Persistence of Wandering Intervals in Self-Similar Affine Interval Exchange Transformations

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by  Xavier Bressaud

PERSISTENCE OF WANDERING INTERVALS IN SELF-SIMILAR
AFFINE INTERVAL EXCHANGE TRANSFORMATIONS
8
0 XAVIERBRESSAUD,PASCALHUBERT,ANDALEJANDROMAASS
0
2
Abstract. Inthisarticleweprovethatgivenaself-similarintervalexchange
n transformation T(λ,π), whose associated matrix verifies a quite general alge-
a braic condition, there exists an affine interval exchange transformation with
J wanderingintervalsthatissemi-conjugatedtoit. Thatis,inthiscontext the
4 existenceofDenjoycounterexamplesoccursveryoften,generalizingtheresult
1 ofM.Coboin[C].

]
S
D
1. Introduction
.
h
Since the work of Denjoy [D] it is known that every C1-diffeomorphism of the
t
a circle such that the logarithm of its derivative is a function of bounded variation
m
has no wandering intervals. There is no analogous result for interval exchange
[ transformations. Levitt in [L] found an example of a non-uniquely ergodic affine

interval exchange transformation with wandering intervals. Latter, Camelier and
1
v Gutierrez[CG],usingRauzyinductiontechniqueexhibitedauniquelyergodicaffine
8 intervalexchangetransformationwithwanderingintervals. Moreover,thisexample
8 is semi-conjugated to a self-similar interval exchange transformation. In geometric
0
language, it means that this inter- val exchange transformation is induced by a
2
pseudo-Anosov diffeomorphism. In combinatorial terms, the symbolic system is
.
1
generated by a substitution
0
Anintervalexchangetransformation(IET)is defined bythe lengthofthe intervals
8
0 λ= (λ1,...,λr) and a permutation π. It is denoted by T(λ,π). To define an affine
: intervalexchangetransformation(AIET)oneadditionalinformationisneeded; the
v
slope of the map on each interval. This is a vector (w ,...,w ) with w > 0 for
i 1 r i
X i = 1,...,r. Camelier and Gutierrez remarked that a necessary condition for an
r AIET to be conjugated to the interval exchange transformation T is that the
(λ,π)
a
vector log(w)=(log(w ),...,log(w )) is orthogonalto λ.
1 r
The conjugacy of an affine interval exchange transformation with an interval ex-
change transformation was studied in details by Cobo [C]. He proved that the
regularity of the conjugacy depends on the position of the vector log(w) in the
flagofthe lyapunovexponents ofthe Rauzy-Veech-Zorichinduction. In particular,
assumethatT isself-similar,whichmeansthatλisaneigenvectorofapositive
(λ,π)
r ×r matrix R obtained by applying Rauzy induction a finite number of times.
Cobo proves that if log(w) belongs to the contracting space of tR then f is C1
conjugated to T . If log(w) is orthogonal to λ and is not in the contracting
(λ,π)
Date:July7,2007.
1991 Mathematics Subject Classification. Primary: 37C15;Secondary: 37B10.
Key words and phrases. interval exchange transformations, substitutive systems, wandering
sets.
1
2 XavierBressaud,PascalHubert,AlejandroMaass
space of tR then any conjugacy between f and T is not an absolutely contin-
(λ,π)
uous function. Moreover, Camelier and Guttierez example shows that conjugacy
between f and T does not always exist.
(λ,π)
In this paper, we prove the following result:
Theorem 1. Let T be a self-similar interval exchange transformation and R
(λ,π)
the associated matrix obtained by Rauzy induction. Let θ be the Perron-Frobenius
1
eigenvalue of R. Assume that R has an eigenvalue θ such that
2
(1) θ is a conjugate of θ ,
2 1
(2) θ is a real number,
2
(3) 1<θ (<θ ).
2 1
Then there exists an affine interval exchange transformation f with wandering in-
tervals that is semi-conjugated to T .
(λ,π)
This result means that Denjoy counterexamples occur very often (see section 5).
1.1. Reader’s guide. Camelier-Gutierrez [CG] and Cobo [C] developed an strat-
egy to prove the existence of a wandering interval in an affine interval exchange
transformationf which is semi-conjugated with a given IET. We explain it in sec-
tion 4. This strategy allowed them to achieve a first concrete example. Here we
explore the limits of this method in order to consider a large (and in some sense
abstract) family of IET. Let T be a self-similar interval exchange transforma-
(λ,π)
tion with associated matrix R. Let γ =(γ ,...,γ ) be the vector of the logarithm
1 r
of the slopes of the affine interval exchange transformation f. If f admits a wan-
dering interval I, the length |f(I)| is equal to eγj|I| if I is contained in interval
j. Roughly speaking, to create a wandering interval from the interval exchange
transformation T , one blows up an orbit of T . The difficulty is to insure
(λ,π) (λ,π)
that the total length remains finite. More precisely, if the symbolic coding of the
orbit is x=(xn)n∈Z, we have to check that the series
(1.1) e−γ(x0)−...−γ(xn−1) and eγ(x−n)+...+γ(x−1)
nX≥1 nX≥1
converge. This is certainly not true for a generic point x of the symbolic system
associated to T . Let ℓ(x) be the broken line with vertices (n,γ(x )+...+
(λ,π) 0
γ(xn−1))n∈N and (n,γ(x−n)+...+γ(x−1))n≥1. Since γ is orthogonal to λ, for a
generic point x, the line ℓ(x) oscillates around 0 as predicted by Ha´lasz’s Theorem
([Ha]). If the vector γ is not in the contracting space of tR the amplitude of the
oscillations tends to infinity with speed
nlog(θ1)/log(θ2).
It is hoped that the series (1.1) convergeif the y-coordinateof the brokenline ℓ(x)
is always positive and tends to infinity fast enough as n tends to ±∞. Points with
this property are called minimal points. Those are the main tool of the paper.
Thisanalysisapplies toaverylargeclassofsubstitutions andnotonlytosubstitu-
tions arising from interval exchange transformations. Section 3 gives an algorithm
to construct minimal points. We prove that the prefix-suffix decomposition of any
minimal point is ultimately periodic. From this analysis, we deduce that for any
minimal point x one has
γ(x )+...+γ(x ) −γ(x )−...−γ(x )
0 n −n −1
(1.2) liminf >0 and liminf >0
n→∞ log(θ2) n→∞ log(θ2)
nlog(θ1) nlog(θ1)
Persistenceofwanderingintervalsinself-similarAIET 3
Formulas in (1.2) imply immediately the convergence of the series in (1.1). More-
over, formulas in (1.2) has its own interest. It is a strengthening of a result by
Adamczewski [Ad] about discrepancy of substitutive systems.
Evenif the fractalcurves studied by Dumont and Thomas in [DT1], [DT2] are not
consideredexplicitlyinthearticle,theywereasourceofinspirationfortheauthors.
Thesecurvescorrespondtotherenormalizationofthebrokenlinesℓ(x)andappear
in subsection 3.3 in another language.
In section 5, we discuss the hypothesis of the main result in a geometric language.
We exhibit many examples where our hypothesis on the matrix R are fulfilled.
2. Preliminaries
2.1. Words and sequences. Let A be a finite set. One calls it an alphabet and
its elements symbols. A word is a finite sequence of symbols in A, w =w ...w .
0 ℓ−1
The length of w is denoted |w| = ℓ. One also defines the empty word ε. The
set of words in the alphabet A is denoted A∗ and A+ = A∗ \{ε}. We will need
to consider words indexed by integer numbers, that is, w = w ...w .w ...w
−m −1 0 ℓ
where ℓ,m ∈ N and the dot separates negative and non-negative coordinates. If
necessary we call them dotted words.
The set of one-sided infinite sequences x =(xi)i∈N in A is denoted by AN. Analo-
gously, AZ is the set of two-sided infinite sequences x=(xi)i∈Z.
GivenasequencexinA+,AN orAZ onedenotesx[i,j]thesub-wordofxappearing
between indexes i and j. Similarly one defines x(−∞,i] and x[i,∞). Let w =
w ...w .w ...w be a (dotted) word in A. One defines the cylinder set [w] as
−m −1 0 ℓ
{x∈AZ :x[−m,ℓ]=w}.
The shift map T : AZ → AZ or T : AN → AN is given by T(x) = (xi+1)i∈N for
x=(xi)i∈N. Asubshift is any shiftinvariantandclosed(for the producttopology)
subset of AZ or AN. A subshift is minimal if all of its orbits by the shift are dense.
Inwhatfollowswewillusetheshiftmapinseveralcontexts,inparticularrestricted
to a subshift. To simplify notations we keep the name T all the time.
2.2. Substitutionsandminimalpoints. Wereferto[Qu]and[F]andreferences
therein for the general theory of substitutions.
A substitution is a map σ : A → A+. It naturally extends to A+, AN and AZ; for
x=(xi)i∈Z ∈AZ the extension is given by
σ(x)=...σ(x )σ(x ).σ(x )σ(x )...
−2 −1 0 1
where the central dot separates negative and non-negative coordinates of x. A
further natural convention is that the image of the empty word ε is ε.
LetM be the matrix with indices inA suchthat M is the number of times letter
ab
b appears in σ(a) for any a,b ∈ A. The substitution is primitive if there is N > 0
such that for any a ∈ A, σN(a) contains any other letter of A (here σN means
N consecutive iterations of σ). Under primitivity one can assume without loss of
generality that M >0.
Let X ⊆ AZ be the subshift defined from σ. That is, x ∈ X if and only if any
σ σ
subword of x is a subword of σN(a) for some N ∈N and a∈A.
Assume σ is primitive. Given a point x ∈ X there exists a unique sequence
σ
(pi,ci,si)i∈N ∈(A∗×A×A∗)N such that for each i∈N: σ(ci+1)=picisi and
...σ3(p )σ2(p )σ1(p )p .c s σ1(s )σ2(s )σ3(s )...
3 2 1 0 0 0 1 2 3
4 XavierBressaud,PascalHubert,AlejandroMaass
is the central part of x, where the dot separates negative and non-negative coordi-
nates. This sequence is called the prefix-suffix decomposition of x (see for instance
[CS]).
If only finitely many suffixes s are nonempty, then there exists a ∈ A and non-
i
negative integers ℓ and q such that
x[0,∞)=c s σ1(s )...σℓ(s ) lim σnq(a)
0 0 1 ℓ
n→∞
Analogously, if only finitely many p are non empty, then
i
x(−∞,−1]= lim σnp(b)σm(p )...σ1(p )p
m 1 0
n→∞
for some b∈A and non-negative integers p and m.
Let θ be the Perron-Frobenius eigenvalue of M. Let λ = (λ(a) : a ∈ A)t be a
1
strictly positive right eigenvector of M associated to θ . We will also assume the
1
following algebraic property that we call (AH): M has an eigenvalue θ which is a
2
conjugateofθ . Noticethatthispropertycoincideswithhypothesis(1)ofTheorem
1
1.
The following lemma are important consequences of the algebraic property (AH).
Lemma 2. Let η : Q[θ ] → Q[θ ] be the field homomorphism that sends θ to θ .
1 2 1 2
The vector γ =η(λ)=(η(λ(a)):a∈A)t is an eigenvector of M associated to θ .
2
Proof. The field homomorphism η naturally extends to Q[θ ]|A|. Since λ belongs
1
to Q[θ ]|A| (up to normalization), then one deduces that Mη(λ) = θ η(λ). Thus,
1 2
η(λ) is an eigenvector of M associated to θ . (cid:3)
2
Lemma 3. Let γ be the eigenvector of M associated to θ as in Lemma 2. Then
2
for any |A|-tuple of non-negative integers (n : a ∈ A), n γ(a) = 0 implies
a a∈A a
na =0 for any a∈A. P
Proof. Assume n γ(a) = 0. Since γ = η(λ), applying η−1 one gets that
a∈A a
a∈Anaλ(a) =P0. This equality implies that na = 0 for every a ∈ A because the
cPoordinates of λ are positive. (cid:3)
Let γ = η(λ) as in Lemma 2. For w = w ...w ∈ A+ denote γ(w) = γ(w )+
0 l−1 0
...+γ(w ).
l−1
Let x ∈ X . Define γ (x) = 0, γ (x) = n−1γ(x ) for n > 0 and γ (x) =
σ 0 n i=0 i n
−1 γ(x ) for n < 0. Put Γ(x) = {γ (x)P: n ∈ Z}. In a similar way, given a
i=n i n
(Pdotted)wordw =w ...w ...w onedefinesγ (w)=0,γ (w)= n−1γ(w )
−m 0 l−1 0 n i=0 i
for 0<n≤l, γ (w)= −1 γ(w ) for −m≤n<0 and the set Γ(w). P
n i=n i
The best occurrence ofPa symbol a ∈ A in w is −m ≤ i < l such that wi = a and
γ (w) = min{γ (w) : −m ≤ j < l,w = a}. By Lemma 3, under hypotheses
i+1 j+1 j
(AH) this number is well defined and unique.
One says x is minimal if γ (x) ≥ 0 for any n ∈ Z. The set of minimal points for
n
σ is denoted by M(σ). It is important to mention that if x is a minimal point of
a substitution satisfying hypothesis (AH) then, by Lemma 3, γ (x) > 0 whenever
n
n6=0.
2.3. Affine interval exchange transformations. Let 0 = a < a < ... <
0 1
a <a =1 and A={1,...,r}.
r−1 r
An affine interval exchange transformation (AIET) is a bijective map f : [0,1) →
[0,1) of the form f(t) = w t + v if t ∈ [a ,a ) for i ∈ A. The vector w =
i i i−1 i
Persistenceofwanderingintervalsinself-similarAIET 5
(w ,...,w ) is called the slope of f. We assume furthermore the slope is strictly
1 r
positive.
An interval exchange transformation (IET) is an AIET with slope w = (1,...,1).
Commonly an IET is given by a vector λ=(λ ,...,λ ) such that λ =|a −a |
1 r i i i−1
for i ∈ A and a permutation π of A which indicates the way intervals [a ,a )’s
i−1 i
are rearranged by the IET. Clearly, a = i λ . We use T to refer to the
i j=1 j (λ,π)
IET associated to λ and π. P
OnesaystheAIETf issemi-conjugatedwiththeIETT ifthereisamonotonic,
(λ,π)
surjective and continuous map h:[0,1)→[0,1) such that h◦f =T ◦h.
(λ,π)
Let T be an interval exchange transformation. There is a natural symbolic
(λ,π)
coding of the orbit of any point t ∈ [0,1) by T . Consider the partition α =
(λ,π)
{[0,a1),...,[ai−1,ai),...,[ar−1,1)} and define φ(t) = (xi)i∈Z ∈ AZ by ti = j if
and only if Ti (t) ∈ [a ,a ). The set φ([0,1)) is invariant for the shift but
(λ,π) j−1 j
it is not necessarily closed, then one considers its closure X = φ([0,1)). This
procedure produces a semi-conjugacy (factor map) ϕ : (X,T) → ([0,1),T ).
(λ,π)
If t is not in the orbit of the extreme points 0,a ,...,1, then it has a unique
1
preimage by ϕ. If not, it has at most two preimages corresponding to the coding
of (T(iλ,π)(lims→t−s))i∈Z.
We use freely concepts related to Rauzy-Zorich-Veech induction. Rauzy induction
wasdefinedin[Ra],extendedtozipperedrectanglesbyVeech[Ve],andaccelerated
byZorich[Zo]. ForacompletedescriptionabouttheRauzy-Veech-Zorichinduction
see also the expository papers by Zorich [Zo2] and Yoccoz [Yo].
An IET T is self-similar if it can be recovered from itself after finitely many
(λ,π)
stepsofRauzyinductions(uptonormalization). Moreprecisely,thereexistsaloop
in the Rauzy diagram and an associated Perron-Frobenius matrix R such that
θ λ=Rλ
1
with θ the dominant eigenvalue of R.
1
Foraself-similarIETT thereisadirectrelationbetweenthesubshiftX andthe
(λ,π)
matrix R associatedto T . Indeed, there exists a substitution σ :A→A+ with
(λ,π)
associated matrix M = tR such that X = X (see [CG] and references therein).
σ
If the IET T is minimal then the subshift X is minimal too. In the sequel,
(λ,π) σ
we will use the fact that the substitution σ is primitive which implies that X is
σ
minimal. Nevertheless, no specific property of substitutions obtained from T
(λ,π)
will be needed for our purpose.
The relation between self-similar IET and pseudo-Anosov diffeomorphisms is ex-
plained in [Ve].
3. Construction of minimal points
Letσ :A→A+ be a primitive substitution with associatedmatrixM >0. Let θ ,
1
θ ,λandγ beasinsubsection2.2. Inaddition,assumeθ verifiesthehypothesesof
2 2
Theorem1. ByPerron-Frobeniustheorem,γ hasnegativeandpositivecoordinates.
Themainobjectiveofthesectionistogiveacombinatorialconstructionofminimal
points in this case.
6 XavierBressaud,PascalHubert,AlejandroMaass
3.1. Existence of minimal points.
Lemma4. Leta∈Asuchthatγ(a)>0andn∈N. Writeσn(a)=p s wherethe
n n
minimum of Γ(σn(a)) is attained at γ (σn(a)) and i=|p |. Then γ(s )≥θnγ(a).
i n n 2
In particular |s | grows exponentially fast with n.
n
Proof. Observe that γ(p )+γ(s )=θnγ(a) and γ(p )≤0. (cid:3)
n n 2 n
Lemma 5. M(σ)6=∅
Proof. Sinceγ haspositiveandnegativecoordinatesandX isminimal,thenthere
σ
exist b,c ∈ A such that bc is a subword of a point in X and γ(b) < 0,γ(c) > 0
σ
holds.
Let n ≥ 0 and define u = σn(b).σn(c). The sequence Γ(u ) attains its minimum
n n
at some N ∈ {−|σn(b)|,...,−1,0,...,|σn(c)|}. Define the (dotted)word v =
n n
u [−|σn(b)|,N −1].u [N ,|σn(c)|−1]=v−.v+. TheminimumofΓ(v )isattained
n n n n n n n
at coordinate 0, and is equal to 0.
By Lemma 4 there is a subsequence (ni)i∈N such that
lim |v−|= lim |v+|=∞
i→∞ ni i→∞ ni
Bycompactnessandeventuallytakingonceagainasubsequencethereexistsx∈X
such that for any m ∈ N there is i ∈ N with n ≥ m and x ∈ [v−.v+]. Thus
i ni ni
Γ(x[−m,m]) ⊆ R+ and its minimum is zero at zero coordinate. This implies
x∈M(σ). (cid:3)
3.2. The best strategy algorithm. Inwhatfollowswedevelopaproceduretocon-
struct minimal points that will become useful in next subsections.
The following two lemma follow directly from equality Mγ = θ γ. Their simple
2
proofs are left to the reader.
Lemma 6. Let m∈N and w ∈A+. Then γ(σm(w))=θmγ(w).
2
Lemma 7. Let w = w ...w ∈ A+. Write σ(w) = σ(w )...σ(w ). The
0 l−1 0 l−1
minimumofΓ(σ(w))isattainedinacoordinatecorrespondingtosomeσ(w ),where
i
w is the best occurrence of this symbol in w.
i
3.2.1. The basic procedure. The following procedure will allow to construct the
prefix-suffix decomposition of a minimal point.
Step0: Foreacha∈Awriteσ(a)=pa,0ca,0sa,0whereΓ(σ(a))attainsitsminimum
0 0 0
at γ (σ(a)).
|p0(a)|
Step 1: Let a ∈ A. By Lemma 7, the minimum of Γ(σ2(a)) comes from σ(b) for
someb∈Ainitsbestoccurrenceinσ(a). Writeσ(a)=pa,1ca,1sa,1whereca,1 =bis
1 1 1 1
the best occurrence of b in σ(a). Put w (a)=σ(pa,1)pb,0.cb,0sb,0σ(sa,1), where the
1 1 0 0 0 1
dot separates negative and non-negative coordinates. Let pa,1 = pb,0, ca,1 = cb,0
0 0 0 0
and sa,1 = sb,0. The sequence (pa,1,ca,1,sa,1)1 is called the best strategy for
0 0 i i i i=0
symbol a at step 1. By construction Γ(w (a)) ⊆ R+ and the minimum is equal to
1
zero at coordinate zero.
Step n+1: assume in previous step we have constructed for each symbol a ∈ A
the best strategy (pa,n,ca,n,sa,n)n . This sequence verifies:
i i i i=0
(i)for0≤i≤n,σ(ca,n)=pa,nca,nsa,n (here ca,n =a). Moreover,eachca,n isthe
i+1 i i i n+1 i
best occurrence of this symbol in σ(ca,n).
i+1
Persistenceofwanderingintervalsinself-similarAIET 7
(ii) Γ(w (a))⊆R+ and its minimum is zero at zero coordinate, where
n
w (a)=σn+1(a)=σn(pa,n)...σ(pa,n)pa,n.ca,nsa,nσ(sa,n)...σn(sa,n)
n n 1 0 0 0 1 n
Now we proceed as in step 1. Consider a ∈ A. By Lemma 7, the minimum of
Γ(σn+2(a))comesfromσn+1(b)forsomeb∈Ainitsbestoccurrenceinσ(a). Write
σ(a) = pa,n+1ca,n+1sa,n+1 where ca,n+1 = b is the best occurrence of b in σ(a).
n+1 n+1 n+1 n+1
The finite sequence (pa,n+1,ca,n+1,sa,n+1)n+1 where (pa,n+1,ca,n+1,sa,n+1) =
i i i i=0 i i i
(pb,n,cb,n,sb,n) for 0 ≤ i ≤ n is a best strategy for a at step n+1 and verifies
i i i
conditions (i) and (ii) by construction.
3.2.2. Finitely many minimal points. For each a ∈ A and n ∈ N consider the
cylinder set Ca,n = [w (a)], where w (a) is the dotted word defined in previous
n n
subsection. It isclearfromthe basic procedurethatfor anya∈A andn∈N there
exists a unique b∈A such that Ca,n+1 ⊆Cb,n. Thus, by compactness, there exist
at most |A| infinite decreasing sequences of the form (Can,n)n∈N. Let C1,...,Cℓ
with ℓ≤|A| be the collectionofintersections of suchsequences. Remark that such
sets are finite.
Given a minimal point x ∈ X with prefix-suffix decomposition (pi,ci,si)i∈N and
n ∈ N, there is a ∈ A such that (p ,c ,s ) = (pan,n,can,n,san,n) for 0 ≤ i ≤ n.
n i i i i i i
Therefore, x∈Ci = n∈NCan,n for some 1≤i≤ℓ.
The following proposTition is plain.
Proposition 8. There are finitely many minimal points.
Wewillseelaterthatminimalpointshaveultimatelyperiodicprefix-suffixedecom-
position. This fact yields to an alternative proof of previous proposition.
3.3. Serie associated to a minimal point. Define S = {(pi,ci,si)i∈N : ∀ i >
0, σ(ci) = pi−1ci−1si−1} and S = {(pi,ci,si)i∈N : ∀ i ≥ 0, σ(ci) = pi+1ci+1si+1}.
Observe that finite sequences taken from sequences in S and S coincide once re-
versed.
Let a∈A and n≥1. Then σn(a) can be decomposed as
σn(a)=σn−1(p )...σ(p )p c s σ(s )...σn−1(s )
1 n−1 n n n n−1 1
where for all 1≤i≤n, σ(c )=p c s (we have considered c =a). This decom-
i−1 i i i 0
positionisnotunique. Toaandthefinite sequence(p ,c ,s )n oneassociatesthe
i i i i=1
finite sum:
n
v(a;(p ,c ,s )n )= θ−iγ(p )
i i i i=1 2 i
Xi=1
Clearly, given x=(pxi,cxi,sxi)i∈N ∈S with cx0 =a, the series
v(a;x)= lim v(a;(px,cx,sx)n )= θ−iγ(px)
i i i i=1 2 i
n→∞
Xi≥1
exists.
Let v(a) = min{v(a;x) :x ∈ S with cx = a}. A sequence x ∈ S with cx = a such
0 0
that v(a;x)=v(a) is said to be minimal for a.
The best strategy for symbol a at step n ≥ 1 given by the algorithm produces a
finite sequence (pa,n,ca,n,sa,n)n . Set v (a) = n θ−n+i−1γ(pa,n). It follows
i i i i=0 n i=0 i
that vn(a)=v(a;(pan,−ni,can,−ni,san,−ni)ni=0). P
8 XavierBressaud,PascalHubert,AlejandroMaass
Lemma 9. For every a ∈ A and n ≥ 1, v (a) is minimal among the
n
v(a;(p ,c ,s )n+1) and v(a)=lim v (a).
i i i i=1 n→∞ n
Proof. Thefirstfactisanalogoustosaythat(pa,n,ca,n,sa,n)n isthebeststrategy.
i i i i=0
Moreover,|v (a)−v(a)|≤Kθ−nforsomeconstantK >0. Thisimpliesthedesired
n 2
result. (cid:3)
Lemma 10. Let a∈A. Assume there is a finite sequence (p¯ ,c¯ ,s¯ )l such that
j j j j=1
for infinitely many n ∈ N, (pa,n ,ca,n ,sa,n )l = (p¯ ,c¯ ,s¯ )l . Then,
n−j+1 n−j+1 n−j+1 j=1 j j j j=1
there exists y = (pyi,cyi,syi)i∈N ∈ S such that (yj)lj=1 = (p¯j,c¯j,s¯j)lj=1, cy0 = a and
v(a)=v(cy;y).
0
Proof. For any n∈N where the property of the lemma holds consider the point
y(n) =y(n)...y(n) =(p,a,s)(pa,n,ca,n,sa,n)...(pa,n,ca,n,sa,n)
0 n n n n 0 0 0
where σ(b)=pas for some b∈A.
Let y = (pyi,cyi,syi)i∈N be the limit of a subsequence (y(ni))i∈N. It follows by
construction that (y )l = (p¯ ,c¯ ,s¯ )l , cy = a and σ(cy) = py cy sy for
j j=1 j j j j=1 0 i i+1 i+1 i+1
any i≥0. Also, cy is the best occurrence of this symbol in σ(cy).
i+1 i
Let ǫ>0 and i ∈N such that |v(a)−v (a)|≤ǫ/2 for i≥i . Let L∈N be such
0 ni 0
that θ−L ≤ ǫ/4C where C > 0 is such that |γ(pc,n)|/(θ −1) ≤ C for any c ∈ A
2 i 2
and n ∈ N. Thus for i enough large, (py,cy,sy) = (p ,c ,s ) for
j j j ni−j+1 ni−j+1 ni−j+1
0 ≤ j < L and |v(a)− θ−iγ(py)| ≤ ǫ. Since ǫ is arbitrary one concludes
i≥1 2 i
v(cy)=v(a)= θ−iγP(py). (cid:3)
0 i≥1 2 i
P
One says that a point y = (pyi,cyi,syi)i∈N ∈ S verifies the continuation property if
v(cy) = v(cy;Ti(y)) for all i ≥ 0, where T is the shift map. It is clear that Ti(y)
i i
hasthecontinuationpropertytoo,foranyi∈N. Infacttosatisfythecontinuation
property it is enough to be minimal for cy.
0
Lemma 11. If y=(pyi,cyi,syi)i∈N ∈S is minimal for cy0 (that is, v(cy0)=v(cy0;y))
then y verifies the continuation property.
Proof. Let b = cy1 and z = (pzi,czi,szi)i∈N ∈ S with cz0 = b and v(b;z) = v(b)
given by Lemma 10 (considering l = 0). The sequence w = y y T(z) belongs
0 1
to S and verifies v(a;w) = θ−1γ(py)+θ−1v(b). Thus, if v(b;T(y)) > v(b), from
2 1 2
v(a)=v(a;y)=θ−1γ(py)+θ−1v(b;T(y)), one deduces that v(a;w)<v(a) which
2 1 2
is a contradiction. (cid:3)
This lemma proves that sequences y constructed in Lemma 10 verifies the contin-
uation property.
3.4. Minimalpointsare ultimatelyperiodic. Inthissectionweprovethatany
minimalpointx∈X hasultimately periodic prefix-suffixdecomposition. Thatis,
σ
if x¯=(pi,ci,si)i∈N is the prefix-suffix decomposition of x, then Tp+q(x¯)=Tqx¯ for
some p>q ≥0. If q =0 one says x is a periodic minimal point.
Lemma 12. For every a ∈ A there exists a ultimately periodic point
x(a) = (pxi(a),cxi(a),sxi(a))i∈N ∈ S with cx0(a) = a and v(a;x(a)) = v(a) (so, x(a)
has the continuation property).
Persistenceofwanderingintervalsinself-similarAIET 9
Proof. Let a∈A and y=(pyi,cyi,syi)i∈N ∈S with cy0 =a and v(a;y)=v(a) given
by Lemma 10 (considering l = 0). We are going to construct another one with
ultimately periodic decomposition.
Let0<q <pbesuchthaty =y andcy =cy =b. Thepreperiodicsequence
q p q−1 p−1
x=y ...y y ...y y ...y ...∈S since σ(cy )=pycysy by hypothesis.
0 q−1 q p−1 q p−1 p−1 q q q
We are going to prove that v(a;x)=v(a).
Observe that, by Lemma 10,
v(b)= θ−(i−q+1)γ(py) and v(b)= θ−(i−p+1)γ(py) .
2 i 2 i
Xi≥q Xi≥p
Thus,v(b)= p−1θ−(i−q+1)γ(py)+ θ−(i−q+1)γ(py)= p−1θ−(i−q+1)γ(py)+
i=q 2 i i≥p 2 i i=q 2 i
θ−(p−q)v(b). IPf we denote B = p−1Pθ−(i−q+1)γ(py), then v(Pb)=B θ−(p−q)i.
2 i=q 2 i i≥0 2
Consequently, P P
q−1
v(a)= θ−iγ(py)+θ−(q−1)B θ−(p−q)i
2 i 2 2
Xi=1 Xi≥0
On the other hand, a direct computation yields to
q−1
v(a;x)= θ−iγ(py)+θ−(q−1)( θ−(p−q)iB) ,
2 i 2 2
Xi=1 Xi≥0
which implies, v(a;x)=v(a). (cid:3)
Toeachpreperiodicsequencex(a)constructedinpreviouslemmaonecanassociate
a point x in the symbolic space X with periodic prefix-suffix decomposition of
σ
period
(p ,c ,s ),...,(p ,c ,s )=(px(a),cx(a),sx(a)),...,(px(a),cx(a),sx(a)) .
0 0 0 p−q p−q p−q p−1 p−1 p−1 q q q
Evenif, by construction,this pointis associatedto the minimalvalue v(b), there is
no reason for it to be a minimal point.
Without loss of generality we will do the following simplification. By iterating σ
enough times one can assume that all ultimately periodic sequences constructed
in Lemma 12 are of period 1 and of preperiod 1. That is, for each letter a ∈ A,
cx(a) = a and x = (p(a),aˆ,s(a)) for all i ≥ 1. The letter a ∈ A is periodic if
0 i
aˆ = a and one denotes Aˆ the subset of periodic letters. Since, the construction of
Lemma 12 implies that v(cx(a))=v(a;Ti(x(a))) for 0≤i≤p−1, then under this
i
simplification v(aˆ)=v(aˆ;T(x(a))).
Lemma 13. Let y ∈ S verifying the continuation property. Then, for any i ≥ 1
the point y(i) =y ...y T(x(cy)) has the continuation property too.
0 i i
Proof. Let i≥ 1 and 1≤ j ≤i. From the continuation property one deduces that
v(cy) = i−j θ−kγ(py )+θ−(i−j)v(cy). But, v(cy) = v(cy;x(cy)) and v(cˆy) =
j k=1 2 k+j 2 i i i i i
v(cˆy;T(xP(cy))), then y(i) =y ...y T(x(cy)) has the continuationproperty too.
i i 0 i−1 i
(cid:3)
Lemma 14. Let x,y ∈ S such that (x ) = (y ) and cx = cy = a. If
i i≥l+1 i i≥l+1 0 0
v(a;x)=v(a;y) then (x ) =(y ) .
i i≥1 i i≥1
10 XavierBressaud,PascalHubert,AlejandroMaass
Proof. Let x = (pxi,cxi,sxi)i∈N and y = (pyi,cxi,syi)i∈N. From the hypothesis one
deduces that
l l
θ−iγ(px)= θ−iγ(py)
2 i 2 i
Xi=1 Xi=1
and consequently
γ(σl−1(px)...px)=γ(σl−1(py)...py) .
1 l 1 l
But words σl−1(px)...px and σl−1(py)...py are prefixes of σl(a). Then, by the
1 l 1 l
algebraic condition (Lemma 3) they must be the same. This implies (px,cx,sx)=
i i i
(py,cy,sy) for 1≤i≤l. (cid:3)
i i i
Theorem 15. The prefix-suffix decomposition of any minimal point is ultimately
periodic.
Proof. Letx∈Xσ beminimalpointwithprefix-suffixedecomposition(pi,ci,si)i∈N.
There exists a finite sequence (p¯ ,c¯ ,s¯ )l such that (p¯ ,c¯ ,s¯ ) = (p¯,c¯,s¯) and
j j j j=0 0 0 0 l l l
for infinitely many i∈N, (p ,c ,s )l =(p¯ ,c¯ ,s¯ )l .
i−j i−j i−j j=0 j j j j=0
Let a = c¯ = c¯. By Lemma 10, there is a point y ∈ S verifying the continuation
0 l
property such that (y )l = (p¯ ,c¯ ,s¯ )l . In particular, v(a;y) = v(a) and
j j=0 j j j j=0
v(a;Tl(y)) = v(a). Since, v(a) = v(a;x(a)), then by Lemma 13 the sequence z =
y ...y T(x(a))hasthe continuationpropertyandv(a)=v(a;z)holds. Therefore,
0 l
by Lemma 14, one concludes that (x(a)) =(z ) .
i≥1 i i≥1
Wehaveprovedthata∈Aˆ,thatisa=aˆ,andthattheword(p(a),a,s(a))(p(a),a,s(a))
appearsinfinitelymanytimesintheprefix-suffixedecompositionofx. Nowweprove
that (pi,ci,si)i∈N is ultimately periodic with period (p(a),a,s(a)).
Assume this result does not hold. Then there is b6=a in A such that
(p ,c ,s )(p ,c ,s )(p ,c ,s )=(p(a),a,s(a))(p(a),a,s(a))(p,b,s)
i i i i−1 i−1 i−1 i−2 i−2 i−2
for infinitely many i∈N.
ByLemma 10, thereis a pointw∈S verifyingthe continuationpropertyandsuch
that w w w = (p(a),a,s(a))(p(a),a,s(a))(p,b,s). Since v(b) = v(b;T2(w)) and
0 1 2
v(b)=v(b;x(b)),byLemma13,thepointsu=w w w T(x(b))andv=x(a) x(b)
0 1 2 0
havethecontinuationproperty. Sinceuandvareultimatelyequal,then,byLemma
14, one concludes a=b which is a contradiction. This proves the theorem. (cid:3)
We stress the fact that it is possible to construct examples with minimal points
having ultimately periodic but not periodic prefix-suffix decomposition.
3.5. Convergence of series associated to minimal points.
Lemma 16. Let y ∈ S such that cy = a ∈ Aˆ and v(a;y) = v(a). Then, y =
0 1
(p(a),a,s(a)).
Proof. Putcy =a. Firstweprovethatv(cy)=v(cy;T(y)). Letz=y y T(x(cy))∈
0 1 1 0 1 1
S. If the assertion is not true then
v(a)=θ−1(γ(py)+v(cy;T(y)))>θ−1(γ(py)+v(cy))=v(a;z)≥v(a)
2 1 1 2 1 1
whichisacontradiction. Thus,v(cy)=v(cy;T(y))andfurthermorev(a)=v(a;z).
1 1
Then, the point w = (p(a),a,s(a))(p(a),a,s(a))T(z) verifies v(a) = v(a;w). But w
and x(a) are ultimately equal, then by Lemma 14, y =(p(a),a,s(a)).
1
(cid:3)