Central sets defined by words of low factor complexity

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by  Michelangelo Bucci

CENTRAL SETS DEFINED BY WORDS OF LOW FACTOR COMPLEXITY
MICHELANGELOBUCCI,SVETLANAPUZYNINA,ANDLUCAQ.ZAMBONI
1 ABSTRACT. A subset A of N is called an IP-set if A containsall finite sums of distinct terms of
1
0 some infinite sequence (xn)n∈N of naturalnumbers. Central sets, first introducedby Furstenberg
2 usingnotionsfromtopologicaldynamics,constituteaspecialclassofIP-setspossessingadditional
nicecombinatorialproperties:Eachcentralsetcontainsarbitrarilylongarithmeticprogressions,and
t
c solutionsto allpartition regularsystems of homogeneouslinear equations. In thispaperwe show
O
how certainfamilies of aperiodicwordsof low factor complexitymay be used to generatea wide
9 assortment of central sets having additional nice properties inherited from the rich combinatorial
1 structure of the underlying word. We consider Sturmian words and their extensions to higher al-
phabets(so-calledArnoux-Rauzywords),aswellaswordsgeneratedbysubstitutionrulesincluding

] the famousThue-Morseword. We also describe a connectionbetween centralsets and the strong
O
coincidenceconditionforfixedpointsofprimitivesubstitutionswhichrepresentsanewapproachto
C
the strong coincidenceconjecturefor irreduciblePisot substitutions. Our methodssimultaneously
.
h exploitthegeneraltheoryofcombinatoricsonwords,thearithmeticpropertiesofabstractnumera-
t tionsystemsdefinedbysubstitutionrules,notionsfromtopologicaldynamicsincludingproximality
a
m andequicontinuity,thespectraltheoryofsymbolicdynamicalsystems,andthebeautifulandelegant
theory,developedbyN.Hindman,D.Straussandothers,linkingIP-setstothealgebraic/topological
[
propertiesoftheStone-CˇechcompactificationofN.Usingthekeynotionofp-limn,regardedasa
1 mappingfromwords to words, we apply ideas fromcombinatoricson words in the frameworkof
v
ultrafilters.
5
2
2
4 CONTENTS
.
0
1. Introduction 2
1
1 2. Words andsubstitutions 6
1 2.1. Words& subshifts 6
:
v 2.2. Substitutions 7
i
X 2.3. Sturmianwords& generalizations 8
r 3. Ultrafilters,IP-sets andcentral sets 9
a
3.1. Stone-Cˇech compactification 9
3.2. IP-sets and central sets 10
3.3. Limitsofultrafilters 12
4. A first analysisofsomeconcreteexamples 14
4.1. TheFibonacciword 14
4.2. Them-bonacci word 17
5. Sturmianpartitions& central sets 18
6. Othercentral partitionsdefined by substitutions 21
7. Infinitecentral partitionsofN 22
2000MathematicsSubjectClassification. Primary68R15&05D10.
Key words and phrases. Sturmian words, numeration systems, IP-sets, central sets and the Stone-Cˇech
compactification.
1
2 M.BUCCI,S.PUZYNINA,ANDL.Q.ZAMBONI
8. Strong coincidencecondition 24
8.1. Abstractnumerationsystemsdefined by substitutions 24
8.2. Proofof(3)inCorollary 8.1 26
References 27
1. INTRODUCTION
Let N = {0,1,2,3,...} denote the set of natural numbers, and Fin(N) the set of all non-empty
finitesubsetsofN.
Definition 1.1. A subset A of N is called an IP-set if A contains { x |F ∈ Fin(N)} for
n F n
someinfinitesequenceofnaturalnumbersx < x < x ··· . Asubset A∈⊆ N iscalled an IP -set
0 1 2 ∗
ifA∩B 6= ∅ foreveryIP-set B ⊆ N. P
ByacelebratedresultofN.Hindman[23],givenanyfinitepartitionofN,atleastoneelementof
thepartitionisanIP-set. ItfollowsfromHindman’stheoremthateveryIP -setisanIP-set,butthe
∗
converse is in general not true. In fact, more generally Hindman showsthat given any finite parti-
tionofanIP-set,atleastoneelementofthepartitionisagainanIP-set. Inotherwordstheproperty
of being an IP-set is partition regular, i.e., cannot be destroyed via a finite partitioning. Other
examples of partition regularity are given by the pigeonhole principle, sets having positive upper
density, and sets having arbitrarily long arithmetic progressions (Van der Waerden’s theorem). In
[22], Furstenberg introduced a special class of IP-sets, called central sets, having a substantial
combinatorial structure. The property of being central is also partition regular. Central sets were
originallydefined intermsoftopologicaldynamics:
Definition1.2. AsubsetA ⊂ Niscalledcentralifthereexistsacompactmetricspace(X,d)and
a continuousmapT : X → X, pointsx,y ∈ X anda neighborhoodU of y suchthat
• y is auniformlyrecurrentpointin X,
• xand y areproximal,
• A = {n ∈ N|Tn(x) ∈ U}.
WesayA ⊂ Nis central ifA∩B 6= ∅ foreverycentralset B ⊆ N.
∗
Recall that x is said to be uniformly recurrent in X if for every neighborhood V of x the set
{n|Tn(x) ∈ V} is syndetic, i.e., of bounded gap. Two points x,y ∈ X are said to be proximal if
for every ǫ > 0 there existsn ∈ N such thatd(Tn(x),Tn(y)) < ǫ. We remark that from theabove
definition,itisnotat allevidentthatcentralsets areIP-sets. Welatergivean alternativedefinition
(see Definition 3.5) which makes this point clear. The equivalence between the two definitions is
duetoBergelson and Hindman[7].
The question of determining whether a given subset A ⊆ N is an IP-set or a central set is typi-
cally quitedifficult,even ifforevery A, eitherAor itscomplementis an IP-set (resp. central set).
ItturnsoutthatineachcasethisquestionmaybereformulatedintermsofwhetherornotthesetA
belongstoacertainclassofultrafiltersonN(seeTheorem5.12in[26]inthecaseofIP-setsand[7]
inthecaseofcentralsets). Butthequestionofbelongingornottoagiven(non-principal)ultrafilter
is generally equally mysterious. An equivalent word combinatorial reformulation of this question
is as follows: Given a binary word ω = ω ω ω ... ∈ {0,1} , put ω = {n ∈ N|ω = 0}
0 1 2 ∞ 0 n
and ω = {n ∈ N|ω = 1}. The question is then to determine whether the set ω or ω is an
1 n (cid:12) 0 1
(cid:12)
(cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)
CENTRALSETSDEFINEDBYWORDSOFLOWFACTORCOMPLEXITY 3
IP-setorcentralset. Ofcourseingeneral,thisreformulationisasdifficultastheoriginalquestion.
However, should the word ω be characterized by some rich combinatorial properties, or be gener-
ated by some “simple” combinatorial or geometric algorithm (such as a substitution rule, a finite
state automaton, a Toeplitz rule...) or arise as a natural coding of a reasonably simple symbolic
dynamical system, then the underlying rigid combinatorial structure of the word may provide in-
sight to our previousquestion. Furthermore, such families of words may be used to obtain simple
constructions of central sets having additional nice properties inherited from the rich underlying
combinatorialstructure. Oneofourobjectiveshereis toillustratethislatterpoint.
Let A denote a finite non-empty set (called the alphabet) and ω = ω ω ω ... ∈ AN. For each
0 1 2
finitewordu onthealphabetA weset
ω = {n ∈ N|ω ω ...ω = u}.
u n n+1 n+u 1
| |−
In otherwords,ω denotes th(cid:12)eset ofalloccurrences ofuin ω.
u (cid:12)
InthispaperweinvestigatepartitionsofNbysetsoftheformω definedbywordsω oflowfactor
(cid:12) u
complexity. Our g(cid:12)oal is to study these partitions in the framework of IP-sets and central sets. All
infinite words ω ∈ AN considered in this paper are uniformly(cid:12)(cid:12)recurrent. As we shall see, in our
frameworkIP-sets andcentral sets areoneand thesame:
Theorem 1. Let ω ∈ AN be uniformlyrecurrent. Then theset ω is an IP-set if and onlyif it is a
u
centralset.
(cid:12)
Theabovetheoremallowsustosimultaneouslystateourresultsi(cid:12)ntermsofIP-setsandcentralsets.
We begin by considering the simplest aperiodic infinite words, namely Sturmian words. Stur-
mian words are infinite words over a binary alphabet having exactly n+1 factors of length n for
each n ≥ 0. Their origin can be traced back to the astronomer J. Bernoulli III in 1772. A funda-
mental result due to Morse and Hedlund [31] states that each aperiodic (meaning non-ultimately
periodic) infinite word must contain at least n + 1 factors of each length n ≥ 0. Thus Sturmian
wordsarethoseaperiodicwordsoflowestfactorcomplexity. Theyarisenaturallyinmanydifferent
areas of mathematics includingcombinatorics, algebra, numbertheory, ergodic theory, dynamical
systems and differential equations. Sturmian words are also of great importance in theoretical
physics and in theoretical computer science and are used in computer graphics as digital approxi-
mationofstraightlines.
Let ω ∈ {0,1}N be a Sturmian word, and let Ω denote the shift orbit closure of ω. Then Ω
contains a unique word ω˜ (called the characteristicword) having the property that both 0ω˜,1ω˜ ∈
Ω.In orderto stateourresults,wemustdistinguishbetween twocases:
Definition 1.3. A Sturmian word ω is called nonsingular if it does not contain the characteristic
word ω˜ as apropertail. Otherwiseit issaidtobesingular.
Theorem2. Letω ∈ ΩbeanonsingularSturmianword,anduafactorofω.Thenω isanIP-set
u
(resp. central set) if and only if u is a prefix of ω. In other words, for every prefix u of ω, the set
(cid:12)
ω is anIP∗-set(resp. central∗-set). (cid:12)
u
As acorollarywededucethat
(cid:12)
(cid:12)
Corollary1. Let ω ∈ Ωbea nonsingularSturmianword. Foreveryfactorv ofω andn ∈ ω the
v
set ω −nisa central set.
v ∗ (cid:12)
(cid:12)
(cid:12)
(cid:12)
4 M.BUCCI,S.PUZYNINA,ANDL.Q.ZAMBONI
We note that in general the property of being an IP -set is not translation invariant. See also
∗
Theorem 1.1in[8]. As an immediateconsequenceto thepreviouscorollary,wehave
Corollary 2. For each r ≥ 1 there exists a partition of N into sets A ,A ,...,A such that for
0 1 r
each 0 ≤ i ≤ r and n ∈ N, exactly one of the sets {A − n,A − n,...,A − n} is an IP -set
0 1 r ∗
(resp. central set).
∗
In fact, given r ≥ 1, let ω be any nonsingular Sturmian word (for instance the Fibonacci word)
andletF (r)denotethesetofallfactorsofω oflengthr.Thenther+1setsω withu ∈ F (r)
ω u ω
definea partitionofN withtherequiredproperty.
(cid:12)
(cid:12)
ForsingularSturmianwordsω wehave
Theorem 3. Let ω ∈ Ω be a Sturmian word such that Tn0(ω) = ω˜ with n ≥ 1. Then ω is an
0 u
IP-set (resp. central set) if and only if either u is a prefix of ω or a prefix of ω where ω is the
′ ′
(cid:12)
uniqueotherelement ofΩ withTn0(ω ) = ω˜. (cid:12)
′
Some (but not all) of the results on Sturmian partitions extend to so-called Arnoux-Rauzy words,
which may be regarded as natural combinatorialextensionsof Sturmian words to larger alphabets
[1].
We also consider partitions defined by words generated by substitution rules. For instance, by
consideringpartitionsofNdefinedbywordsgeneratedbythegeneralizedThue-Morsesubstitution
toan alphabet ofsizer ≥ 2,weshowthat
Theorem 4. Foreachpairofpositiveintegersr andN thereexistsa partitionof
N = A ∪A ∪···∪A
1 2 r
suchthat
• A −nis acentralset foreach1 ≤ i ≤ r and1 ≤ n ≤ N.
i
• Foreach n > N,exactlyoneofthesets{A −n,A −n,...,A −n} is acentralset.
1 2 r
ThesecondassertionofTheorem4reliesonthefactthateachfixedpointofthegeneralizedThue-
Morse substitution is distal. At least in the case of the Thue-Morse substitution itself this may
already beknown,buttheauthorshavebeenunabletolocatethisresultanywhereintheliterature.
Our proof of this fact uses a result of V. Baker, M. Barge and J. Kwapisz which states that for
subshifts (X,T) generated by primitive substitutions of Pisot type, the maximal equicontinuous
factorπ : X → Xeq is finitetoone[3].
Byconsideringpartitionsdefinedbywordsgeneratingminimalsubshiftswhicharetopologically
weakmixing(forexamplethesubshiftgeneratedbythesubstitution0 7→ 001and1 7→ 11001)we
provethat
Theorem 5. For each positive integer r there exists a partition of N = A ∪A ∪···∪A such
1 2 r
thatforeach 1 ≤ i ≤ r andn ≥ 0,thesetA −n isa centralset.
i
We also consider words on infinite alphabets. Via iterated palindromic closures (see Defini-
tion 7.1), we construct a uniformly recurrent infinite word ω on an infinite alphabet A which
givesrisetoan infinitepartitionofN intocentral sets:
CENTRALSETSDEFINEDBYWORDSOFLOWFACTORCOMPLEXITY 5
Theorem 6. Let ∆bea rightinfinitewordon afiniteor infinitealphabetAwiththepropertythat
each letter a ∈ A occurs in ∆ an infinite number of times. Let ψ denote the iterated palindromic
operatorandset ω = ψ(∆).Then
(1) ω is uniformly recurrent and closed under reversal, i.e., if v = v v ...v is a factor of ω,
1 2 k
then soisitsmirrorimagev ...v v .
k 2 1
(2) Theset ω +1 isa centralsetfor eachlettera ∈ A.
a
In particular if we take the word ∆ to be on an infinite alphabet, the sets {ω + 1} form a
(cid:12) a a
countablyinfinite(cid:12)collectionof pairwisedisjointcentralsubsetsof N.1 ∈A
(cid:12)
(cid:12)
An important open problem in the theory of substitutions is the so-called strong coincidence
conjecturewhichstatesthateachpairoffixedpointsxandyofanirreducibleprimitivesubstitution
ofPisottypesatisfythefollowingconditioncalledthestrongcoincidencecondition: Thereexista
letter a and a pair of Abelian equivalent words s,t, such that sa is a prefix of x and ta is a prefix
of y. This combinatorial condition, originally due to P. Arnoux and S. Ito, is an extension of a
similar condition considered by F.M. Dekking in [14] in the case of uniform substitutions. In this
caseDekkingprovesthattheconditionissatisfiedifandonlyiftheassociatedsubstitutivesubshift
has pure discrete spectrum, i.e., is metrically isomorphic with translation on a compact Abelian
group. The strong coincidence conjecture has been verified for irreducible primitivesubstitutions
of Pisot type on a binary alphabet by M. Barge and B. Diamond [4]. The following establishes a
linkbetween thestrongcoincidenceconjectureand central sets:
Theorem 7. Let τ bea primitivesubstitutionverifyingthestrongcoincidencecondition. Then for
anypairoffixed pointsxandy,and anyprefixuof y,wehavethatx isa centralset.
u
Our proof of Theorem 7 makes use of the so-called Dumont-Thomas numeration systems de-
(cid:12)
(cid:12)
fined by substitutions,and constitutesanew approach tothestrongcoincidenceconjecture.
The main results in this paper rely on various interactions between different areas of mathe-
matics, some of which had not previously been directly linked: They include the general theory
of combinatorics on words, the arithmetic properties of abstract numeration systems defined by
substitutions, topological dynamics, the spectral theory of symbolic dynamical systems, and the
beautiful theory, developed by Hindman, Strauss and others, linking IP-sets and central sets to
the algebraic/topological properties of the Stone-Cˇech compactification βN. We regard βN as the
collection of all ultrafilters on N. An ultrafilter may be thought of as a {0,1}-valued finitely addi-
tive probability measure defined on all subsets of N. This notion of measure induces a notion of
convergence (p-lim ) for sequences indexed by N, which we regard as a mapping from words to
n
words. This key notion of convergence allows us to apply ideas from combinatorics on words in
theframeworkofultrafilters.
The paper is organized as follows: In §2 we present some of the basic ideas and tools from
combinatoricsonwordswhichwillbeusedthroughoutthepaper. In§3weoutlinethekeyfeatures
of the algebraic and topological properties of the Stone-Cˇech compactification βN in connection
with IP-sets and central sets. Since the material in §2 may be unfamiliar to specialists in topolog-
ical semigroups and vice-versa, we take some care to explain both topics in an attempt to make
the paper moreaccessible. In §4 we analyze someconcrete exampleswhich illustratesomeof the
1ThisisaspecialcaseofapriorresultofHindman,LeaderandStrauss[25]inwhichtheyshowthateverycentral
setinNisacountableunionofpairwisedisjointcentralsets.
6 M.BUCCI,S.PUZYNINA,ANDL.Q.ZAMBONI
results mentioned abovein Theorems 2 and 3. We use nothing more than the combinatorial prop-
erties of the words considered (all generated by substitutions)and the arithmetic properties of the
underlying Dumont-Thomasnumeration system. In §5 we extend the results in §4 to all Sturmian
words,inparticularthosenotgenerated bysubstitutions. Herewemakeuseofthealgebraicprop-
erties of the semigroup βN. In §6 we consider partitions defined by the generalized Thue-Morse
substitutionandproveTheorem4. Alsoin§6weproveTheorem5byconsideringsubshiftswhich
aretopologicallyweak mixing. In §7 weconsidersomeinfinitewordson aninfinitealphabetgen-
erated by iteration of the palindromic closure operator. Using these words we construct infinite
partitions of N and prove Theorem 6. Finally in §8, after a brief review of the Dumont-Thomas
numerationsystemsdefinedbysubstitutions,wediscussaconnectionbetweencentralsetsandthe
strongcoincidenceconditionforsubstitutions.
Acknowledgements. TheauthorswouldliketothankV.BergelsonandY.Sonformanyinsightful
e-mail exchanges pertaining particularly to the last section of the paper, and for pointing out to us
thekeyfeatureusedintheproofofTheorem5relatingtopologicallyweakmixingwithproximality.
WearealsoextremelygratefultoM.Barge forpointingouttoushisjointworkwithV. Bakerand
J.KwapiszusedintheproofofTheorem4. FinallywewishtothankN.Hindmanforhiscomments
and suggestions on a preliminary version of the paper. The third author is partially supported by
a grant from the Academy of Finland and by grant no. 090038011 from the Icelandic Research
Fund.
2. WORDS AND SUBSTITUTIONS
In this section we give a brief summary of some of the basic background in combinatorics on
words.
2.1. Words &subshifts. Givenafinitenon-emptyset A (calledthealphabet),wedenoteby A ,
∗
AN and AZ respectively the set of finite words, the set of (right) infinite words, and the set of
bi-infinitewordsoverthealphabetA. Givenafiniteword u = a a ...a withn ≥ 1 anda ∈ A,
1 2 n i
we denote the length n of u by |u|. The empty word will be denoted by ε and we set |ε| = 0. We
put A+ = A −{ε}.Foreach a ∈ A,we let |u| denotethenumberofoccurrences ofthe lettera
∗ a
in u. Two words u and v in A are said to be Abelian equivalent, denoted u ∼ v, if and only if
∗ ab
|u| = |v| forall a ∈ A.It is readilyverified that∼ defines an equivalencerelationon A .
a a ab ∗
Givenaninfinitewordω ∈ AN,awordu ∈ A+ iscalledafactorofω ifu = ω ω ···ω for
i i+1 i+n
somenatural numbersiand n. WedenotebyF (n) theset ofall factors ofω oflengthn, and set
ω
F = F (n).
ω ω
n N
[∈
A factor u of ω is called right special if both ua and ub are factors of ω for some pair of distinct
lettersa,b ∈ A.Similarlyuiscalledleftspecialifbothauandbuarefactorsofω forsomepairof
distinct letters a,b ∈ A. The factor u is called bispecial if it is both right special and left special.
Foreach factoru ∈ F set
ω
ω = {n ∈ N|ω ω ...ω = u}.
u n n+1 n+u 1
| |−
Wesayω isrecurrentifforev(cid:12)eryu ∈ F thesetω isinfinite. Wesayω isuniformlyrecurrentif
(cid:12) ω u
foreveryu ∈ F theset ω is syndedic,i.e., ofboundedgap.
ω u (cid:12)
(cid:12)
(cid:12)
(cid:12)
CENTRALSETSDEFINEDBYWORDSOFLOWFACTORCOMPLEXITY 7
WeendowAN withthetopologygenerated by themetric
1
d(x,y) = where n = inf{k : x 6= y }
2n k k
whenever x = (xn)n N and y = (yn)n N are two elements of AN. Let T : AN → AN denote the
∈ ∈
shift transformation defined by T : (xn)n N 7→ (xn+1)n N. By a subshift on A we mean a pair
(X,T) where X is aclosed and T-invarian∈tsubset ofAN.∈A subshift(X,T) is said to beminimal
whenever X and the empty set are the only T-invariant closed subsets of X. To each ω ∈ AN is
associated the subshift (X,T) where X is the shift orbit closure of ω. If ω is uniformly recurrent,
thentheassociatedsubshift(X,T)isminimal. Thusanytwowordsxandy inX haveexactlythe
same set of factors, i.e., F = F . In this case we denote by F the set of factors of any word
x y X
x ∈ X.
Two points x,y in X are said to be proximal if and only if for each N > 0 there exists n ∈ N
such that
x x ...x = y y ...y .
n n+1 n+N n n+1 n+N
Two points x,y ∈ X are said to be regionallyproximal if for every prefix u of x and v of y, there
exist points x,y ∈ X with x beginning in u and y beginning in v and with x proximal to y .
′ ′ ′ ′ ′ ′
Clearly if two points in X are proximal, then they are regionally proximal. A point x ∈ X is
called distal if the only point in X proximal to x is x itself. A minimal subshift (X,T) is said to
be topologicallymixing iffor everyany pair offactors u,v ∈ F there existsa positiveintegerN
X
such that for each n ≥ N, there exists a block of the form uWv ∈ F with |W| = n. A minimal
X
subshift (X,T) is said to be topologicallyweak mixing if for every any pair of factors u,v ∈ F
X
theset
{n ∈ N|uAnv ∩F 6= ∅}
X
isthick,i.e., foreverypositiveintegerN,theset containsN consecutivepositiveintegers.
2.2. Substitutions. Many of the words and subshifts considered in this paper are generated by
substitutions. A substitution τ on an alphabet A is a mapping τ : A → A+. The mapping τ
extends by concatenation to maps (also denoted τ) A → A and AN → AN. The Abelianization
∗ ∗
ofτ isthesquarematrixM whoseij-thentryisequalto|τ(j)| ,i.e.,thenumberofoccurrencesof
τ i
iinτ(j).Asubstitutionτ issaidtobeprimitiveifthereisapositiveintegernsuchthatforeachpair
(i,j) ∈ A×A,theletterioccursinτn(j).EquivalentlyifalltheentriesofMn arestrictlypositive.
τ
In thiscaseitiswell knownthatthematrixM has asimplepositivePerron-Frobenius eigenvalue
τ
called the dilation of τ. A substitution τ is said to be irreducible if the minimal polynomial of its
dilationis equal to the characteristicpolynomialofits AbelianizationM .A substitutionτ is said
τ
tobeofPisottypeifitsdilationisaPisotnumber. RecallthataPisotnumberisanalgebraicinteger
greaterthan 1 allofwhosealgebraic conjugatesliestrictlyinsidetheunitcircle.
Let τ be a primitivesubstitutionon A. A word ω ∈ AN is called a fixed pointof τ if τ(ω) = ω,
and is called a periodic point if τm(ω) = ω for some m > 0. Although τ may fail to have a
fixed point, it has at least one periodic point. Associated to τ is the topological dynamical system
(X,T), where X is the shift orbit closure of a periodic point ω of τ. The primitivity of τ implies
that(X,T) isindependentofthechoiceofperiodicpointandis minimal.
An important example of a primitivesubstitutionis the Thue-Morse substitutiondefined by the
morphism0 7→ 01and 1 7→ 10.It has twofixed points
u = 011010011001011010010110011010...
8 M.BUCCI,S.PUZYNINA,ANDL.Q.ZAMBONI
and
v = 100101100110100101101001100101...
where u = 1 − v for every n ≥ 0. Alternatively, it can be shown that u is equal to 0 if and
n n n
only if the binary expansion of n contains an even number of 1s. For example, u = u = 0,
5 6
and in fact 5 = 101 and 6 = 110 expressed in base 2. Two other primitive substitutions we will
make reference to, first introduced some thirty years ago by F.M. Dekking and M. Keane, are the
substitutions0 7→ 001,1 7→ 11100and0 7→ 001,1 7→ 11001.Bothhavetwofixedpoints,andhave
the same Abelianization. It is shown in [15] that the subshift generated by the first substitution is
topologicallymixing,butnotthesecond. But bothare topologicallyweak mixing.
2.3. Sturmian words & generalizations. Let ω ∈ AN and set
ρ (n) = Card(F (n)).
ω ω
The function ρ : N → N is called the factor complexity function of ω. Given a minimal subshift
ω
(X,T)onA,wehaveFω(n) = Fω′(n)forallω,ω′ ∈ X andn ∈ N.Thuswecandefinethefactor
complexityρ (n) ofaminimalsubshift(X,T) by
(X,T)
ρ (n) = ρ (n)
(X,T) ω
foranyω ∈ X.
A word ω ∈ AN is periodic if there exists a positive integer p such that ω = ω for all
i+p i
indices i, and it is ultimately periodic if ω = ω for all sufficiently large i. An infinite word is
i+p i
aperiodic if it is not ultimately periodic. By a celebrated result due to Hedlund and Morse [31], a
wordisultimatelyperiodicifandonlyifitsfactorcomplexityisuniformlybounded. Inparticular,
p (n) < n for all n sufficiently large. Words whose factor complexity ρ (n) = n + 1 for all
ω ω
n ≥ 0 are called Sturmian words. Thus, Sturmian words are those aperiodic words having the
lowest complexity. Since ρ (1) = 2, it follows that Sturmian words are binary words. The most
ω
extensivelystudiedSturmianword istheso-calledFibonacci word
f = 01001010010010100101001001010010010100101001001010010···
fixed by the morphism0 7→ 01 and 1 7→ 0. Let ω ∈ {0,1}N be a Sturmian word, and let Ω denote
theshiftorbitclosureofω.Theconditionρ (n) = n+1impliestheexistenceofexactlyoneright
ω
specialandoneleft specialfactorofeach length. Clearly,givenanytwoleftspecialfactors,oneis
necessarilyaprefix oftheother. ItfollowsthatΩcontainsauniquewordall ofwhoseprefixesare
leftspecialfactorsofω.Suchawordiscalledthecharacteristicwordanddenotedω˜.Itfollowsthat
both 0ω˜,1ω˜ ∈ Ω. It is readily verified that the Fibonacci word above is a characteristic Sturmian
word. A Sturmian word ω is called singularif Tn(ω) = ω˜ for somen ≥ 1. Otherwiseit is said to
benonsingular.
Sturmianwordsadmitvarioustypesofcharacterizationsofgeometricandcombinatorialnature.
We give two such characterizations which will be used in the paper: as irrational rotations on
the unit circle and as mechanical words. In [31] Hedlund and Morse showed that each Sturmian
word may be realized measure-theoretically by an irrational rotation on the circle. That is, every
Sturmianwordisobtainedbycodingthesymbolicorbitofapointxonthecircle(ofcircumference
one) under a rotation R by an irrational angle α, 0 < α < 1, where the circle is partitioned into
α
two complementary intervals, one of length α and the other of length 1 − α. And conversely
each such coding gives rise to a Sturmian word. The quantity α is called the slope. Namely, the
rotationby angle α is the mapping R from [0,1) (identified with the unit circle) to itself defined
α
CENTRALSETSDEFINEDBYWORDSOFLOWFACTORCOMPLEXITY 9
by R (x) = {x+α}, where {x} = x−[x] is the fractional part of x. Considering a partition of
α
[0,1)intoI = [0,1−α),I = [1−α,1),defineaword
0 1
0, ifRn(ρ) = {ρ+nα} ∈ I ,
s (n) = α 0
α,ρ 1, ifRn(ρ) = {ρ+nα} ∈ I
( α 1
One can also define I = (0,1−α], I = (1− α,1], the corresponding word is denoted by s .
0′ 1′ ′α,ρ
ForaSturmianword w ofslopeα itssubshiftΩ isgivenby Ω = {s ,s |ρ ∈ [0,1)}.
α,ρ ′α,ρ
Astraightforward computationshowsthat
s (n) = ⌊α(n+1)+ρ⌋−⌊αn+ρ⌋,
α,ρ
s (n) = ⌈α(n+1)+ρ⌉−⌈αn+ρ⌉;
′α,ρ
s and s arecalled theupperand lower mechanicalwords(ofslopeα)based at ρ.
α,ρ ′α,ρ
In [1] Arnoux and Rauzy introduced a class of uniformly recurrent (minimal) sequences ω on
a m-letter alphabet of complexity ρ (n) = (m− 1)n+ 1 characterized by the following combi-
ω
natorial criterionknownas the⋆ condition: ω admitsexactly onerightspecial and oneleft special
factor of each length. We call them Arnoux-Rauzy sequences. This condition distinguishes them
from other sequences of complexity (m − 1)n + 1 such as those obtained by coding trajectories
of m-interval exchange transformations. These words are generally regarded as natural combi-
natorial generalizations of Sturmian words to higher alphabets. In particular, the Fibonacci word
generalizes tothem-bonacci word fixed bythesubstitution
σ : {0,1,...,m−1} → {0,1,...,m−1}
m ∗
givenby
0(i+1) for 0 ≤ i < m−1
σ (i) =
m 0 for i = m−1
(cid:26)
However, many of the dynamical and geometrical interpretations of Sturmian words do not
extendtothisnew classofwords (see[12]forexample).
In the subsequent sections we will considerpartitions of N defined by words. Let ω ∈ AN, and
letF denotethesetoffactorsofω.AfinitesubsetX iscalledaF-prefixcodeifX ⊂ F andgiven
anytwodistinctelementsofX,neitheroneisaprefix oftheother. A F-prefixcodeisF-maximal
if it is not properly contained in any other F-prefix code. The simplest example of a F-maximal
prefix code is the set of all elements of F of somefixed length d. Each F-maximalprefix code X
defines apartition
N = ω
u
u X
[∈ (cid:12)
Ifω is aSturmianword, thenthecorrespondingpartit(cid:12)ionis calleda Sturmianpartition.
3. ULTRAFILTERS, IP-SETS AND CENTRAL SETS
3.1. Stone-Cˇech compactification. Many of our results rely on the algebraic/topological prop-
erties of the Stone-Cˇech compactification of N. The Stone-Cˇech compactification βN of N is one
of many compactifications of N. It is in fact the largest compact Hausdorff space generated by
N. More precisely βN is a compact and Hausdorff space together with a continuous injection
i : N ֒→ βN satisfying the following universal property: any continuous map f : N → X into a
compact Hausdorff space X lifts uniquely to a continuous map βf : βN → X, i.e., f = βf ◦ i.
10 M.BUCCI,S.PUZYNINA,ANDL.Q.ZAMBONI
This universal property characterizes βN uniquely up to homeomorphism. While there are differ-
ent methods for constructing the Stone-Cˇech compactification of N, we shall regard βN as the set
ofallultrafilters onN withtheStonetopology.
Recall thata setU ofsubsetsofN iscalled an ultrafilterifthefollowingconditionshold:
• ∅ ∈/ U.
• IfA ∈ U and A ⊆ B,then B ∈ U.
• A∩B ∈ U wheneverbothAand B belongto U.
• ForeveryA ⊆ N eitherA ∈ U orAc ∈ U where Ac denotes thecomplementofA.
For every natural numbern ∈ N, the set U = {A ⊆ N|n ∈ A} is an exampleof an ultrafilter.
n
Thisdefinesaninjectioni : N ֒→ βNby: n 7→ U .Anultrafilterofthisformissaidtobeprincipal.
n
By way ofZorn’slemma,onecan showtheexistenceofnon-principal(orfree)ultrafilters.
It is customary to denote elements of βN by letters p,q,r.... For each set A ⊆ N, we set
A = {p ∈ βN|A ∈ p}. Then the set B = {A |A ⊆ N} forms a basis for the open sets (as well
◦ ◦
as a basis for the closed sets) of βN and defines a topology on βN with respect to which βN is
both compact and Hausdorff.2 It is not difficultto see that theinjection i : N ֒→ βN is continuous
and satisfies the required universal property. In fact, given a continuous map f : N → X with X
compact Hausdorff, for each ultrafilter p ∈ βN, the pushfoward f(p) = {f(n)|n ∈ p} defines an
ultrafilteronX havingauniquelimitpointβf(p).
Thereisanaturalextensionoftheoperationofaddition+onNtoβNmakingβNacompactleft-
topological semigroup. More precisely we define addition of two ultrafilters p,q by the following
rule:
p+q = {A ⊆ N|{n ∈ N|A−n ∈ p} ∈ q}.
It is readily verified that p + q is once again an ultrafilter and that for each fixed p ∈ βN, the
mappingq 7→ p+q definesacontinuousmapfromβNintoitself.3 TheoperationofadditioninβN
isassociativeandforprincipalultrafilterswehaveU +U = U .Howeveringeneraladdition
m n m+n
ofultrafiltersishighlynon-commutative. Infact itcanbeshownthatthecenterispreciselytheset
ofallprincipalultrafilters [26].
3.2. IP-setsandcentral sets. Let(S,+)beasemigroup. Anelementp ∈ S iscalledan idempo-
tentifp+p = p. We recall thefollowingresult ofEllis[20]:
Theorem 3.1 (Ellis [20]). Let (S,+) be a compact left-topological semigroup (i.e., ∀x ∈ S the
mappingy 7→ x+y is continuous). Then S containsan idempotent.
It follows that βN contains a non-principal ultrafilter p satisfying p + p = p. In fact, we could
simply apply Ellis’s result to the semigroup N−{0}. This would then exclude the only principal
idempotentultrafilter,namelyU .Fromhereon,byanidempotentultrafilterinβNwemeanafree
0
idempotentultrafilter.
We will make use of the following striking result due to Hindman linking IP-sets and idempo-
tentsin βN :
2AlthoughtheexistenceoffreeultrafiltersrequiresZorn’slemma,thecardinalityofβNis22N fromwhichitfollows
thatβNisnotmetrizable.
3Ourdefinitionofadditionofultrafiltersisthesameasthatgivenin[6]butisthereverseofthatgivenin [26]in
which A ∈ p+q if and only if {n ∈ N|A−n ∈ q} ∈ p}. In this case, βN becomesa compactright-topological
semigroup.