Central sets and substitutive dynamical systems

0.18 MB·English

by  Marcy Barge

CENTRAL SETS AND SUBSTITUTIVE DYNAMICAL SYSTEMS
MARCYBARGEANDLUCAQ.ZAMBONI
3 ABSTRACT. Inthispaperweestablishanewconnectionbetweencentralsetsandthestrongcoin-
1 cidenceconjectureforfixedpointsofirreducibleprimitivesubstitutionsofPisottype. Centralsets,
0
firstintroducedbyFurstenbergusingnotionsfromtopologicaldynamics,constituteaspecialclass
2
ofsubsetsofNpossessingstrongcombinatorialproperties:Eachcentralsetcontainsarbitrarilylong

n
arithmeticprogressions,andsolutionstoallpartitionregularsystemsofhomogeneouslinearequa-
a
tions. We giveanequivalentreformulationofthe strongcoincidenceconditionin termsofcentral
J
setsandminimalidempotentultrafiltersintheStone-CˇechcompactificationβN.Thisprovidesanew
4
2 arithmeticalapproachtoanoutstandingconjectureintilingtheory,thePisotsubstitutionconjecture.
Theresultsinthispaperrelyoninteractionsbetweendifferentareasofmathematics,someofwhich

] hadnotpreviouslybeendirectlylinked:Theyincludethegeneraltheoryofcombinatoricsonwords,
O
abstractnumerationsystems,tilings,topologicaldynamicsandthealgebraic/topologicalproperties
C ofStone-CˇechcompactificationofN.
.
h
t
a 1. INTRODUCTION
m
An important open problem in the theory of substitutions is the so-called strong coincidence
[
conjecture: Itstatesthateachpairoffixedpointsxandy ofanirreducibleprimitivesubstitutionof
1
Pisottypearestronglycoincident: ThereexistaletteraandapairofAbelianequivalentwordss,t,
v
5 such that sa is a prefix of x and ta is a prefix of y.This combinatorialcondition,originallydueto
4
P. Arnoux and S. Ito, is an extensionofa similarconditionconsidered by F.M. Dekkingin [15]in
7
5 thecaseofuniform substitutions. In thiscaseDekking provesthat theconditionis satisfied bythe
. “pure base” of the substitution if and only if the associated substitutivesubshift has pure discrete
1
0 spectrum, i.e., is metrically isomorphic with translation on a compact Abelian group. The strong
3
coincidence conjecture has been verified for irreducible primitive substitutions of Pisot type on a
1
: binaryalphabet in[2]and isotherwisestillopen.
v
Thestrongcoincidenceconjecture is linkedto diffraction properties ofone-dimensionalatomic
i
X
arrangements in the following way. It is shown in [18] and [27] that an atomic arrangement de-
r
a termined by a substitution has pure point diffraction spectrum (i.e., is a perfect quasicrystal) if
and only if the tiling system associated with the substitution has pure discrete dynamical spec-
trum. The Pisot substitution conjecture asserts that the dynamical spectrum of the tiling system
associated with an irreducible Pisot substitution has pure discrete dynamical spectrum. For the
latter to hold, it is necessary, and conjecturally sufficient, for the substitution to satisfy the strong
coincidencecondition.
Inthispaperweestablishalinkbetweenthestrongcoincidenceconjectureandcentralsets,orig-
inallyintroducedbyFurstenbergin[20]. Moreprecisely,weobtainanequivalentreformulationof
theconjecturein termsofminimalidempotentsin theStone-Cˇech compactificationβN.
Date:April30,2012.
2000MathematicsSubjectClassification. Primary68R15&05D10.
Key words and phrases. Sturmian words, abstract numeration systems, IP-sets, central sets and the Stone-Cˇech
compactification.
1
2 M.BARGEANDL.Q.ZAMBONI
Let N = {0,1,2,3,...} denote the set of natural numbers, and Fin(N) the set of all non-empty
finite subsets of N. A subset A of N is called an IP-set if A contains { x |F ∈ Fin(N)}
for some infinite sequence of natural numbers x < x < x ··· . A suPbsent∈FA ⊆n N is called an
0 1 2
IP∗-setifA∩B 6= ∅foreveryIP-set B ⊆ N.In[20],FurstenbergintroducedaspecialclassofIP-
sets, called central sets, having a substantial combinatorial structure. Central sets were originally
defined in termsoftopologicaldynamics:
Definition1.1. 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
forevery ǫ > 0 thereexistsn ∈ N suchthat d(Tn(x),Tn(y)) < ǫ.
It is not evident from the above definition that central sets are IP-sets. The connection between
thetwoliesinthealgebraicandtopologicalpropertiesoftheStone-CˇechcompactificationβN.We
regard βN as thecollection of all ultrafilters on N.There is a natural extensionof theoperation of
addition + on N to βN making βN a compact left-topological semigroup. Via a celebrated result
ofEllis[19], βN containsidempotents,i.e., ultrafilters p ∈ βNsatisfyingp+p = p.
A striking result due to Hindman links IP-sets and idempotents in βN : A subset A ⊆ N is an
IP-set if and only ifA ∈ p forsomeidempotent p ∈ βN (see Theorem 5.12 in [24]). Thus A is an
IP∗-set ifandonlyifA ∈ p foreveryidempotent p ∈ βN (seeTheorem2.15 in[7]).
ItfollowsthatgivenanyfinitepartitionofanIP-set,atleastoneelementofthepartitionisagainan
IP-set. In otherwordsthepropertyofbeingan IP-set is partitionregular,i.e.,cannotbedestroyed
via a finite partitioning. Other examples of partition regularity are given by the pigeonhole prin-
ciple, sets having positive upper density, and sets having arbitrarily long arithmetic progressions
(Van derWaerden’s theorem).
In [8], Bergelson and Hindman showed that central sets too may alternatively be defined in
terms of a special class of ultrafilters, called minimal idempotents. Every compact Hausdorff left-
topological semigroup S admits a smallest two sided ideal K(S) which is at the same time the
unionofallminimalrightidealsofS andtheunionofallminimalleftidealsofS (seeforinstance
[24]). It is readily verified that the intersection of any minimal left ideal with any minimal right
idealisagroup. Inparticular,thereareidempotentsin K(S).Suchidempotentsarecalledminimal
and their elements are called central sets, i.e., A ⊂ N is a central set if it is a member of some
minimalidempotentin βN.
ItnowfollowsthateverycentralsetisanIP-setandthatthepropertyofbeingcentralispartition
regular. Central sets are known to have substantial combinatorial structure. For example, any
central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular
systemsofhomogeneouslinearequations(seeforexample[9]).
An ultrafiltermay bethoughtofas a {0,1}-valuedfinitely additiveprobabilitymeasure defined
onallsubsetsofN.Thisnotionofmeasureinducesanotionofconvergence(p-lim )forsequences
n
indexedbyN,whichweregardasamappingfromwordstowords. Thiskeynotionofconvergence
allowsustoreformulatethestrongcoincidenceconjecture in termsofcentral sets:
CENTRALSETSANDSUBSTITUTIVEDYNAMICALSYSTEMS 3
Theorem 1. Letτ beanirreducibleprimitivesubstitutionofPisottype. Thenforanypairoffixed
pointsx andy of τ thefollowingareequivalent:
(1) xand y arestronglycoincident.
(2) There exists a minimal idempotent p ∈ βN such that y = p-lim Tn(x) where T denotes
n
theshiftmap.
(3) Foranyprefixuof y,theset ofoccurrences ofuin xis acentralset.
Theorem1assertsthatforanirreducibleprimitivesubstitutionofPisottype,thestrongcoincidence
condition is equivalent to the condition that the idempotent ultrafilters in βN permute the fixed
pointsofthesubstitution.
In the context of uniformly recurrent words, IP-sets and central sets are one and the same (see
Theorem 4.9provedin[13]); thusweobtain:
Corollary1. Letτ beanirreducibleprimitivesubstitutionofPisottype. Thenforanypairoffixed
points x and y of τ, x and y are strongly coincident if and only if for any prefix u of y, the set of
occurrences of uinx isan IP-set.
Since IP-sets may be defined arithmetically in terms of finite sums of distinct terms of infinite
sequences (xn)n∈N of natural numbers, Corollary 1 provides an arithmetical approach to solving
the strong coincidence conjecture. To this end, we show that certain abstract numeration systems
firstintroducedbyJ.-M.DumontandA.Thomasin[16,17]provideausefularithmetictooltothe
conjecture.
Acknowledgements. ThesecondauthorissupportedinpartbyaFiDiPrograntfrom theAcademy
ofFinland.
2. WORDS AND SUBSTITUTIONS
Givenafinitenon-emptysetA(calledthealphabet),wedenotebyA∗,AN andAZ respectively
the set of finite words, the set of (right) infinite words, and the set of bi-infinite words over the
alphabetA. Givenafinitewordu = a a ...a withn ≥ 1anda ∈ A,wedenotethelengthnof
1 2 n i
u by |u|. The empty word will be denoted by ε and we set |ε| = 0. We put A+ = A∗ −{ε}. For
each a ∈ A, we let |u| denote the numberof occurrences of the letter a in u. Two words u and v
a
in A∗ are said to be Abelian equivalent, denoted u ∼ v, if and only if |u| = |v| forall a ∈ A.
ab a a
It isreadily verifiedthat ∼ defines an equivalencerelationon A∗.
ab
Givenaninfinitewordω ∈ AN,awordu ∈ A+ iscalledafactorofω ifu = ω ω ···ω for
i i+1 i+n
somenatural numbers iand n. WedenotebyF (n) theset ofall factors of ω oflengthn, and set
ω
F = F (n).
ω [ ω
n∈N
Foreach finiteword uon thealphabet Aweset
ω = {n ∈ N|ω ω ...ω = u}.
(cid:12)u n n+1 n+|u|−1
(cid:12)
In otherwords, ω denotes theset ofalloccurrences of uin ω.
(cid:12)u
Wesayω isrec(cid:12)urrentifforeveryu ∈ F thesetω isinfinite. Wesayω isuniformlyrecurrent
ω (cid:12)u
ifforevery u ∈ F thesetω issyndedic,i.e., ofbo(cid:12)undedgap.
ω (cid:12)u
(cid:12)
4 M.BARGEANDL.Q.ZAMBONI
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-invariantsubset of AN.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. Thusanytwowords xandy 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
A point x ∈ X is called distalif the only point in X proximal to x is x itself. A minimalsubshift
(X,T) is said to be topologically mixing if for every any pair of factors u,v ∈ F there exists a
X
positive integer N such that for each n ≥ N, there exists a block of the form uWv ∈ F with
X
|W| = n. A minimal subshift (X,T) is said to be topologically weak mixing if for every pair of
factors u,v ∈ F theset
X
{n ∈ N|uAnv ∩F 6= ∅}
X
isthick,i.e., foreverypositiveinteger N,theset containsN consecutivepositiveintegers.
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 of τ is
the square matrix M whose ij-th entry is equal to |τ(j)| , i.e., the number of occurrences of i in
τ i
τ(j). A substitution τ is said to be primitive if there is a positiveinteger n such that for each pair
(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.
3. ULTRAFILTERS, IP-SETS AND CENTRAL SETS
3.1. Stone-Cˇech compactification. 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. This
universal property characterizes βN uniquely up to homeomorphism. While there are different
CENTRALSETSANDSUBSTITUTIVEDYNAMICALSYSTEMS 5
methods for constructing the Stone-Cˇech compactification of N, we shall regard βN as the set of
all ultrafilterson NwiththeStonetopology.
Recall thata setU ofsubsetsofN iscalled an ultrafilterifthefollowingconditionshold:
• ∅ ∈/ U.
• IfA ∈ U and A ⊆ B,then B ∈ U.
• A∩B ∈ U wheneverboth Aand B belongto U.
• Forevery A ⊆ N eitherA ∈ U orAc ∈ U where Ac denotes thecomplementofA.
Forevery natural number n ∈ 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.1 It is not difficultto see that theinjectioni : N ֒→ βN is continuous
and satisfies the required universal property. In fact, given a continuous map f : N → X with X
compact Hausdorff, foreach ultrafilter p ∈ βN,the pushfoward f(p) = {f(A)|A ∈ p} generates
auniqueultrafilter βf(p)on X.
Thereisanaturalextensionoftheoperationofaddition+onNtoβNmakingβNacompactleft-
topological semigroup. More precisely addition of two ultrafilters p,q is defined 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.2 TheoperationofadditioninβN
isassociativeandforprincipalultrafilterswehave U +U = U .Howeveringeneraladdition
m n m+n
ofultrafiltersishighlynon-commutative. Infact itcanbeshownthatthecenterispreciselytheset
ofallprincipalultrafilters [24].
3.2. IP-setsandcentral sets. Let(S,+)beasemigroup. Anelementp ∈ S iscalledan idempo-
tentifp+p = p. We recall thefollowingresult ofEllis[19]:
Theorem 3.1 (Ellis [19]). 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 βN−U . This would then exclude the only principal
0
idempotentultrafilter,namelyU .Fromhereon,byanidempotentultrafilterin βNwemeanafree
0
idempotent ultrafilter. The following striking result due to Hindman establishes a link between
IP-sets and idempotentultrafilters:
Theorem 3.2 (Theorem 5.12 in [24]). A subset A ⊆ N is an IP-set if and only if A ∈ p for some
idempotentp ∈ βN.
1AlthoughtheexistenceoffreeultrafiltersrequiresZorn’slemma,thecardinalityofβNis22N fromwhichitfollows
thatβNisnotmetrizable.
2Ourdefinitionofadditionofultrafiltersisthesameasthatgivenin[7]butisthereverseofthatgivenin [24]in
which A ∈ p+q if and only if {n ∈ N|A−n ∈ q} ∈ p}. In this case, βN becomesa compactright-topological
semigroup.
6 M.BARGEANDL.Q.ZAMBONI
A subset I ⊆ S is called a right (resp. left) ideal if I +S ⊆ I (resp. S +I ⊆ I). It is called
a two sided ideal if it is both a left and right ideal. A right (resp. left) ideal I is called minimal if
everyright(resp. left)ideal J includedinI coincides withI.
Minimalright/leftidealsdonotnecessarilyexiste.g. the commutativesemigroup(N,+)hasno
minimal right/left ideals (the ideals in N are all of the form I = [n,+∞) = {m ∈ N|m ≥ n}.)
n
However,every compactHausdorffleft-topologicalsemigroupS (e.g., βN) admitsa smallesttwo
sidedidealK(S)whichisatthesametimetheunionofallminimalrightidealsofS andtheunion
of all minimal left ideals of S (see for instance [24]). It is readily verified that the intersection of
anyminimalleft idealwithany minimalrightidealis agroup. In particular,thereare idempotents
inK(S). Such idempotentsare called minimaland theirelementsare called centralsets:
Definition 3.3. Anidempotentp iscalleda minimalidempotentof S if itbelongstoK(S).
Definition 3.4. A subset A ⊂ N is called central if it is a member of some minimal idempotent in
βN.Itis calleda central∗-set ifitbelongsto everyminimalidempotentinβN.
The equivalence between definitions 1.1 and 3.4 is due to Bergelson and Hindman in [8]. It
follows from the above definition that every central set is an IP-set and that the property of being
central is partition regular. Central sets are known to have substantial combinatorial structure.
For example, any central set contains arbitrarily long arithmetic progressions, and solutions to
all partition regular systems of homogeneous linear equations (see for example [9]). Many of
the rich properties of central sets are a consequence of the Central Sets Theorem first proved by
Furstenberg in Proposition 8.21 in [20] (see also [14, 9, 25]). Furstenberg pointed out that as
an immediate consequence of the Central Sets Theorem one has that whenever N is divided into
finitely many classes, and a sequence (xn)n∈N is given, one of the classes must contain arbitrarily
longarithmeticprogressionswhoseincrementbelongsto { x |F ∈ Fin(N)}.
Pn∈F n
3.3. Limits of ultrafilters. It is often convenient to think of an ultrafilter p as a {0,1}-valued,
finitelyadditiveprobabilitymeasureonthepowersetof N.Moreprecisely,foranysubset A ⊆ N,
we say A has p-measure 1, or is p-large if A ∈ p. This notion of measure gives rise to a notion of
convergenceofsequencesindexedbyNdenotedp-lim .Fromourpointofview,itismorenatural
n
to consider it as a mapping p∗ from words to words. More precisely, let A denote a non-empty
finiteset.
Definition 3.5. For each p ∈ βN and ω ∈ AN,we define p∗(ω) ∈ AN by thecondition: u ∈ A∗ is
a prefixofp∗(ω)⇐⇒ω ∈ p.
(cid:12)u
(cid:12)
Wenotethatifu,v ∈ A∗,ω ,ω ∈ pand|v| ≥ |u|,thenuisaprefixofv.Infact,ifv′denotesthe
(cid:12)u (cid:12)v
prefix of v of length |u| then(cid:12) as ω(cid:12) ⊆ ω , it follows that ω ∈ p and hence u = v′. Thus p∗(ω)
(cid:12)v (cid:12)v′ (cid:12)v′
is well defined. It follows immed(cid:12)iately f(cid:12)rom the definition o(cid:12)f p∗, Definition 3.4 and Theorem 3.2
that
Lemma3.6. Thesetω isanIP-set(resp. centralset)ifandonlyifuisaprefixofp∗(ω)forsome
(cid:12)u
idempotent(resp. minim(cid:12) alidempotent)p ∈ βN.
Itisreadilyverifiedthatourdefinitionofp∗ coincideswiththatofp-lim .Moreprecisely,given
n
a sequence (xn)n∈N in a topological space and an ultrafilter p ∈ βN, we write p-limnxn = y if
for every neighborhood U of y one has {n|x ∈ U } ∈ p. In our case we have p∗(ω) = p-
y n y
lim (Tn(ω))(see[22]). With thisin mind,we obtain(see forinstance[11, 22]):
n
Lemma 3.7. Let p,q ∈ βNand ω ∈ AN.Then
CENTRALSETSANDSUBSTITUTIVEDYNAMICALSYSTEMS 7
• p∗(T(ω)) = T(p∗(ω))whereT : AN → AN denotestheshiftmap.
• (p+q)∗(ω) = q∗(p∗(ω)).In particular,if pis anidempotent,thenp∗(p∗(ω)) = p∗(ω).
We will make use of the following key result in [24] (see also Theorem 1 in [11] and Theorem
3.4in [7]):
Theorem 3.8(Theorem19.26in[24]). Let(X,T)beatopologicaldynamicalsystem. Theniftwo
points x,y ∈ X are proximal with y uniformly recurrent, then there exists a minimal idempotent
p ∈ βNsuchthatp∗(x) = y.
4. STRONG COINCIDENCE CONDITION
Letn ≥ 2beapositiveintegerandsetA = {1,2,...,n}.Aprimitivesubstitutionτ : A → A+
is said to satisfy the strong coincidence condition if and only if any pair of fixed points x and y
are strongly coincident, i.e., we can write x = scx′, and y = tcy′ for some s,t ∈ A+, c ∈ A,
and x′,y′ ∈ A∞ with s ∼ t. This combinatorial condition, originally due to P. Arnoux and
ab
S. Ito, is an extension of a similar condition considered by F.M. Dekking in [15] in the case of
constant length substitutions, i.e., when |τ(a)| = |τ(b)| for all a,b ∈ A. Every such substitution
τ has an algorithmically determined “pure base” substitution and Dekking proves that the strong
coincidenceconditionissatisfiedbythepurebaseifandonlyifthesubstitutivesubshiftassociated
with τ has pure discrete spectrum, i.e., is metrically isomorphic with translation on a compact
Abelian group. The Thue-Morse substitutionis equal to its pure base and clearly does not satisfy
the strong coincidence condition - in fact the two fixed points disagree in each coordinate. It is
conjectured howeverthat if τ is an irreducibleprimitivesubstitutionof Pisot type, then τ satisfies
the strong coincidence condition. This conjecture is established for binary primitivesubstitutions
of Pisot type in [2]. Otherwise the conjecture remains open for substitutionsdefined on alphabets
of size greater than two. Substitutions of Pisot type provide a framework for non-constant length
substitutionsinwhichthestrongcoincidenceconditionisnecessary(and,conjecturally,sufficient)
for pure discrete spectrum (see [2, 3, 4, 5]). We now establish the following reformulation of the
strongcoincidenceconditioninterms ofcentral sets:
Theorem 4.1. Let τ be an irreducible primitive substitution of Pisot type. Then for any pair of
fixed pointsx andy of τ thefollowingareequivalent:
(1) xand y arestronglycoincident.
(2) xand y areproximal.
(3) There existsa minimalidempotentp ∈ βNsuchthaty = p∗(x).
(4) Foranyprefixuof y,theset x isa centralset.
(cid:12)u
(cid:12)
Remark 4.2. For a general primitive substitution we always have that (1) =⇒ (2) =⇒ (3) =⇒
(4). But in general in the non-Pisot setting, these conditions need not be equivalent: For instance,
the two fixed points of the uniform substitution a 7→ aaab, b 7→ bbab are proximal but do not
satisfy the strong coincidence condition. V. Bergelson and Y. Son [10] showed that the fixed
points of a 7→ aab, b 7→ bbaab satisfy (4) but not (1), (2) and (3). It would be interesting to
understandingeneralunderwhatconditionsdotheidempotentultrafilterspermutethefixedpoints
ofsubstitutions.
Proof. We first show that (1) =⇒ (2) =⇒ (3) =⇒ (4). Clearly (2) is immediate from the defini-
tion of strong coincidence. By Theorem 3.8 we have that (2) implies (3) and hence (4). In what
follows we will show that (4) =⇒ (1).To this end, we introduce the machinery of “strand space”
8 M.BARGEANDL.Q.ZAMBONI
(a convenient presentation of tiling space) which will allow us to apply results developed for the
R-action onstrandspace totheshiftactionon words.
Letτ bean irreducibleprimitivesubstitutionofPisottypeonthealphabetA = {1,...,n}with
AbelianizationM, dilationλ and normalizedpositiverighteigenvector w = (w ,...,w )t:
1 n
Mw = λw; ||w|| = w2 +···+w2 = 1.
q 1 n
Wedenoteby(X,T)thesubshiftofAN obtainedbytakingtheshiftorbitclosureofaright-infinite
τ-periodic word x and by (X¯,T) the subshift of AZ on the shift orbit closure of a bi-infinite τ-
periodicword x¯. (We requirethat x¯ x¯ beafactorof x.)
−1 0
From the irreducible Pisot hypothesis, the spectrum of M is nonzero and disjoint from the unit
circle. Thus M isalinearisomorphismand Rn decomposesas an invariantdirect sum
Rn = Eu ⊕Es
with Eu = {tw : t ∈ R} and Mkv → 0 as k → ∞ for all v ∈ Es. Let pru : Rn → Eu and
prs : Rn → Es denote the corresponding projections. We denote by e the standard unit vector
i
e = (0,...,1,...,0)t and by σ the arc σ = {te : 0 ≤ t ≤ 1}, i = 1,...,n. By a segment, we
i i i i
will mean an arc of the form v+σ with v ∈ Rn: i is its type, v is its initialvertex, and v+e is
i i
its terminal vertex. A strand is a collection of segments {vi + σxi}i∈Z with the property that the
terminal vertex of v + σ equals the initial vertex of v + σ for all i ∈ Z. Such a strand
i xi i+1 xi+1
follows pattern x = (x ) ∈ AZ. (Note that a strand is a collection, rather than a sequence, so if a
i
strand follows pattern x, it also follows pattern Tk(x) for all k ∈ Z.) Let S denote the collection
ofallstrands in Rn.
Givenasegmentσ = v+σ , let
i
Σ (σ) = {Mv+σ ,Mv+e +σ ,...,Mv+e +···+e +σ },
τ i1 i1 i2 i1 il−1 il
provided τ(i) = i ...i , and defineΣ : S → S by
1 l τ
Σ (S) = ∪ Σ (σ).
τ σ∈S τ
Notethat ifS followspattern x, then Σ (S) followspattern τ(x). ForR > 0, let CR = {v ∈ Rn :
τ
||prs(v)|| < R} denote the R-cylinder about Eu and let SR = {S ∈ S : σ ⊂ CR forall σ ∈ S}
denotethecollectionofallstrands in CR.
Forthefollowing,seeLemma5.1 in[4].
Lemma 4.3. There is R sothat Σ (SR) ⊂ SR forallR ≥ R . Furthermore,if S ∈ SR forsome
0 τ 0
R, thenthereis k ∈ N so thatΣk(S) ∈ SR0.
τ
WithR as in thelemma,thestrandspaceassociatedwith τ is theglobalattractor
0
S∗ = ∩ Σk(SR0).
τ k≥0 τ
Thereis anaturalmetrictopologyon S∗ in whichstandsS and S′ arecloseifthere is v ∈ Rn and
τ
r > 0, ||v||small and r large, so that σ ∈ S and pru(σ) ⊂ {tw : |t| ≤ r} =⇒ v+σ ∈ S′. That
is, S and S′ are close if, after a small translation, S and S′ agree in a large neighborhood of the
origin. Let d denote a metric inducing this topology. There is a continuous R-action on S∗ given
τ
by
(S,t) 7→ S −tw = {σ −tw : σ ∈ S}.
CENTRALSETSANDSUBSTITUTIVEDYNAMICALSYSTEMS 9
This R-action may not be minimal. To clean things up a bit, we define the minimal strand space
associatedwithτ, S , tobetheω-limitset ofanyS ∈ S∗:
τ τ
S = ∩ cl{S −tw : t ≥ T}.
τ T≥0
The resulting space does not depend on which S is chosen, and the R-action on S is minimal.
τ
Furthermore, Σ restricted to S isahomeomorphismand thedynamicsinteractby
τ τ
Σ (S −tw) = Σ (S)−λtw.
τ τ
Everystrand intheminimalstrandspacefollowsthepattern ofsomeword in X¯.
StrandsS,S′ ∈ Sτ areproximal(withrespecttotheR-action),denotedS ∼p S′,ifinft∈Rd(S−
tw,S′ −tw) = 0.
Lemma 4.4. There is B ∈ N so that ♯{S′ ∈ S : S′ ∼ S} ≤ B, for all S ∈ S . If lim d(S −
τ p τ i→∞
t w,S′ − t w) = 0, then there is r → ∞ so that S − t w and S′ − t w have exactly the same
i i i i i
segmentsin ther -ballcentered at0.
i
Proof. Theorem 4.2 of [1] asserts that the map g : S → X onto the maximal equicontinuous
τ max
factoroftheR-actiononS isboundedlyfinite-to-one. Asproximallityistrivialforequicontinuous
τ
actions,{S′ ∈ S : S′ ∼ S} ⊂ g−1(g(S)).
τ p
The second assertion follows from [6]: S is a Pisot family substitution tiling space and in all
τ
such spaces proximality and strong proximality (the condition that the strands exactly agree in
largeballs)arethesamerelation. (cid:3)
Lemma 4.5. SupposethatS ∈ S follows patternx ∈ X¯. Then thereis a uniqueS′ ∈ S such that
Σ (S′) = S and S′ followsapatterninX¯.
τ
Proof. It follows from Mosse´’s recognizability result ([28]), that there is x′ ∈ X¯ so that τ(x′) =
Tk(x) for some k ∈ Z, and such an x′ is unique, up to shift. Let S be any strand that follows
1
pattern x′. Then Σ (S ) follows pattern x so there is v ∈ Rn so that Σ (S ) = S + v : let
τ 1 1 τ 1 1
S′ = S − M−1v . If S′′ is a strand that follows a pattern in X¯ and Σ (S′′) = S, that pattern
1 1 τ
must be (a shift of) x′, so S′′ = S′ +v forsome v ∈ Rn. Then S = S +Mv, and it must be that
Mv = 0,sincexis notT-periodic(see[26]). Thus v = 0 and S′ isunique. (cid:3)
Lemma 4.6. Given x = (xi) ∈ X¯ there is a unique S = Sx = {vi + σxi}i∈Z ∈ Sτ with the
properties: S followspatternx; andv ∈ Es.
0
Proof. Let y = (y ) ∈ X¯ be τ-periodic, say τm(y) = y with m > 0. Let S be the strand
i y
S = {...,−e −e +σ ,−e +σ }∪{σ ,e + σ ,e + e +σ ,...}. Then
y y−2 y−1 y−2 y−1 y−1 y0 y0 y1 y0 y1 y2
Σkm(S ) = S forall k ∈ N, so S ∈ S , and S follows pattern y. Since X¯ is minimalunderthe
τ y y y τ y
shift, thereare k ∈ N with Tki(y) → x. Let t = ni−1w . Then S −t w → S ∈ S with S
i i Pj=0 yj y i x τ x
as desired.
SupposethatS′ ∈ S alsofollowspatternxandhasinitialvertexon Es. ThenS′ = S +v for
x τ x x
somev ∈ Es. ItfollowsfromLemma4.6that,foreachk ∈ N,Σ−k(S′) = Σ−k(S )+M−kv ∈ S .
τ x τ x τ
Ifvisnot0,thereisk largeenoughsothat||M−kv|| > 2R . ButthenΣ−k(S′)andΣ−k(S )can’t
0 τ x τ x
bothbeinS ⊂ SR0. ThusS isunique. (cid:3)
τ x
Lemma 4.7. There is K ∈ N with the property: if x = (xi),x′ = (x′i) ∈ X¯ and Sx,Sx′ ∈ Sτ are
as in Lemma 4.6 with initial vertices v ,v′ ∈ Es, and kth vertices v = v + k−1e ,v′ =
0 0 k 0 Pj=0 xj k
v0′ +Pjk=−01ex′j, k ∈ N, then ♯{vk −vk′ : k ∈ N} ≤ K, and, furthermore, ♯{pru(vk −vk′ ) : k ∈
N,x,x′ ∈ X} ≤ K, wherein thislastboundthex,x′ varyover allofX.
10 M.BARGEANDL.Q.ZAMBONI
Proof. Let P = maxi∈{1,...,n}||prs(z)||. The strands Sx −v0 and Sx′ −v0′ are in S2R0 and vk −
v ,v′ −v′ ∈ Zn ∩{z = (z ,...,z )t : n z = k}. Let¯z be the closest point in Zn ∩{z =
(z0,.k..,z )0t : n z = k}1to Eu,nThenP||pi=r1s(¯zi )|| ≤ P, v k−v −¯z ,v′ −v′ −¯z ∈ {z =
(z1,...,zn)t ∈PZni=:1 i n z = 0,||z|| ≤ 2R +Pk}, and pru(kv −0v′ ) =k prku((v0−vk−¯z )−
(v1′ −v′ n−¯z )). ThPusi=K1 =i (♯{z = (z ,...0,z )t ∈ Zn : nk z =k0,||z|| ≤ 2kR +0P})2kwill
k 0 k 1 n i=1 i 0
work. P (cid:3)
Proposition4.8. Let x,y ∈ X. Then:
(1) {x′ ∈ X : x′ is proximalwithx} isfinite;and
(2) ifx andy areproximaland fixed byτ, thenx andy arestronglycoincident.
Proof. Suppose x′ is proximal with x. Extend x and x′ to bi-infinite words in X¯ - call these
extensions x and x′. There is k → ∞ so that the jth coordinates of Tki(x) and Tki(x′) are
i
the same for all |j| ≤ i. Let Sx,Sx′ ∈ Sτ be as in Lemma 4.6. By Lemma 4.7, there is a
subsequencekij sothatvkij −vk′ij ≡ visconstant(here vk,vk′ denotethekthverticesofSx,Sx′).
Let pru(v ) = t w and pru(v) = tw. After passing to a subsequence, we may assume that
k j
ij
Sx −tjw → S ∈ Sτ and Sx′ −(ti −t)w → S′ ∈ Sτ. Then there is y ∈ X¯ so that S and S′ both
follow pattern y and have initial vertices on Es. By the uniqueness in Lemma 4.6, these initial
vertices, which differ by prs(v), must be the same. Thus, v = pru(v), and Sx ∼p Sx′ +v in Sτ.
ByLemma4.6,thereareonlyfinitelymanyvthatcanarisethisway,andbyLemma4.4,thereare
only finitely many strands in S proximal with S . Thus, theset {x′ ∈ X : x′ isproximalwith x}
τ x
isalso finite.
For (2), we may take extensions of x and y in X¯ that are τ-periodic: say τm(x) = x and
τm(y) = y withm > 0. ThestrandsS andS inS haveinitialverticesattheorigin,arefixedby
x y τ
Σm,and,bytheabove,thereistsothatS ∼ S +tw. ThenS = Σkm(S ) ∼ Σkm(S +tw) =
τ x p y x τ x p τ y
Σkm(S ) + λkmtw = S + λkmtw. Since there are only finitely many strands proximal with S
τ y y x
(and y is not T-periodic, again by [26]), t = 0. Thus S ∼ S and there is k ∈ N so that S and
x p y x
S notonlyhavethesamekthvertex,butalsosharetheirkthsegment(Lemma4.4). Thatis,xand
y
y are stronglycoincident. (cid:3)
We return to the proof that (4) =⇒ (1) in Theorem 4.1. Let x and y be fixed points of τ and
suppose that for every prefix u of y the set x is a central set. This means that for every prefix u
(cid:12)u
ofy thesetx belongstosomeminimalidem(cid:12) potentp ∈ βN.Thecollection
(cid:12)u u
(cid:12) P = {p∗(x)|u isaprefix of y}
u
consists of infinite words in X each proximal to x. By (1) of Proposition 4.8, the set P is finite.
Moreoversince p∗(x) → y as |u| → +∞ (since u is a prefix of p∗(x)), it follows that y ∈ P and
u u
hence y is proximal to x. Whence by (2) of Proposition 4.8 we deduce that x and y are strongly
coincident. (cid:3)
Wenowrecall thefollowingresultfrom [13]:
Theorem 4.9. Let ω ∈ AN be uniformlyrecurrent. Then theset ω is an IP-set if and only if it is
(cid:12)u
a centralset. (cid:12)
Combiningtheorems4.1 and 4.9weobtain
Corollary 4.10. Let τ be an irreducible primitive substitution of Pisot type. Then for any pair of
fixed pointsxand y of τ thefollowingareequivalent, x and y are stronglycoincident. if andonly
ifforanyprefixu ofy,theset x isan IP-set.
(cid:12)u
(cid:12)