Substitution dynamical systems : algebraic

characterization of eigenvalues

Sebastien FerencziLaboratoire de Mathematiques Discretes

CNRS - UPR 9016163 av. de Luminy, F13288 Marseille Cedex 9

France

Christian MauduitLaboratoire de Mathematiques Discretes

CNRS - UPR 9016

163 av. de Luminy, F13288 Marseille Cedex 9France

Arnaldo Nogueira

Instituto de MatematicaUniversidade Federal de Rio de Janeiro

Caixa postal 68.530, 21.945-970 Rio de Janeiro RJBrazil

May 11, 2005

November 7, 1994

Abstract

We give a necessary and sufficient condition allowing to compute

explicitly the eigenvalues of the dynamical system associated to any

0

primitive substitution ; this yields a simple criterion to determine

wether a substitution is weakly mixing ; we apply these results to

examples where the matrix has two expanding and two contracting

eigenvalues.

1

Primitive substitutions, or morphisms of languages on a finite alphabet,form extensively studied examples of dynamical systems, see [QUE]. Thecomputation of eigenvalues of the associated spectral operator (or dynamicalsystem) is the first step towards the understanding of the geometrical struc-ture of this system. They are well known in several classes of examples, suchas substitutions of constant length ([DEK]), or the ones where the dominanteigenvalue of the matrix of the substitution is a Pisot number.

The computation for the general case has been the object of a number ofpapers : in [HOS], Host proved the continuity of the eigenfunctions, and gavea necessary and sufficient condition for a complex number to be an eigenvalueof the system ; a similar condition was given simultaneously by Livshits([LIV], see also [LIV-VER]), though he used the language of adic systemsrather than the one of harmonic analysis. In both cases, the fundamentalrole played by the eigenvalues of the matrix of the substitution was apparent,but explicit conditions (in the sense that they are algorithmically computablefor a given substitution) were given only for some limited classes of examples.Other examples were given by Solomyak ([SOL1], [SOL2]) ; in the particularcase where the characteristic polynomial of the matrix of the substitutionis irreducible, Solomyak gave in [SOL3] an explicit sufficient condition for acomplex number to be an eigenvalue of the system, and an explicit necessaryand sufficient condition for the system to be weakly mixing.

The aim of this work is to solve this problem for any primitive substi-tution with a non-periodical fixed point : in the present paper, we give anexplicit necessary and sufficient condition allowing us to compute the eigen-values of the system. This uses another reformulation of the condition ofHost, in the language of finite rank, with a more geometrical proof, and thenotion of Pisot family, with techniques developped by Mauduit in his studyof normal sets associated to substitutive sequences of integers ([MAU]). Itis easier to express and to apply when the expanding eigenvalues of the ma-trix are simple, which is a weaker condition than the irreducibility of thecharacteristic polynomial ; it is more complicated in the general case, buttakes a simple form if we want only to know the direction of the irrationaleigenvalues, or to determine whether the system is weakly mixing (the lasttwo criterions could be deducted from [MAU], but the proof we give hereis more complete). We then proceed to use our condition on two examples,where the matrix has two expanding and two contracting eigenvalues (hencethe dominant eigenvalue is not a Pisot number) : one is proved to be weakly

2

mixing, while for the other, the system has eigenvalues and we give them allexplicitely (a very similar system has been studied in [SOL3], but without acomplete description of the eigenvalues).

The authors wish to thank Bernard Host for many helpful discussionsduring their work.

1 Basic definitions and notations

Let A = (a1, ..., as) be a finite alphabet, and A⋆ the set of finite words on A.For a word w, we denote by |w| its length. The concatenation of two wordsv and w is denoted by vw.

Let σ be a substitution on A, that is an application from A to A⋆, whichextends into a morphism of A⋆ by the rule σ(vw) = σ(v)σ(w).

In all what follows, we assume that σ has a fixed point, denoted by u (ifσ has more than one fixed point, we chose one) ; T is the shift defined onAN by (Tx)n = xn+1 ; X is the closed orbit of u under T . The substitutionσ extends into a continuous map from X to X.

By the eigenvalues and eigenvectors of the dynamical system (X, T ),we mean complex numbers λ and (borelian) measurable functions f suchthat

f ◦ T = λf.

If there are no eigenvectors except the constant functions, we say that thesystem is weakly mixing.

We denote by [a] the cylinder of X defined by (x0 = a) ; for a wordw = w0...wr, we denote by [w] the cylinder (x0 = w0, ..., xr = wr). Note thatσ[a] ⊂ [σa], but this inclusion may be strict : for σ0 = 010 et σ1 = 01, wecheck that σ[0] is the cylinder [0100] and that σ[1] is the cylinder [0101].

σ is said to be primitive if there exists k > 0 such that σka contains bfor each couple (a, b) of elements in A.

If σ is primitive, the dynamical system (X, T ) is uniquely ergodic, that isadmits only one invariant probability measure, which we denote by µ. Fur-thermore, there exists an at most countable set D, invariant under T and σ,

3

such that T is a homeomorphism of X/D.

Through all this paper, σ will be a primitive substitution, and u a non-periodical fixed point. The matrix of σ is defined by M = ((mi,j)) where mi,j

is the number of times the letter aj appears in the word σai : then the vector|σna1|, ...|σnas| is computed simply by applying the matrix Mn to the vectore = (1, ...1). Let P be the characteristic polynomial of M , and θ1, ...θt itseigenvalues (not to be confused with the eigenvalues of the dynamical system); let di be the multiplicity of θi. We will denote by I the s×s-identity matrix.

The property of bilateral recognizability is proved in [MOS] :

Lemma 1 Let σ be primitive and u a non-periodical fixed point ; let E bethe subset of N defined by

E = (0) ∪ (|σ (u0...up−1)| , p ∈ N) .

Then there exists l > 0 such that, if n ∈ E and un−l...un+l−1 = um−l...um+l−1,then m ∈ E.

We call l the index of recognizability of σ and E the set of bars of σ.

2 Host’s criterion revisited

2.1

The following lemma is proved in [QUE] under a stronger hypothesis, butthe proof remains the same.

Lemma 2 Let σ be primitive and u a non-periodical fixed point ; for integersn, p and q and letters a and b in A, if 0 ≤ p < |σna| and 0 ≤ q < |σnb|, then

T pσn[a] ∩ T qσn[b] ∩ X/D = ∅

except if p = q and σna = σnb.

Proof

Suppose that T pσn[a]∩T qσn[b]∩X/D 6= ∅, with q > p ; then σnx = T q−pσnyfor a point x in [a] and a point y in [b] ; as all these points are in X/D, we

4

have still T−lσnx = T−l+q−pσny, if l is the index of recognizability of σn. LetE be the set of bars of σn.

We have y = lim T miu and x = lim T piu ; let si = |σn(u0...umi−1)| and ti =|σn(u0...upi−1)| ; si and ti are in E, and u−l+si

...ul+si−1 = u−l+ti+q−p...ul+ti+q−p−1,hence the ti + q − p are still in E. But the hypotheses on q and p and thedefinition of E force q = p.

If p = q and σna 6= σnb while the considered intersection is nonempty,there cannot exist any j such that (σna)j 6= (σnb)j , and then for example σnais a strict prefix of σnb, and a new application of the recognizability propertygives the conclusion. CQFD

We need now to define some particular sequences of integers, associatedto σ and explicitly computable for each given σ, which shall play the keyrole in the characterization of eigenvalues of the dynamical system ; thefollowing lemma, whose proof is straightforward from the definitions, thedecomposition of the matrix M under Jordan form, and the expression ofthe projectors on the eigenspaces, sums up what we need to know aboutthem :

Lemma 3 Let σ be a primitive substitution, u a non-periodical fixed pointof σ and (X, T ) the associated dynamical system ; then there exist an integer1 < r ≤ s and an integer N such that, for every n ≥ N , the set σnA hasexactly r elements.

We call a return word any word b1...bz−1 appearing in u and satisfying

∀n ≥ N, σnbz = σnb1, σnbj 6= σnb1 ∀1 < j < z.

For a given return word C = b1...bz−1, we define the associated return time

sequence byrn(C) = |σnb1| + ... + |σnbz−1|

for all n ≥ 1.Then there exists only a finite collection of return words C1, .., Cq, which

all appear inside the words of the set (σkaσkb, a ∈ A, b ∈ A), where k isgiven in the definition of primitivity. And there exists an s × q-matrix L,explicitly computable for any given σ, such that, if Rn denotes the vector(rn(C1), ..., rn(Cq)), we have

Rn = LMne.

5

Hence

rn(Cj) =t∑

i=1

di−1∑

h=0

Wi,h,Cj(θi)Vh(n)θn

i ,

, for some polynomials Wi,h,Cj∈ Q[X] and Vh(n) = n(n − 1)...(n − h + 1),

V0(n) = 1. Whenever θi and θk are algebraically conjugate, then di = dk, andWi,h,C = Wk,h,C for any 0 ≤ h ≤ di − 1 and any return word C. If θi is asimple eigenvalue, di = 1 and Wi,0,Cj

(θi) is the j-th coordinate of the vector1

P ′(θi)L∏

v 6=i(M − θvI)e.

2.2

We are now ready to give a new version of Host’s criterion :

Proposition 1 The complex number λ of modulus 1 is an eigenvalue of thedynamical system (X, T ) if and only if

λrn(C) → 1 (1)

when n → +∞ , for every return word C.

Proof

To simplify notations, we suppose first of all that N = 1 and r = s, that isσ and its iterates are injective on letters.

Stacks

We build a sequence of Rokhlin stacks generating the system (X, T ). Atstage n, there are s stacks, of bases Fn,a = σn[a] and heights hn,a = |σna| ,for every element a in A. The sets T iFn,a, a ∈ A, 0 ≤ i ≤ hn,a − 1, form,because of lemma 2, a partition Qn of X (up to a set of measure zero). Bylemma 6 of [HOS], these partitions increase towards the σ-algebra of X ; thesystem (X, T ) is of finite rank, and the sets

τn,a = ∪hn,a−1i=0 T iFn,a

are the Rokhlin stacks of the system, the τn,a being referred to as then-stacks ; the Fn,a are the bases and the T iFn,a the levels of the stacks.

The Rokhlin stacks can be naturally built by recurrence in the followingway : if

σai = a(i, 1)....a(i, c(i))

6

for 1 ≤ i ≤ r and if d(j) is the number of couples (i, k), 1 ≤ i ≤ r, 1 ≤k ≤ c(i) such that a(i, k) = aj, we cut each base Fn,aj

into d(j) pieces(Dn,aj ,z, z ∈ U(j)). The stack τn+1,ai

has as a base some Dn,a(i,1),z(i,1), andits successive levels are : its hn,a(i,1) − 1 first iterates, some Dn,a(i,2),z(i,2), itshn,a(i,2) − 1 iterates,..., some Dn,a(i,c(i)),z(i,c(i)), its hn,a(i,c(i)) − 1 iterates, thez(i, j), 1 ≤ j ≤ r, taking exactly d(i) different values. At the first stage, eachstack τ0,a is made with only one level, the cylinder [a]. The constraints

µ(Dn,a(i,k),z(i,k)) = µ(Dn,a(i,l),z(i,l))

for all 1 ≤ k ≤ c(i), 1 ≤ l ≤ c(i), and

∑

z∈U(j)

µ(Dn,aj ,z) = µ(Fn,aj)

for all j, ensure that the measures of the Dn,aj ,z, and hence of the new stacks,are determined recursively.

The set ∪hn,aj

k=O T kDn,aj ,z, for any fixed z, is called a column of the stackτn,aj

. Note that the measures of the stacks, and hence of the columns (pro-vided they are always numbered in the same order) are independent of n.

Necessary condition

Let f be an eigenvector of T for the eigenvalue λ ; for fixed ǫ and for n largeenough, there exists a function fn, constant on each atom of the partitionQn, such that

‖ fn − f ‖2< ǫ.

Let C = b1...bz−1 be a return word, appearing in one σpa. When we builtgeometrically the stack τn+p,a it includes a column of τn,b1 , topped with acolumn of τn,b2 ,..., etc, until we finish with a column of τn,bz

, which is thesame as τn,b1 . And this pattern will appear, by primitivity, in all the m-stacks for m large enough.

Hence we found a full column Dn of τn,b1 , of fixed measure, µ(Dn) = γ,such that, for each point x of Dn, T rn(C)x is again in Dn, and on the samelevel.

But then∫

Dn

∣

∣

∣T rn(C)fn − λrn(C)fn

∣

∣

∣

2<∫

Dn

∣

∣

∣T rn(C)fn − T rn(C)f∣

∣

∣

2+∫

Dn

∣

∣

∣λrn(C)f − λrn(C)fn

∣

∣

∣

2< 2ǫ.

7

And, on Dn, T rn(C)fn = fn as fn is constant on the levels of τn,b1 , so we get

∣

∣

∣λrn(C) − 1∣

∣

∣ <4ǫ

γ ‖ f ‖2

for all n large enough, hence the criterion is necessary.

Sufficient condition

Suppose that λ is a complex number of modulus one satisfying (1). Then,by Lemma 1 of [HOS], for each return time sequence rn(C), we have

∣

∣

∣λrn(C) − 1∣

∣

∣ < Kρn

for some ρ < 1 and some constant K.We can then define fn in the following way :

• fn = 1 on the basis of τn,a1 ,

• for each b 6= a1, we chose a word a1b1...bqb appearing in u, independentof n, and we put fn = λ|σna1|+|σnb1|+...|σnbi| on the basis of τn,b,

• fn is defined on the other levels of the n-stacks by f(Tx) = λf(x) forevery x except those on the top levels.

Then we have|fn+1(x) − fn(x)| <

∣

∣

∣λsn(x) − 1∣

∣

∣ ,

where sn(x) is a finite sum of return time sequences. Hence

‖ fn+1 − fn ‖∞< Cρn,

and the fn converge uniformly to a function which will be an eigenfunctionfor λ ; hence the criterion is sufficient.

General case for N and rIn that case, let (A1, ...Ar) be the partition of A according to the differentvalues of σna for all n ≥ N ; this partition is independent of n. Then then-stacks will have as bases F ′

n,i = ∪a∈Aiσn[a]. The geometrical construction

has to be made in two steps : the stacks τn,a are built recursively in exactlythe same way as before, but the stacks generating the system are the τ ′

n,i =∪a∈Ai

τn,a ; the columns Dn used in the proof are of course columns of theτ ′n,i. Everything else in the proof is unchanged. QED

8

2.3

We would like to mention that the technics developed here may be appliedto self-inducing interval exchange transformations.

Let π be a permutation on m letters and ξ = (ξ1, ..., ξm) be a vector inthe positive cone R+m. Set ξπ = (ξπ−11, ..., ξπ−1m), |ξ| =

∑mi=1 ξj, a0(ξ) = 0,

ai(ξ) =∑i

j=1 ξj, Ii(ξ) = [ai−1, ai[ for 1 ≤ i ≤ m. Let T = T(π,ξ) be the mapdefined by

Tx = x + aπi−1(ξπ) − ai−1(ξ)

for x ∈ Ii.T is called an interval exchange transformation, as it translates the

interval Ii(ξ) to the interval Iπi(ξπ) for each 1 ≤ i ≤ m.

Let T = T(π,ξ), and I = [0, b[ be a subinterval of [0, |ξ|[. Let S be thePoincare first return map induced by T on I, namely

Sx = T n(x)x

where n(x) = min (n > 0; T nx ∈ I). For suitably chosen I, S will be anexchange of the same number of intervals as T ; in this case, we write T =T(π′,ξ′), with ξ′ = Bξ where B is a positive integer-valued m × m matrix.

We say that T is a self-inducing interval exchange transformation

if there exists a subinterval I such that

π′ = π and ξ′ = βξ

where 0 < β < 1.

For basic references on the subject, we mention the works of Keane [KEA]and Veech [VEE]. The weak mixing property of interval exchange transfor-mations is still conjectural, although Nogueira and Rudolph [NOG-RUD]have proved that, for m ≥ 3, when π is an irreducible permutation, then forLebesgue-almost all ξ in R+m the interval exchange transformation T(π,ξ) istopologically weakly mixing, which means that all continuous eigenfunctionsof T are constant.

As for self-inducing interval exchange transformations, they are metricallyisomorphic to substitutions, and hence all the results in this paper may be

9

applied to them ; but also, it follows straight from [NOG-RUD] that, if T isa self-inducing interval-exchange trnsformation and if

λ = e2πiα, for α ∈ R

is an eigenvalue for T(π,ξ), then there exists an integer-valued vector (depend-ing on α) v = (v1, ..., vm) ∈ Zm such that

limn→+∞

(tB)n(αe − v) = 0

where B is the matrix in the definition of the induced map, and satisfiesξ = B(βξ). This implies that

α =v1ξ1 + ... + vmξm

|ξ| .

3 Algebraic characterization

3.1

Back to substitutions, we begin by giving an easy characterization of therational eigenvalues of the dynamical system.

Proposition 2 A number of the form e2πip

q is an eigenvalue of the dynamicalsystem (X, T ) if and only if q divides rn(C) for any return word C and anyn large enough.

Proof

It follows immediately from proposition 1. QED

3.2

We now give the characterization of the eigenvalues of the dynamical sys-tem in its fullest generality. For this, we need the following lemma aboutPisot families, first introduced and studied in [MAU], using the argumentsdevelopped by Pisot in [PIS], see also [SAL] :

10

Lemma 4 Let (η1, ...ηk) be algebraic numbers of modulus greater or equal toone, which are roots of multiplicity di of the polynomial

P (X) = Xd + r1Xd−1 + ... + rd =

l∏

i=1

(X − ηi)di ∈ Z[X].

Let

R(X) = XdP(

1

X

)

,

and d′ = sup1≤i≤k di. Then the two following properties are equivalent :

(i)k∑

i=1

di−1∑

h=0

βi,hVh(n)ηni → 0 mod 1

when n → +∞, for βi,h ∈ C, 1 ≤ i ≤ k, 0 ≤ h ≤ di − 1, with βi,di6= 0

for each i, and Vh ∈ Z[X]⋆, of degree h.

(ii) a) whenever η is conjugate to some ηi and η 6= η1,...,η 6= ηk, then|η| < 1,((η1, ...ηk) are then said to form a Pisot family),and

b) there exists Q in Z[X] such that , for any 1 ≤ i ≤ k, the rationalfractions

Q(X)

R(X)d′

anddi−1∑

h=0

βi,h

Πh(ηiX)

(1 − ηiX)h+1

have the same simple elements of denominator (X − 1ηi

)h+1 forany h ≥ 0, where Πh is defined by

+∞∑

n=0

Vh(n)zn =Πh(z)

(1 − z)h+1.

Proof

By Lemma 1 of [MAU], Πh is in Z[X], has degree h and satisfies Πh(1) 6= 0.

11

(i) implies (ii) :We have

k∑

i=1

di−1∑

h=0

βi,hVh(n)ηni = an + en,

where an ∈ Z and en → 0 when n → +∞.. Hence

f(z) =+∞∑

n=0

enzn

is defined and analytic for |z| < 1, and has no poles of modulus one (here infact it is defined on |z| ≤ 1 as the convergence is geometric).

Also, for all i, P (ηi) = 0, hence, if γj, 0 ≤ j ≤ d”, are the coefficients ofthe polynomial P d′, we have

d”∑

j=0

γjηji =

d”∑

j=0

jγjηji = ... =

d”∑

j=0

jd′−1γjηji = 0.

Hencek∑

i=1

di−1∑

h=0

d”∑

j=0

γjβi,hVh(n + j)ηn+ji = 0.

Then, for all n big enough,

d”∑

j=0

γd”−jan+j = 0,

and hence+∞∑

n=0

anzn =Q(z)

R(z)d′,

where Q is a polynomial with integer coefficients and R is as defined above.

Thus+∞∑

n=0

k∑

i=1

di−1∑

h=0

βi,hVh(n)ηni zn =

Q(z)

R(z)d′+ f(z)

12

for all z such that |z| ≤ 1, which is equivalent to

k∑

i=1

di−1∑

h=0

βi,h

Πh(ηiz)

(1 − ηiz)h+1=

Q(z)

R(z)d′+ f(z).

As the poles of 1R(z)

of modulus smaller or equal to one are all the ( 1η, η ∈

H, |η| ≥ 1), we get that each of these must be equal to some 1ηi

, which is

(ii)a).And, for each 1 ≤ i ≤ k, by multiplying the equality above by (1−ηiz)di , andmaking z → 1

ηi, we must get the same limit li ; then we substract li

(1−ηiz)di,

multiply by (1 − ηiz)di−1 and so on, and finally we get (ii)b)

(ii) implies (i) :We put

f(z) =k∑

i=1

di−1∑

h=0

βi,h

Πh(ηiz)

(1 − ηiz)h+1− Q(z)

R(z)d′. (2)

Because of (ii)b), f has no pole at 1ηi

for 1 ≤ i ≤ k ; hence (ii)a) ensures thatevery pole of f has modulus strictly greater than one. Hence

f(z) =+∞∑

n=0

enzn

for |z| ≤ 1, where en → 0 if n → +∞.

Now, if H is the set of all roots of P , the multiplicity of each root η beingdenoted by d(η),

1

R(z)=

1∏

η∈H(1 − ηz)d(η)=∏

η∈H

(+∞∑

n=0

ηnzn)d(η),

and, as d(η) is the same for all elements of a given class of algebraic conjugacy,we have,

Q(z)

R(z)d′=

+∞∑

n=0

anzn

13

for |z| ≤ 1, with an ∈ Z.And, by identifying the coefficients of zn in (2), we get

k∑

i=1

di−1∑

h=0

βi,hVh(n)ηni = an + en.

QED

Our main result follows now immediately from Lemmas 3 and 4 andProposition 1 :

Proposition 3 Let σ be a primitive substitution, u a nonperiodical fixedpoint of σ, θ1, ...θt the eigenvalues of its matrix, P its characteristic polyno-mial, D the (finite) set of its return words, the (rn(C), C ∈ D) its returntime sequences.

For C ∈ D, we define A(C) to be the set of 1 ≤ i ≤ s such that |θi| ≥ 1and that Wi,h,C(θi) 6= 0 for at least one 0 ≤ h ≤ di − 1.Let B(C) be the set of i such that (θi, i ∈ B(C)) is the closure under algebraicconjugacy of (θi, i ∈ A(C)).For i ∈ A(C), let di,C be the biggest h ≤ di such that Wi,h−1,C(θi) 6= 0. Letd′(C) = supi∈A(C) di,C.Let

RC(X) =∏

i∈B(C)

(1 − θiX)di,C .

Let Vh and Wi,h,C be as defined in Lemma 3, Πh as defined in Lemma 4. Letvh,i,C = Uh,i,C(θi), Uh,i,C ∈ Q[X], be the numerator of the simple element ofthe rational fraction

di,C−1∑

h=0

Wh,i,C(θi)Πh(θiX)

(1 − θiX)h+1

of denominator (X − 1θi

)h+1.

Then λ = e2πiα is an eigenvalue of the dynamical system associated to σif and only if, for every C ∈ D, there exists a polynomial QC ∈ Z[X] suchthat, for every i ∈ A(C), the numerator of the simple element of the rational

fraction QC(X)

RC(X)d′(C) of denominator (X − 1θi

)h+1 is αUi,h,C(θi) for 0 ≤ h ≤di,C − 1, and 0 for h ≥ di,C.

14

Remark

Each (θi, i ∈ A(C)) is the intersection of the set of expanding eigenvaluesof M with a set closed under algebraic conjugacy, and contains the Perron-Frobenius eigenvalue.

3.3

An important particular case is the one where every expanding eigenvalue ofthe matrix is simple ; then each d′(C) is equal to 1, and, because of Lemma3 and the expression of the simple elements related to simple poles, we have

Proposition 4 Let σ be a primitive substitution such that every eigenvalueof modulus greater or equal to one of its matrix M is a simple eigenvalue, ua nonperiodical fixed point of σ, θ1, ...θt the eigenvalues of its matrix, D the(finite) set of its return words, the (rn(C), C ∈ D) its return time sequences.For Cj ∈ D, let h(i, Cj) be the j-th coordinate of the vector L

∏

v 6=i(M−θvI)e,and let

A(Cj) = (i ∈ (1, ...t); |θi| ≥ 1 and h(i, Cj) 6= 0).

Then λ = e2πiα is an eigenvalue of the dynamical system associated to σ ifand only if, for every C ∈ D, there exists a polynomial QC ∈ Z[X] such that

α =1

h(i, C)θs−1

i QC

(

1

θi

)

for every i ∈ A(C).

Remarks

When P is irreducible over Q[X], each set (θi, i ∈ A(C)) is just the set ofexpanding eigenvalues of M , for any C.Note also that Z(θi) ⊂ Z( 1

θi) ⊂ Q(θi), the inclusions being strict in general.

3.4

The criterion in Proposition 3 has to be a little complicated (though stillexplicitely computable if we know σ) as we want the exact values of α ;however, if we are satisfied by knowing only the direction Zα, it takes amuch simpler form :

15

Proposition 5 Under the hypotheses and with the notations of Proposition3, let A = ∪C∈DA(C), and F = (E1, ...Ez) be the set of algebraic conjugacyclasses of the θi, where Ek = (θi, i ∈ Gk).If e2πiα is an eigenvalue of the dynamical system associated to σ, then, forevery E = (θi, i ∈ G) ∈ F , there exists a polynomial SE ∈ Q[X] such that

α = SE(θi)

for every i ∈ A ∩ G.If, for every E = (θi, i ∈ G) ∈ F , there exists a polynomial SE ∈ Q[X] suchthat

α = SE(θi)

for every i ∈ A∩G, then there exists b ∈ Z such that e2πibα is an eigenvalueof the dynamical system associated to σ.

Proof

All the elements in the formulas giving α in Proposition 3 are rational poly-nomials in θi, and, if θi and θj are in the same algebraic conjugacy class, allthese elements involve the same rational polynomials in θi and θj ; recipro-cally, if each bWi,h,C(θi)α is an integer polynomial in θi, which remains thesame if we replace θi by one of its algebraic conjugates, then (1) is satisfied.QED

Corollary 1 The dynamical system has irrational eigenvalues if and onlyif for every algebraic conjugacy class E = (θi, i ∈ G) such that A ∩ G 6= ∅,there exists a polynomial SE ∈ Q[X] such that SE(θ) takes the same irrationalvalue for every θ ∈ E.

Corollary 1, together with Proposition 2, gives an easy necessary andsufficient condition for a substitution to be weakly mixing.

4 Examples

4.1

a → abbbccccccccccddddddddb → bcccc → dd → a

16

M =

1 3 10 80 1 3 00 0 0 11 0 0 0

P (X) = X4−2X3−7X2−2X+1 = (X2−(1+√

10)X+1)(X2−(1−√

10)X+1)

θ1 =1 +

√10 +

√

7 + 2√

10

2

θ2 =1 +

√10 −

√

7 + 2√

10

2

θ3 =1 −

√10 +

√

7 − 2√

10

2

θ4 =1 −

√10 −

√

7 − 2√

10

2We have θ1 > 1, θ2 < 1, −1 < θ3 < 0, and θ4 < −1, M has two expandingand two contracting eigenvalues.Among the return words, we find b, c, d, and a, hence the convergence ofevery rn(C)α to zero modulo 1 is equivalent to the convergence ot everyl(σnaj)α to zero modulo 1, j = 1, ...4 ; hence we may take L = I.The four groups of conditions in Proposition 3 are equivalent ; the simplestto write is for j = 3, which gives :

α =θ31

(1 +√

10)θ1 + 11 −√

10Q(

1

θ1

)

=θ34

(1 −√

10)θ4 + 11 +√

10Q(

1

θ4

)

,

and computations prove that this gives only integer values for α, or, equiv-alently, that there are no rational eigenvalues by Proposition 2 and that thehypotheses of Corollary 1 are satisfied : the dynamical system associated

to this substitution is weakly mixing.

4.2

a → abddb → bcc → dd → a

17

M =

1 1 0 20 1 1 00 0 0 11 0 0 0

P (X) = X4 − 2X3 −X2 + 2X − 1 = (X2 −X − 1−√

2)(X2 −X − 1 +√

2)

θ1 =1 +

√

5 + 4√

2

2

θ2 =1 −

√

5 + 4√

2

2

θ3 =1 + i

√

−5 + 4√

2

2

θ4 =1 − i

√

−5 + 4√

2

2

We have θ1 > 1, θ2 < −1, |θ3| = |θ4| =√

2 − 1 < 1, M has two expandingand two contracting eigenvalues.Among the return words, we find d, a, dab, and bcaa, hence the convergenceof every rn(C)α to zero modulo 1 is equivalent to the convergence ot everyl(σnaj)α to zero modulo 1, j = 1, ...4 ; hence we may take L = I.The four groups of conditions in Proposition 3 are equivalent ; the simplestto write is for j = 4, which gives :

α =1√

2θ1 + 3θ31Q

(

1

θ1

)

=1√

2θ2 + 3θ32Q

(

1

θ2

)

,

and the second equation is satisfied if and only if

(

θ1 −1

2+

√2

4

)

Q(

1

θ1

)

=

(

θ2 −1

2+

√2

4

)

Q(

1

θ2

)

;

taking into account the expression of θi and the facts that, for x = θ1 orx = θ2, we have

√2 = x2 − x− 1 and x = 2 + 1

x− 2

x2 + 1x3 , we check that this

is nontrivially satisfied if and only if

Q(

1

θi

)

= k

(

θi −1

2−

√2

4

)

+l

(

θi − 12

+√

24

)

18

for i = 1, 2, and (k, l) such that the above expression is an integer polynomialin 1

θi, or, equivalently, in θi ; computations show that this is the case if and

only if l ∈ 474Z and k ∈ 12l + 4Z ; we get

α =k

4(4 + 3

√2) +

l

47(12 + 5

√2),

and we conclude that the eigenvalues of the dynamical system asso-

ciated to this substitution are the eikπ√

2 for k ∈ Z. Note that we dohave S(θ1) = S(θ2) = 1 +

√2, with the polynomial S(X) = X2 − X ∈ Z[X]

playing a key role in the equalities above ; α = (S(θi))k, for k ∈ Z, is a

sufficient condition for e2πiα to be an eigenvalue of the system (this appearsin [SOL3], with a slightly different substitution, where the matrix has thesame eigenvalues), but the necessary condition is strictly weaker.

References

[DEK] M. DEKKING : The spectrum of dynamical systems arising fromsubstitutions of constant length, Zeit. Wahrsch. verw. Geb. 41 (1978),p. 221-239.

[HOS] B. HOST : Valeurs propres des systemes dynamiques definis par dessubstitutions de longueur variable, Ergodic Th. Dyn. Syst. 6 (1986), p.529-540.

[KEA] M.S. KEANE : Interval exchange transformations, Math. Zeitsch. 141(1975), p. 25-31.

[LIV] A. N. LIVSHITS : A sufficient condition for weak mixing of substitu-tions and stationary adic transformations, Mat. Zametki 44 (1988), p.785-793, translated in Math. Notes 44 (1988), p. 920-925.

[LIV-VER] : A. N. LIVSHITS, A. M. VERSHIK : Adic models of ergodictransformations, spectral theory, substitutions, and related topics, Adv.

19

Sov. Mat. 9 (1992), p. 185-204.

[MAU] C. MAUDUIT : Caracterisation des ensembles normaux substitutifs,Inventiones Math. 95 (1989), p. 133-147.

[MOS] B. MOSSE : Notions de reconnaissabilt pour les substitutions et com-plexit des suites automatiques, submitted.

[NOG-RUD] A. NOGUEIRA, D.J. RUDOLPH : Topological weak mixing ofinterval exchange maps, preprint, Maryland University (1991).

[PIS] C. PISOT : La repartition modulo 1 et les nombres algebriques, Ann.Sc. Norm. Super. Pisa Cl. Sci. , IV Ser. (1938), p. 205-248.

[QUE] M. QUEFFELEC : Substitution dynamical systems - Spectral anal-ysis, Lecture Notes in Math. vol. 1294 (1987), Springer-Verlag.

[SAL] R. SALEM : Algebraic numbers and Fourier analysis, Heath Mathe-matical Monographs (1963), Heath and Co.

[SOL1] B. SOLOMYAK : Substitutions, adic transformations, and beta-expansions, Contemporary Mat. 135 (1992), p. 361-372.

[SOL2] B. SOLOMYAK : On simultaneous action of Markov shift and adictransformation, Adv. Sov. Mat. 9 (1992), p. 231-239.

[SOL3] B. SOLOMYAK : On the spectral theory of adic transformations,Adv. Sov. Mat. 9 (1992), p. 217-230.

20

[VEE] W. VEECH : Gauss measures for transformations on spaces of intervalexchange maps Annals of Math. 115 (1982), p. 201-242.

21