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Khinchin theorem for interval exchange transformations
MARCHESE, Luca
Abstract
We define a diophantine condition for interval exchange transformations (i.e.t.). When the number of intervals is two, that is for rotations on the circle, our condition coincides with the one in the classical Khinchin theorem, modulo the identification of a rotation with its rotation number. We prove that for i.e.t.s we have the same dichotomy of Khinchin theorem.
MARCHESE, Luca. Khinchin theorem for interval exchange transformations.
arxiv : 1003.5883v1
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http://archive-ouverte.unige.ch/unige:11907
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arXiv:1003.5883v1 [math.DS] 30 Mar 2010
KHINCHIN THEOREM FOR INTERVAL EXCHANGE TRANSFORMATIONS.
LUCA MARCHESE
Abstract. We define a diophantine condition for interval exchange transfor- mations (i.e.t.). When the number of intervals is two, that is for rotations on the circle, our condition coincides with the one in the classical Khinchin theorem, modulo the identification of a rotation with its rotation number. We prove that for i.e.t.s we have the same dichotomy of Khinchin theorem.
Contents
1. Introduction. 1
1.1. Interval exchange transformations. 2
1.2. Decomposition of the problem. 4
1.3. Contents of this article. 6
Acknowledgements 7
2. Background theory. 7
2.1. The algorithm of Rauzy and Veech. 7
2.2. Reduction of Rauzy classes. 11
2.3. Lebesgue measure and distortion underQ.b 13
3. The convergent case. 14
3.1. A necessary condition for the orbits ofQ. 16
3.2. A global estimate. 17
3.3. A local uniform estimate. 18
4. The divergent case. 21
4.1. Sufficient condition for the orbits ofQ. 21
4.2. Sufficient shrinking target criterion. 25
4.3. Refined shrinking target. 29
4.4. Borel-Cantelli formulation and end of the proof. 33
5. Main technical results. 38
5.1. Main combinatorial property. 38
5.2. Main estimate. 43
References 49
1. Introduction.
A rotation on the circle is completely described by its rotation number and we can establish a very precise dictionary between the dynamical properties of the transformations in this class and the arithmetical properties of the rotation number.
For example a rotation is periodic if and only if its rotation number is rational and conversely all irrational rotations are minimal. Besides this first coarse dichotomy
1
we can classify real numbers according to their diophantine properties, which in terms of rotations on the circle corresponds to give a measure of recurrence. In particular, forϑ∈[0,1) we can define the most general diophantine condition by
(1.1) {nϑ}< ϕ(n),
where {x} := min{x−k;k ∈ Z, k ≤ x}, and where ϕ : N → R+ is a positive (vanishing) sequence. IfRϑis the rotation with rotation numberϑ, then a solution n∈Nof equation (1.1) corresponds to an iterateRnϑ at (small) distanceϕ(n) from the identity. Khinchin theorem is a classical result in arithmetic and establish a dichotomy between two opposite behaviors (see [K]).
Theorem (Khinchin). Let ϕ:N→R+ be a positive sequence such that nϕ(n)is monotone decreasing.
• If P∞
n=1ϕ(n) < ∞ then equation (1.1) has just finitely many solutions n∈Nfor almost any ϑ.
• IfP∞
n=1ϕ(n) =∞then for almost anyϑequation (1.1) has infinitely many solutions n∈N.
1.1. Interval exchange transformations. An alphabet is a finite set A with d≥2 elements. Aninterval exchange transformation (also called i.e.t.) is a mapT from an interval Ito itself such that Iadmits two finite partitions Pt:={Iαt}α∈A andPb:={Iβb}β∈Aintodopen intervals and for anyα∈ Athe restriction of T to the interval Iαt is a translation with image the interval Iαb. The mapT :I →I is therefore defined by rearranging (via translations) the intervals of the partitionPt
in a new order insideI given byPb and is entirely defined by the following data:
(1) The lengths of the intervals,
(2) The order before and after rearranging.
The first are calledlength data, and are given by a vectorλ∈RA+, whereλαdenotes the length ofIαt (which is equal to the length ofIαb) for anyα∈ A. The second are calledcombinatorial data and are given by a pair of bijectionsπ= (πt, πb) fromA to{1, .., d}. The meaning ofπis that for anyα∈ A, if we count starting from the left, the intervalIαt is inπt(α)-th position inPtandIαb is inπb(α)-th position inPb. For any combinatorial datumπlet us call ∆π:={π} ×RA+ the set of all i.e.t.s with combinatorial datum π. Let us consider any T ∈∆π and let us writeT = (π, λ), whereλis the corresponding length datum. For anyα∈ Awithπt(α)>1 we call utαthe left endpoint ofIαt. In generalT is not continuous atutα. Similarly for any β∈ Awithπb(β)>1 we callubβthe left endpoint ofIβb. In general the inverseT−1 ofT is not continuous atubβ. If we identify the intervalI with (0,P
α∈Aλα) then the position of the singularitiesutα andubβ is given by
utα:= X
πt(α′)<πt(α)
λα′ and ubβ:= X
πb(β′)<πb(β)
λβ′.
We say that the combinatorial datum π isadmissible if there is no proper subset A′⊂ Awithk < delements such thatπt(A′) =πb(A′) ={1, .., k}. In the following always consider admissible combinatorial data. A connection for T : I → I is a triple (β, α, n) with πb(β) > 1, πt(α) > 1 and n ∈ N such that Tnubβ = utα. In particular, ifT has no connections thenπis admissible.
Veech [Ve] and then Zorich [Z1] introduced a re-normalization map acting on i.e.t.s which generalizes theGauss map. It happens that theinfinitely re-normalizable
i.e.t.s are those without connections, exactly as irrational real numbers are the points where the Gauss map can be iterated infinitely many times. This reason lead us to think to i.e.t.s with connections as rational numbers. Another reason to trust this philosophy is Keane’s theorem (see [Ke]), which says that if T has no connections, then it is minimal (we remark anyway that there is not an true dichotomy as for rotations, since there exist minimal i.e.t.s with connections).
Khincin type condition for i.e.t.s. In order to define the diophantine condition that we study in this paper let us consider a positive sequence ϕ:N→ R+ such that nϕ(n) is decreasing monotone. LetT :I→Ibe an i.e.t. without connections with combinatorial datum π. We consider triples (β, α, n) with πb(β) > 1,πt(α) > 1 andn∈Nsuch that
(1.2) Tn(ubβ)−utα< ϕ(n).
For any triple (β, α, n) as above let us callI(β, α, n) the open subinterval ofIwhose endpoints areTn(ubβ) andutα(their reciprocal order does not matter).
Definition 1.1. Let T be an i.e.t. with admissible combinatorial datum π and (β, α, n) be a triple with πb(β) > 1, πt(α) > 1 and n ∈ N. We say that (β, α, n) is a reduced triple for T if for any k ∈ {0, ..n} the pre-image T−k(I(β, α, n)) of I(β, α, n)does not contain in its interior any singularityutα′forTor any singularity ubβ′ for T−1 (whereα′, β′∈ A andπb(β′)>1, πt(α′)>1).
Definition 1.2. Given a function ϕ:N→R+ as before and an admissible π, an interval exchange transformationT ∈∆π is said
• modϕ-Diophantine if equation (1.2) has just finitely many solutions.
• modϕ-Liouville if for any pair of lettersβ, α withπb(β)>1andπt(α)>1 there exists infinitely many triples(β, α, n)reduced forT which are solution of equation (1.2).
We proved the following generalization of Khinchin theorem.
Theorem 1.3. Let us consider a positive sequenceϕ:N→R+ such that nϕ(n) : N→R+ is decreasing monotone. For any admissible combinatorial datum π0 we have the following dichotomy:
a: IfP+∞
n=1ϕ(n)<+∞then almost any i.e.t. T ∈∆π0 ismodϕ-Diophantine.
b: If P+∞
n=1ϕ(n) = +∞then almost any i.e.t. T ∈∆π0 ismodϕ-Liouville.
I.e.t.s are related to translation surfaces and to the Teichm¨uller flow on their moduli space (see [Ve], [Ma1] and [Z2]). In particular in [Mar2] we prove a version of theorem 1.3 for translation surfaces which implies a refinement of Masur’slog- arithmic law for the Teichm¨uller flow onstrata of the moduli space of translation surfaces (see [Ma2] for Masur’s original result).
Linear involutions are a natural generalization of i.e.t.s introduced in [DaNo] by Danthony and Nogueira. In [BoLa] Boissy and Lanneau related linear involutions to half-translation surfaces, which are the general contest where Masur’s logarithmic law holds. We believe that the techniques introduced in this paper can be extended to linear involutions and we ask if a generalization of theorem 1.3 can be proved for them (more precisely for the subclass of linear involutions which are relevant for half-translation surfaces, as it is explained in [BoLa]). Our guess is also motivated by the paper of Avila and Resende ([A,R]), where the authors generalize some results of [A,G,Y] which play an important rule in the proof of theorem 1.3.
Boshernitzan and Chaika studied shrinking target properties for i.e.t.s related to our diophantine condition in definition 1.2 (see [B,Ch] and [Ch]). Their results are obtained independently from this article and use different techniques. In particular in [Ch] it is proven the following result.
Theorem (Chaika). Let ϕ : N → R+ be a positive sequence such that nϕ(n) is decreasing monotone and P∞
n=1ϕ(n) =∞. Then for almost any i.e.t. T :I →I with admissible combinatorial datum, for any x in I and for almost any y ∈ I, there are infinitely manyn∈Nsuch that
|Tn(x)−y|< ϕ(n).
1.2. Decomposition of the problem. Theorem 1.3 is an example of application to a dynamical problem of the so calledBorel-Cantelli lemma (see [Bi]).
Theorem. Let (X,P)be a probability space and let(Xn)n∈N be a countable family of events inX.
• If P∞
n=1P(Xn) < +∞ then almost any x ∈ X belongs to finitely many eventsXn.
• On the other hand, ifP∞
n=1P(Xn) = +∞and the eventsXnare each other independent, then almost anyx∈X belongs to infinitely manyXn. In applications to dynamics the events Xn are usually related to the iterates of some map on the space X. In this case they are almost never independent in the probabilistic sense and the main difficulty is to develop some weak form of independence for them in order to make the Borel-Cantelli argument work.
In our case, for any triple (β, α, n) as in theorem 1.3, the set of those T such that (β, α, n) is reduced forT and satisfies equation (1.2) defines an event in ∆π0. Roughly speaking our strategy is to prove that the probability of such event is of the same order ofϕ(n) and that, when the triple (β, α, n) varies, we have some weak form of independence for the family of the associated events. Even if this program is perfectly meaningful without any notion of dynamics, in order to pursue it we need to define a dynamical system on the space of all i.e.t.s with the same number d of intervals. The dynamic we refer to is the so-calledRauzy-Veech map, which has been introduced in [Ve] and then modified in [Z1] in order to have better ergodic properties. Unfortunately, for a genericT, we are not able to translate the properties of being modϕ-Diophantine or modϕ-Liouville into equivalent ones for its orbit. We can just establish a necessary condition for the orbits of a modϕ- DiophantineT which holds for the convergent case (i.e. when P∞
n=1ϕ(n)<+∞) and conversely, in the divergent case (i.e. whenP∞
n=1ϕ(n) = +∞), we can establish a sufficient condition for the orbits of T in order to have that that T is modϕ- Liouville. The techniques in the two cases are different, therefore the proof of part a) and of part b) of theorem 1.3 have to be treated separately. We also have to introduce a normalization on length data of i.e.t.s. A first reason is that the Borel-Cantelli lemma requires a probabilistic setting, whereas for any combinatorial datumπthe cone ∆π has infinite lebesgue measure. On the other hand the Rauzy- Veech map has interesting recurrence properties just at projective level, that is on the space of rays in ∆π. For the reasons above we will consider i.e.t.sT acting on an interval I with length one. On this codimension-one subspace of ∆π0 we will state separately two criterions. The first is proposition 1.4, which is sufficent to get part a) of theorem 1.3, the second is proposition 1.5, which is sufficient to get part b) of theorem 1.3.
1.2.1. Normalization of lengths. For any vectorλ∈RA+ we introduce the notation kλk:= X
α∈A
λα and bλ:= λ kλk.
If the i.e.t. T is defined by the data (π, λ), we callTbthe i.e.t. whose combinatorial and length data are (π,bλ). For any admissible combinatorial datumπwe introduce the (d−1)−simplex ∆(1)π :={T = (π, λ)∈∆π;kλk= 1}. The mapT 7→(T ,b kλk) establishes an homeomorphism between ∆π and ∆(1)π ×R+and by Fubini’s theorem thed−dimensional lebesgue measure Lebd on ∆π0 is equivalent to the product of the (d−1)−dimendional lebesgue measure Lebd−1on ∆(1)π0 with Leb1 onR+
We fix an admissible combinatorial datumπ0 as in theorem 1.3, we consider an i.e.t. T = (π0, λ) in ∆π0 and the corresponding normalized i.e.t. Tb = (π0,bλ) in
∆(1)π0. Ifubβandutαare the singularities forT we denoteubbβandbutαthe corresponding singularities forTb. We have|Tbnubbβ−ubtα|=kλk−1|Tnubβ−utα|, therefore
|Tnubβ−utα|< ϕ(n)⇔ |Tbnbubβ−butα|<kλk−1ϕ(n).
1.2.2. Convergent case. For any positive integerMwe consider the set ∆π0(>1/M) of thoseT ∈∆π0 whose length datumλ satisfieskλk >1/M. If a triple (β, α, n) is solution of equation (1.2) for T ∈ ∆π0(>1/M), then for the normalizedTb we necessarily have|Tbnubbβ−ubtα|< M ϕ(n). We observe that ifP∞
n=1ϕ(n)<+∞then the same holds for the re-scaled sequence M ϕ(n). If we assume that part a) of theorem 1.3 holds on ∆(1)π0 with respect to the measure Lebd−1, then the argument above, together with Fubini’s theorem, implies that it holds on ∆π0(>1/M) with respect to the measure Lebd. Since M is arbitrary we have that the following proposition is a sufficient condition to get part a) of theorem 1.3.
Proposition 1.4. Let π0, (β, α) and ϕ be respectively a combinatorial datum, a pair of letters and a positive sequence as in theorem 1.3. If P∞
n=1ϕ(n) < +∞, then for almost anyT ∈∆(1)π0 there exists just finitely many triples (β, α, n)which satisfy equation (1.2).
1.2.3. Divergent case. For any positive integer M we consider the set ∆π0(< M) of those T ∈∆π0 whose length datumλsatisfies kλk< M. We first observe that a triple (β, α, n) is reduced for T if and only if it is reduced for T. Moreover if ab reduced (β, α, n) satisfies|Tbnbubβ−ubtα|<(1/M)ϕ(n), then it is a reduced solution of equation (1.2) for T ∈ ∆π0(> 1/M). As in the convergent case, the re-scaled sequence (1/M)ϕhas divergent series if and only ifϕhas divergent series. Arguing as before, if we assume that part b) of theorem 1.3 holds on ∆(1)π0 with respect to the measure Lebd−1, then it holds on ∆π0(< M) with respect to the measure Lebd. SinceM is arbitrary we have that the following proposition is a sufficient condition to get part b) of theorem 1.3.
Proposition 1.5. Let π0, (β, α) and ϕ be respectively a combinatorial datum, a pair of letters and a positive sequence as in theorem 1.3. If P∞
n=1ϕ(n) =∞, then for almost anyT ∈∆(1)π0 there exists just finitely many triples(β, α, n)which satisfy equation (1.2).
1.3. Contents of this article. This article is devoted to the proof of theorem 1.3.
As we explained, the convergent case is treated independently from the divergent one and we are reduced to prove two independent statements: proposition 1.4 for the former and proposition 1.5 for the latter.
InSection 2. we recall the basic theory of i.e.t.s. In paragraph 2.1 we describe the Rauzy-Veech map and the Zorich’s acceleration, in particular we introduce Rauzy classes. In paragraph 2.2 we describe a combinatorial operation on Rauzy classes introduced in [A,G,Y] and calledreduction, the formalism introduced in this paragraph is used in paragraphs 4.3, 4.4, 5.1 and 5.2. The normalized Rauzy-Veech map is a piecewise liner-projective map, in paragraph 2.3 we describe the connected components of its domains and we give a formula for their volume. The volume has unbounded distortion under iteration of the map and in paragraph 2.3.1 we state a result proved in [A,G,Y] on the control of the distortion. We also explain how the distortion can be interpreted in terms of conditional probability.
Section 3. treats the convergent case, that is the proof of proposition 1.4. We first give a short argument in order to reduce ourselves to the case of reduced triples, then for any (β, α, n) as in theorem 1.3 we introduce the setI(π0, β, α, n) of thoseT in ∆(1)π0 such that the triple (β, α, n) is reduced forT and equation (1.2) is satisfied. The main task is to prove that Lebd−1(I(π0, β, α, n)) is of the same order ofϕ(n). Lemma 3.2 in paragraph 3.1 is one of the main steps of the proof:
it says that for a triple (β, α, n) reduced for T the quantity|Tnubβ−utα|equals to
|u(r),bβ −u(r),tα |, whereu(r),bβ andu(r),tα are the singularities of someT(r)in the orbit ofT under the Rauzy-Veech map. In paragraph 3.2 we show that the locus of those T such that u(r),bβ =u(r),tα is a codimension-one subset of ∆(1)π with a complicated shape and we reduce the estimation of Lebd−1(I(π0, β, α, n)) to an estimation of the (d−2)-volume of this codimension one subset. The estimate is proved in lemma 3.6 in paragraph 3.3.
InSection 4. we treat the divergent case, that is we prove proposition 1.5. The proof is obtained by a sequence of tree sufficient criterions, each one implying the precedent. The main idea in paragraph 4.1 is to consider the orbit (T(r))r∈N of T under the Rauzy-Veech map and to find good instantsr such that|Tnubβ−utα| equals the length of some interval ofT(r). The idea is developed in lemmas 4.3 and 4.4. They work in parallel, since they hold under some combinatorial conditions on the pair (β, α) (explained in definition 4.1) and one applies when the other fails and vice-versa. The final result is proposition 4.5, which is the first sufficient condition.
In paragraph 4.2 we convert proposition 4.5 into a shrinking target criterion: to any pair (β, α) as in theorem 1.3 we associate a mapFη, which essentially coincides with thefirst returnof the Rauzy-Veech map to a proper section ∆(1)η (see definition 4.10 and equation (4.5)). The property that we require is that almost any orbit under Fη enters infinitely many times into a sequence of sets, or targets, whose (d−1)-volume shrinks to zero. In paragraph 4.4 we state and prove our final sufficient criterion, that is proposition 4.24. It is formulated as the divergent part of the Borel-Cantelli lemma, where the eventsXk have the formFη−k(Ek) and the setsEk are subsets of the shrinking targets in proposition 4.12. It is important that anyEkis measurable with respect to the sigma-algebra generated by the connected components of the domain of the mapFη. This is quite a delicate point and it is treated in paragraphs 4.3 and 4.4.1.
In Section 5. we state and prove two general results for i.e.t.s that we had to develop to prove proposition 1.5. The first is theorem 5.1, which affirms that for any pair of letters (β, α) as in theorem 1.3, any Rauzy class contains an element π where the two letters are in some required reciprocal position. Theorem 5.1 provides the combinatorial property that we need to apply proposition 4.5. The second general result is theorem 5.2: we cut any ∆(1)π along an hyper-plane parallel to any (d−2)-hyper-face, at any height, then consider the band in ∆(1)π delimited by the hyper-plane and the hyper-face. The theorem says that the total volume of those connected component of the domain of Fη that are included in the band is proportional to the volume of the band. Theorem 5.2 plays a crucial role passing from proposition 4.12 to proposition 4.24.
Acknowledgements. The results in this paper were obtained in my Ph-D thesis.
I would like to thank Jean-Christohpe Yoccoz for many discussions and for his help in revising this work. I am also grateful to Stefano Marmi for many discussions and to Giovanni Forni and Pascal Hubert for many precious remarks.
2. Background theory.
2.1. The algorithm of Rauzy and Veech. In this paragraph we give a brief survey of the basic properties of the algorithm of Rauzy and Veech and we develop the notation that we use in the following. We follow [M,M,Y] and [A,G,Y].
Let π = (πt, πb) and λdefine an interval exchange transformation T : I → I.
Letǫ∈ {t, b}, where the lettert stands fortop and the letterbforbottom. Ifǫ=t we put 1−ǫ:=band ifǫ=bwe put 1−ǫ:=t. Let us callαtandαb the two letters in Asuch that respectivelyπt(αt) =dandπb(αb) =d. The rightmost singularity ofT is thereforeutαt and the rightmost singularity ofT−1isubαb. We suppose that
(2.1) utαt 6=ubαb
and we consider the value ofǫ∈ {t, b}such that uǫαǫ < u1−ǫα1−ǫ.
With this definition ofǫwe say thatT is oftype ǫ. We also say that the letterαǫ
is thewinner ofT andα1−ǫis theloser. We consider the subinterval ofI I˜:=I∩(0, u1−ǫα1−ǫ)
and we define ˜T : ˜I→I˜as the first return map ofT to ˜I. It is easy to check that T˜is an interval exchange transformation. The combinatorial datum ˜π= (˜πt,˜πb) of T˜ is given by:
(2.2)
˜
πǫ(α) =πǫ(α)∀α∈ A
˜
π1−ǫ(α) =π1−ǫ(α) ifπ1−ǫ(α)≤π1−ǫ(αǫ)
˜
π1−ǫ(α1−ǫ) =π1−ǫ(αǫ) + 1
˜
π1−ǫ(α) =π1−ǫ(α) + 1 ifπ1−ǫ(αǫ)< π1−ǫ(α)< d.
The length datum ˜λof ˜T is given by:
(2.3) λ˜α=λα ifα6=αǫ
λ˜αǫ=λαǫ−λα1−ǫ.
WhenT = (π, λ) satisfies condition (2.1), equations (2.2) and (2.3) define a map T 7→ Q(T) := ˜T that we call theRauzy-Veech induction map. We introduce two operations Rt and Rb on the set of admissible combinatorial data as follows: if ǫ is the type ofT and πis its combinatorial datum, then we set Rǫ(π) := ˜π, where
˜
π is the combinatorial datum of ˜T. It is easy to check that ifπ is an admissible combinatorial datum then bothRt(π) andRb(π) are admissible.
Definition 2.1. Let us call S the set of all the admissible combinatorial data π over some alphabet A. The mapsRtandRb from S to itself are called the Rauzy elementary operations.
• A Rauzy class is a minimal non-empty subset R of S which is invariant under Rt andRb.
• A Rauzy diagram is a connected oriented graph D whose vertexes are the elements of R and whose oriented arcs, or arrows, correspond to Rauzy elementary operationsπ7→Rε(π)between elements of R.
• An arrow corresponding toRt is called a toparrows and we say thatαtis its winner andαb is its loser. Conversely an arrow corresponding toRb is called a bottomarrow and we say that αt is its loser andαb is its winner.
• A concatenation of compatible arrows in a Rauzy diagram is called a Rauzy path. The set of all Rauzy paths connecting elements ofRis denotedΠ(R).
If a pathγ is concatenation ofrsimple arrows, we say that γhas lengthr.
Length one paths are arrows, we also identify the elements ofRwith trivial paths, that is length zero paths.
• A partial ordering≺is defined onΠ(R)saying thatν≺γiffγ begins with ν. A subfamily Γof Π(R) is called disjoint iff for any two elements η and ν ofΓ we have neitherη ≺γ nor γ≺η.
With the notation above, recalling that for any combinatorial datumπwe defined
∆π={π} ×RA+, we denote ∆(R) :=F
π∈R∆πthe set of all the intervals exchange transformations with combinatorial datum in the Rauzy classR.
2.1.1. Linear action. For any Rauzy classRand any path γ ∈Π(R) we define a linear mapBγ ∈SL(d,Z) as follows. If γ is trivial thenBγ =id. Ifγ is an arrow with winner α and loser β then we set Bγeα = eα+eβ and Bγeξ = eξ for all ξ∈ A \ {α}, where{eξ}ξ∈Ais the canonical basis ofRA. We extend the definition to paths so thatBγ1γ2=Bγ2Bγ1.
Let us fix some elementπ in the Rauzy classR. For any γ∈Π(R) starting at πwe define the simplicial sub-cone ∆γ ⊂∆π by
∆γ={π} ×tBγ(RA+),
wheretBγ denotes the trasposed of the matrixBγ defined above. For the sameγ we also define the vectorqγ ∈NAby
qγ :=Bγ~1,
where~1 denotes the vector ofNAthat has all entries equal to 1.
2.1.2. Iteration of the algorithm. WhenT ∈∆(R) is such that ther-th iterated of Qis defined, we have an explicit formula forT(r):=Qr(T).
Lemma 2.2. Let γ∈Π(R)be a path in the Rauzy diagram with length r and let Bγ and∆γ be respectively the matrix and the simplicial cone defined in paragraph
2.1.1. Then for anyT ∈∆γ ther-th iterated of Qis defined and the length datum λ(r) of Qr(T)is given by the formula
λ(r)=tBγ−1λ.
Proof: Let us first consider an arrowγ. We callπits starting point,αits winner andβ its loser. We suppose thatγis of type top, the other case being identic. Let us considerT = (π, λ) in ∆γ. By the definition of the matrixBγ we have
Bγ =id+Eβ,α
where Eβ,αis the matrix whose entry in the columnαand row β is 1 and all the others are 0. It follows that tBγRd+ its the open half cone of Rd+ of those λsuch that λα > λβ. Since πt(α) = πb(β) = d this last condition is equivalent to the condition (2.1) and thereforeQis defined onT. Moreover equation (2.3) says that for the length datum λ(1) ofQ(T) we haveλ(1)=tBγ−1λ.
The proof continues by induction onr. Let us suppose that the lemma is proved for any concatenationγ1...γr−1ofr−1 arrows. We consider a pathγ=γ1...γr−1γr
starting at π, where γr is an arrow which can be concatenated to γr−1. Let us consider anyT = (π, λ)∈∆γ ={π} ×tBγRd+. SinceBγ =BγrBγ1...γr−1 it follows that ∆γ is a sub-cone of ∆γ1...γr−1, therefore the inductive hypothesis applies, that is T(r−1) = Qr−1(T) is defined and its length datum is given by λ(r−1) =t Bγ−11...γr−1λ. Moreover we have T(r−1)∈∆γr, thus we can apply the argument in the beginning and get that T(r) = Q(T(r−1)) is defined an its length datum is λ(r)=tBγ−1rλ(r−1)=tB−1γ λ. The lemma is proved.
We denote ∆1(R) the domain ofQ, whose connected components are the simpli- cial cones ∆γ associated to arrowsγ∈Π(R). One easily checks that ifγis an arrow starting atπand ending atπ′then the matrixtBγ−1establishes an homeomorphism between ∆γ and ∆π′. Let ∆r(R) be the domain of Qr. Lemma 2.2 says that the connected components of ∆r(R) are naturally labeled by pathsγin Π(R) of length r. Ifγ is such a path ending atπ′ thenQr: ∆γ →∆π′ is a homeomorphism. The set ∆∞(R) :=T
k∈N∆k(R) is the set of those i.e.t. T such that the mapQcan be applied infinitely many times. Being the intersection of countably many sets of full lebesgue measure, ∆∞(R) has full lebesgue measure.
The complement of ∆∞(R) is the set of those i.e.t. T such that for some r∈N ther−th elementT(r) of theQ-orbit violates the condition in equation (2.1), that is the algorithm stops. For the complement the following characterization holds.
Lemma. When applied to an i.e.t. T the Rauzy algorithm Q eventually stops if and only if T has a connection.
2.1.3. Return times. Let γ be a finite Rauzy path of length r starting at π and let T : I → I be an i.e.t. defined by the data (π, λ) ∈ ∆γ. We consider the i.e.t. T(r)=Qr(T) and we denoteI(r)⊂I the domain ofT(r). We recall that by definitionT(r)is the first return map ofTtoI(r). For anyα∈ AletIα(r),tbe the sub- interval of I(r) whereT(r)acts as a translation. We have T(r)(Iα(r),t) =TR(Iα(r),t) for some integerR=R(α, k) and fori∈ {1, .., R−1}the iteratesTi(Iα(r),t) are never contained inI(r)and never meet the sets of singularities{utα′}α′∈Aand{ubβ′}β′∈A
respectively forT and forT−1. In particular they are all disjoint (Rstands for first return time). Since the length datumλof T is linked to the length datumλ(r) of
T(r)byλ=tBγλ(r), it follows that for anyβ∈ Athere are exactly [Bγ]β,αintegers i ∈ {0, .., R} such thatTi(Iα(r),t)⊂Iβt, where [Bγ]β,α is the entry ofBγ in rowα and columnβ. It follows that the return timeR ofIα(r),t toI(r) under iteration of T is given by the entryqαγ of the vectorqγ =Bγ~1.
Letu(r),tα andu(r),bβ be singularities forT(r)and (T(r))−1. SinceT(r)is the first return map of T to I(r), then for n∈Nwe have Tnubβ ⊂I(r)if and only if there exists somel such thatTnubβ = (T(r))lu(r),bα . In particular, denotingl=l(r, β) the first time such thatTlubβ∈I(r), we have
u(r),bβ =Tlubβ.
Foru(r),tα the discussion is the same. Ifh=h(r, α) is the smallest integer such that T−hutα∈I(r)then
u(r),tα =T−hutα.
For any α ∈ A the definition of the integer h(r, α) implies Th(r,α)(Iα(r),t) ⊂ Iαt. Then applying T we haveTh(r,α)+1(Iα(r),t)⊂Iαb. Finally iteratingl(r, α) times we getTh(r,α)+1+l(r,α)(Iα(r),t) =Iα(r),b. For anyα∈ A the positive integersl(r, α) and h(r, α) satisfy
h(r, α) + 1 +l(r, α) =qγα.
2.1.4. Normalized Rauzy Veech algorithm and Zorich’s acceleration. The Rauzy- Veech algorithm has interesting recurrence properties just at projective level. We introduce a normalization on the sum of the lengths of the intervals. Forλ∈RA+ we recall the notation kλk :=P
α∈Aλα and bλ:=kλk−1λ. For any combinatorial datumπin someRwe write
∆(1)π :={(π, λ)∈∆π;kλk= 1}.
The normalized length datum of an i.e.t. T ∈∆(1)π will be often denotedbλ. For any Rauzy classR the set of all normalized i.e.t.s with combinatorial datum in R is denoted ∆(1)(R) :=F
π∈R∆(1)π . From now on, when applying the mapQ, we will not worry about its domain and we will say that it is defined on ∆(1)(R) modulo a set of measure zero.
Definition 2.3. Let R be a Rauzy class over an alphabet A. The normalized Rauzy-Veech algorithm is the mapQb: ∆(1)(R)→∆(1)(R) defined by
Q(π, λ) := (b π,e λe keλk),
where(eπ,eλ) =Q(π, λ)is the Rauzy-Veech algorithm introduced in paragraph 2.1.
IfT ∈∆(1)(R) is an i.e.t. without connections, for anyr∈Nwe denoteTb(r):=
Qbr(T). For anyrletγrbe the simple arrow associated to the stepTb(r)=Q(b Tb(r−1)) of the algorithm. We obtain a sequenceγ1, γ2, .., γr, ...of simple arrows. We denote γ(T, r) the concatenation γ1...γr of the first r arrows in the sequence. We have γ(T, r) ≺ γ(T, r+ 1) with respect to the ordering ≺ in definition 2.1. Then we defineγ(T,∞) as the half infinite path in Π(R) such thatγ(T, r)≺γ(T,∞) for all r >0.
Veech proved thatQb has an unique invariant measure which is absolutely con- tinuous with respect to the lebesgue measure, nevertheless this measure is not finite (see [Ve]). Zorich introduced anacceleration of Qb with a finite invariant measure (see [Z1]). For an i.e.t. T without connections we define the integer N(T) as the minimum of thoser∈Nsuch that the type ofT is different from the type ofQbr(T).
Definition 2.4. The Zorich’s acceleration is the map Z : ∆(1)(R) → ∆(1)(R) defined by
Z(T) :=QbN(T)(T),
whereQN is theN-th iterated of the Rauzy-Veech algorithm introduced in paragraph 2.1 andQbN is its normalized version.
The following is one of the main results in the ergodic theory of i.e.t.s ([Z1]).
Theorem(Zorich). For any Rauzy classRthe Zorich mapZ in definition 2.4 has an unique invariant measure µ which is absolutely continuous with respect to the lebesgue measure on∆(1)(R). Moreoverµ is finite and ergodic.
2.2. Reduction of Rauzy classes. In this paragraph we describe a combinatorial operation on Rauzy classes calledreduction, which has been introduced in [A,G,Y]
generalizing a previous simpler version appearing in [A,V]. We closely follow§5 of [A,G,Y].
2.2.1. Decorated Rauzy classes. LetRbe a Rauzy class with alphabetAandA′⊂ Abe a proper subset. An arrow is calledA′-colored if its winner belongs toA′. A pathγ∈Π(R) isA′-colored if it is a concatenation ofA′−colored arrows.
For an element π∈ R we say thatπ isA′-trivial if the last letters on both the top and the bottom rows ofπ do not belong toA′,πisA′-intermediate if exactly one of those letters belongs toA′and finallyπisA′-essential if both letters belong to A′. AnA′-decorated Rauzy class R∗ ⊂ R is a maximal subset whose elements can be joined by an A′− colored path. We let Π∗(R∗) be the set of A′−colored paths starting (and ending) at permutations in R∗.
A decorated Rauzy class is calledtrivial if it contains a trivial elementπ, in this caseR∗={π}and Π∗(R∗) ={π}, recalling that vertices are identified with zero- length paths. A decorated Rauzy class is calledessential if it contains an essential element. Any essential decorated Rauzy class contains intermediate elements.
LetR∗be an essential decorated Rauzy class and letRess∗ ⊂ R∗be the subset of essential elements. Let Πess∗ (R∗) be the set of paths in Π∗(R∗) starting and ending at elements ofRess∗ . Anarcis a minimal non-trivial path inRess∗ , all arrows in the same arc are of the same type and have the same winner, so winner and type of an arc are well defined. The losers in an arc are all distinct, moreover the first loser is inA′ and the others are not. Any element inRess∗ is the starting point of a top and of a bottom arc and also the ending point of a top and a bottom arc.
Ifγ∈Π∗(R∗) is an arrow then there exist unique pathsγsandγein Π∗(R∗) such thatγsγγeis an arc, called thecompletionofγ. Ifπis intermediate the completion of theA′−colored arrow starting atπis the only arc passing throughπ.
If π ∈ R∗ we define πess as follows. If π is essential then πess = π, if π is intermediate letπess be the end of the arc passing throughπ.
Toγ∈Π∗(R∗) we associate an elementγess∈Πess∗ (R∗) as follows. For a trivial pathπ∈ R∗ we use the previous definition of πess. Assuming thatγ is an arrow we distinguish two cases:
(1) If γ starts at an essential element, we letγess be the completion ofγ.
(2) Otherwise, we letγess be the endpoint of the completion ofγ.
We extend the definition to paths γ ∈ Π∗(R∗) by concatenation. Notice that if γ∈Πess∗ (R∗) thenγess=γ.
2.2.2. Reduction of Rauzy classes. Given a permutationπon the alphabetA, even not admissible, whose top and bottom rows end with differen letters, we obtain the admissible end of πby deleting as many letters from the top and bottom rows of π as necessary to obtain an admissible permutation. The resulting permutation belongs to some Rauzy classR′′ on some alphabetA′′⊂ A.
Let R∗ be an essential decorated Rauzy class, and let π ∈ Ress∗ . Delete all the letters not belonging toA′ from the top and bottom rows ofπ. The resulting permutationπ′ is not necessary admissible, but sinceπ is essential the letters in the end of the top and bottom rows ofπ′ are distinct. Letπred be the admissible end ofπ′. We call πred thereduction of π. We extend the operation of reduction fromRess∗ toR∗defining the reduction ofπ∈ R∗ as the reduction ofπess.
Ifγ∈Πess∗ (R∗) is an arc starting atπsand ending inπe, then the reductions of πsand πe belong to the same Rauzy class and we define γredas the arrow which joinsπrede withπreds . γredhas the same type and the same winner of the arcγand its loser is the first loser ofγ. It follows that the set of reductions of allπ∈ R∗ is a Rauzy classRred on some alphabetA′′⊂ A′ ⊂ A. We define the reduction of a pathγ∈Π∗(R∗) as follows. Ifγis a trivial (zero-length) path or an arc, it is defined as above. We extend the definition to the caseγ∈Πess∗ (R∗) by concatenation. In general we let the reduction ofγto be equal to the reduction ofγess. Restricted to essential elements the operation of reduction give a bijectionred:Ress∗ → Rred. If we think to elements π∈ R as trivial paths we can extend the previous operation to a bijectionred: Πess∗ (R∗)→Π(Rred) compatible with concatenation on the set of arcs.
2.2.3. Drift in essential decorated Rauzy classes. LetR∗ ⊂ Rbe an essential A′- decorated Rauzy class. Forπ∈ R∗letαt(π) (respectivelyαb(π)) be the rightmost letter in the top (respectively in the bottom) row of π that belongs to A \ A′. Letdt(π) (respectivelydb(π)) be the position ofαt(π) (respectively ofαb(π)). Let d(π) :=dt(π)+db(π). An essential element ofR∗is thus someπsuch thatdt(π)< d anddb(π)< d. Ifπs is an essential element ofR∗ andγis an arrow starting atπs
and ending atπethen
(1) dt(πe) =dt(πs) ordt(πe) =dt(πs) + 1, the second possibility happening if and only ifγis a bottom whose winner precedesαt(πs) in the top ofπs. (2) db(πe) =db(πs) ordb(πe) =db(πs) + 1, the second possibility happening if
and only ifγis a top whose winner precedesαb(πs) in the bottom ofπs. In particulard(πe) =d(πs) ord(πe) =d(πs) + 1. In the second case we say thatγ isdrifting. LetRred be the reduction ofR∗ and letA′′⊂ A′⊂ Abe the alphabet ofRred. Ifπ∈ R∗is essential, then there exists someα∈ A′′which either precedes αt(π) in the top row ofπor precedesαb(π) in the bottom row ofπ, we call such an αgood. Indeed, ifγ∈Π∗(R∗) is a path starting atπ, ending with a drifting arrow and minimal with this property, then the winner of the last arrow ofγ belongs to A′′ and either precedesαt(π) in the top ofπ (if the drifting arrow is a bottom) or precedesαb(π) in the bottom ofπ(if the drifting arrow is a top).
Note that ifγ∈Πess∗ (R∗) is an arrow starting and ending at essential elements πs, πe then a good letter for πs is also a good letter for πe. Moreover, if γ is not drifting then the winner ofγ is not a good letter forπs.
2.2.4. Standard decomposition of separated paths. An arrow is called (A \ A′)- separated if both its winner and its loser belong to A′. A path γ ∈ Π∗(R∗) is (A \ A′)-separated if it is a concatenation of (A \ A′)−separated arrows. We also say that a Rauzy pathγ iscomplete (orA-complete) if for any letter α∈ Athere exists an arrow composingγhavingαas winner.
Ifγ ∈Π(R) is a non-trivial maximal (A \ A′)-separated path then there exists an essential A′-decorated Rauzy classR∗ ⊂ Rsuch that γ ∈Π∗(R∗). Moreover, ifγ =γ1...γn then any arrow γi starts at an essential element πi ∈ Ress∗ (andγn
ends at an intermediate element ofR∗ by maximality).
Remark 2.5. Let r := d(πn)−d(π1). Let γ = γ(1)γ1...γ(r)γr, where the γi are drifting arrows andγ(i)are (possibly trivial) concatenation of non drifting arrows.
Ifαis a good letter forπ1, then it follows thatαis not the winner of any arrow in anyγ(i). The reduction of anyγ(i)are therefore non-complete paths in Π(Rred).
2.3. Lebesgue measure and distortion underQb. Letπbe an admissible com- binatorial datum in some Rauzy class R over the alphabet A and consider the associated simplex ∆(1)π of those T ∈ ∆π with kλk = 1. We call Lebd−1 the lebesgue measure on ∆(1)π normalized in order to give measure one to it. For any finite pathγ∈Π(R) starting atπ we define a sub-simplex of ∆(1)π by
∆(1)γ := ∆γ∩∆(1)π .
We can identify ∆(1)π with the standard simplex ∆(1):={λ∈Rd+;kλk= 1}, modulo this identification the vertices of ∆(1)γ are the vectors{(1/qγα)tBγeα}α∈A. Recalling thatBγ ∈SL(d,Z) for anyγ∈Π(R), we have the nice formula
(2.4) Lebd−1(∆(1)γ ) = Y
α∈A
(qγα)−1.
Let Γ be a family of disjoint Rauzy paths starting atπ. Disjointness of Γ is equiv- alent to say that the simplices ∆(1)γ forγ ∈Γ are each other disjoint. In this case we have
Lebd−1([
γ∈Γ
∆(1)γ ) =X
γ∈Γ
Lebd−1(∆(1)γ ).
Given a proper sub-alphabetA′ ofAwithd′ letters (thus 2≤d′≤d−1) we call
∆b(1)π,A′ the hyper-face of ∆(1)π whose extremal points are the vectorseαwithα∈ A′. It is a (d′−1)−simplex. We call Lebd′−1the lebesgue measure on it normalized in order to have Lebd′−1(∆b(1)π,A′) = 1. Ifγ is a Rauzy path starting at π we denote
∆b(1)γ,A′ the hyper-face of ∆(1)γ spanned by the vectors{(1/qαγ)tBγeα}α∈A′. With the normalization introduced above we have
Lebd′−1(∆(1)γ ) = Y
α∈A′
(qαγ)−1.
2.3.1. Probabilistic interpretation and distortion estimate. Let us consider π in some Rauzy classRand a pathγ∈Π(R) starting at π. In view of lemma 2.2 we can interpret Lebd−1(∆(1)γ ) as the probability of the event{T ∈∆(1)π ;γ≺γ(T,∞)}.
We may look at orbits of Qb not only from the beginning, but also from their middle. This is to say, for some fixed integer i0 > 0, we split any orbit into two parts, that is {Tb(i)}i<i0 and{Tb(i)}i≥i0. Then we fixπ∈ Rand we consider those orbits such that Tb(i0) belongs to ∆(1)π . Restarting the algorithm Qb at Tb(i0) we consider {Tb(i)}i<i0 as its past and {Tb(i)}i≥i0 as its future. Ifν ∈Π(R) is a finite Rauzy path ending in π, it is natural to consider the probability, forTb(i0)∈∆(1)π , that γ(Tbi0,∞) begins with γonce we know that γ(T, i0) ends withν. We denote Pν(∆(1)γ ) such probability. According to lemma 2.2, all the obits such thatγ(T, i0) ends with ν are characterized by the condition Tb(i−1) ∈ ∆(1)ν for some common instanti−1< i0. Then they all follow the steps of the algorithm prescribed by the path ν till they arrive in ∆(1)π at the instant i0 (in particular i0−i−1 coincides with the length ofν). The condition on the futureγ≺γ(Tb(i0),∞) is equivalent to Tb(i0)∈∆(1)γ , that is to sayTb(i−1)∈∆(1)νγ, thus forPν(∆(1)γ ) we have the formula (2.5) Pν(∆(1)γ ) =Lebd−1(∆(1)νγ)
Lebd−1(∆(1)ν ) = Q
α∈Aqνα Q
α∈Aqνγα
.
Observing thatBνγ =BγBν, and thereforeqνγ=Bγqν, we are led to consider the following generalization. For any vectorq in RA+ we write N(q) :=Q
α∈Aqα, then we define
Pq(∆(1)γ ) := N(q) N(Bγq).
ForA′⊂ Aandq∈RA+ we putMA′(q) := maxα∈A′qα. In the trivial caseA′=A we simply denoteM(q) :=MA(q). We also call Ππ(R) the set of those γ∈Π(R) which start atπ. In [A,G,Y] it is proved the following theorem.
Theorem (Avila-Gouezel-Yoccoz). There exist a pair of constantC >0 andθ >
1, depending only on the number of intervals d, with the following property. Let A′ ⊂ Abe a non-empty proper subset,m andM be integers with 0≤m≤M and qbe any vector in RA+. Then we have the estimates
(2.6)
Pq{γ∈Ππ(R);M(Bγq)>2MM(q), MA′(Bγq)<2M−mM(q)} ≤C(m+ 1)θ2−m, (2.7)
Pq{γ∈Ππ(R);γ is not complete ;MA′(Bγq)>2MM(q)} ≤C(M + 1)θ2−M. Note. If fact the complete result in [A,G,Y] contains two more distortion esti- mates similar to these ones, but we don’t need them in our work.
3. The convergent case.
This section is devoted to the proof of proposition 1.4. Let π0 and (β, α) be respectively a combinatorial datum and a pair as in theorem 1.3. For anyn∈Nwe callI(π0, β, α, n) the set of thoseT in ∆(1)π0 such that the triple (β, α, n) is reduced forT and equation (1.2) is satisfied, that is such that|Tnubβ−utα|< ϕ(n).
Consider T ∈ ∆(1)π0 such that the triple (β, α, n) satisfies equation (1.2). If (β, α, n) is not reduced forT then there exists an integerk∈ {0, .., n} and a letter
Z∈ Asuch thatT−kI(β, α, n) contains in its interior eitherutZ orubZ, moreover we can suppose thatkis minimal with one of the two properties. The first case implies
|Tn−kubβ−utZ|<|Tnubβ−utα|, thus|Tn−kubβ−utZ|< ϕ(n−k), sinceϕis monotone.
Similarly the second case implies|ubZ−T−kutα|<|Tnubβ−utα|, so|TkubZ−utα|< ϕ(k) by minimality ofk. In both the two cases we pass from (β, α, n) to an other triple (β′, α′, n′) satisfying equation (1.2) withn′< n. Applying iteratively the argument we get a triple reduced forT which still satisfies equation (1.2).
Now let us suppose that for the i.e.t. T there exist infinitely many triples which are solutions of equation (1.2), but which are not necessarily reduced. With the argument above we get a sequence of reduced solutions {(βk, αk, nk)}k∈N for T. Finally there exist at least one pair (β′, α′) appearing infinitely many times in the sequence.
According to the first part of the Borel-Cantelli lemma, if for any pair (β, α) as in theorem 1.3 we show that P∞
n=1I(π0, β, α, n) < +∞, then for almost any T in ∆(1)π0 there exist just finitely many triples (β, α, n) which are reduced for T and satisfy (1.2). Then it follows form the discussion above that the following proposition holds.
Proposition 3.1. In order to prove proposition 1.4 it is enough to prove that there exists a positive constantC =C(d) depending only on the number of letters of A such that for anyn∈Nwe have
(3.1) Lebd−1(I(π0, β, α, n))< Cϕ(n).
3.0.2. Notation. Let us consider an admissible combinatorial datumπin the same Rauzy classRofπ0. LettingL vary inA we denoteeL the correspondent vectors of the standard basis ofRd. For a pair of lettersβ andαsuch thatπb(β)>1 and πt(α)>1 we define the vectors with integer coordinates
wβb(π) := X
πb(β′)<πb(β)
eβ′ and wtα(π) := X
πt(α′)<πt(α)
eα′,
then we set wβ,α(π) := wbβ(π)−wtα(π). The property of these vectors is that the singularities for the i.e.t. T = (π, λ) are given by ubβ =hwbβ(π), λi and utα = hwαt(π), λi. Therefore
ubβ−utα=hwβ,α(π), λi
and in particular we have that the set of those T ∈ ∆(1)π such that ubβ = utα coincides with ∆(1)π ∩(wβ,α(π))⊥, where (wβ,α(π))⊥ denotes the hyperplane normal towβ,α(π). We observe that forL∈ Awe havehwβ,α(π), eLi= 0 or±1 and all the tree values are attained. Therefore, if for some letterβ′ we havehwβ,α(π), eβ′i 6= 0 then there exists some other letterα′ such thathwβ,α(π), eβ′+eα′i= 0. It follows that ∆(1)π ∩(wβ,α(π))⊥ is a convex sub-set of ∆(1)π with dimensiond−2.
Finally we introduce the antisymmetric matrix Ωπ ∈hom(Zd,Zd) defined by (Ωπ)β,α=−1 if πt(β)< πt(α), πb(β)> πb(α)
(Ωπ)β,α= 1 if πt(β)> πt(α), πb(β)< πb(α) (Ωπ)β,α= 0 otherwise.
The property of the matrix Ωπis the following. Let us consider any i.e.t. T = (π, λ) and let us define the vectorδ=δ(π, λ) := Ωπλ. ThenT acts by
T|It
L(x) =x+δL.
