Divergent directions in some periodic wind-tree models

The periodic wind-tree model is a family T(a,b) of billiards in the plane in which identical rectangular scatterers of size axb are disposed at each integer point. It was proven by P. Hubert, S. Leli\`evre and S. Troubetzkoy (arXiv:0912.2891v1) that for a residual set of parameters (a,b) the billiard flow in T(a,b) is recurrent in almost every direction. We prove that for many parameters (a,b) there exists a set S of angles of positive Hausdorff dimension such that every billiard trajectory in T(a,b) with initial angle in S is self-avoiding. In particular, the flow in a direction of S is divergent.


Introduction
We study periodic versions of the wind-tree model introduced by P. Ehrenfest and T. Ehrenfest in 1912 [EhEh90]. A point ("wind") moves in the plane and collides with rectangular scatterers ("trees") with the usual law of reflexion. In the periodic version of the wind-tree model, due to J. Hardy and J. Weber [HaWe80], the scatterers are identical rectangular obstacles located periodically in the plane, every obstacle centered at each point of Z 2 . The scatterers are rectangles of size a × b, with 0 < a < 1, 0 < b < 1. We denote by T (a, b) the subset of the plane obtained by removing the obstacles and name its billiard the wind-tree model. Our aim is to understand some of its dynamical properties (see Figure 1 for two different behaviors in the golden wind-tree table T (Φ, Φ)). The phase space of the billiard is naturally T (a, b) × S 1 . Each barrier in T (a, b) is either horizontal or vertical. Hence, for the point (x, θ) ∈ T (a, b) × S 1 and for every time t, the possible angles for the orbit of (x, θ) in T (a, b) at time t are θ, −θ, π/2 − θ or −π/2 + θ. Let τ h : θ → −θ and τ v : θ → π/2 − θ be respectively the horizontal and vertical reflexions and K = τ h , τ v Z/2 × Z/2 the group they generate. We define the billiard flow in direction θ in T (a, b) to be the map φ θ t : T (a, b) × K → T (a, b) × K which is defined as follows. Let (x 0 , κ 0 ) be an element of T (a, b) × K, then (x t , κ t ) = φ θ t (x, κ) is such that if a ball has an initial position x 0 and an initial angle κ 0 (θ) then after time t it has position x t and direction κ t (θ). We will often consider the quantity φ θ t (x, τ ) as an element of T (a, b) and write φ θ t (x) for φ θ t (x, id). Let d be the Euclidean distance in R 2 . We say that the flow in direction θ is recurrent, if for almost all points x in T (a, b) we have lim inf t→∞ d(x, φ θ t (x)) = 0. We say that it is divergent if for almost all points x ∈ T (a, b) lim inf t→∞ d(x, φ θ t (x)) = +∞. P. Hubert, S. Lelièvre and S. Troubetkoy [HuLeTr] exhibit a residual set E ⊂ (0, 1) × (0, 1) such that for any parameters (a, b) ∈ E for almost all θ ∈ S 1 (with respect to the Lebesgue measure on S 1 ) the flow in T (a, b) in direction θ is recurrent. In the present paper we study the opposite behavior: the set of parameters (a, b, θ) for which the flow in T (a, b) in direction θ is divergent. As a consequence of our main result (Theorem 2) we obtain Theorem 1. If a and b are either rational or quadratic of the form 1/(1 − a) = x + y √ D and 1/(1 − b) = (1 − x) + y √ D with x and y rationals and D > 1 a square-free integer, then there exists a dense set Λ ⊂ S 1 of Hausdorff dimension not smaller than 1/2 such that for every θ ∈ Λ and every point x in T (a, b) with infinite forward orbit lim inf d(x, φ θ t (x)) = ∞. In particular the flow φ θ t is divergent.
The subset Λ which appears in the above theorem is made explicit in Proposition 6. The strategy to prove Theorem 1 is very similar to the one used in [HuLeTr]. We explain the idea for the special case T (1/2, 1/2). For every angle θ ∈ (0, π/2) for which the slope tan θ is rational, the billiard flow in T (a, b) in direction θ has a periodic behavior. Two important types of slopes are of interest for our purpose: half-divergent slope and periodic slope (see Figure 1 for an example of a half-divergent slope). We explain the definition on two examples. In the horizontal direction θ = 0 in T (1/2, 1/2), there is a bunch of trajectories which reflect between two consecutive scatterers spaced by (1, 0) while the others go to infinity: 0 is a half-divergent slope. On the contrary, in the direction θ = π/4, all trajectories are periodic with the same period: the slope tan(π/4) = 1 is periodic. To prove recurrence, the strategy of [HuLeTr] consists in approximate a generic slopes by rational ones which correspond to directions of periodic type in T (1/2, 1/2). To build divergent trajectories in the same billiard table, we use slopes are in a sense badly approximate by slopes of periodic type.
The proof of our main result uses a renormalization algorithm due to S. Ferenczi and L. Zamboni [FeZa10,FeZa]. Their induction operate on interval exchange transformations and we give a geometric interpretation on translation surfaces using suspensions. Similar geometric renormalization is described by C. Ulcigraï and J. Smillie for the regular "octagon" [SmUl11]. The geometric interpretation we use was known in greater generality by P. Hubert and C. Ulcigraï [HuUl].
We first consider a discretization of the flow φ θ t in T (a, b) and prove that the distance d(x, φ θ t (x)) corresponds to a Birkhoff sum of a function over an interval exchange transformation g = g a,b,θ . Then we build a set of parameters (a, b, θ) by imposing some conditions in the Ferenczi-Zamboni induction of g. For those parameters, we have a very simple continued fraction algorithm-like which is define as follows. For a 4-tuple of positive real numbers Z = (x 1 , x 2 , y 1 , y 2 ) define where . designs the floor. If Z satisfies x 1 + x 2 > y 1 > x 2 and y 1 + y 2 > x 1 > y 2 (1) 3 then F 2 (Z) = Z. We say that the quadruple Z is F -renormalizable if for all k ≥ 0, F k (Z) does not satisfy (1). To a F -renormalizable quadruple Z we associate an infinite sequence of 2-tuples ((m k , n k )) k∈N defined by m k (Z) = m(F k (Z)) and n k (Z) = n(F k (Z)). We call the sequence ((m k (Z), n k (Z)) k≥0 the F -convergents of Z. The set of F -renormalizable quadruples defines an uncountable set of zero Lebesgue measure.
Proposition 1. If Z = (x 1 , x 2 , y 1 , y 2 ) is F -renormalizable then -for k ≥ 0, if m k = 0 then m k+1 = 0 and n k+1 = 0, -for infinitely many k, m 2k = 0. The same is true for m 2k+1 , n 2k and n 2k+1 . Conversely, if ((m k , n k )) k≥0 is a sequence of 2-tuples of non negative integers that satisfy the above condition, then there exists at least one quadruple Z such that for all k, m k (Z) = m k and n k (Z) = n k .
Using the above description of F -renormalizable slopes, our main result is Theorem 2. Let Z = (x 1 , x 2 , y 1 , y 2 ) ∈ R 4 + and (a, b, θ) ∈ (0, 1) 2 × [0, π/2) be related by Assume that Z is F -renormalizable and let ((m k , n k )) k≥0 be its F -convergents. If for all k, n k ≡ 0 mod 2, then any infinite forward trajectory in direction The infinite billiard T (a, b) can be considered as a particular case of Z 2 -periodic translation surface (with finite quotient). For Z-periodic translation surfaces the recurrence of the flow follows from general results on 1-dimensional cocycles. For highly symmetric examples, the ergodicity of Z-periodic translation surface the flow can be established (see [HuSc10], [HuWe], [HoHuWe]). The main difficulty of the wind-tree model comes from dimension 2.
We mention other results on the wind-tree-model. As we explained above, it is proven in [HuLeTr] that for a residual set of parameters (a, b) for almost all angles θ the flow φ θ t in T (a, b) is recurrent. The problem of diffusion is studied in [DeHuLe] (it is proven that for almost all parameters and almost all angles θ the polynomial growth of d(p, φ θ t (p)) is 2/3). The ergodic decomposition for irrational parameters (a, b) and some angles θ with rational slopes is done in [CoGu]. We would also mention that the main result in [Ho] gives the following result Theorem 3 ( [Ho]). If a = p/q and b = r/s are rationals with p, r odd and r, s even, then there exists a dense set Λ ⊂ S 1 of Hausdorff dimension not less than 1/2 such that for every θ ∈ S 1 the billiard flow φ θ t is ergodic.
The paper is organized as follows. In Section 2 we define translation surfaces and interval exchange transformations. We build the discretization of the flow φ θ t as a Z 2 -cocycle over an interval exchange transformation. Next in Section 3 we recall the Ferenczi-Zamboni induction and see the relation with the map F defined above. The proof that Theorem 2 implies Theorem 1 is left to Section 3.4. The proof uses the classification of Veech surfaces in genus 2 by K. Calta [Ca04] and C. McMullen [Mc03]. The proof of Theorem 2 is postponed to Section 3.5.

Aknowledgments:
The author would like to thank Pascal Hubert and Samuel Lelièvre for introducing him to the known results about the wind-tree model T (p/q, r/s) and more generally to the theory of finite and infinite square tiled surfaces. Many experimentations have been done with the math software Sage [Sa, SaC]. The script (for computations and drawings) as well as a collection of pictures are available on the web page of the author.

The wind-tree cocycle
In this section we build a discretization of the billiard flow in T (a, b).

Translation surface and Poincare maps of the linear flow
A flat surface is a compact oriented surface X endowed with a flat metric defined on X\Σ where Σ ⊂ X is a finite set of points which are conic type singularities for the metric. It is a translation surface if moreover the holonomy given by parallel transport in X\Σ is trivial. Concretely, any translation surface can be built from a finite set of polygons P i in R 2 and identifying pairs of edges with translations. We refer to the survey of A. Zorich [Zo06] and the notes of M. Viana [Vi] for the latter construction and other equivalent definitions of translation surfaces.
Let X be a translation surface, the absence of holonomy implies that directions are globally defined. The geodesic flow on the tangent bundle of X preserves directions and can be defined on X as soon as we specify the direction and the speed. We assume that in a translation surface a fixed direction is given which we call vertical. The linear flow of X is the unit speed geodesic flow in the vertical direction on X. The flow in a rational billiards can be transformed into the linear flow of a translation surface. We will use this construction to define the wind-tree cocycle (see Section 2.5). In this paper we mainly focus on the family of translation surfaces L(a, b) where 0 < a < 1 and 0 < b < 1 are two parameters. The surface L(a, b) is built as follows (see also Figure 2 belled v 2 ). The translation surface L(a, b) is a genus 2 surface that has one conic singularity of angle 6π. The equivalence classes of translation surfaces with a single conic singularity of angle 6π form a moduli space denoted H(2). The number 2 in H(2) does not refer to the genus but to the degree of the singularity. Any surface in H(2) can be built from a polygon which is L-shaped but non necessarily right-angled (see [Ca04], [HuLe06] and [Mc03]).
Let X be a translation surface and I ⊂ X an horizontal segment (or any segment transverse to the linear flow of X). The first return map on I is defined for every point in I for which the orbit under the linear flow returns to I before reaching a singularity of the metric. There is a natural Lebesgue measure on I induced from the flat metric of X which is preserved by the first return map. Out of the discontinuities, the first return map on I is a translation. For our purpose, it will be easier to work with Poincare maps obtained on transversal made of more than one interval.

Interval exchange transformations and quadrangulations
We mainly follows the presentation of [FeZa,FeZa10] and give a geometric point of view on their induction procedure. We use here the letters (for left) and r (for right) whereas in [FeZa,FeZa10] the letters m (for minus) and p (for plus) are used.
Let λ = ((λ 1, , λ 1,r ), . . . , (λ d, , λ d,r )) be a vector of d pairs of real numbers where λ i, < 0 and The combinatorics of T depends on a pair of permutations π = (π , π r ) ∈ S d × S d such that the group they generate acts transitively on {1, . . . , d}. We build two decompositions of each interval E i as follows.
. We assume implicitly here, that the length-vector λ satisfies the train-track relations We denote by T π,λ the application constructed above from the data λ and π = (π , π r ) (see the top picture of Figure 4). We call the map T : E → E an interval exchange transformation. We warn the reader that our definition of interval exchange transformation does not correspond to the standard one in which one interval is cut in several pieces. In our case many intervals are cut in only two pieces. Let T π,λ : E → E be an interval exchange transformation. There is a natural way to code the orbits of the dynamical system associated to T . To each point x in E we associate a label in A = {1, . . . , d} × { , r} corresponding to the interval E i, or E i,r it belongs. To an infinite orbit x, T (x), T 2 (x), . . . we associate an infinite sequence w ∈ A N in such way that w k is the label associated to the point T k (x). The orbit of x starts with the finite word Similarly to Veech zippered-rectangle construction [Ve82], we define zippered rectangles for the map T .

Definition 1. A suspension datum for an interval exchange transformation
To a suspension datum ζ of (π, λ) we associate a translation surface in the following way. For i = 1, 2, . . . , d, let R i be the quadrilateral with vertices (in trigonometric order) ζ i,l , 0, ζ i,r , ζ i,r + ζ πr(i), = ζ i, + ζ π (i),r . The suspension S(π, ζ) is the disjoint union of the rectangles R i for i = 1, . . . , d in which we identify the sides which have the same label (i, ) or (i, r) (see the second picture in Figure 4).

Definition 2.
Let X be a translation surface. A quadrangulation of X is a simplicial decomposition of X for which the vertices are the conic singularities of X, the edges are geodesics and every face is a quadrilateral which does not contain any singularity. A quadrangulation is admissible if every face has exactly two adjacent edges for which the linear flow (in the vertical direction) is incoming.
In the suspension S(π, ζ) of an interval exchange transformation, the rectangles R i naturally define an admissible quadrangulation. Reciprocally any admissible quadrangulation gives rise to a train-track: we associate to a quadrangulation the first return map on its sides (see the third picture of Figure 4).
Let X be a translation surface with an admissible quadrangulation. To an orbit of the linear flow, we associate its natural cutting sequence made of the ordered list of edges meet by the orbit. The cutting sequences in a suspension S(π, ζ) of an interval exchange transformation T π,λ are in bijections with the coding of the orbits in T π,λ . 1 r 3 r 3 l 1 r 2 l 2 r 1 l 3 r 3 l 1 r 3 r 3 l 1 r 2 l 2 r . . . Figure 3: A geodesic in L(a, b) and its cutting sequence.

The quadrangulation of L(a, b)
We are now interested in the cutting sequences of the linear flow in the surfaces r −θ ·L(a, b) defined in Section 2.1 and where r −θ denotes the rotation by an angle θ. The linear flow in the surface r −θ · L(a, b) is the geodesic flow of L(a, b) in the direction π/2 + θ. The next proposition asserts that there is a one to one correspondence between length parameters ((λ 1, , λ 1,r ), (λ 2, , λ 2,r ), (λ 3, , λ 3,r )) satisfying the train-track relations for π = (1, 3) and π r = (1, 2) and parameters (a, b, θ) of the flat surface r −θ · L(a, b) (up to rescaling).
Proposition 2. Let π l = (1, 2)(3), π r = (1, 3)(2) and λ satisfy the train track relations (2). Then there exists a unique suspension datum ζ of (π, λ) such that the suspension S(π, ζ) is isomorphic, up to horizontal and vertical rescaling, to a surface of the form r θ · L(a, b) with θ ∈]0, π/2[ endowed with its natural quadrangulation. Moreover, a, b and θ are deduced from the length-vector λ and by the following relations Proof. For i = 1, 2, 3, the real parts ζ i, and ζ i,r of the suspension datum ζ are respectively λ i, and λ i,r . We are looking for the imaginary parts such that the suspension S(π, ζ) is isomorphic to a translation surface of the form r θ · S(π, ζ).
There are two independent equations on the imaginary parts imposed by the train track relations ζ 1, = ζ 2, and ζ 1,r = ζ 3,r . ( Furthermore, as in any surface r θ · L(a, b) the sides of the quadrangulation are orthogonal, there are three other independent equations The equations (3) and (4) give 5 independent linear relations for our six parameters Im(ζ i,m ), Im(ζ i,p ).
Hence there is exactly one solution up to rescaling.

L(a, b) with extra symmetries: Calta-McMullen L's
Let X be a translation surface with singularities Σ ⊂ X. An affine diffeomorphism of X is an orientation preserving homeomorphism of X that permutes the singularities of the flat metric and acts affinely on the flat structure of X. We denote by Aff(X) the group of affine diffeomorphisms of X\Σ. Let denote by Γ(X) the image of the derivative map d : which is called the Veech group. For surfaces in H(2), the affine group is isomorphic to the Veech group under the derivative map (see Proposition 4.4 in [HuLe06]). Next, we identify the Veech group and the affine group for surfaces in H(2). A translation surface X is called a Veech surface (or lattice surface) if Γ(X) is a lattice in SL(2, R). Veech surfaces were introduced in [Ve89] for dynamical purposes. Let X be a Veech surface. For an angle θ ∈ S 1 if there exists an orbit of the linear flow which joins two singularities (saddle connection) then the linear flow φ θ t : X → X in direction θ is parabolic: any geodesic in direction θ is either a saddle connection or a loop and moreover there exists a non trivial element φ ∈ Aff(X) which stabilizes all geodesics in that direction [Ve89]. The name parabolic comes from the fact that dφ ∈ SL(2, R) is a parabolic matrix.
In the surface L(a, b) ∈ H(2) the horizontal and vertical directions are completely periodic: all trajectories are either saddle connections or closed loops. A necessary condition for L(a, b) to be a Veech surface is that those two directions are parabolic, in other words there exists non identity matrices of the form 1 * 0 1 and 1 0 * 1 in Γ(X). It turns out that these conditions are equivalent (which is a miracle of genus 2 translation surfaces).
Now, we explicit the form of the parabolic elements in horizontal and vertical directions for parameters that satisfy condition 3 in the above Theorem. Those elements will be used to find paths in the Ferenczi-Zamboni induction. Let 0 < a < 1 and 0 < b < 1. Then, the stabilizers in SL(2, R) of, respectively, the bottom and top cylinders in the horizontal decomposition of L(a, b) are the two parabolic subgroups of SL(2, R) generated by Remark that the matrix P bot (resp. P top ) acts as a Dehn twist around the circumference of the bottom (resp. top) cylinder. The intersection P bot ∩ P top is non trivial (different from In particular, L(a, a) is a Veech surface if and only if there exists D = n/m ∈ Q with For a parameter a as above, the affine multi-twists parameters (m h , m v , n h , n v ) of L(a, a) are m h = m v = m and n h = n v = n.

From L(a, b) to T (a, b): the wind-tree cocycle
Now, we describe a discretization of the billiard flow in T (a, b) as a Z 2 -cocycle over an interval exchange transformation.
Given a rational billiard, there is a classical procedure to get a translation surface called Katok-Zemliakov construction or unfolding procedure (see the original articles [FoKe36] and [KaZm75] or the surveys [Ta95] or [MaTa02]). The unfolding procedure consists in taking reflected copies of the billiard instead of considering a reflected trajectory. The construction applies to the infinite billiard table T (a, b) and is made of four copies associated to the four directions that a trajectory may take with a given initial angle. We denote by X ∞ (a, b) the translation surface obtained by unfolding the billiard table T (a, b). X ∞ (a, b) is a normal cover of L(a, b) and the Deck group Deck(X ∞ (a, b), L(a, b)) is isomorphic to the semi-direct product The proof of the proposition is elementary and we refer to [DeHuLe]. Now, we consider how a geodesic in X ∞ (a, b) can be built from the ones in L(a, b). The Klein four group K = Z/2 × Z/2 in the proposition naturally identifies with the group generated by the vertical and horizontal reflexions denoted respectively τ v and τ h .

Proposition 4 ([DeHuLe]). The infinite translation surface
In Section 2.2, we defined a symbolic coding of geodesics in L(a, b) on the alphabet A = {1, 2, 3} × { , r}. The preimage of the quadrangulation of L(a, b) in X(a, b) gives a symbolic coding on the alphabet A × K. Given a geodesic γ (finite or infinite) in L(a, b) and its coding w = (w i ) i (in A * or A N ) it has four lifts in X(a, b) and hence four possible codings. The canonical one that we denotew = ((w i , κ i )) i is the one which starts in the copy labelled id ∈ K. Let g : A → K be defined by Then the canonical liftw = ((w i , κ i )) of w can be defined by κ 0 = 1 and for i ≥ 0, κ i+1 = κ i g(w i ).
The Klein four group K acts transitively on the four lifts of w. The three other cutting sequences for the lifts are ((a i , τ v κ i )) i , ((a i , τ h κ i )) i and ((a i , τ v τ h κ i )) i . In Figure 5 we give an example of the lift of a geodesic in L(a, b). To simplify notations we use ++ (resp. −+, +− and −−) for the element id ∈ K (resp. τ v , τ h and τ v τ h ). The surface X ∞ (a, b) is a Z 2 -cover of X (a, b). Hence, the preimage of the quadrangulation of X(a, b) determines a quadrangulation in X ∞ (a, b). We fix an origin in X ∞ (a, b) and consider a bijection between the faces of the quadrangulation and {1, 2, 3}×K ×Z 2 . To lift the cutting sequence of a geodesic in X(a, b) to X ∞ (a, b) there is another cocycle which is defined on the copies {1, 2, 3} × {id} by f ((2 , id)) = f ((2 r , id)) = f ((3 , id)) = f ((3 r , id)) = (0, 0) f ((1 , id)) = (1, 0) and f ((1 r , id)) = (0, 1) where τ h and τ v acts on Z 2 by reflexion X ∞ (a, b), w its cutting sequence on {1, 2, 3} × K × Z 2 and γ its image in T (a, b). Then for all t > 0 we have

Proposition 5 ([DeHuLe]). Let γ be a geodesic in
. . . f (w n−1 ) and n is such that the geodesic from x has cut n sides of the quadrangulation before time t.
As the cover X ∞ (a, b) → L(a, b) is normal, we can build a non-commutative cocycle to lift the cutting sequences of geodesics in X(a, b) to X ∞ (a, b).
To simplify notations, we use a direct description from L(a, b) to X ∞ (a, b). Let D ∞ = Z Z/2 be the infinite dihedral group (where Z/2 acts by multiplication by −1 on Z). We use the following notation for G = D 2 ∞ . The generators of K = Z/2 × Z/2 are denoted by τ v and τ h . We use multiplicative notations and write the product rule in D 2 ∞ as follows. For (x 1 , y 1 ), (x 2 , y 2 ), (x, y) ∈ Z 2 we have The cocycle which describe a cutting sequence in X ∞ (a, b) from one in L(a, b) is the map f : A → G defined by The cocycle f can be viewed as a function φ on the domain E = E 1 E 2 E 3 of an interval exchange transformation T = T (λ,π) with π l = (1, 3) and π r = (1, 2) which is constant on each interval E i, (resp. E i,r ). Its value on E i, is given by f (i ) (resp. by f (i r )).

Divergent wind-tree cocycles
We recall the renormalization procedure of S. Ferenczi and L. Zamboni [FeZa,FeZa10] for linear flow on translation surfaces. The induction was used to obtain fine properties of interval exchanges in the hyperelliptic classes and many examples of exotic ergodic behaviors of interval exchange transformations. We use their induction to control the wind-tree cocycle. An other induction procedure for interval exchange transformations is the one of G. Rauzy [Ra79] which seems less adapted to our situation.

Ferenczi-Zamboni induction in hyperelliptic strata
We now recall the induction procedure introduced in [FeZa10]. In next sections, we restrict our study to the case of the stratum H(2) which is the subject of [FeZa] and corresponds to our surface L(a, b).
Let X be a translation surface with an admissible quadrangulation. The general principle of the Ferenczi-Zamboni induction consists in looking at a sequence of admissible quadrangulations of the surface such that the quadrilaterals become more and more flat in the direction of the linear flow. In the case of hyperelliptic strata, we consider only quadrangulations which are stable under the hyperelliptic involution. This restriction guarantees the existence of an induction procedure. The main point is that the hyperelliptic involution simplifies the train-track relations (2) (see Definition 2.4 and the discussion which follows in [FeZa10]). Now, we describe the induction. Let T = T π,λ : E → E be an interval exchange transformation on d intervals. We assume that E = E 1 . . . E d is stable under the hyperelliptic involution. We want to define a new interval exchange transformation which corresponds to a first return map on a union of d subintervals E = E 1 . . . E d ⊂ E where for each i = 1, . . . , d, E i ⊂ E i and E i contains the origin of E i . For each i, we define its state (see Figure 6): i is in left state if λ π (i),r > λ i, (or equivalently 0 ∈ E i,l ), i is in right state if λ πr(i), > λ i,r (or equivalently 0 ∈ E i,r ). Knowing the state of each level we want to define T by choosing among the following choices -if i is in a left state either we choose Now, let T = T π,λ be an interval exchange transformation in a hyperelliptic strata such that it is stable under the hyperelliptic involution. As shown in [FeZa10] there are canonical choices which ensure that first of all the induction is always possible. Secondly, the induced interval exchange transformation is also stable under the hyperelliptic involution. The choice is done as follows. We consider the cycles in the disjoint cycle decomposition of the permutations π and π r . If there exists a cycle c of π for which each element of c are in left state then we allow to perform a left induction for all of them. Such a cycle is called a left branch of induction. Formally a left induction step for c on π = (π , π r ) and λ = ( , r) gives the combinatorial data π = (π , π r ) and λ = (λ , λ r ) where -for all i in c, λ i,r = λ i,r + λ πr(i), and π r (i) = π r • π (i), -for all i not in c, λ i,r = λ i,r and π r (i) = π r (i). The definition of induction can easily be extended to suspensions by replacing λ i, and λ i,r in the formulas above by respectively ζ i, and ζ i,r .
The following general theorem can be checked by hand for H(2) by building the so called "graph of graphs"(see Figure 7).

saddle connection. Then, T admits at least one induction branch and any map induced from T by cutting all intervals in an induction branch is an hyperelliptic interval exchange transformation. Moreover, for any choice of infinite sequence of inductions for T such that
-if i is not in the induction branch at stage n then the state of i at step n and n + 1 is the same -each interval E i for i = 1, 2, . . . , d is cut infinitely many times on its left and on its right. Conversely, given an infinite path of inductions in the graph of graphs starting from π that satisfies the two conditions above, there exists at least one parameter λ for which the interval exchange T π,λ has no saddle connection and from which we can perform these steps of induction.
We use the following multiplicative algorithm similar to the Gauss map for coding geodesics in the torus.

Description of the language in terms of induction
We now follow [FeZa] to describe the language of an interval exchange transformation in terms of one of its induction. Let T = T π,λ be a symmetric interval exchange transformation in H(2). For i = 1, 2, 3, we note L i = (π l (i), r) and R i = (π r (i), ) and L = (L 1 , L 2 , L 3 ) and R = (R 1 , R 2 , R 3 ). The words L i and R i are the two possible continuations of a letter of the form (i, * ).
Let c be a union of left (or a union of right) admissible branch of induction for T and let T be the interval exchange transformation obtained from T by inducing with respect to c.
Then the words L i and R i on this new interval exchange transformation seen from T can be described by the following rules left induction on c right induction on c Starting from a symmetric interval exchange transformation and performing successively left and right inductions, we build a sequence of words L  Figure 7. From the definition of our multiplicative algorithm, at odd step 2k+1 for k ≥ 0, we perform a right induction and we have R . Similarly at even step 2k + k for k ≥ 0, we perform left inductions and we have L (1, 2, 3) (3, 1)(2) (1, 2, 3) (2, 3)(1)

A subset of parameters (a, b, θ) defined from the induction
We restrict our attention to a subset of inductions which simplifies considerably the form of the infinite words we obtain. This subset of possible inductions is similar to the one used in [FeZa] Section 5. For parameters (a, b, θ) associated to these inductions, we will be able to control the billiard orbits of T (a, b) in direction θ.
Our graph of induction consists of the unique state π = (π , π r ) with π = (1, 3) and π r = (1, 2) which corresponds to the quadrangulation of L (a, b). We consider as induction steps -the induction which are a succession of left inductions only that go from π to π, -the induction which are a succession of right inductions only that go from π to π. The subgraph of inductions is the part of the graph of graphs in the rectangle in Figure 7. There are two possible left induction from π associated respectively to the states · and · ·. As two left inductions commute, each step of the multiplicative algorithm (Definition 3) corresponds to a 2-tuple of integers (m, n) where m corresponds to the multiplicity of the loop of length two associated to state · and n is the multiplicity of the loop associated to the state · ·. A 2-tuple also encodes the left inductions. The induction algorithm is then a shift on sequences of two-tuples of integers ((m k , n k )) k≥0 .
The language of an interval exchange transformation that admits an induction which belongs to the subgraph is defined by two families of substitutions indexed by a 2-tuple (m, n) of integers Because of the train track relations for π, namely λ 1, = λ 2, and λ 1,r = λ 3,r , the possible vector-lengths are described by the 3-tuple Z = (x 1 , x 2 , y 1 , y 2 ) = (|λ 2, |, |λ 3, |, |λ 3,r |, |λ 2,r |) ∈ P 3 (R). The application of one step of the above algorithm corresponds, at the level of vectorlengths, to the application of one of the two projective maps below F (m, n) : The multiplicative induction algorithm on the subgraph corresponds exactly to the map F defined in the introduction. We emphasize that not all vector-lengths parameters Z = (λ 2, , λ 3, , λ 3,r , λ 2,r ) admit a continued fraction expansion with respect to this algorithm (the domain is a Cantor set). We recall that we name F -renormalizable a quadruple of length parameters Z for which the induction is exactly prescribed by our subgraph with one vertex at π = ((1, 3), (1, 2)). From Theorem 5, an F -renormalizable 3-tuple determines a unique sequence of 2-tuples ((m k , n k )) k≥0 such that -for each k ≥ 1 either m k = 0 or n k = 0, -if m k = 0, then m k+1 = 0 and n k+1 = 0 -for infinitely many i, m 2k = 0 (resp. m 2k+1 = 0), -for infinitely many i, n 2k = 0 (resp. n 2k+1 = 0). Reciprocally, from Proposition 5, we know that every sequence of 2-tuples of integers that satisfy the above conditions gives a F -renormalizable vector-lengths.

Renormalizable slopes in Veech L(a, b)
In this section, to a Veech surface of the form L(a, b) (see Proposition 4 for a characterization) we build a set of slopes Λ ⊂ S 1 for which the corresponding vector lengths of the interval exchange transformation F -renormalizable. Proof. It is more convenient to consider coordinates in P 1 (R) instead of θ ∈ (0, π/2). To the angle θ we associate the (oriented) slope x = tan(θ) −1 = (cos(θ) : sin(θ)) ∈ P 1 (R).
As the Veech group of L(a, b) is a lattice s h s v ≥ 1, otherwise the group generated by ρ and ρ r won't be discrete. In particular, if x = x(θ) has the form given in the statement, then the sequence (a k ) k associated to x ∈ P 1 (R) is unique. More precisely, the expansion of x is defined from a modified continued fraction algorithm.
As s h s v ≥ 1 the domain of ψ and ψ r are disjoint and we define ψ to be the map that equals ψ on (0, 1/s v ) and ψ r on (s h , ∞). The map ψ is associated to the shift on the sequence (a k ) k that defines x = tan(θ): if x is defined by the sequence (a 0 , a 1 , a 2 , . . .) then ψ(x) is defined either by the sequence (0, a 1 , a 2 , . . .) if a 0 = 0 or (a 2 , a 3 , . . .) if a 0 = 0. The maps ψ and ψ r correspond to the standard projective action of powers of the inverses of two matrices ρ and ρ r that corresponds to the horizontal and vertical multi-twist in L(a, b): Let θ ∈ S 1 be such that x = tan(θ) −1 admits an infinite expansion with respect to ψ and S(π, ζ) be the suspension associated to r π/2−θ · L(a, b) as in Proposition 2. Assume that a 0 = 0, then we can perform a left induction with parameters (a 0 m h , a 0 n h ). Let S(π, ζ ) be the surface obtained after this step of left induction. Then the vertical direction in S(π, ζ ) corresponds to the direction x in ρ a 0 h · L(a, b) or equivalently, to the direction ψ(x) in L(a, b).
We now estimate the Hausdorff dimension of the set of renormalizable slope in a Veech surface of the form L(a, b).
Proposition 7. Let s and t be positive real numbers with st ≥ 1 and consider R(s, t) ⊂ R the set of real numbers x of the form x = a 0 s + 1 where a 0 ≥ 0 and a k ≥ 1 for k ≥ 1 are integers. Then -for any λ > 0, Hdim(R(s h , s v )) = Hdim(R(λs, λ −1 t)), -the map s → Hdim(R(s, s)) is decreasing and not smaller than 1/2.
The fact that s → Hdim R(s, s) is decreasing is immediate from the construction. In the following, we fix s > 1 and show that the Hausdorff dimension of R(s, s) is not smaller than 1/2. It follows from [He] Chapter 9 that the Hausdorff dimension can be computed with the canonical covers. More precisely, for a tuple v = (a 1 , . . . , a k ) we define the s-convergents as follows p 0 = 1 p 1 = s a 2 p k+1 = s a k p k + p k−1 q 0 = s a 1 q 1 = s 2 a 2 a 1 + 1 q k+1 = s a k q k + q k−1 .

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Then the quantity [a 1 , a 2 . . . , a k ] s,s = p k /q k converges to the real number
We first describe the strategy of the proof. Let T be an interval exchange transformation without saddle connection that is F -renormalizable. Then, for each step k ∈ N, the k-th step of the Ferenczi-Zamboni induction can be used to decompose the coding of a trajectory with the six pieces L (k) i and R (k) i for i = 1, 2, 3. The size of the pieces grows with k and more precisely, the pieces at step k + 1 are concatenations of the pieces at step k. The rule to glue the pieces is given by the substitutions defined in Section 3.2 and depends on the convergents ((m k , n k )) k≥0 of the restricted Ferenczi-Zamboni induction of T . Now assume that T satisfies the statement of Theorem 2. Then we prove that for each k ≥ 0 the wind-tree cocycle over T has no "local self-intersection". More precisely, for i = 1, 2, 3 and k ≥ 0, let L by an element of G = (Z Z/2) 2 . We prove that for k > 2, the values of level k + 1 are built in such way that each part from level k do not intersect each other. The reason why we need k > 2 is due to the fact that for step 1 (resp. step 2) the trajectory can rebound between two vertical scatterers (resp. horizontal scatterers) which implies that the values of the cocycle during this period take only two values (x, y) and (x, y)τ v (resp. (x, y) and (x, y)τ h ). In particular we prove the stronger statement that the trajectory in the wind-tree model are "self-avoiding".
We fix for the remaining of the section a triple (a, b, θ) that fulfill the hypothesis of Theorem 2 and consider the associated interval exchange transformation T . We denote by ((m k , n k )) k≥0 the convergents of the restricted Ferenczi-Zamboni induction of T and (L (k) , R (k) ) the 6-tuple of words that describe the coding of the orbits in T . For i = 1, 2, 3, let L  , 2 r , 1 r ), (2 , 1 , 3 )).  The six words L (k) i and R (k) i for i = 1, 2, 3 are defined recursively by the substitutions σ and σ r in (5). More precisely denoting L = L (k−1) , L = L (k) , R = R (k−1) and R = R (k) we have for odd steps k for even steps k R = R L = L L 1 = (R 1 R 2 ) m k L 1 R 1 = (L 1 L 3 ) m k R 1 L 2 = (R 2 R 1 ) m k L 2 R 2 = (L 2 ) n k R 2 L 3 = (R 3 ) n k L 3 R 3 = (L 3 L 1 ) m k R 3 In order to simplify notations, we use the above notations in many proofs: R and R (resp. L and L) for R (k+1) and R (k) (resp. for L (k+1) and L (k) ). The first step of the proof consists in analyzing the value of the cocycle at the endpoints of each of the pieces L (k) and R (k) . In Lemma 1, we prove that the endpoints are always oriented in the same way for all k, more precisely the value of the cocycle g with value in K is constant. Then, using this property, we prove Lemma 2 which gives an explicit values for theses endpoints.
As it was defined in Section 2.5, the wind-tree cocycle f decomposes into two parts. The first one g with values in K = Z/2×Z/2 and the other one with values in Z 2 . Let g : A * → K be the composition of f with the projection G → K. There is a natural lift of K into G and we set for w ∈ A * , f (w) = f (w)g(w) ∈ Z 2 . Lemma 1. We have for any k ≥ 0 g L (k) 1 = g(3 r ) = 1 g L (k) 2 = g(2 r ) = τ h g L (k) 3 = g(1 r ) = 1 g R (k) 1 the right, say it is blocked at N , then it means that in the future the orbit of p encounters a singularity of the interval exchange transformation after N steps.
We now refine the argument at small scales to prove that the trajectory in the infinite billiard is self-avoiding. It follows from the combinatorics of the surface L(a, b), that two consecutive pieces of level k may intersect in only few cases (see Figure 3) which corresponds to block on which the cocycle remains constant: -either in the word w = 3 r 3 n where n ≥ 0, -or in the form w = 2 2 n r where n ≥ 0. In the first case (resp. the second) the word is followed by 1 r (resp. 1 ). In the billiard table, the cutting sequence w lift to a piece of trajectory which reflects between two horizontally (resp. vertically) consecutive scatterers but does not reflect vertically (resp. horizontally). In particular, the trajectory is self-avoiding in w.

An example
We now consider the example of the periodic sequence of convergents ((1, 2), (1, 2), . . .) associated to the square tiled surface L(1/2, 1/2) and the slope √ 2 − 1. For this slope, the Ferenczi-Zamboni induction is periodic or in other words the interval exchange transformation is self-similar. The Figure 8 shows a three level of boxes for an associated orbit in the infinite billiard table T (1/2, 1/2).