Central sets generated by uniformly recurrent words
aa r X i v : . [ m a t h . C O ] J a n CENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS
MICHELANGELO BUCCI, SVETLANA PUZYNINA, AND LUCA Q. ZAMBONIA
BSTRACT . A subset A of N is called an IP-set if A contains all finite sums of distinct terms of someinfinite sequence ( x n ) n ∈ N of natural numbers. Central sets, first introduced by Furstenberg usingnotions from topological dynamics, constitute a special class of IP-sets possessing rich combinatorialproperties: Each central set contains arbitrarily long arithmetic progressions, and solutions to allpartition regular systems of homogeneous linear equations. In this paper we investigate central setsin the framework of combinatorics on words. Using various families of uniformly recurrent words,including Sturmian words, the Thue-Morse word and fixed points of weak mixing substitutions, wegenerate an assortment of central sets which reflect the rich combinatorial structure of the underlyingwords. The results in this paper rely on interactions between different areas of mathematics, someof which had not previously been directly linked. They include the general theory of combinatoricson words, abstract numeration systems, and the beautiful theory, developed by Hindman, Straussand others, linking IP-sets and central sets to the algebraic/topological properties of the Stone- ˇCechcompactification of N .
1. I
NTRODUCTION
Let N = { , , , , . . . } denote the set of natural numbers, and Fin ( N ) the set of all non-emptyfinite subsets of N . Definition 1.1.
A subset A of N is called an IP-set if A contains { P n ∈ F x n | F ∈ Fin ( N ) } forsome infinite sequence of natural numbers x < x < x · · · . A subset A ⊆ N is called an IP ∗ -setif A ∩ B = ∅ for every IP-set B ⊆ N . By a celebrated result of N. Hindman [21], given any finite partition of N , at least one element ofthe partition is an IP-set. It follows from Hindman’s theorem that every IP ∗ -set is an IP-set, but theconverse is in general not true. In fact, more generally Hindman shows that given any finite parti-tion of an IP-set, at least one element of the partition is again an IP-set. In other words the propertyof being an IP-set is partition regular , i.e., cannot be destroyed via a finite partitioning. Otherexamples of partition regularity are given by the pigeonhole principle, sets having positive upperdensity, and sets having arbitrarily long arithmetic progressions (Van der Waerden’s theorem). In[20], Furstenberg introduced a special class of IP-sets, called central sets, having a substantialcombinatorial structure. The property of being central is also partition regular. Central sets wereoriginally defined in terms of topological dynamics: Definition 1.2.
A subset A ⊂ N is called central if there exists a compact metric space ( X, d ) anda continuous map T : X → X, points x, y ∈ X and a neighborhood U of y such that • y is a uniformly recurrent point in X, • x and y are proximal, Date : April 30, 2012.2000
Mathematics Subject Classification.
Primary 68R15 & 05D10.
Key words and phrases.
Sturmian words, Stone- ˇCech compactification, IP-sets, and central sets. • A = { n ∈ N | T n ( x ) ∈ U } . We say A ⊂ N is central ∗ if A ∩ B = ∅ for every central set B ⊆ N . Recall that x is said to be uniformly recurrent in X if for every neighborhood V of x the set { n | T n ( x ) ∈ V } is syndetic, i.e., of bounded gap. Two points x, y ∈ X are said to be proximal iffor every ǫ > there exists n ∈ N such that d ( T n ( x ) , T n ( y )) < ǫ. We remark that from the abovedefinition, it is not at all evident that central sets are IP-sets. We later give an alternative definition(see Definition 3.4) which makes this point clear. The equivalence between the two definitions isdue to Bergelson and Hindman [5].The question of determining whether a given subset A ⊆ N is an IP-set or a central set is typi-cally quite difficult, even if for every A, either A or its complement is an IP-set (resp. central set).It turns out that in each case this question may be reformulated in terms of whether or not the set A belongs to a certain class of ultrafilters on N (see Theorem 5.12 in [24] in the case of IP-sets and [5]in the case of central sets). But the question of belonging or not to a given (non-principal) ultrafilteris generally equally mysterious. An equivalent word combinatorial reformulation of this questionis as follows: Given a binary word ω = ω ω ω . . . ∈ { , } ∞ , put ω (cid:12)(cid:12) = { n ∈ N | ω n = 0 } and ω (cid:12)(cid:12) = { n ∈ N | ω n = 1 } . The question is then to determine whether the set ω (cid:12)(cid:12) or ω (cid:12)(cid:12) is anIP-set or central set. Of course in general, this reformulation is as difficult as the original question.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 finitestate automaton, a Toeplitz rule...) or arise as a natural coding of a reasonably simple symbolicdynamical system, then the underlying rigid combinatorial structure of the word may provide in-sight to our previous question. Furthermore, such families of words may be used to obtain simpleconstructions of central sets having additional nice properties inherited from the rich underlyingcombinatorial structure. One of our objectives here is to illustrate this latter point.Let A denote a finite non-empty set (called the alphabet) and ω = ω ω ω . . . ∈ A N . For eachfinite word u on the alphabet A we set ω (cid:12)(cid:12) u = { n ∈ N | ω n ω n +1 . . . ω n + | u |− = u } . In other words, ω (cid:12)(cid:12) u denotes the set of all occurrences of u in ω. In this paper we investigate partitions of N by sets of the form ω (cid:12)(cid:12) u defined by a uniformly recurrentword ω. Our goal is to study these partitions in the framework of IP-sets and central sets. We beginby showing that in this framework IP-sets and central sets are one and the same:
Theorem 1.
Let ω ∈ A N be uniformly recurrent. Then the set ω (cid:12)(cid:12) u is an IP-set if and only if it is acentral set. This allows us to simultaneously state our results in terms of IP-sets and central sets.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 foreach n ≥ . Their origin can be traced back to the astronomer J. Bernoulli III in 1772. A funda-mental result due to Morse and Hedlund [29] states that each aperiodic (meaning non-ultimatelyperiodic) infinite word must contain at least n + 1 factors of each length n ≥ . Thus Sturmianwords are those aperiodic words of lowest factor complexity. They arise naturally in many different
ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 3 areas of mathematics including combinatorics, algebra, number theory, ergodic theory, dynamicalsystems and differential equations. Sturmian words are also of great importance in theoreticalphysics and in theoretical computer science and are used in computer graphics as digital approxi-mation of straight lines.The next two theorems give a complete characterization of those factors u of a Sturmian word ω ∈ { , } N for which ω (cid:12)(cid:12) u is an IP-set (respectively central set). First, a Sturmian word ω iscalled singular if T n ( ω ) = ˜ ω for some n ≥ , where T denotes the shift map and ˜ ω denotesthe characteristic Sturmian word in the shift orbit closure of ω (see § nonsingular. Theorem 2.
Let ω ∈ Ω be a nonsingular Sturmian word, and u a factor of ω. Then ω (cid:12)(cid:12) u is an IP-set(resp. central set) if and only if u is a prefix of ω. Hence for every prefix v of ω and n ∈ ω (cid:12)(cid:12) v the set ω (cid:12)(cid:12) v − n is an IP ∗ -set (resp. central ∗ set). Theorem 3.
Let ω ∈ Ω be a Sturmian word such that T n ( ω ) = ˜ ω with n ≥ . Then ω (cid:12)(cid:12) u is anIP-set (or central set) if and only if either u is a prefix of ω or a prefix of ω ′ where ω ′ is the uniqueother element of Ω with T n ( ω ′ ) = ˜ ω. Some (but not all) of the results on Sturmian partitions extend to the class of Arnoux-Rauzy words,which may be regarded as natural combinatorial extensions of Sturmian words to larger alphabets[1].Using ω -bonacci and the iterated palindromic closure operator, we construct infinite partitions of N into central sets having special translation invariant properties.We also consider partitions defined by words generated by substitution rules. For instance, byconsidering partitions of N defined by words generated by the generalized Thue-Morse substitutionto an alphabet of size r ≥ , we show that Theorem 4.
For each pair of positive integers r and N there exists a partition of N = A ∪ A ∪ · · · ∪ A r such that • A i − n is a central set for each ≤ i ≤ r and ≤ n ≤ N. • For each n > N, exactly one of the sets { A − n, A − n, . . . , A r − n } is a central set. The second assertion of Theorem 4 relies on the fact that each fixed point of the generalized Thue-Morse substitution is distal.By considering partitions defined by words generating minimal subshifts which are topologicallyweak mixing (for example the subshift generated by the substitution and weprove that Theorem 5.
For each positive integer r there exists a partition of N = A ∪ A ∪ · · · ∪ A r suchthat for each ≤ i ≤ r and n ≥ , the set A i − n is a central set. The results in this paper rely on various interactions between combinatorics on words, topo-logical dynamics and the algebraic and topological properties of the Stone- ˇCech compactification β N . We regard β N as the collection of all ultrafilters on N . An ultrafilter may be thought of asa { , } -valued finitely additive probability measure defined on all subsets of N . This notion of
M. BUCCI, S. PUZYNINA, AND L.Q. ZAMBONI measure induces a notion of convergence ( p - lim n ) for sequences indexed by N , which we regardas a mapping p ∗ from words to words. This key notion of convergence allows us to apply ideasfrom combinatorics on words in the framework of ultrafilters. Acknowledgements.
The authors would like to thank V. Bergelson and Y. Son for many insightfule-mail exchanges and in particular for pointing out to us the key feature used in the proof ofTheorem 5 relating topologically weak mixing with proximality. We are also extremely gratefulto N. Hindman for his comments and suggestions on a preliminary version of this paper. The thirdauthor is partially supported by a grant from the Academy of Finland.2. W
ORDS AND SUBSTITUTIONS
In this section we give a brief summary of some of the basic background in combinatorics onwords.2.1.
Words & subshifts.
Given a finite non-empty set A (called the alphabet ), we denote by A ∗ , A N and A Z respectively the set of finite words, the set of (right) infinite words, and the set ofbi-infinite words over the alphabet A . Given a finite word u = a a . . . a n with n ≥ and a i ∈ A , we denote the length n of u by | u | . The empty word will be denoted by ε and we set | ε | = 0 . Weput A + = A ∗ − { ε } . For each a ∈ A , we let | u | a denote the number of occurrences of the letter a in u. Given an infinite word ω ∈ A N , a word u ∈ A + is called a factor of ω if u = ω i ω i +1 · · · ω i + n forsome natural numbers i and n. We denote by F ω ( n ) the set of all factors of ω of length n, and set F ω = [ n ∈ N F ω ( n ) . A factor u of ω is called right special if both ua and ub are factors of ω for some pair of distinctletters a, b ∈ A . Similarly u is called left special if both au and bu are factors of ω for some pair ofdistinct letters a, b ∈ A . The factor u is called bispecial if it is both right special and left special.For each factor u ∈ F ω set ω (cid:12)(cid:12) u = { n ∈ N | ω n ω n +1 . . . ω n + | u |− = u } . We say ω is recurrent if for every u ∈ F ω the set ω (cid:12)(cid:12) u is infinite. We say ω is uniformly recurrent iffor every u ∈ F ω the set ω (cid:12)(cid:12) u is syndedic, i.e., of bounded gap.We endow A N with the topology generated by the metric d ( x, y ) = 12 n where n = inf { k : x k = y k } whenever x = ( x n ) n ∈ N and y = ( y n ) n ∈ N are two elements of A N . Let T : A N → A N denote the shift transformation defined by T : ( x n ) n ∈ N ( x n +1 ) n ∈ N . By a subshift on A we mean a pair ( X, T ) where X is a closed and T -invariant subset of A N . A subshift ( X, T ) is said to be minimal whenever X and the empty set are the only T -invariant closed subsets of X. To each ω ∈ A N isassociated the subshift ( X, T ) where X is the shift orbit closure of ω. If ω is uniformly recurrent,then the associated subshift ( X, T ) is minimal. Thus any two words x and y in X have exactly thesame set of factors, i.e., F x = F y . In this case we denote by F X the set of factors of any word x ∈ X. ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 5
Two points x, y in X are said to be proximal if and only if for each N > there exists n ∈ N such that x n x n +1 . . . x n + N = y n y n +1 . . . y n + N . Two points x, y ∈ X are said to be regionally proximal if for every prefix u of x and v of y, thereexist 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 iscalled distal if the only point in X proximal to x is x itself. A minimal subshift ( X, T ) is said tobe topologically mixing if for every any pair of factors u, v ∈ F X there exists a positive integer N such that for each n ≥ N, there exists a block of the form uW v ∈ F X with | W | = n. A minimalsubshift ( X, T ) is said to be topologically weak mixing if for every pair of factors u, v ∈ F X theset { n ∈ N | u A n v ∩ F X = ∅} is thick, i.e., for every positive integer N, the set contains N consecutive positive integers.Many of the words and subshifts considered in this paper are generated by substitutions. A sub-stitution τ on an alphabet A is a mapping τ : A → A + . The mapping τ extends by concatenationto maps (also denoted τ ) A ∗ → A ∗ and A N → A N . Let τ be a primitive substitution on A . A word ω ∈ A N is called a fixed point of τ if τ ( ω ) = ω, and is called a periodic point if τ m ( ω ) = ω for some m > . Although τ may fail to have afixed 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 impliesthat ( X, T ) is independent of the choice of periodic point and is minimal.2.2. Sturmian words & generalizations.
Let ω ∈ A N and set ρ ω ( n ) = Card ( F ω ( n )) . The function ρ ω : N → N is called the factor complexity function of ω. Given a minimal subshift ( X, T ) on A, we have F ω ( n ) = F ω ′ ( n ) for all ω, ω ′ ∈ X and n ∈ N . Thus we can define the factorcomplexity ρ ( X,T ) ( n ) of a minimal subshift ( X, T ) by ρ ( X,T ) ( n ) = ρ ω ( n ) for any ω ∈ X. A word ω ∈ A N is periodic if there exists a positive integer p such that ω i + p = ω i for allindices i , and it is ultimately periodic if ω i + p = ω i for all sufficiently large i . An infinite word is aperiodic if it is not ultimately periodic. By a celebrated result due to Hedlund and Morse [29], aword is ultimately periodic if and only if its factor complexity is uniformly bounded. In particular, p ω ( n ) < n for all n sufficiently large. Words whose factor complexity ρ ω ( n ) = n + 1 for all n ≥ are called Sturmian words . Thus, Sturmian words are those aperiodic words having thelowest complexity. Since ρ ω (1) = 2 , it follows that Sturmian words are binary words. The mostextensively studied Sturmian word is the so-called Fibonacci word f = 01001010010010100101001001010010010100101001001010010 · · · fixed by the morphism and . Let ω ∈ { , } N be a Sturmian word, and let Ω denotethe shift orbit closure of ω. The condition ρ ω ( n ) = n + 1 implies the existence of exactly one rightspecial and one left special factor of each length. Clearly, given any two left special factors, one isnecessarily a prefix of the other. It follows that Ω contains a unique word all of whose prefixes areleft special factors of ω. Such a word is called the characteristic word and denoted ˜ ω. It follows thatboth ω, ω ∈ Ω . It is readily verified that the Fibonacci word above is a characteristic Sturmian
M. BUCCI, S. PUZYNINA, AND L.Q. ZAMBONI word. A Sturmian word ω is called singular if T n ( ω ) = ˜ ω for some n ≥ . Otherwise it is said tobe nonsingular.
Sturmian words admit various types of characterizations of geometric and combinatorial nature.We give two such characterizations which will be used in the paper: as irrational rotations onthe unit circle and as mechanical words. In [29] Hedlund and Morse showed that each Sturmianword may be realized measure-theoretically by an irrational rotation on the circle. That is, everySturmian word is obtained by coding the symbolic orbit of a point x on the circle (of circumferenceone) under a rotation R α by an irrational angle α , < α < , where the circle is partitioned intotwo complementary intervals, one of length α and the other of length − α. And converselyeach such coding gives rise to a Sturmian word. The quantity α is called the slope . Namely, the rotation by angle α is the mapping R α from [0 , (identified with the unit circle) to itself definedby R α ( x ) = { x + α } , where { x } = x − [ x ] is the fractional part of x . Considering a partition of [0 , into I = [0 , − α ) , I = [1 − α, , define a word s α,ρ ( n ) = ( , if R nα ( ρ ) = { ρ + nα } ∈ I , , if R nα ( ρ ) = { ρ + nα } ∈ I One can also define I ′ = (0 , − α ] , I ′ = (1 − α, , the corresponding word is denoted by s ′ α,ρ .For a Sturmian word w of slope α its subshift Ω is given by Ω = { s α,ρ , s ′ α,ρ | ρ ∈ [0 , } .A straightforward computation shows that s α,ρ ( n ) = ⌊ α ( n + 1) + ρ ⌋ − ⌊ αn + ρ ⌋ ,s ′ α,ρ ( n ) = ⌈ α ( n + 1) + ρ ⌉ − ⌈ αn + ρ ⌉ ; s α,ρ and s ′ α,ρ are called the upper and lower mechanical words (of slope α ) based at ρ .In [1] Arnoux and Rauzy introduced a class of uniformly recurrent (minimal) sequences ω ona m -letter alphabet of complexity ρ ω ( n ) = ( m − n + 1 characterized by the following combi-natorial criterion known as the ⋆ condition: ω admits exactly one right special and one left specialfactor of each length. We call them Arnoux-Rauzy sequences . This condition distinguishes themfrom other sequences of complexity ( m − n + 1 such as those obtained by coding trajectoriesof m -interval exchange transformations. These words are generally regarded as natural combi-natorial generalizations of Sturmian words to higher alphabets. In particular, the Fibonacci wordgeneralizes to the m -bonacci word fixed by the substitution σ m : { , , . . . , m − } → { , , . . . , m − } ∗ given by σ m ( i ) = (cid:26) i + 1) for ≤ i < m − for i = m − However, many of the dynamical and geometrical interpretations of Sturmian words do notextend to this new class of words (see [10] for example).In the subsequent sections we will consider partitions of N defined by words. Let ω ∈ A N , andlet F denote the set of factors of ω. A finite subset X is called a F - prefix code if X ⊂ F and givenany two distinct elements of X, neither one is a prefix of the other. A F -prefix code is F - maximal if it is not properly contained in any other F -prefix code. The simplest example of a F -maximalprefix code is the set of all elements of F of some fixed length d. Each F -maximal prefix code X defines a partition ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 7 N = [ u ∈ X ω (cid:12)(cid:12) u If ω is a Sturmian word, then the corresponding partition is called a Sturmian partition .3. U
LTRAFILTERS , IP-
SETS AND CENTRAL SETS
Stone- ˇCech compactification.
Many of our results rely on the algebraic/topological proper-ties of the Stone- ˇCech compactification of N , denoted β N . We regard β N as the set of all ultrafilterson N with the Stone topology.
Recall that a set U of subsets of N is called an ultrafilter if the following conditions hold: • ∅ / ∈ U . • If A ∈ U and A ⊆ B, then B ∈ U . • A ∩ B ∈ U whenever both A and B belong to U . • For every A ⊆ N either A ∈ U or A c ∈ U where A c denotes the complement of A. For every natural number n ∈ N , the set U n = { A ⊆ N | n ∈ A } is an example of an ultrafilter.This defines an injection i : N ֒ → β N by: n
7→ U n . An ultrafilter of this form is said to be principal.
By way of Zorn’s lemma, one can show the existence of non-principal (or free ) ultrafilters.It is customary to denote elements of β N by letters p, q, r . . . . For each set A ⊆ N , we set A ◦ = { p ∈ β N | A ∈ p } . Then the set B = { A ◦ | A ⊆ N } forms a basis for the open sets (as well asa basis for the closed sets) of β N and defines a topology on β N with respect to which β N is bothcompact and Hausdorff. There is a natural extension of the operation of addition + on N to β N making β N a compact left-topological semigroup. More precisely we define addition of two ultrafilters p, q by the followingrule: 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 ∈ β N , themapping q p + q defines a continuous map from β N into itself. The operation of addition in β N is associative and for principal ultrafilters we have U m + U n = U m + n . However in general additionof ultrafilters is highly non-commutative. In fact it can be shown that the center is precisely the setof all principal ultrafilters [24].3.2.
IP-sets and central sets.
Let ( S , +) be a semigroup. An element p ∈ S is called an idempo-tent if p + p = p. We recall the following result of Ellis [18]:
Theorem 3.1 (Ellis [18]) . Let ( S , +) be a compact left-topological semigroup (i.e., ∀ x ∈ S themapping y x + y is continuous). Then S contains an idempotent. It follows that β N contains a non-principal ultrafilter p satisfying p + p = p. In fact, we couldsimply apply Ellis’s result to the semigroup β N − U . This would then exclude the only principal Although the existence of free ultrafilters requires Zorn’s lemma, the cardinality of β N is N from which it followsthat β N is not metrizable. Our definition of addition of ultrafilters is the same as that given in [4] but is the reverse of that given in [24] inwhich A ∈ p + q if and only if { n ∈ N | A − n ∈ q } ∈ p } . In this case, β N becomes a compact right-topologicalsemigroup. M. BUCCI, S. PUZYNINA, AND L.Q. ZAMBONI idempotent ultrafilter, namely U . From here on, by an idempotent ultrafilter in β N we mean a freeidempotent ultrafilter.We will make use of the following striking result due to Hindman linking IP-sets and idempo-tents in β N : Theorem 3.2 (Theorem 5.12 in [24]) . A subset A ⊆ N is an IP-set if and only if A ∈ p for someidempotent p ∈ β N . It follows immediately that A is an IP ∗ -set if and only if A ∈ p for every idempotent p ∈ β N (seeTheorem 2.15 in [4]). We also note that the property of being an IP-set is partition regular.In [20], Furstenberg introduced a special class of IP-sets, called central sets, having additionalrich combinatorial properties. They were originally defined in terms of topological dynamics (seeDefinition 1.2). As in the case of IP-sets, they may be alternatively defined in terms of belongingto a special class of free ultrafilters, called minimal idempotents . To define a minimal idempotentwe must first review some basic properties concerning ideals in β N .Let ( S , +) be any semigroup. Recall that 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. Aright (resp. left) ideal I is called minimal if every right (resp. left) ideal J included in I coincideswith I . Minimal right/left ideals do not necessarily exist e.g. the commutative semigroup ( N , +) has nominimal right/left ideals (the ideals in N are all of the form I n = [ n, + ∞ ) = { m ∈ N | m ≥ n } . ) However, every compact Hausdorff left-topological semigroup S (e.g., β N ) admits a smallest twosided ideal K ( S ) which is at the same time the union of all minimal right ideals of S and the unionof all minimal left ideals of S (see for instance [24]). It is readily verified that the intersection ofany minimal left ideal with any minimal right ideal is a group. In particular, there are idempotentsin K ( S ) . Such idempotents are called minimal and their elements are called central sets:
Definition 3.3.
An idempotent p is called a minimal idempotent of S if it belongs to K ( S ) . Definition 3.4.
A subset A ⊂ N is called central if it is a member of some minimal idempotent in β N . It is called a central ∗ -set if it belongs to every minimal idempotent in β N . The equivalence between definitions 1.2 and 3.4 is due to Bergelson and Hindman in [5]. Itfollows from the above definition that every central set is an IP-set and that the property of beingcentral is partition regular. Central sets are known to have substantial combinatorial structure.For example, any central set contains arbitrarily long arithmetic progressions, and solutions toall partition regular systems of homogeneous linear equations (see for example [6]). Many ofthe rich properties of central sets are a consequence of the
Central Sets Theorem first proved byFurstenberg in Proposition 8.21 in [20] (see also [11, 6, 25]). Furstenberg pointed out that asan immediate consequence of the Central Sets Theorem one has that whenever N is divided intofinitely many classes, and a sequence ( x n ) n ∈ N is given, one of the classes must contain arbitrarilylong arithmetic progressions whose increment belongs to { P n ∈ F x n | F ∈ Fin ( N ) } . Limits of ultrafilters.
It is often convenient to think of an ultrafilter p as a { , } -valued,finitely additive probability measure on the power set of N . More precisely, for any subset A ⊆ N , we say A has p -measure , or is p -large if A ∈ p. This notion of measure gives rise to a notion ofconvergence of sequences indexed by N which is the key tool in allowing us to apply ideas from The equivalence between the two definitions is due to Bergelson and Hindman [5].
ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 9 combinatorics on words to the framework of ultrafilters. However, from our point of view, it ismore natural to define it alternatively as a mapping from words to words (see Remark 3.10). Let A denote a non-empty finite set. Then each ultrafilter p ∈ β N naturally defines a mapping p ∗ : A N → A N as follows: Definition 3.5.
For each p ∈ β N and ω ∈ A N , we define p ∗ ( ω ) ∈ A N by the condition: u ∈ A ∗ isa prefix of p ∗ ( ω ) ⇐⇒ ω (cid:12)(cid:12) u ∈ p. We note that if u, v ∈ A ∗ , ω (cid:12)(cid:12) u , ω (cid:12)(cid:12) v ∈ p and | v | ≥ | u | , then u is a prefix of v. In fact, if v ′ denotesthe prefix of v of length | u | then as ω (cid:12)(cid:12) v ⊆ ω (cid:12)(cid:12) v ′ , it follows that ω (cid:12)(cid:12) v ′ ∈ p and hence u = v ′ . Thus p ∗ ( ω ) is well defined.We note that if ω, ν ∈ A N and if each prefix u of ν is a factor of ω, then there exists an ultrafilter p ∈ β N such that p ∗ ( ω ) = ν. In fact, the set C = { ω (cid:12)(cid:12) u | u is a prefix of ν } satisfies the finite intersection property, and hence by a routine argument involving Zorn’s lemmait follows that there exists a p ∈ β N with C ⊆ p. It follows immediately from the definition of p ∗ , Definition 3.4 and Theorem 3.2 that
Lemma 3.6.
The set ω (cid:12)(cid:12) u is an IP-set (resp. central set) if and only if u is a prefix of p ∗ ( ω ) for someidempotent (resp. minimal idempotent) p ∈ β N . Lemma 3.7.
For each p ∈ β N , ω ∈ A N and u ∈ A ∗ we have p ∗ ( ω ) (cid:12)(cid:12) u = { m ∈ N | ω (cid:12)(cid:12) u − m ∈ p } where ω (cid:12)(cid:12) u − m is defined as the set of all n ∈ N such that n + m ∈ ω (cid:12)(cid:12) u . Proof.
Suppose m ∈ p ∗ ( ω ) (cid:12)(cid:12) u . Then by definition u occurs in position m in p ∗ ( ω ) . Let v denote theprefix of p ∗ ( ω ) of length | v | = m + | u | . Then, as u is a suffix of v we have ω (cid:12)(cid:12) v + m ⊆ ω (cid:12)(cid:12) u andhence ω (cid:12)(cid:12) v ⊆ ω (cid:12)(cid:12) u − m. But as v is a prefix of p ∗ ( ω ) we have ω (cid:12)(cid:12) v ∈ p and hence ω (cid:12)(cid:12) u − m ∈ p asrequired.Conversely, fix m ∈ N such that ω (cid:12)(cid:12) u − m ∈ p. Let Z be the set of all factors v of ω of length | v | = m + | u | ending in u. Then ω (cid:12)(cid:12) u − m ⊆ [ v ∈ Z ω (cid:12)(cid:12) v . It follows that there exists v ∈ Z such that ω (cid:12)(cid:12) v ∈ p. In other words, there exists v ∈ Z such that v is a prefix of p ∗ ( ω ) . It follows that u occurs in position m in p ∗ ( ω ) . (cid:3) Lemma 3.8.
For p, q ∈ β N and ω ∈ A N , we have ( p + q ) ∗ ( ω ) = q ∗ ( p ∗ ( ω )) . In particular, if p isan idempotent, then p ∗ ( p ∗ ( ω )) = p ∗ ( ω ) . Proof.
For each word u ∈ A ∗ we have that u is a prefix of ( p + q ) ∗ ( ω ) if and only if ω (cid:12)(cid:12) u ∈ p + q ⇐⇒ { m ∈ N | ω (cid:12)(cid:12) u − m ∈ p } ∈ q. On the other hand, u is a prefix of q ∗ ( p ∗ ( ω )) if and only if p ∗ ( ω ) (cid:12)(cid:12) u ∈ q. The result now followsimmediately from the preceding lemma. (cid:3)
Lemma 3.9.
For each p ∈ β N and ω ∈ A N we have p ∗ ( T ( ω )) = T ( p ∗ ( ω )) where T : A N → A N denotes the shift map.Proof. Assume u ∈ A ∗ is a prefix of p ∗ ( T ( ω )) . Then T ( ω ) (cid:12)(cid:12) u ∈ p. But T ( ω ) (cid:12)(cid:12) u = [ a ∈A ω (cid:12)(cid:12) au . It follows that there exists a ∈ A such that ω (cid:12)(cid:12) au ∈ p. Thus au is a prefix of p ∗ ( ω ) and hence u is aprefix of T ( p ∗ ( ω )) . (cid:3) Remark 3.10.
It is readily verified that our definition of p ∗ coincides with that of p - lim n . Moreprecisely, given a sequence ( x n ) n ∈ N in a topological space and an ultrafilter p ∈ β N , we write p - lim n x n = y if for every neighborhood U y of y one has { n | x n ∈ U y } ∈ p. In our case we have p ∗ ( ω ) = p - lim n ( T n ( ω )) (see [22]). With this in mind, the preceding two lemmas are well known(see for instance [8, 22]). However, our defining condition of p ∗ in Definition 3.5 does not directlyrely on the topology and so may be applied in other general settings. For instance, let Ω ⊆ A N bea subshift, and N = { n < n < n < · · · } an infinite sequence of natural numbers. For each ω ∈ Ω we put X N k = { ω n + n ω n + n . . . ω n + n k − | n ≥ } ⊆ A k . For each u ∈ X N k we define the set ω N (cid:12)(cid:12) u = { n ∈ N | ω n + n ω n + n . . . ω n + n k − = u } . Then the sets ω N (cid:12)(cid:12) u with u ∈ X N k partition N . So, given p ∈ β N , for each k ≥ there exists aunique u ∈ X N k with ω N (cid:12)(cid:12) u ∈ p. Moreover if v ∈ X N k +1 and ω N (cid:12)(cid:12) v ∈ p, then u is a prefix of v. So using the condition in Definition 3.5, each infinite sequence N and ultrafilter p ∈ β N definesa mapping Ω → Ω . Of particular interest is the case in which Ω is a uniform set in the sense of T.Kamae and N is chosen such that ω [ N ] is a super-stationary set (see [26, 27]).Another situation in which the defining condition of Definition 3.5 applies is in the context ofinfinite permutations [19]. By an infinite permutation π we mean a linear ordering on N . Then foreach finite permutation u of { , , . . . , n } we say that u occurs in position m of π if the restrictionof π to { m, m + 1 , . . . , m + n − } is equal to u. Thus we may define the set π (cid:12)(cid:12) u as the set ofall m ∈ N such that u occurs in position m in π, and again the sets π (cid:12)(cid:12) u (over all permutations u of { , , . . . , n } ) determine a partition of N . Hence each p ∈ β N defines a map from the set of allinfinite permutations into itself.In what follows, we will make use of the following key result in [24] (see also Theorem 1 in [8]and Theorem 3.4 in [4]): Theorem 3.11 (Theorem 19.26 in [24]) . Let ( X, T ) be a topological dynamical system. Then if twopoints x, y ∈ X are proximal with y uniformly recurrent, then there exists a minimal idempotent p ∈ β N such that p ∗ ( x ) = y. As a consequence we have
Theorem 3.12.
Let ω ∈ A N be a uniformly recurrent word, and let u ∈ A + . Then ω (cid:12)(cid:12) u is an IP-setif and only if ω (cid:12)(cid:12) u is a central set. ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 11
Proof.
For any A ⊂ N we have that if A is central then A belongs to some minimal idempotent p ∈ β N and hence in particular A belongs to an idempotent in β N . Hence by Theorem 3.2 wehave that A is an IP-set. Now suppose that ω (cid:12)(cid:12) u is an IP-set. Then ω (cid:12)(cid:12) u belongs to some idempotent p ∈ β N . Set ν = p ∗ ( ω ) . Then u is a prefix of ν. Also, since p is idempotent we have p ∗ ( ν ) = p ∗ ( p ∗ ( ω )) = p ∗ ( ω ) = ν. Hence for every prefix v of ν we have that ν (cid:12)(cid:12) v ∈ p and ω (cid:12)(cid:12) v ∈ p and hence ν (cid:12)(cid:12) v ∩ ω (cid:12)(cid:12) v ∈ p. In particular ν (cid:12)(cid:12) v ∩ ω (cid:12)(cid:12) v = ∅ . Hence ω and ν are proximal. Since ω is uniformlyrecurrent, it follows that ν is also uniformly recurrent. Hence by Theorem 3.11 there exists aminimal idempotent q with q ∗ ( ω ) = ν. Hence ω (cid:12)(cid:12) u ∈ q, whence ω (cid:12)(cid:12) u is central. (cid:3) Remark 3.13.
A special case of Theorem 3.11 states that if x and y are uniformly recurrent infinitewords, then x and y are proximal if and only if p ∗ ( x ) = y for some idempotent ultrafilter p ∈ β N . In the case of binary words, we could consider the following alternative notion: We say that x and y are anti-proximal if the set { n ∈ N | x n = y n } is thick. For example the two fixed points t and t of the Thue-Morse morphism are anti-proximal. In [9], together with N. Hindman we show thatfor every prefix u of t , the set t (cid:12)(cid:12) u is finite FS-big. We recall that A ⊆ N is finite FS-big if ∀ k there exists ( x i ) ki =1 such that FS ( x i ) ki =1 ⊆ A whereFS ( x i ) ki =1 = { X i ∈ F x i | F ⊆ { , , . . . , k }} . As in the case of IP-sets, the property of being finite FS-big is partition regular, i.e., if A ⊆ N isfinite FS-big and A = S ri =1 A i , then some A i is finite FS-big (see [9]). In the context of binarywords, the notions of proximality and anti-proximality are somewhat similar in the sense that inboth cases the behavior of one word is strongly affected by the behavior of the other: In case x and y are proximal, then x does as y on a thick set while if x and y are anti-proximal, then x and y play opposites on a thick set. One might ask the question of finding an analogue of Theorem 3.11characterizing anti-proximality.4. A FIRST ANALYSIS OF SOME CONCRETE EXAMPLES
The Fibonacci word.
While most of the proofs of the results announced in the Introductionrely on the algebraic and topological properties of ultrafilters on N and their links to IP-sets, webegin by analyzing concretely a few examples generated by simple substitution rules. To establishthat certain subsets of N are IP-sets, we will use nothing more than the definition of IP-sets andthe abstract numeration systems defined by substitutions first introduced by J.-M. Dumont and A.Thomas [15, 16].Let us begin with the Fibonacci infinite word f = f f f . . . ∈ { , } N given by f = 01001010010010100101001001010010010100101001001010010 · · · We set f (cid:12)(cid:12) = { n ∈ N | f n = 0 } and f (cid:12)(cid:12) = { n ∈ N | f n = 1 } . So f (cid:12)(cid:12) = { , , , , , , , , , , , . . . } and f (cid:12)(cid:12) = { , , , , , , , . . . } . This definesthe Sturmian partition N = f (cid:12)(cid:12) ∪ f (cid:12)(cid:12) . Let us denote by F n the n th Fibonacci number so that F =1 , F = 2 , F = 3 , . . . . It is well known that each positive integer n has one or more representationswhen expressed as a sum of distinct Fibonacci numbers, i.e., n = P ki =0 t i F i with t i ∈ { , } and t k = 1 . We call the associated { , } -word t k t k − · · · t a representation of n. For example, for n = 50 we obtain the following representations (arranged in decreasing lexicographic order): The lexicographically largest representation is obtained by applying the greedy algorithm . Thisgives rise to a representation of n of the form n = P ki =0 t i F i with t i +1 t i = 11 for each ≤ i ≤ k − . This representation of n is called the Zeckendorff representation [30] (a special case of theDumont-Thomas numeration system [15, 16]). We shall write Z ( n ) = t k t k − . . . t . It followsimmediately that Z ( F n ) = 10 n . The connection between Z ( n ) and the entry f n of the Fibonacciword f is given by the following well known fact: f n = 0 whenever Z ( n ) ends in and f n = 1 whenever Z ( n ) ends in . Thus f (cid:12)(cid:12) = { n ∈ N | Z ( n ) ends in } and f (cid:12)(cid:12) = { n ∈ N | Z ( n ) ends in } . We now consider the sequence ( x n ) n ∈ N given by x n = F n +1 . It is readily verified that for each A ∈ Fin ( N ) , the Zeckendorff representation of P n ∈ A x n ends in m +1 where m = min ( A ) . In fact, the symbolic sum of the individual Zeckendorff representations of each x n occurring in P n ∈ A x n does not involve any carry overs. Moreover the resulting expression does not containany occurrences of and hence is equal to the Zeckendorff representation of P n ∈ A x n . Thusevery finite sum of the form P n ∈ A x n with A ∈ Fin ( N ) belongs to f (cid:12)(cid:12) . Thus we have shown that f (cid:12)(cid:12) is an IP-set.We next verify that f (cid:12)(cid:12) is not an IP-set, and hence f (cid:12)(cid:12) is an IP ∗ -set. We will use the follow-ing general observation. Consider a subset A ⊂ N partitioned into k > non-intersecting sets: A = A ∪ A ∪ · · · ∪ A k . Suppose that for each ≤ j ≤ k there exists a positive integer N (which may depend on j ) such that whenever m , m , . . . , m N are distinct elements of A j , we have P Ni =1 m i / ∈ A . Then A is not an IP-set. In fact, if A were an IP-set, then for some ≤ j ≤ k, there would exist a sequence x < x < x < · · · contained in A j such that { P n ∈ F x n | F ∈ Fin ( N ) } ⊂ A. Let α = −√ . Then the Fibonacci word f is the orbit of the point α under irrational rotation R α on the unit circle by α. Let I be the interval [1 − α, (the interval coded by ). So n ∈ f (cid:12)(cid:12) if andonly if R nα ( α ) = { α + nα } = { ( n + 1) α } ∈ I .Fix (1 − α ) / ≤ α ′ ≤ (1 − α ) / and put I = [1 − α, − α ′ ) and I = [1 − α ′ , . Since α ′ ≤ (1 − α ) / it follows that α ′ < α. Also for j = 1 , set A j = { n ∈ N | R n ( α ) ∈ I j } . ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 13
Thus A , A partitions the set f (cid:12)(cid:12) . We now show that f (cid:12)(cid:12) is not an IP-set by showing that the sumof any three elements of A belongs to f (cid:12)(cid:12) and that the sum of any two elements of A belongs to f (cid:12)(cid:12) . Now take any n , n , n ∈ A and set x = { ( n + 1) α } , x = { ( n + 1) α } , x = { ( n + 1) α } . Then x , x , x ∈ [1 − α, − α ′ ) and n + n + n corresponds to the point { ( n + n + n + 1) α } = { x + x + x − α } . Since x , x , x ∈ [1 − α, − α ′ ) , we have { x + x + x − α } ∈ [ { − α } , { − α ′ − α } ) . Since α ′ ≥ − α it follows that − α ′ − α ≤ − α, and hence { − α ′ − α } ≤ − α, which gives { − α ′ − α } ≤ − α as required.Similarly take any n , n ∈ A . Set x = { ( n + 1) α } , x = { ( n + 1) α } so that x , x ∈ [1 − α ′ , . Then n + n corresponds to the point { ( n + n + 1) α } = { x + x − α } . Since x , x ∈ [1 − α ′ , , we have { x + x − α } ∈ [ { − α ′ − α } , − α ) . Since α ′ ≤ − α it follows that { − α ′ − α } ≥ , and hence { − α ′ − α } ≥ . The above arguments may be generalized to show that f (cid:12)(cid:12) u is an IP ∗ -set for every prefix u of f . In contrast, let us consider the sets g (cid:12)(cid:12) and g (cid:12)(cid:12) where g = 0 f = 001001010010010 . . . . Thus, g (cid:12)(cid:12) = { n ∈ N | g n = 0 } = { } ∪ { n ≥ | f n − = 0 } . Consider the sequence ( y n ) n ∈ N defined by y n = F n +2 . It is readily verified that Z ( y n −
1) =(10) n +1 and hence each y n belongs to g (cid:12)(cid:12) . Now fix A ∈ Fin ( N ) . Since the Zeckendorff represen-tation of P n ∈ A y n ends in m +2 where m = min ( A ) , it follows that Z ( P n ∈ A y n − ends in (10) m +1 , and hence P n ∈ A y n ∈ g (cid:12)(cid:12) . Thus, g (cid:12)(cid:12) is an IP-set. Similarly, it is readily verified thatfor each A ∈ Fin ( N ) , we have that P n ∈ A x n ∈ g (cid:12)(cid:12) where x n = F n +1 . Thus this time we obtainthe Sturmian decomposition N = g (cid:12)(cid:12) ∪ g (cid:12)(cid:12) in which both sets g (cid:12)(cid:12) and g (cid:12)(cid:12) are IP-sets, and hence central sets. In this case, neither g (cid:12)(cid:12) nor g (cid:12)(cid:12) is an IP ∗ -set. Once again, these arguments may beextended to show that both g (cid:12)(cid:12) u and g (cid:12)(cid:12) u are central sets for any prefix u of f and hence neither setis an IP ∗ -set.In summary, by Theorem 3.12 we have: Proposition 4.1.
Let f denote the Fibonacci word. Then for every prefix u of f the set f (cid:12)(cid:12) u is anIP ∗ -set (and hence a central ∗ set). Setting g = 0 f we have that for every prefix u of f the sets g (cid:12)(cid:12) u and g (cid:12)(cid:12) u are both IP-sets (resp. central sets). The m -bonacci word. The above analysis extends more generally to the so-called m -bonacciword. Fix a positive integer m ≥ , and let t = t t t . . . ∈ { , , . . . , m − } N denote the m -bonacci infinite word fixed by the substitution σ m : { , , . . . , m − } → { , , . . . , m − } ∗ given by σ m ( i ) = (cid:26) i + 1) for ≤ i < m − for i = m − Using the associated Dumont-Thomas numeration system, we will show:
Proposition 4.2.
Let m ≥ , and consider the partition of N given by N = [ ≤ k ≤ m − g (cid:12)(cid:12) k where g = 0 t ∈ { , , . . . , m − } N . Then for each ≤ k ≤ m − the set g (cid:12)(cid:12) k is an IP-set (resp.central set). The proof is a simple extension of the ideas outlined above in the case of the Fibonacci word.For each m ≥ , we define the m -bonacci numbers by T k = 2 k for ≤ k ≤ m − and T k = T k − + T k − + · · · + T k − m for k ≥ m. When m = 2 , these are the usual Fibonacci numbers.Each positive integer n may be written in one or more ways in the form n = P ki =1 t i T k − i where t i ∈ { , } and t = 1 . By applying the greedy algorithm, one obtains a representation of n of the form w = t t · · · t k with the property that w does not contain m consecutive ’s. Sucha representation of n is necessarily unique and is called the m - Zeckendorff representation of n, denoted Z m ( n ) (see [17]). Thus Z m ( T n ) = 10 n for n ≥ . Proof.
Fix ≤ k ≤ m − . We will show that the set g (cid:12)(cid:12) k is an IP-set. It is well known that t n = k if and only if Z m ( n ) ends in k . Hence g (cid:12)(cid:12) k = { n ∈ N | g n = k } = { n ∈ N | t n − = k } = { n ∈ N | Z m ( n − ends in k } . Consider the sequence ( x n ) n ∈ N given by x n = T mn + k . It is readily verified for any finite subset A ⊂ N , the m -Zeckendorff representation of the finite sum s = P n ∈ A x n ends in mr + k where r = min ( A ) and hence the m -Zeckendorff representation of s − ends in (1 m − r k and hence s ∈ g (cid:12)(cid:12) k as required.Having established that each of the sets g (cid:12)(cid:12) k is a central set (for ≤ k ≤ m − , it follows that no g (cid:12)(cid:12) k is an IP ∗ -set. (cid:3) ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 15
5. S
TURMIAN PARTITIONS & CENTRAL SETS
We now study more generally partitions of N generated by Sturmian words and prove theorems 2and 3. Throughout this section ω = ω ω ω . . . ∈ { , } N will denote a Sturmian word, F the setof all factors of ω, and (Ω , T ) the subshift generated by ω, where T denotes the shift map. Wedenote by ˜ ω ∈ Ω the characteristic word. Lemma 5.1. If ω, ω ′ , ω ′′ ∈ Ω are such that T n ( ω ) = T n ( ω ′ ) = T n ( ω ′′ ) , then Card { ω, ω ′ , ω ′′ } ≤ . Proof.
This follows immediately from the fact that Ω contains a unique characteristic word andthat this word is aperiodic. (cid:3) We will make use of the following key lemma which essentially says that two distinct Sturmianwords ω and ω ′ are proximal if and only if T n ( ω ) = T n ( ω ′ ) = ˜ ω for some n ≥ . Lemma 5.2.
Let ω and ω ′ be distinct elements of Ω . Then either T n ( ω ) = T n ( ω ′ ) = ˜ ω for some n ≥ , or there exists N > such that ω n ω n +1 . . . ω n + N = ω ′ n ω ′ n +1 . . . ω ′ n + N for every n ∈ N . Proof.
We will use a definition of Sturmian words via rotations, which we recalled in Section 2.Notice that ˜ ω = s α,α = s ′ α,α , and singular words correspond to the case when the orbit of a pointunder rotation map goes through the point α . If s α,ρ is non-singular, then s α,ρ = s ′ α,ρ . If w = w ′ aresingular words defined by rotations of the same point, i. e., w = s α,ρ , w ′ = s ′ α,ρ , then they differonly when they pass through − α and , i. e., in maximum two points, so there exists n ≥ such that T n ( ω ) = T n ( ω ′ ) = ˜ ω .Now consider the case when w , w ′ are defined by rotations of two different points ρ , ρ ′ , ≤ ρ <ρ ′ < . To be definite, let us consider the interval exchange of I and I for both w and w ′ . Weshould prove that there there exists N > such that ω n ω n +1 . . . ω n + N = ω ′ n ω ′ n +1 . . . ω ′ n + N for every n ∈ N . We have w i = w ′ i if and only if w i ∈ I , w ′ i ∈ I or w i ∈ I , w ′ i ∈ I . Thiscondition is equivalent to w i ∈ [1 − α − ( ρ ′ − ρ ) , − α ) ∪ [1 − ( ρ ′ − ρ ) , . The distribution of points from the orbit of any point θ under rotation by α is dense, it meansthat for every ǫ there exists N ( ǫ ) , such that after N ( ǫ ) iterations points split the interval [0 , intointervals of length less than ǫ . Putting ǫ = ρ ′ − ρ , we get that every N = N ( ǫ ) consecutive iterationsthere will be a point in every interval of length ρ ′ − ρ , so there are points in [1 − α − ( ρ ′ − ρ ) , − α ) and [1 − ( ρ ′ − ρ ) , every N iterations, and hence for every n there exists i ∈ [ n, n + N − with w i = w ′ i . (cid:3) We first consider the case of nonsingular Sturmian words:
Lemma 5.3.
Let ω ∈ { , } N be a nonsingular Sturmian word and p ∈ β N an idempotent ultrafil-ter. Then p ∗ ( ω ) = ω. Proof.
Suppose to the contrary that p ∗ ( ω ) = ω. Then since ω is nonsingular, Lemma 5.2 impliesthat for all sufficiently long factors u of ω, we have that ω (cid:12)(cid:12) u ∩ p ∗ ( ω ) (cid:12)(cid:12) u = ∅ . But, by Lemma 3.8we have p ∗ ( p ∗ ( ω )) = p ∗ ( ω ) , that is the image under p ∗ of ω and p ∗ ( ω ) coincides. It follows bydefinition of p ∗ that for every prefix u of p ∗ ( ω ) we have ω (cid:12)(cid:12) u ∈ p and p ∗ ( ω ) (cid:12)(cid:12) u ∈ p and hence ω (cid:12)(cid:12) u ∩ p ∗ ( ω ) (cid:12)(cid:12) u ∈ p, a contradiction. (cid:3) Proof of Theorem 2.
Let ω be a nonsingular Sturmian word, u a prefix of ω, and p ∈ β N anidempotent ultrafilter. Then by Lemma 5.3 u is a prefix of p ∗ ( ω ) and hence ω (cid:12)(cid:12) u ∈ p. Thus for eachprefix u of ω the set ω (cid:12)(cid:12) u belongs to every idempotent ultrafilter and hence is an IP ∗ -set. It followsthat if v ∈ F is not a prefix of ω, then ω (cid:12)(cid:12) v is not an IP-set. Finally, let v be any factor of ω and n ∈ N . Then ω (cid:12)(cid:12) v − n = T n ( ω ) (cid:12)(cid:12) v . If n ∈ ω (cid:12)(cid:12) v , then v is a prefix of T n ( ω ) from which it follows that ω (cid:12)(cid:12) v − n = T n ( ω ) (cid:12)(cid:12) v ∈ p. Hence ω (cid:12)(cid:12) v − n is an IP ∗ -set (cid:3) As a consequence of the above theorem we have
Corollary 5.4.
Let ω and ω ′ be two nonsingular Sturmian words, not necessarily of the same slope.Then for every prefix u of ω and every prefix u ′ of ω ′ we have that ω (cid:12)(cid:12) u ∩ ω ′ (cid:12)(cid:12) u ′ is an IP ∗ -set (resp.central ∗ set), in particular the intersection is infinite. We note that the assumption that ω and ω ′ be nonsingular is necessary, as for example if weconsider ω = 0 f and ω ′ = 1 f with f the Fibonacci word, then ω (cid:12)(cid:12) ∩ ω ′ (cid:12)(cid:12) = { } . Proof.
Let ω and ω ′ be two nonsingular Sturmian words, u a prefix of ω, u ′ a prefix of ω ′ , and p ∈ β N an idempotent ultrafilter. Then by Corollary ?? we have that ω (cid:12)(cid:12) u ∈ p and ω (cid:12)(cid:12) u ′ ∈ p andhence ω (cid:12)(cid:12) u ∩ ω (cid:12)(cid:12) u ′ ∈ p. Thus ω (cid:12)(cid:12) u ∩ ω (cid:12)(cid:12) u ′ belongs to every idempotent and hence is an IP ∗ -set. (cid:3) We next consider singular Sturmian words.
Lemma 5.5.
Let ω, ω ′ ∈ Ω be distinct Sturmian words such that T n ( ω ) = T n ( ω ′ ) = ˜ ω for some n ≥ . Then for every u ∈ F and every non-principal ultrafilter p ∈ β N we have ω (cid:12)(cid:12) u ∈ p ⇐⇒ ω ′ (cid:12)(cid:12) u ∈ p. In particular, p ∗ ( ω ) = p ∗ ( ω ′ ) . Proof.
Since p is a non-principal ultrafilter, we have that ω (cid:12)(cid:12) u ∈ p ⇐⇒ ω (cid:12)(cid:12) u ∩ [ N, + ∞ ) ∈ p for all N ≥ . Similarly ω ′ (cid:12)(cid:12) u ∈ p ⇐⇒ ω ′ (cid:12)(cid:12) u ∩ [ N, + ∞ ) ∈ p for all N ≥ . But for each u ∈ F , we have ω (cid:12)(cid:12) u ∩ [ n , + ∞ ) = ω ′ (cid:12)(cid:12) u ∩ [ n , + ∞ ) . The result now follows. (cid:3)
Lemma 5.6.
Let ω, ω ′ ∈ Ω be as in the previous lemma, and let p ∈ β N be an idempotentultrafilter. Then p ∗ ( ω ) = p ∗ ( ω ′ ) ∈ { ω, ω ′ } . Proof.
That p ∗ ( ω ) = p ∗ ( ω ′ ) follows from the previous lemma and the fact that idempotent ultra-filters are non-principal (see for instance [4]). By Lemma 3.9, p ∗ commutes with the shift map T, and hence T n p ∗ ( ω ) = p ∗ ( T n ω ) = p ∗ (˜ ω ) = ˜ ω where the last equality follows from Lemma 5.3. By Lemma 5.1 applied to ω ′′ = p ∗ ( ω ) it followsthat p ∗ ( ω ) = ω or p ∗ ( ω ) = ω ′ . (cid:3) Proof of Theorem 3.
Let ω ∈ Ω and n be as in the statement of the theorem. Then there exists aunique ω ′ ∈ Ω with ω ′ = ω and with T n ( ω ′ ) = ˜ ω. Suppose that ω (cid:12)(cid:12) u is an IP-set for some u ∈ F . Then by Lemma 3.6 it follows that u is a prefix of p ∗ ( ω ) for some idempotent ultrafilter p ∈ β N . It follows from Lemma 5.6 that u is a prefix of ω or a prefix of ω ′ . This proves one direction.To establish the other direction, we must show that ω (cid:12)(cid:12) u is a central set for each prefix u of ω or of ω ′ . By Theorem 3.11, there exist minimal idempotent ultrafilters p , p ∈ β N such that p ∗ ( ω ) = ω and p ∗ ( ω ) = ω ′ . The result now follows. (cid:3)
ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 17
Remark 5.7.
V. Bergelson [7] suggested to us that the above result may be related to a previouslyknown partition of N into two central sets X = { [ mx ] , m ∈ N } and Y = { [ my ] , m ∈ N } , where x and y are two irrational numbers satisfying /x + 1 /y = 1 . In fact, this partition preciselycorresponds to our partition of N into two IP-sets ω (cid:12)(cid:12) and ω (cid:12)(cid:12) where ω is of the form ω and ˜ ω isa characteristic Sturmian.This could be seen using the definition of Sturmian words via mechanical words (see Section 2for notation). For a slope α we have s α, = 0˜ ω . Let α = 1 /x and /y = 1 − α ; then s α, ( n ) = 1 ifand only if there exists an integer k such that α ( n + 1) ≥ k and αn < k . It is easy to see that thispair of equations is equivalent to n < kx ≤ n + 1 , which implies n ∈ X . We have s α, ( n ) = 0 ifand only if there exists an integer k such that α ( n + 1) < k + 1 and αn ≥ k . It is not difficult tosee that this pair of equations is equivalent to n ≤ ( n − k ) y < n + 1 , which implies n ∈ Y . Remark 5.8.
We do not know if the above results on Sturmian partitions extend to the broader classof Arnoux-Rauzy words. In fact, our proof of Lemma 5.2 relies on the geometric interpretationof Sturmian words as codings of orbits under an irrational rotation on the circle. It was shownin [10] that there exist Arnoux-Rauzy words which are not measure-theoretically conjugate to arotation on the n -torus. In this case, we do not understand which pairs of Arnoux-Rauzy words inthe subshift are proximal. 6. P ROOFS OF T HEOREMS topological flow we mean a pair ( X, f ) consisting of a compact set X together with a homeomorphism f of X. Inour framework we will consider X to be a set consisting of bi-infinite words on a finite alphabetand f the shift map. A topological flow ( X, f ) is said to be equicontinuous if for every ǫ > , there exists a δ > , such that for all x, y ∈ X, if d ( x, y ) < δ then d ( f n ( x ) , f n ( y )) < ǫ for every n ∈ Z . A topological flow ( Y, g ) is called a factor of ( X, f ) if there exists a continuous surjection π : X → Y such that π ◦ f = g ◦ π. It is well known (for instance by way of Zorn’s lemma) that every topologicalflow ( X, f ) has a maximal equicontinuous factor ( Y, g ) i.e., ( Y, g ) is an equicontinuous factor of ( X, f ) and any equicontinuous factor ( Z, h ) of ( X, f ) is also a factor of ( Y, g ) . It is also wellknown that if π : X → Y is the maximal equicontinuous factor, then for any two points x, y ∈ X we have that π ( x ) = π ( y ) if and only if x and y are regionally proximal (see [2] ). Proof of Theorem 4.
Let us fix positive integers r and N. Consider the constant length substitution τ : { , , . . . , r } → { , , . . . , r } + given by · · · r, · · · r , · · · r , . . . , r r · · · r − . In case r = 2 we have the Thue-Morse substitution on the alphabet { , } . For ≤ i ≤ r, let x ( i ) denote the i thfixed point of τ beginning in the letter i. As in the case of Thue-Morse, for i = j the words x ( i ) and x ( j ) never coincide, i.e., x ( i ) n = x ( j ) n for each n ∈ N . Let ( X, T ) denote the one-sided minimalsubshift generated by the primitive substitution τ. We will now show that each of the fixed points x ( i ) is distal. Lemma 6.1.
Let x denote any one of the fixed points x ( i ) of the substitution τ above. Then x isdistal. In particular, the two fixed points of the Thue-Morse substitution are each distal. Proof.
Let ( ˜
X, T ) denote the two-sided subshift generated by τ, and let π : ˜ X → Y denote themaximal equicontinuous factor. The substitution τ above is of Pisot type, in fact, the dilation of τ is r and all other eigenvalues are equal to . (Note that τ is not an irreducible substitution). It isproved in [3] that, for a primitive substitution of Pisot type (irreducible or not), the mapping ontothe maximal equicontinuous factor is finite to one. Thus there exists a constant C such that forany z ∈ ˜ X, there are at most C points z ′ ∈ ˜ X which are regionally proximal to z In particular, forany z ∈ ˜ X, there are at most C points z ′ ∈ ˜ X which are proximal to z. Now suppose y ∈ X is proximal to x. We will show that y = x. It is easy to see that thebi-infinite word z = x rev · x ∈ ˜ X where x rev denotes the reversal or mirror image of x, andwhere · denotes the origin. Similarly, let y ′ denote a left infinite word such that the concatenation z ′ = y ′ · y ∈ ˜ X. Since x and y are proximal, it follows that z and z ′ are proximal. Set σ = τ r . Since τ, and hence σ, are of constant length, it follows that σ ( z ′ ) is proximal to σ ( z ) . But σ ( z ) = z. Hence ( σ n ( z ′ )) n ≥ defines an infinite sequence of points in ˜ X each of which is proximal to z, andwhich in the limit tends to x ( i ) rev · x ( j ) where i is the first (meaning rightmost) letter of y ′ and j is thefirst letter of y. But since there are only finitely many points in ˜ X which are proximal to z it followsthat σ n ( z ′ ) = x ( i ) rev · x ( j ) for some n ≥ . Hence by de-substituting we obtain z ′ = x ( i ) rev · x ( j ) fromwhich it follows that y = x ( j ) . Thus both x and y are fixed points of τ which are assumed proximal.It follows that y = x and hence x is distal as required. (cid:3) Put x = x (1) . Since x is distal, so is T n ( x ) for each n ≥ . On the other hand, it is easyto see that for each positive integer n we have u ( i ) [ n ] x ∈ X, where u ( i ) [ n ] denotes the reversalof the prefix of x ( i ) of length n. Thus the r words { u (1) [ n ] x, u (2) [ n ] x, . . . , u ( r ) [ n ] x } are pairwiseproximal and each begin in distinct letters (this is because the fixed points never coincide). Finallylet ω = u (1) [ N + 1] x, and set A i = ω (cid:12)(cid:12) i for each ≤ i ≤ r. Then each A i is a central set. For each ≤ n ≤ N, we have that A i − n = T n ( ω ) (cid:12)(cid:12) i = u (1) [ N + 1 − n ] x (cid:12)(cid:12) i is a central set. But for k ≥ , we have that A i − ( N + k ) = T k − ( x ) (cid:12)(cid:12) i which is a central set if and only if T k − ( x ) begins in i. (cid:3) Proof of Theorem 5.
Fix a positive integer r. Let τ be a primitive substitution whose associatedsubshift Ω is topologically weak mixing. For instance we may take the substitution and or and (see [13]). Let ω ∈ Ω . Fix m such that ρ ω ( m ) ≥ r, and put s = ρ ω ( m ) . Let u , u , . . . , u s denote the factors of ω of length m. As pointed out to usby V. Bergelson and Y. Son [7], the weak mixing implies that the set of points in Ω proximal to ω is dense in Ω (see for instance page 184 of [20]). Thus for each factor u i there exists a word x i ∈ Ω beginning in u i and which is proximal to ω. Hence by Theorem 3.11 there exists a minimalidempotent ultrafilter p i ∈ β N such that p ∗ i ( ω ) = x i . Hence for each ≤ i ≤ s we have that ω (cid:12)(cid:12) u i ∈ p i and hence ω (cid:12)(cid:12) u i is a central set. Finally, for each positive integer n and for each ≤ i ≤ s we have that ω (cid:12)(cid:12) u i − n = T n ( ω ) (cid:12)(cid:12) u i . Again the weak mixing implies that there exists a word x ∈ Ω beginning in u i and proximal to T n ( ω ) . Hence there exists a minimal idempotent p ∈ β N such that p ∗ ( T n ( ω )) = x from which it The authors study the maximal equicontinuous factor of -dimensional substitutive real tiling spaces. To applytheir finiteness result (Theorem 4.2 in [3]), we use the fact that in our setting all the tiles have the same length, andhence proximality of points in X with respect to the shift map T implies proximality of the corresponding tilings underthe R − action. ENTRAL SETS GENERATED BY UNIFORMLY RECURRENT WORDS 19 follows that ω (cid:12)(cid:12) u i − n ∈ p and hence ω (cid:12)(cid:12) u i − n is a central set. Thus we obtain a partition of NN = s [ i =1 ω (cid:12)(cid:12) u i into s -many central sets and for each positive integer n and ≤ i ≤ s we have that ω (cid:12)(cid:12) u i − n isagain a central set. Thus, setting A i = ω (cid:12)(cid:12) u i for i = 1 , . . . , r − , and A r = s [ i = r − ω (cid:12)(cid:12) u i we obtain the desired partition of N . (cid:3)
7. I
NFINITE CENTRAL PARTITIONS OF N In this section we construct infinite partitions of N into central sets by using words on an infinitealphabet. Our construction makes use of the notion of iterated palindromic closure operator (firstintroduced in [14]): Definition 7.1.
The iterated palindromic operator ψ is defined inductively as follows: • ψ ( ε ) = ε , • For any word w and any letter a , ψ ( wa ) = ( ψ ( w ) a ) (+) .We denote with w (+) the right palindromic closure of the word w , i.e., the shortest palindromewhich has w as a prefix. For example, ψ ( aaba ) = aabaaabaa. The operator ψ has been extensively studied for its centralrole in constructing standard Sturmian and episturmian words. It follows immediately from thedefinition that if u is a prefix of v, then ψ ( u ) is a prefix of ψ ( v ) . Thus, given an infinite word ω = ω ω ω . . . on the alphabet A we can define ψ ( ω ) = lim n →∞ ψ ( ω ω ω . . . ω n ) . The following lemma summarizes the properties of ψ needed. Lemma 7.2.
Let ∆ be a right infinite word over the (finite or infinite) alphabet A and let ω = ψ (∆) .Then the following statements hold: (1) The word ω is closed under reversal, i.e., if v = v v . . . v k is a factor of ω , then so is itsmirror image v k . . . v v . (2) The word ω is uniformly recurrent. (3) If each letter a ∈ A appears in ∆ an infinite number of times, then for each prefix u of ω and each a ∈ A, we have au is a factor of ω. Proof.
Since any factor of ω is contained in some ψ ( v ) for a sufficiently long prefix v of ∆ , and ψ ( v ) is by definition a palindrome (and hence closed under reversal), the first statement is proved.The second statement is easily derived from the fact that for any finite prefix va of ∆ ( a being aletter), we have that | ψ ( va ) | ≤ | ψ ( v ) | + 1 and moreover ψ ( va ) begins and ends in ψ ( v ) . It followsthat any factor of length (for example) | ψ ( v ) | contains an occurrence of ψ ( v ) .Finally suppose each a ∈ A appears infinitely many times in ∆ . Thus for any letter a and anyprefix v of ∆ there exists a prefix of ∆ of the form vv ′ a . From the definition of ψ we then have that ψ ( vv ′ ) a is a prefix of ω and ψ ( vv ′ ) ends in ψ ( v ) , so ψ ( v ) a is a factor of ω . Since ψ ( v ) is apalindrome and ω is closed under reversal, we obtain that for any prefix v of ∆ and for any letter a , the word aψ ( v ) is a factor of ω and the third statement easily follows. (cid:3) With the preceding Lemma, we are now able to construct infinite partitions of N such that eachelement of the partition is an IP-set. Proposition 7.3.
Let ω = ψ (∆) where ∆ is a right infinite word on an infinite alphabet A with theproperty that each letter a ∈ A occurs in ∆ an infinite number of times. Then, for any a ∈ A , theset aω (cid:12)(cid:12) a is a central set, thus { ω (cid:12)(cid:12) a + 1 } a ∈A is an infinite partition of N − { } into central sets .Proof. From 7.2 we have that ω is uniformly recurrent and closed under reversal. Furthermore,since each a ∈ A occurs in ∆ an infinite number of times, (3) of 7.2 implies that the set of factorsof aω coincides with that of ω. It follows therefore that aω is uniformly recurrent as well. Let usdenote by π a the image of ω under the morphism µ a defined as follows: • µ a ( a ) = 0 , • µ a ( x ) = 1 if x = a .Since aω is uniformly recurrent for any a , it is clear that also π a is uniformly recurrent for any a . From Theorem 3.11, we then have that for any a there exists a minimal idempotent ultrafilter p a such that p ∗ a (0 π a ) = 0 π a . In particular, this means, by Lemma 3.6, that π a (cid:12)(cid:12) (which clearlycoincides with aω (cid:12)(cid:12) a by definition) is a central set for any a . The statement can then be easilyderived from the fact that aω (cid:12)(cid:12) a − ω (cid:12)(cid:12) a . (cid:3) R EFERENCES [1] P. Arnoux and G. Rauzy,
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