then it is regular.

Next we dene the pressure of an upper semi- ontinuous potential, the notion of an

equilibriummeasure and re all several resultsthat hara terize the equilibriummeasures

fora ertain lass of potentials.

Denition 15. The pressure of a upper semi- ontinuous potential

ϕ : Σ

d

p

A

q Ñ

R

at inverse temperature

β

is the value

P

p

βϕ

q

:

sup

µ

P

M

σ

p

Σ

d

p

A

qq "

h

p

µ

q »

βϕdµ

*

.

(27)

Denition 16. An equilibrium measure for a potential

ϕ

at inverse temperature

β

is a measure

µ

βϕ

P

M

σ

p

Σ

d

p

A

qqsu h that

P

p

βϕ

q

h

p

µ

βϕ

q »

βϕdµ

βϕ

.

An important hara terization for the set of equilibrium measures for a regular lo al

potential is that it is exa tly the set of invariant Gibbs measures. In order to state this

result, we present one possible denition of Gibbs measures based on[22℄.

Remark 2. Here we will dene all these notions and results for the full shift over a

nite alphabet, but these denitions and results are also valid for a more general lass

of subshifts, for instan e Muir [31℄ works with a ountable alphabetin multidimensional

subshifts and Israel [21℄ extended to general ompa t spin spa es and quantum systems

for the full shift.

Consider

ϕ

a regularpotentialon

Σ

d

p

A

qand denote

ϕ

n

:

¸

g

P

Λ

n

ϕ

σ

g

where

Λ

n

J

n, n

K

d

. Weareinterestedinhow

ψ

n

p

w

qwill hangeifwealternitelymany sites. For that, we will introdu e, as in Keller [22℄, a lass of lo alhomeomorphisms on

Σ

d

p

A

q.

Denition 17. Let

ϕ

be a regular potential dened over

Σ

d

p

A

q. We denote by

ε

n

the set ofall maps

τ : Σ

d

p

A

qÑ

Σ

d

p

A

qsu h that p

τ

p

w

qq

i

#

τ

i

p

w

i

q

, i

P

Λ

n

w

i

,

i

R

Λ

n

where

τ

i

: A

Ñ

A

are permutations in the state spa e. We denote by

ε :

n

¡

0 ε

n

the

set ofall homeomorphisms in

Σ

d

p

A

q that hange onlynitely many oordinates.

Lemma 4. (Keller[22℄) Let

ϕ

be aregular potentialand

τ

P

ε

. For

n

¡

0

dene

Ψ

n

_τ

: Σ

d

p

A

qÑ

R

,

Ψ

n

τ

:

ϕ

n

τ

1 ϕ

n

.

Then the limit

Ψ

τ

:

lim

n

Ñ 8

Ψ

n

_τ

exists uniformlyon

Σ

d

p

A

q.

(28)

µ

P

M

1

p

Σ

d

p

A

qqis a Gibbs measure for the potential

ϕ

if

τ

µ

e

Ψ

τ

forea h

τ

P

ε

.

The previous denition goesba k toCapo a ia [9℄ and does not involve onditional

measures asin a more lassi al denition of Gibbs measure [17,32℄.

Assaidbefore,thereareseveral hara terizationsforaGibbsmeasure(seeGeorgii[17℄

and Ruelle [32℄) and several results for the equivalen e between these denitions (see

Kimura [25℄ and Keller[22℄) even for potentialsdened overmore general subshifts.

The next theorem from Keller [22℄ gives a important hara terization of the set of

invariantGibbs measures for a regularlo alpotential.

Theorem 6. Let

Σ

d

p

A

q

A

Z

d

be the full shift and

ϕ : Σ

d

p

A

q Ñ

R

be a regular lo al potential. The set of equilibriummeasures for

ϕ

is nonempty, ompa t, onvex subset of

M

_σ

p

Σ

d

p

A

qqand everyequilibriummeasureisalsoaGibbs invariantprobability measure.

Given apotential

βϕ

atinverse temperature

β

and

ϕ

aregularlo alpotential,theset of equilibriummeasures is exa tlythe set of Gibbs invariantmeasures for

βϕ

.

2.4 Turing Ma hines and the Simulation Theorem

We present here the basi on epts of a Turing ma hine and how we an hara terize a

language based on its omputability. The automaton that we all Turing ma hine was

rst introdu ed by Alan Turing in 1936 and is similar to a nite automaton but with

unlimitedand unrestri ted memory. This modelworks onan innite tape and therefore

has unlimitedmemory. Thereis a head of al ulation whi h an read and write symbols

on the tape and move over the tape, both forward and ba kward. We will introdu e a

formaldenition of aTuringma hine as inSipser [34℄.

Denition 19. A Turing ma hine

M

is a

7

tuple p

Q, A, T , δ, q

0 , q

a

, q

r

q,where

• Q

is a niteset of states of the head of al ulation;

• A

isthe inputalphabetwhi h does not ontain the blank symbol 7;

• T

isthe tapealphabet whi h ontains the blank symbol7 and

A

T

;

• δ : Q

T

Ñ

Q

T

t

1,

1

u isthe transitionfun tion;

• q

0

is the initialstate of the head of al ulation;

(29)

• q

r

P

Q

isthe reje t state.

The ma hine works on aninnitetape divided into dis reteboxeson whi h the head

willa t. Ifwethinkof

Z

asabi-innitetapelled withsymbolsof

T

, we anexpress the Turing ma hine

M

by des ribing the state of the head and in whi h box the head is.

We always start the al ulation over a word dened on the alphabet

A

that will be written on the tape of the ma hine. The other boxes of the innite tape are lled with

the blank symbols 7. The head will start on the leftmost symbol of the word with the

initialstate

q

0

. Atea hstep of its al ulationthe head a ts (read/write)onlyonthe box where the head islo ated. Basedonthe symbolthat the head reads and the state of the

head, the transition fun tion will give us whi h symbol the head must write in the box,

the new state of the head and in whi h dire tion the head should move,

1

if it should move for the left box or

1

if it should move for the right box. It is possible to dene the transition fun tion with the possibility of the head staying in the same box after a

al ulation, but the denitions are equivalent.

One wayofrepresenting thetransitionfun tionisby adire ted graphwhereea hnode

represents a state of the head of al ulation and the arrows are tagged with the rules of

the transitionfun tion. See the transitionrepresented below.

PSfrag repla ements

q

m

x

q

n

Ñ

y,

1 y

Ñ

y,

1

Figure 2.1: Dire ted graphrepresenting two rules of some transitionfun tion

δ

.

If the head of al ulation is inthe state

q

m

and it reads the symbol

x

, then the head repla es this symbol by

y

, hange of state to

q

n

and move to the box to the right. If instead the head is in the state

q

m

and reads the symbol

y

, then the head keeps the symbol

y

in that box, does not hange the state and moves tothe box onthe right.

The al ulationof aTuring ma hine stops when the head rea hes the a ept state

q

a

orthe reje tstate

q

r

. Ifthe ma hinenever rea hes one ofthese statesthe al ulationwill never stop. As said before, the al ulation of a Turing ma hine starts over a nite word