Coincidence for substitutions of Pisot type

COINCIDENCE FOR SUBSTITUTIONS OF PISOT TYPE

by Marcy Barge & Beverly Diamond

Abstract. — Letϕbe a substitution of Pisot type on the alphabetA={1,2, . . . , d}; ϕsatisfies thestrong coincidence conditionif foreveryi, j∈ A, there are integersk, n such thatϕⁿ(i) andϕⁿ(j) have the samek-th letter, and the prefixes of lengthk−1 of ϕⁿ(i) and ϕⁿ(j) have the same image underthe abelianization map. We prove that the strong coincidence condition is satisfied ifd= 2 and provide a partial result ford≥2.

R´esum´e(Co¨ıncidence pour les substitutions de type Pisot). — Soitϕune substitu- tion de type Pisot surun alphabetA={1,2, . . . , d}; on dit queϕsatisfait lacondition de co¨ıncidence forte si pourtouti, j ∈ A, il existe des entiersk, ntels queϕⁿ(i) et ϕⁿ(j) aient la mˆemek-i`eme lettre et les pr´efixes de longueurk−1 deϕⁿ(i) etϕⁿ(j) aient la mˆeme image parl’application d’ab´elianisation. Nous montrons que la condi- tion de co¨ıncidence forte est satisfaite pourd= 2 et nous donnons un r´esultat partiel pourd≥2.

A substitution ϕ on a finite alphabet A= {1,2, . . . , d} satisfies the strong coincidence condition if for everyi, j∈ A, there are integersk, nsuch that

(i)ϕⁿ(i) andϕⁿ(j) have the same k-th letter and

Texte re¸cu le 21novembre 2001, accept´e le 14 mai 2002

Marcy Barge, Department of Mathematics, Montana State University, Bozeman MT 59717 (USA) • E-mail :barge@math.montana.edu

Beverly Diamond, Department of Mathematics, College of Charleston, Charleston SC 29424 (USA) • E-mail :diamondb@cofc.edu

2000 Mathematics Subject Classification. — 37B10.

Key words and phrases. — Substitution, dynamical system, Pisot, coincidence conjecture, pure discrete spectrum.

The second author was partially supported by a research incentive grant from the South Carolina Commission on Higher Education.

BULLETIN DE LA SOCI ´ET ´E MATH ´EMATIQUE DE FRANCE 0037-9484/2002/619/$ 5.00 c Soci´et´e Math´ematique de France

(ii) the prefixes of length k−1 of ϕⁿ(i) and ϕⁿ(j) have the same image under the abelianization map (that is, these prefixes contain the same number of occurrences of eachi∈ A).

This condition (without the requirement on prefixes) was first introduced for substitutions of constant length by F.M. Dekking [5], who proved that for such substitutions, the condition is satisfied if and only if the associated substitutive dynamical system has pure discrete spectrum (equivalently, the substitutive system is metrically isomorphic with translation on a compact abelian group). In its current form, the strong coincidence condition is due to Arnoux and Ito. The equivalence of the strong coincidence condition and pure discrete spectrum also holds for substitutions of nonconstant length of Pisot type on two letters (unpublished work of B. Host, [8], [9], [6], and [7]). Accord- ing to [4], Host also proved that ifϕis a Pisot type substitution on two letters which is unimodular (its transition matrix has determinant +1 or −1) and which satisfies the strong coincidence condition, then the substitutive system associated withϕis metrically isomorphic with both an interval exchange and a one-dimensional toral rotation. Arnoux and Ito [1] have recently shown that every unimodular Pisot substitution ond≥2 letters satisfying the strong coin- cidence condition is metrically isomorphic with an exchange of domains inR^d−1 and is semi-conjugate to a rotation on thed−1-torus (see also [11]). At present, an additional condition is needed to improve the semi-conjugacy to metric iso- morphism. For examples of nonunimodular Pisot substitutive systems that are metrically isomorphic with translations on p-adic groups andp-adic solenoids, see [10].

No examples of Pisot type substitutions which do not satisfy the strong co- incidence condition are known, and the statement that all such substitutions satisfy this condition is known as the Strong Coincidence Conjecture. In this paper, we prove that the Strong Coincidence Conjecture holds for Pisot substi- tutions on two letters and provide a partial result for substitutions ondletters where d >2.

Theorem 1. — Letϕbe a Pisot substitution on an alphabetA={1,2, . . . , d}. There are distinct letters i, j ∈ A for which there are integers k, n such that ϕⁿ(i)andϕⁿ(j)have the same k-th letter, and the prefixes of lengthk−1 of ϕⁿ(i)andϕⁿ(j)have the same image under the abelianization map.

We introduce terminology necessary for the proof of the theorem.

Forj= 1 to d,Ij will denote the interval of length 1 inR^d given by Ij={0} × {0} × · · · ×[0,1]× · · · × {0},

where [0,1] appears in thej-th position.

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Similarly, for 1 ≤j ≤d, ej will denote the unit vector (0,0, . . . ,1, . . . ,0), where 1 appears in the j-th position. Ifx∈Z^d, the integer lattice inR^d, the vector from the origin to xis denoted by x.

Forv= (v₁, v₂, . . . , v_d),

Ij+v={v1} × {v2} × · · · ×[vj, vj+ 1]× · · · × {vd}.

The collection M of line segments joining ‘adjacent’ elements of the integer lattice is then

M=

Ij+x: 1≤j≤d, x∈Z^d . The termsegmentwill refer to an element ofM.

Let A = {1,2, . . . , d} be a finite alphabet and A^∗ the collection of finite non-empty words formed from the alphabetA.

A substitution ϕ is a map ϕ : A → A^∗; ϕ has an associated transition matrix Aϕ = A = (aij)i,j∈A in which aij is the number of occurrences of i in the word ϕ(j). (In what follows, we will also use A to denote the linear transformation on vectors ofR^dand the linear function acting on points ofR^d.) The substitutionϕisprimitiveif there isnso that for eachi, j∈ A,jappears in ϕⁿ(i);ϕ isof Pisot type, or simplyPisot, if all eigenvalues of Aother than the Perron-Frobenius eigenvalue have modulus strictly between 0 and 1.

If ϕ is Pisot, then ϕ is primitive and the (hyperbolic) linear map on R^d defined by the matrixA has stable spaceE^s of dimensiond−1 and unstable spaceE^uof dimension 1 spanned by a positive Perron-Frobenius eigenvectorvu. Also, neitherE^snorE^u contain elements of the integer lattice other than the origin. (See Chapter 1 of [3], for instance.)

Assume for the remainder of this paper that ϕis Pisot.

We define an order on Z^d as follows. Given P1, P2 ∈ Z^d, P1 ≤P2 if there are t1 ≤t2∈Rso that P1 ∈E^s+t1vu, P2 ∈E^s+t2vu (we use E^s+tvu to denote the set of pointsxsuch thatx=w+tvufor somew∈E^s). Note that ifP1, P2∈Z^d, and P1≤P2, thenAP1≤AP2.

We employ a geometrical formulation of substitution developed by Arnoux, Ito and Sano [2]. The inflation and substitution mapFϕ is defined on subsets of the setMof line segments in the following manner. Suppose that

ϕ(j) =a(j,1)a(j,2)· · ·a j, n(j)

, for 1≤j≤d.

Define

v_ϕ(j,0) = 0,

vϕ(j, i) = vϕ(j, i−1) +e_a(j,i), 1≤i≤n(j).

Then

Fϕ(Ij) =

I_a(j,i)+vϕ(j, i−1) : 1≤i≤n(j) , Fϕ(Ij+v) =Fϕ(Ij) +Av=

s+Av :s∈Fϕ(Ij) .

Finally, if M ⊆ M,

Fϕ(M) =

F(s) :s∈M .

(Strictly speaking, in the above, we should writeF_ϕ({I_j}) rather thanF_ϕ(I_j), etc., but we abuse notation and use the latter when convenient.) Note that

Fϕⁿ=F_ϕⁿ. Suppose that

S=

s_i =I_s(i)+v_i: 1≤i≤k, 1≤s(i)≤d

is a non-empty finite collection of segments, i.e., a subset of M. The setS is a strandif for each 1≤i≤k−1,

vi+1=vi+e_s(i).

Theword associated withS isw_S =s(1)s(2)· · ·s(k). For such a wordw, l(w) = (w1, w2, . . . , wd)

where for each i ∈ A, w_i = # of occurrences of i in W. The displacement vector for S is thenl(S) =l(wS) and thelength of S is|S|=|l(S)|, where

(v₁, v₂, . . . , v_d)=

1≤j≤d

|v_j|.

The endpoints of the segments belonging to S will be called vertices of S. A strandS has aninitial vertexP and aterminal vertex Qwith the property thatP ≤R≤Qfor all verticesRofS; we say thatS is a strand fromP toQ.

The segments containingP andQare theinitial andterminalsegments ofS, respectively. Note that if S is a strand fromP to Q, thenFϕ(S) is a strand from AP toAQ. Two strands S1 and S2 are coincidentif S1∩S2 =∅(that is, they share a segment), and eventually coincidentifF_ϕ^m(S1)∩F_ϕ^m(S2)=∅ for some m∈N.

The coincidence conjecture can then be stated as follows:

Coincidence Conjecture. — For 1 ≤ j, r ≤ d, Ij and Ir are eventually coincident.

We prove the following:

Theorem 1. — There are 1 ≤ j = r ≤ d so that Ij and Ir are eventually coincident.

In the following, NB(E^u) will denote {x ∈ R^d : dist(x, E^u) < B} where dist(x, y) =|x−y|, the usual Euclidean distance betweenxandy.

Lemma 2. — There areB >0 andn∈Nso that

(i)if a segmentslies inNB(E^u), then the strandF_ϕⁿ(s)lies inNB(E^u)and (ii) for any segments, there isk∈N so thatF_ϕ^nk(s) lies inNB(E^u).

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Proof. — The fact thatϕis Pisot implies that there areλ, 0< λ <1, and a constant C so that|Aⁿv|< Cλⁿ|v| for allv∈E^s. It follows that there aren andλ withλ < λ<1 so thatAⁿ(NB(E^u))⊆NλB(E^u) for allB >0.

Now suppose that s is any segment lying in NB(E^u). The initial point of F_ϕⁿ(s) lies in NλB(E^u) and the entire strand F_ϕⁿ(s) lies in NλB+m(E^u), where m is the maximum of the lengths of ϕⁿ(i) fori ∈ A. ChooseB large enough so that λB+m < λB for someλ, 0< λ <1. For suchB and any segmentsin NB(E^u),F_ϕⁿ(s) lies inNB(E^u).

If s is a segment not in NB(E^u), let B be large enough so that s lies inN_B(E^u). Ifk≥1 is such that (λ)^kB≤B, thenF_ϕ^kn(s) lies inN_B(E^u).

SinceFϕⁿ=F_ϕⁿ,Ij andIr are eventually coincident underFϕif and only if they are eventually coincident under Fϕⁿ. Thus we assume for the remainder of this paper that thenof Lemma 2 isn= 1.

A configuration of segmentsis a collectionC of segments with the property thatE^s+vintersects each element ofCin an interior point for somev. Acon- figuration of strands is a collection C of strands with the property that both the collections of initial segments and final segments form configurations of seg- ments. Thesizeof a configurationC of strands is the number of strands inC. IfP is the largest of the initial vertices of strands ofCandQis the smallest of the terminal vertices of strands ofC, we say thatCis a configuration of strands fromP toQ, or thatCextends fromP toQ, withlengthequal to|Q−P|. Given a configurationC of strands fromP toQ, there is a unique configu- ration of strandsC⁽¹⁾⊆F_ϕ(C) with the following properties:

(i)C⁽¹⁾ has the same size asC;

(ii)C⁽¹⁾ extends fromAP toAQ; and

(iii) for each strandS ∈ C⁽¹⁾, there is a strandS∈ C withS⊆F_ϕ(S).

We callC⁽¹⁾ the first iterate ofC; higher iterates are defined by C⁽²⁾= (C⁽¹⁾)⁽¹⁾, etc.

A configurationCof strands isnot eventually coincidentif each distinct pair of strands ofC is not eventually coincident.

The following is a consequence of the definitions and Lemma 2.

Lemma 3. — Suppose that C is a not eventually coincident configuration of strands of size m. The iteratesC^(k) of C satisfy the following:

(i)C^(k) is a not eventually coincident configuration of strands;

(ii) length(C^(k))→ ∞ ask→ ∞; and

(iii) there is a positive integerK so that C^(k)lies inNB(E^u)for allk≥K.

Lemma 4. — There is an integer M with the following properties:

(i) there exists a configuration of segments of size M that is not eventually coincident; and

(ii)every configuration of segments of size larger thanM is eventually coin- cident.

Proof. — For each t ∈ R, let n(t) be the number of distinct segments lying in NB(E^u) that meetE^s+tvu and letN = sup_t∈Rn(t). ThenN <∞. LetC be a configuration of not eventually coincident segments of sizemand choosek large enough so that C^(k) lies in NB(E^u). For somet, E^s+tvu contains no vertices ofC^(k)and meets a segment from each strand ofC^(k). Since the strands ofC^(k)are not coincident,m≤N. ThenN is an upper bound on the size of not eventually coincident configurations.

Proof of Theorem 1. — LetCbe a not eventually coincident configuration of segments of maximum sizeM as in Lemma 4. Without loss of generality,C^(k) lies in NB(E^u) for allk≥0. For eachk, letP0< P1<· · ·< Pm (Pi =Pi(k)) be the vertices of C^(k) that lie between the initial vertexP₀ and the terminal vertexPm. Lett0< t1<· · ·< tm(ti=ti(k)) be such thatPilies onE^s+tivu. For 1≤i≤m, let

Ci^(k) =

s:sis a segment of some strand ofC^(k) and smeetsE^s+tvu forti−1< t < ti

.

ThenC_i^(k)is a configuration of segments of sizeM (one from each strand ofC^(k)) that is not eventually coincident and that extends fromPi−1 toPi. Let$i≥1 be the number of segments ofCi^(k)that terminate atPi. Then exactly$iof the segments ofCi+1^(k) begin atPi. LetSi+1denote the collection of segments ofC^(k)i+1

issuing fromPi, letSi+1 be any collection of$i(distinct) segments beginning at Pi different fromSi+1, if there is such a collection, and consider the collection

Ci+1 = (Ci+1^(k) \ Si+1)∪ Si+1 .

Suppose that for someSi+1 ,Ci+1 is not eventually coincident. The maximality ofCi+1^(k) implies thatCi+1^(k) ∪ Si+1 is eventually coincident, hence there are

s∈ Si+1, s ∈ Si+1 , s=s, such that{s, s}is eventually coincident. Write

s=Ij+v, s=Ir+v.

It follows thatI_j andI_r are eventually coincident, and the theorem is proved.

Suppose then, for the remainder of this proof, that for each Si+1 =Si+1, Ci+1 is eventually coincident. ThenCi^(k) completely determinesCi+1^(k).

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In the above argument, the location of configurations did not play a role;

only the relative position of segments within configurations was considered.

In particular, if there arepwith 0≤i≤i+p < m andv so that Ci+p^(k) =Ci^(k)+v, then Ci+p+1^(k) =C^(k)i+1+v.

Up to translation, there are only finitely many configurations of segments that lie in NB(E^u); let b be an upper bound on the number of equivalence classes (up to translation) of such configurations of segments. Recall thatC^(k) extends fromP0 toPm(m=m(k)) and thatP0< P1<· · ·< Pm(Pi =Pi(k)) are the vertices ofC^(k)that lie betweenP₀ andP_m. The vectors

P_i+1−P_i, 0≤i≤m−1,

are bounded independently ofm, and length(C^(k)) =|P_m−P₀| → ∞ask→

∞. Thus m(k) → ∞ as k → ∞ and, for m(k) > b, there are i = i(k) ≤ b and $ withi < $ ≤b ≤m(k) such thatC^(k) =Ci^(k)+v for the integer vector v =P−Pi. It follows from the above that

C_i+p(^(k) _−i)=Ci^(k)+pv

for all positive integers p with i+p($−i) ≤m. Since i < $≤ b and v is a nonzero integer vector bounded independently ofk, some such vectorvoccurs for infinitely manyk. For thisv, it must be the case thatpv lies in N2B(E^u) for arbitrarily large p. But then v is a scalar multiple of vu, so that E^u contains a nonzero point of Z^d, an impossibility. This contradiction proves the theorem.

By graphing the fixed words, the reader can verify easily that neither of the (non-Pisot) substitutions

(1) ϕ(1) = 121, ϕ(2) = 221,

(2) τ(1) = 12, τ(2) = 21,

satisfies the Strong Coincidence Conjecture.

Acknowledgements. — We would like to thank P. Arnoux, A. Siegel and C. Holton for introducing us to the Strong Coincidence Conjecture and reading an early draft of this manuscript.

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