Continuous and measurable eigenfunctions of linearly recurrent dynamical Cantor systems

0.19 MB·English

by  Maria Isabel Cortez

CONTINUOUS AND MEASURABLE EIGENFUNCTIONS OF
LINEARLY RECURRENT DYNAMICAL CANTOR SYSTEMS
8
0 MARIAISABELCORTEZ,FABIENDURAND,BERNARDHOST,
0 ANDALEJANDROMAASS
2

n Abstract. The class of linearly recurrent Cantor systems contains the sub-
a stitution subshifts and some odometers. For substitution subshifts measure–
J theoretical and continuous eigenvalues are the same. It is natural to ask
0 whether this rigidityproperty remains true for the class of linearlyrecurrent
3 Cantor systems. Wegivepartialanswerstothisquestion.

]
S
D
1. Introduction
.
h
Let (X,T) be a topological dynamical system and µ a T–invariant probability
t
a measure. Whenameasure-theoreticaleigenvalueλ∈Cofthesystem,thatisf◦T =
m λf forsomef ∈L2(µ)\{0},isassociatedtoacontinuouseigenfunctionf :X →C?
[ InthispaperweareinterestedinconditionsonminimaldynamicalCantorsystems

thatanswerthis question. Our motivationcomes from[Ho]where it is provedthat
1
v all eigenvalues of minimal substitution subshifts are associated to a continuous
6 eigenfunction. Such a question also appears in [NR] where the authors show that
1 generically interval exchange transformations are not topologically weakly mixing
6
(i.e., they donothavenontrivialcontinuouseigenfunctions)andwherethey “fully
4
expect” the same holds for (measure-theoretical) weak mixing (i.e., they do not
.
1
have non trivial eigenfunctions). It is in general not true that all eigenvalues of a
0
minimaldynamicalsystemhaveacontinuouseigenfunctionascanbeseenforsome
8
Toeplitz systems [Iw, DL] and for some interval exchange transformations [FHZ].
0
: In this paper we focus on linearly recurrent dynamical Cantor systems (also
v
calledlinearlyrecurrentsystems). Theynaturallyextendthenotionofsubstitution
i
X subshiftsinthesensetheysharethesamereturntimestructure. Linearlyrecurrent
r subshifts were studied in [DHS, Du1, Du2, Le].
a
The paper is organized as follows. In Section 2 we define linearly recurrent
systems by means of nested sequence of Kakutani-Rokhlin partitions and obtain
some general properties. In particular we prove that these systems are uniquely
ergodic but are not strongly mixing.
In the following section, when the dynamical system (X,T)is linearly recurrent
andµisaT–invariantprobabilitymeasurewegiveanecessaryconditionforacom-
plexnumbertobeaneigenvalue. Wealsoexhibitasufficientconditionforacomplex
number to be a continuous eigenvalue,which involves the underlying matrix struc-
tureofthe nestedsequence ofKakutani-Rokhlinpartitionsdefining (X,T). This is
usedinthelastsectiontoprovefornaturalprobabilityspacesassociatedtofamilies
1991 Mathematics Subject Classification. Primary: 54H20;Secondary: 37B20.
Key words and phrases. minimal Cantor systems, linearly recurrent dynamical systems,
eigenvalues.
1
2 MariaIsabelCortez,FabienDurand,BernardHost,AlejandroMaass
oflinearlyrecurrentsystems,andunderaconditionof“hyperbolicity”,thatalmost
every system of such family has only continuous eigenvalues. We give in Section 4
severalexamples to illustrate the results of the paper.
2. Definitions and background
2.1. Dynamical systems. By a topological dynamical system we mean a couple
(X,T) where X is a compact metric space and T : X → X is a homeomorphism.
WesaythatitisaCantor systemifX isaCantorspace;thatis,X hasacountable
basis of its topology which consists of closed and open sets (clopen sets) and does
not have isolated points. The topological dynamical system (X,T) is minimal
whenever X and the empty set are the only T-invariant closed subsets of X. We
only deal here with minimal Cantor systems. A complex number λ is a continuous
eigenvalue of (X,T) if there exists a continuous function f : X → C, f 6= 0, such
that f ◦T = λf; f is called a continuous eigenfunction associated to λ. If (X,T)
is minimal, then every continuous eigenvalue is of modulus 1 and every continuous
eigenfunction has a constant modulus.
When(X,T)isatopologicaldynamicalsystemandµisaT-invariantprobability
measure, i.e., Tµ = µ, defined on the Borel σ-algebra B of X, we call the triple
X
(X,T,µ) a dynamical system. We do not recall the definitions of ergodicity, weak
mixing and strong mixing (see [Wa] for example). A complex number λ is an
eigenvalue of the dynamical system (X,T,µ) if there exists f ∈L2(µ), f 6=0, such
thatf◦T =λf,µ-a.e.;f iscalledaneigenfunction (associatedtoλ). Ifthe system
is ergodic, then every eigenvalue is of modulus 1, and every eigenfunction has a
constant modulus. By abuse of language we will also say that an eigenvalue is
continuous when the associated eigenfunction is continuous.
In this paper we mainly consider topological dynamical systems (X,T) which
are uniquely ergodic, that is systems that admit a unique invariant probability
measure; this measure is then ergodic.
2.2. Partitions. Sequences of partitions of a minimal Cantor system were used
in [HPS] to build a representation of the system as an adic transformation on an
ordered Bratteli diagram. We recall some definitions and fix some notations we
shall use along this paper.
Let (X,T) be a minimal Cantor system. A clopen Kakutani-Rokhlin partition
(CKR partition) is a partition P of X of the kind:
(2.1) P ={T−jB ;1≤k ≤C, 0≤j <h }
k k
where C is a positive integer, B ,...,B are clopen subsets of X and h ,...,h
1 C 1 k
arepositiveintegers. For1≤k ≤C,thek-thtowerofP isthefamily{T−jB ;0≤
k
j <h }, and the base of P is the set B = B . Let
k 1≤k≤C k
S
(2.2) P(n)={T−jB (n):1≤k ≤C(n), 0≤j <h (n)} ; n∈N
k k
(cid:0) (cid:1)
be a sequence of CKR partitions. For every n we write B(n) for the base of P(n),
and we assume that P(0) is the trivial partition, that is B(0) = X, C(0) = 1 and
h (0)=1.
1
We say that this sequence (P(n);n∈N) is nested if for every n≥0 it satisfies:
(KR1) B(n+1)⊂B(n) and
EigenfunctionsoflinearlyrecurrentdynamicalCantorsystems 3
′
(KR2) P(n+1)(cid:23)P(n); i.e., for all A∈P(n+1) there exists A ∈P(n) such
′
that A⊂A .
We consider mostly nested sequences of CKR partitions which satisfy also the
properties:
(KR3) ∩∞ B(n) consists of a unique point;
n=0
(KR4) the sequence of partitions spans the topology of X;
In [HPS] it is proven that for each minimal Cantor system (X,T) there exists
a nested sequence of CKR partitions fulfilling (KR1)-(KR4) (i.e., (KR1), (KR2),
(KR3) and (KR4)) and the following conditions:
(KR5)foralln≥1,1≤k ≤C(n−1),1≤l ≤C(n), there exists0≤j <h (n)
l
such that T−jB(n)⊂B (n−1);
l k
(KR6) for all n∈N, B(n+1)⊂B (n).
1
To suchasequence ofpartitions weassociatea sequenceofmatrices (M(n);n≥
1),where the matrix M(n)=(m (n);1≤l≤C(n),1≤k ≤C(n−1)) isgivenby
l,k
m (n)=#{0≤j <h (n);T−jB (n)⊂B (n−1)}.
l,k l l k
WenoticethatProperty(KR5)isequivalenttotheconditionthatforeveryn≥1
the matrix M(n) has positive entries. As the sequence of partitions is nested, we
get
C(n−1)
h (n)= m (n)h (n−1), n≥1, 1≤l ≤C(n).
l l,k k
Xk=1
Werewritethisformulainamatrixform. Foreveryn≥0,letH(n)=(h (n);1≤
l
l≤C(n))t,that isthe columnvectorwith entriesh (n),h (n),...,h (n). Then
1 2 C(n)
we have H(n)=M(n)H(n−1) for n>0. For n>m≥0 we define
P(n,m)=M(n)M(n−1)...M(m+1) and P(n)=P(n,1) .
We have:
P (n,m)=# j;0≤j ≤h (n), T−jB(n)⊂B (m)
l,k l l k
and (cid:8) (cid:9)
P(n,m)H(m)=H(n)=P(n)H(1) .
2.3. Linearly recurrent systems.
Definition1. AminimalCantorsystem(X,T)is linearlyrecurrent(withconstant
L)if there exists anestedsequenceof CKR partitions (P(n)={T−jB (n);1≤k ≤
k
C(n),0≤j <h (n)};n∈N) satisfying (KR1)-(KR6) and
k
(LR) there exists L such that for all (l,k) ∈ {1,...,C(n)}×{1,...,C(n−1)}
and all n≥1
h (n)≤L h (n−1) .
l k
The notion of linearly recurrent dynamical Cantor system (also called linearly
recurrent system) is the extension of the concept of linearly recurrent subshift
introduced in [DHS]. Of course it can be proved that linearly recurrent subshifts
are linearly recurrent Cantor systems (see [Du1, Du2]). Examples of such systems
are substitution subshifts [DHS] and Sturmian subshifts whose associated rotation
number has a continued fraction with bounded coefficients [Du1, Du2].
4 MariaIsabelCortez,FabienDurand,BernardHost,AlejandroMaass
Lemma 2. Let (X,T) be a linearly recurrent system and (P(n);n ∈ N) a se-
quence of CKR partitions satisfying Properties (KR1)-(KR6), and Property (LR)
with constant L. Then:
(1) for every n∈N we have C(n)≤L;
(2) for every n ∈ N, 1 ≤ k ≤ C(n) and 1 ≤ k′ ≤ C(n) we have h (n) ≤
k
L hk′(n).
Proof. Property (1) follows directly from the hypotheses (KR5) and (LR). Indeed
C(n)h (n)≤h (n+1)≤Lh (n) where h (n)=min{h (n);1≤k ≤C(n)}.
i 1 i i k
InasimilarwayweproveProperty(2). From(KR5)itcomesthatallh (n+1)are
i
greaterthan C(n)h (n). Consequently,from(LR)weobtainforall1≤k ≤C(n)
j=1 j
and 1≤k′ ≤PC(n)
C(n)
hk(n)≤ hj(n)≤hi(n+1)≤Lhk′(n).
Xj=1
This ends the proof. (cid:3)
From this lemma we deduce that the set {M(n);n≥1} is finite. The following
proposition,whoseproofisleft to the reader,tells us thisis infacta necessaryand
sufficient condition to be linearly recurrent.
Proposition 3. Let (X,T) be a minimal Cantor system. The system (X,T) is
linearly recurrent if and only if there exist a nested sequence of CKR partitions
(P(n);n∈N), satisfying (KR1)-(KR6), and a constant K such that: for all n≥1
and all (l,k)∈{1,...,C(n)}×{1,...,C(n−1)},
1≤m (n)≤K,
l,k
where (M(n)=(m (n);1≤l≤C(n),1≤k ≤C(n−1));n≥1) be the associated
l,k
sequence of matrices.
2.4. Unique ergodicity and absence of strong mixing of linearly recur-
rent systems. In this subsection (X,T) is a linearly recurrent system with a
nested sequence of CKR partitions (P(n) = {T−jB (n);1 ≤ k ≤ C(n),0 ≤
k
j < h (n)};n ∈ N) satisfying (KR1)-(KR6) and (LR) with constant L. Let
k
(M(n) = (m (n);1 ≤ l ≤ C(n),1 ≤ k ≤ C(n −1));n ≥ 1) be the associated
l,k
sequence of matrices.
We notice that for each T–invariant probability measure µ and for every n ≥ 1
and 1≤k ≤C(n−1) we have
C(n)
(2.3) µ(B (n−1))= m (n)µ(B (n))
k l,k l
Xl=1
and
C(n)
(2.4) h (n)µ(B (n))=1 .
k k
kX=1
To prove that linearly recurrent systems are uniquely ergodic we need the fol-
lowing lemma that is used through all this paper.
EigenfunctionsoflinearlyrecurrentdynamicalCantorsystems 5
Lemma 4. Let µ be an invariant measure of (X,T). Then, for all n ∈ N and
1≤k ≤C(n) we have
1
h (n)µ(B (n))≥ .
k k
L
Proof. Fixkwith1≤k ≤C(n). ByEquation(2.3),sincealltheentriesofM(n+1)
are positive, we get
C(n+1)
µ(B (n))≥ µ(B (n+1)) .
k l
Xℓ=1
By (LR), for every l we have h (n)≥h (n+1)/L thus
k l
C(n+1)
h (n+1) 1
l
h (n)µ(B (n))≥ µ(B(n+1))= .
k k l
L L
Xℓ=1
(cid:3)
Proposition 5. Every linearly recurrent system is uniquely ergodic.
Proof. Let (X,T) be a linearly recurrent system. Given a T–invariant probability
measure µ, we define the numbers
µ =µ(B (n)), n≥0, 1≤k ≤C(n) .
n,k k
These nonnegative numbers satisfy the relations
Cn
(2.5) µ =1 and µ = µ m (n) for n≥1, 1≤k ≤C(n−1) .
0,1 n−1,k n,l l,k
Xl=1
Ina matrixform: with V(n)=(µ ,...,µ ), we haveV(n−1)=V(n)M(n).
n,1 n,C(n)
Conversely, let the nonnegative numbers (µ ;n≥0, 1≤k ≤C(n)) satisfy these
n,k
conditions. As the partitions P(n) are clopen and span the topology of X, it is
immediate to check that there exists a unique invariant probability measure µ on
X with µ =µ(B (n)) for every n∈N and k ∈{1,...,C(n)}.
n,k k
From Lemma 4, there exists a constant δ >0 such that
µ ≥δµ
n,i n−1,k
for every n≥1 and (i,k)∈{1,...,C(n)}×{1,...,C(n−1)}, and every invariant
measure µ. Without loss of generality we can assume δ <1/2.
Let µ,µ′ be two invariant measures, and µ ,µ′ be defined as above. We
n,k n,k
define
µ′ µ′ µ′ µ′ S
n,k n,i n,k n,j n
S =max = , s =min = , and r =
n n n
k µn,k µn,i k µn,k µn,j sn
for some i,j. For every k ∈{1,...,C(n−1)} we have:
µ′ = µ′ m (n)+µ′ m (n)
n−1,k n,l l,k n,j j,k
Xl6=j
≤S µ m (n)+s µ m (n)
n n,l l,k n n,j j,k
Xl6=j
=S µ −(S −s )µ m (n)≤µ S −(S −s )µ
n n−1,k n n n,j j,k n−1,k n n n n,j
≤µ s r (1−δ)+δ .
n−1,k n n
(cid:0) (cid:1)
And in similar way, for every k ∈{1,...,C(n−1)} we have
6 MariaIsabelCortez,FabienDurand,BernardHost,AlejandroMaass
µ′ ≥µ s δr +(1−δ) .
n−1,k n−1,k n n
(cid:0) (cid:1)
We deduce that
(1−δ)x+δ
r ≤φ(r ) where φ(x)= .
n−1 n
δx+(1−δ)
The function φ is increasing on [0,+∞) and tends to (1−δ)/δ at +∞. Writing
φm = φ◦···◦φ (m times), for every n,m ∈ N we have 1 ≤ r ≤ φm(r ) ≤
n n+m
φm−1((1−δ)/δ). Taking the limit with m→+∞, we get r =1. (cid:3)
n
From now on we call µ the unique invariant measure on (X,T). Let m≥1 and
0≤k ≤C(m). By unique ergodicity,
1
#{0≤j <N; T−jx∈B (m)}→µ(B (m))
k k
N
uniformly as N →∞. But for n>m, 1≤l ≤C(n), for every x∈B (n) we have
k
#{0≤j <h (n); T−jx∈B (m)}=P (n,m) .
l k l,k
We deduce:
P (n,m)
l,k
(2.6) max −µ B (m) →0 as n→+∞ .
k
1≤l≤C(n)(cid:12)(cid:12) hl(n) (cid:0) (cid:1)(cid:12)(cid:12)
(cid:12) (cid:12)
Proposition 6. Linearly recurrent systems are not strongly mixing.
Proof. Let m be an integer such that µ(B (m))<1/L2 and for n>m let D(n)=
1
B (m)∩Th1(n)B (m). We prove that lim µ(D(n))>µ(B (m))2 which will imply
1 1 n 1
that (X,T,µ) is not strongly mixing.
For n>m we write
E(n)={0≤j <h (n−1); T−jB (n−1)⊂B (m)} .
1 1 1
By hypothesis (KR6) we have B(n)⊂B (n−1) and for j ∈E(n) we get
1
T−j−h1(n)B (n)⊂T−jB(n)⊂T−jB (n−1)⊂B (m)
1 1 1
andT−jB (n)⊂D(n). Itfollowsthatµ(D(n))≥#E(n)·µ(B (n)). But#E(n)=
1 1
P (n−1,m)thus #E(n)/h (n−1) convergesto µ(B (m)) as n→+∞by Equa-
1,1 1 1
tion (2.6). Therefore
limµ(D(n))≥limh (n−1)µ(B (n))µ(B (m))
1 1 1
n n
1
≥lim h (n)µ(B (n))µ(B (m)) (by (LR))
1 1 1
L
n
1
≥ µ(B (m)) (by Lemma 4)
L2 1
and the proof is complete. (cid:3)
It is well-known that there exist substitution subshifts, and a fortiori linearly
recurrent systems, which are weakly-mixing (see [Qu]).
EigenfunctionsoflinearlyrecurrentdynamicalCantorsystems 7
3. Some conditions to be an eigenvalue
Inthissectionwesuppose(X,T,µ)islinearlyrecurrent,thatistosay(P(n);n≥
0) satisfies (KR1)-(KR6) and (LR) (with constant L). Let (M(n);n ≥ 1) be its
associated sequence of matrices.
We give a sufficient condition to be a continuous eigenvalue and a necessary
conditiontobeaneigenvalue. Wedefineforn≥1,1≤k ≤C(n−1),1≤l≤C(n),
J(n,k,l)= 0≤j <h (n); T−jB (n)⊂B (n−1) , J(n)= J(n,k,l).
l l k
(cid:8) (cid:9) 1≤k≤[C(n−1)
1≤l≤C(n)
so that #J(n,k,l)=m (n).
l,k
Proposition 7. Let λ∈C satisfy
∞
max |λhk(n)−1|<∞ .
nX=11≤k≤C(n)
Then λ is a continuous eigenvalue of (X,T,µ).
Proof. For every n∈N, let f be the function on X defined by
n
f (x)=λ−j for x∈T−jB (n),
n k
1≤k ≤C(n) and 0≤j <h (n).
k
We compare f and f . By construction, for every x, f (x)/f (x) belongs
n n−1 n n−1
to the set {λ−j;j ∈J(n)}. But each integer in J(n) is a sum of terms of the form
h (n−1), and this sum contains at most L terms. We get
k
||f −f || ≤L max |λhk(n−1)−1| .
n n−1 ∞
1≤k≤C(n−1)
Byhypothesis,theseries ||f −f || converges. Thusthesequence(f ;n∈
n≥1 n n−1 ∞ n
N)convergesuniformlytoPacontinuousfunctionf,whichisclearlyaneigenfunction
for λ. (cid:3)
Proposition 8. If λ∈C is an eigenvalue of (X,T,µ) then
∞
max λhk(n)−1 2 <∞ .
nX=11≤k≤C(n)(cid:12) (cid:12)
(cid:12) (cid:12)
Proof. We use the sets J(n,k,l) defined above.
Assume that λ = exp(2iπα), α ∈ R, is an eigenvalue, and that f is a corre-
sponding eigenfunction of modulus 1. For every n ∈ N, let f be the conditional
n
expectation of f with respect to the σ-algebra spanned by P . For 1≤ k ≤ C(n),
n
f is constant on B (n), and we write c(n,k) this constant.
n k
The sequence (f ;n ∈ N) is a martingale ([Do]), and converges to f in L2(µ).
n
Moreoverthe functions f −f , n≥1, are mutually orthogonalin L2(µ), hence
n n−1
we have
∞
(3.1) ||f −f ||2 <∞
n n−1 2
nX=1
(see [Do] for the details).
We fix n ≥ 1, 1 ≤ l ≤ C(n) and 1 ≤ k ≤ C(n−1), and we choose some j ∈
J(n,k,l). Lookingatthestructureofthetowers,weseethatj+h (n−1)≤h (n).
k l
For 0≤p< h (n−1) we have j+p< h (n), and T−(j+p)B(n)⊂ T−pB (n−1).
k l l k
8 MariaIsabelCortez,FabienDurand,BernardHost,AlejandroMaass
For x ∈ T−(j+p)B(n), we have f (x) = exp(−2iπ(j +p)α)c(n,l) and f (x) =
l n n−1
exp(−2iπpα)c(n−1,k). We get
||f −f ||2 ≥h (n−1)µ(B(n))|exp(−2iπjα)c(n,l)−c(n−1,k)|2 .
n n−1 2 k l
ByLemma 4and(LR), h (n−1)µ(B (n))≥L−2 ,andfromEquation(3.1)we get
k l
∞
(3.2) max max max |exp(−2iπjα)c(n,l)−c(n−1,k)|2 <∞ .
nX=11≤l≤C(n)1≤k≤C(n−1)j∈J(n,k,l)
We use this bound first with k =1 and an arbitrary l. By (KR6), 0∈J(n,1,l)
and from Equation (3.2) we get
∞
max |c(n,l)−c(n−1,1)|2 <∞ .
nX=11≤l≤C(n)
Using this three times, we get
∞
(3.3) max max |c(n,l)−c(n−1,k)|2 <∞ .
nX=11≤l≤C(n)1≤k≤C(n−1)
For each n ∈ N and 1 ≤ k ≤ C(n), the function |f | is constant and equal to
n
|c(n,k)| on the k-th tower of P(n). By Lemma 4, the measure of this tower is not
less than 1/L. Since kf k → kfk = 1, we get that inf |c(n,k)| converges to 1
n 2 2 k
when n→+∞. Hence from Equations (3.2) and (3.3), we get
∞
max max max |exp(−2iπjα)−1|2 <∞ .
nX=11≤l≤C(n)1≤k≤C(n−1)j∈J(n,l,k)
We use this bound with twoconsecutive elements of the same setJ(n,l,k)and get
the announced result. (cid:3)
The followingsufficientconditionforweakmixingfollowsfromProposition8. A
similar condition appears in [FHZ].
Corollary 9. For every n∈N, let
K =inf |h (n)−h (n)|:1≤i,j ≤C(n), h (n)6=h (n)
n i j i j
(cid:8) (cid:9)
and let K =lim K . If K is finite, then (X,T,µ) has at most K eigenvalues. In
n n
particular, if K =1 then this system is weakly mixing.
Now we restate Proposition 7 and Proposition 8 in terms of matrices.
Notation. For every real number x we write |||x||| for the distance of x to the
nearest integer. For a vector V =(v ,...,v )∈Rm, we write
1 m
kVk= max |v | and |||V|||= max |||v ||| .
j j
1≤j≤m 1≤j≤m
Weusesimilarnotationsforrealmatrices. Withthesenotations,thetwopreceding
Theorems can be written as follows.
Theorem 10. Let α∈R and λ=exp(2iπα).
(1) If λ is an eigenvalue of (X,T,µ) then |||αP(n)H(1)|||2 <∞.
nX≥1
(2) If |||αP(n)H(1)|||<∞ then λ is a continuous eigenvalue of (X,T,µ).
nX≥1
EigenfunctionsoflinearlyrecurrentdynamicalCantorsystems 9
Proposition 11. Let α∈R and λ=exp(2iπα).
If λ is an eigenvalue of (X,T,µ)then it satisfies at least one of the two following
properties:
(1) α is rational, with a denominator dividing gcd(h (m) : 1 ≤ i ≤ C(m)) for
i
some m∈N. In this case λ is a continuous eigenvalue.
(2) There exist m ∈ N and integers w , 1 ≤ j ≤ C(m), such that α =
j
C(m)
w µ(B (m)).
j j
Xj=1
Moreover, if α is rational, with a denominator dividing gcd(h (m):1≤i≤C(m))
i
for some m∈N, then λ is an eigenvalue of (X,T,µ).
The proof of Proposition 11 needs the following lemma.
Lemma 12. Let u be a real vector such that |||P(n)u||| → 0 as n → +∞. Then
there exist m∈N, an integer vector w and a real vector v with
P(m)u=w+v and kP(n,m)vk→0 as n→+∞ .
Proof. By hypothesis, for every n ∈ N we can write P(n)u = v +w , where w
n n n
is an integer vector and v a real vector with kv k→0 as n→+∞. Since all the
n n
matrices M(m) belong to a finite family, kM(m)v −v k converges to 0 as m
m m+1
goes to infinity. But for everym∈N we have P(m+1)u=M(m+1)P(m)u, thus
M(m)v −v =−M(m)w +w
m m+1 m m+1
andM(m)v −v isanintegervector. Thereforethesequence(M(m)v −v )
m m+1 m m+1
is eventually zero. There exists m∈N such that v =P(n,m)v for every n>m.
n m
The vectors v =v and w=w satisfy the announced properties. (cid:3)
m m
Proof of Proposition 11. Let u = αH(1). Since λ is an eigenvalue, |||P(n)u||| → 0
as n → ∞ by Theorem 10. Let m,v and w be as in Lemma 12. We recall that
P(m)u=αH(m). We distinguish two cases.
First we assume that v =0. Then αH(m) is equal to the integer vector w, and
α is rational with a denominator dividing gcd(h (m) : 1 ≤ i ≤ C(m)). For n ≥ m
i
the vector αH(m) has integer entries, thus |||αH(m)||| = 0 and λ is a continuous
eigenvalue by Theorem 10.
Now suppose v 6=0. For n>m we have
C(m) C(m)C(n)
µ(B (m))v = P (n,m)µ(B (n))v by (2.3)
k k l,k l k
kX=1 kX=1 Xl=1
C(n)
= (P(n,m)v) µ(B (n))≤kP(n,m)vk
l l
Xl=1
and the last term converges to 0 as n→+∞, thus
C(m)
µ(B (m))v =0 .
k k
Xk=1
As w =αH(m)−v, that is, w =αh (m)−v for 1≤j ≤C(m), we get
j j j
C(m) C(m)
w µ(B (m))=α h (m)µ(B (m))=α .
j j j j
Xj=1 Xj=1
10 MariaIsabelCortez,FabienDurand,BernardHost,AlejandroMaass
The rest of the proof is left to the reader. (cid:3)
4. Examples
We studysomeexampleswherewecanexplicitlysaythatthe eigenfunctionsare
continuousortheredonotexistnontrivialeigenvalues. Wekeepthenotationsand
hypotheses of the preceding section.
4.1. Example 1: The sequence (M(n);n ≥ 2) is ultimately constant. Let
(M(n);n ≥ 1) be the sequence of matrices associated to the linearly recurrent
system (X,T,µ). We say that (X,T,µ) has a stationary sequence of matrices if
there exist a square matrix M and an integer n ∈N such that M(n)=M for all
0
n≥n . Without loss we can assume that n =2. We have P =Mn−1 for n≥2.
0 0 n
Substitution subshifts andodometerswith constantbasebelongto the family of
linearlyrecurrentsystems with a stationary sequence ofmatrices (see [DHS]). The
followinglemmawasusedin[Ho]toprovethateigenvaluesofsubstitutionsubshifts
are continuous.
Lemma 13. Let M be a matrix with integer entries. If u is a real vector such that
kMnuk → 0 when n → ∞, then the convergence is exponential, i.e., there exist
0≤r <1 and a constant K such that kMnuk≤Krn for all n∈N.
From this Lemma and Theorem 10 we get:
Proposition14. Let(X,T,µ)bealinearlyrecurrentCantorsystemwithastation-
ary sequence of matrices. Then every eigenfunction of this system is continuous.
Moreover all the linearly recurrent Cantor systems with the same stationary se-
quence of matrices have the same eigenvalues.
4.2. Example 2: A family of weakly mixing systems. We build a family
of linearly recurrent systems which are the Cantor analogues of interval exchange
transformations considered in [FHZ] (Theorem 2.2). Let N be a positive integer.
Let (M(n);n≥2) be a sequence of matrices in the family
l k−1 1 l−1 k 1
 l−1 k 1 ,  l k−1 1  : 1≤l,k≤N
 l−1 k−1 1 l k 1 
   
and let (X,T) be a linearly recurrent system with this sequence of matrices. For
anyn andanyv =(v ,v ,v )t,the vectoru=M(n)v satisfies|u −u |=|v −v |.
1 2 3 1 2 1 2
Consequently if we suppose |h (1)−h (1)|=1 then it follows by induction that
1 2
foralln≥1we have|h (n)−h (n)|=1. By Corollary9 the system(X,T,µ)does
1 2
not have non trivial eigenvalues, i.e. it is weakly mixing.
4.3. Example 3: The sequence (M(n);n ≥ 1) has infinitely many rank 1
matrices.
Proposition 15. Let (X,T,µ) be a linearly recurrent Cantor system and let the
associated sequence of matrices be (M(n);n≥1). Suppose that M(n) has rank one
for infinitely many values of n. Then λ = exp(2iπα) is an eigenvalue of (X,T,µ)
if and only if α is rational with a denominator equal to gcd(h (m):1≤i≤C(m))
i
for some m∈N. Moreover every eigenfunction is continuous.