Ergodic directions for billiards in a strip with periodically located obstacles

We study the size of the set of ergodic directions for the directional billiard flows on the infinite band $\R\times [0,h]$ with periodically placed linear barriers of length $0<\lambda<h$. We prove that the set of ergodic directions is always uncountable. Moreover, if $\lambda/h\in(0,1)$ is rational the Hausdorff dimension of the set of ergodic directions is greater than 1/2. In both cases (rational and irrational) we construct explicitly some sets of ergodic directions.


Introduction
In this paper we consider the following infinite periodic billiard, whose ergodic properties have been object of recent investigation (see e. g. [1,3,9]). Let T (h, a, λ) be the billiard table (shown in Figure 1) given by an infinite band R × [0, h] with periodically placed linear barriers of length 0 < λ < h handling from the lower side of the band perpendicularly, that is: A billiard trajectory is the trajectory of a point-mass which moves freely inside the table on segments of straight lines and undergoes elastic collisions (angle of incidence equals to the angle of reflection) when it hits the boundary of the table. The billiard flow (ϕ t ) t∈R is defined on the subset of the phase space T (h, a, λ) × S 1 that consists of the points (x, θ) ∈ T (h, a, λ) × S 1 such that if x belongs to the boundary of T (h, a, λ) then θ is an inward direction. For t ∈ R and (x, θ) in the domain of (ϕ t ) t∈R , ϕ t maps (x, θ) to ϕ t (x, θ) = (x t , θ t ), where x t is the point reached after time t by flowing at unit speed along the billiard trajectory starting at x in direction θ and θ t is the tangent direction to the trajectory at x t . A similar billiard in a semi-infinite band was studied in [1] in the context of perfect retroreflectors. For a survey on billiards in finite and infinite polygons we refer the reader to [5,6].
Denote by Γ the 4-element group of isometries of S 1 generated by the reflections θ → θ, θ → −θ. For every direction θ ∈ S 1 the billiard flow on T (h, a, λ) has the invariant subset T (h, a, λ) × (Γθ) in the phase space. The billiard flow (ϕ θ t ) t∈R Date: August 28, 2012. 2000 Mathematics Subject Classification. 37A40, 37E35. restricted to this set preserves the product of the Lebesgue measure on T (h, a, λ) and the counting measure on the orbit Γθ.
In [3] we proved the following result: Theorem ( [3]). If λ/h is rational or belongs to a set ∆ ⊂ (0, 1) of full Lebesgue measure then for almost every direction θ the billiard flow (ϕ θ t ) t∈R on T (h, a, λ) is not ergodic.
It is hence natural to ask whether there are exceptional ergodic directions. Hubert and Weiss proved in [9] that if λ/h is rational then the set of ergodic directions contains a dense G δ set. The aim of this paper is to prove the existence of ergodic directions for all irrational values of the relative slit length λ/h. In addition, we also study the size of this exceptional set of ergodic directions in the rational case. More precisely, in Section 4, we prove that the set of ergodic directions is uncountable when λ/h is irrational (see Theorem 4.1) and for rational λ/h we prove that its Hausdorff dimension is greater than 1/2, (see Theorem 5.1).
In both cases (rational and irrational) we give an explicit construction of ergodic directions by specifying their continued fraction expansions. The proofs use an ergodicity criterion from [9] (Theorem 2.1) based on an approximation of θ by directions with infinite strips. The main idea is to study the action of SL(2, Z) on homology and to cleverly exploit the symmetries of the system to construct infinite strips.
Combining our results from Section 5 with the approach introduced recently by Hooper in [7] one might be able to describe all invariant ergodic Radon measures for (ϕ θ t ) t∈R whenever λ/h is rational and θ belongs to a set of positive Hausdorff dimension.

Background material
2.1. Directional flows on translation surfaces and Z-covers. The study of directional billiard flows on any rational polygon (not necessary compact) can be reduced via an unfolding procedure (introduced by Katok and Zemlyakov in [10]) to the study of directional flows on a translation surface. The translation surface corresponding to the table T (h, a, λ) will be described in Section 2.2. In this section we briefly recall some basic definitions related to the notion of translation surface.
Let M be an oriented surface (not necessarily compact). A translation surface (M, ω) is a complex structure on M together with an nonzero Abelian differential ω, that is a non-zero holomorphic 1-form. Let Σ = Σ ω ⊂ M be the set of zeros of ω. For every θ ∈ S 1 = R/2πZ denote by X θ = X ω θ the directional vector field in direction θ on M \ Σ, defined by i X θ ω = e iθ . Then the corresponding directional flow (φ θ t ) t∈R = (φ ω,θ t ) t∈R (also known as translation flow ) on M \ Σ preserves the area form ν ω = i 2 ω ∧ ω = ℜ(ω) ∧ ℑ(ω). We will denote by A(ω) := ν ω (M ) the area of the surface.
Let (M, ω) be a compact connected translation surface. Denote by · , · : Recall that a Z-cover of M is a surface M with a free totally discontinuous action of the group Z such that the quotient manifold M /Z is homeomorphic to M . The map p : M → M obtained by composition of the projection M → M /Z and the homeomorphism M /Z → M is called a covering map. Denote by ω the pullback of the form ω by the map p. Then ( M , ω) is a translation surface as well. The translation flow on ( M , ω) in direction θ will be denoted by ( φ θ t ) t∈R . All Z-covers of M (up to isomorphism) are in one-to-one correspondence with homology classes in H 1 (M, Z). The Z-cover M γ determined by γ ∈ H 1 (M, Z), under this correspondence, has the following properties. If σ is a close curve in M and [σ] ∈ H 1 (M, Z), then σ lifts to a path σ : [t 0 , t 1 ] → M γ such that σ(t 1 ) = n · σ(t 0 ), where n := γ, [σ] ∈ Z and · denotes the action of Z on ( M γ , ω γ ) by deck transformations.

2.2.
From billiard flows to translation flows on translation surfaces. Fix parameters (h, a, λ) and a direction θ ∈ S 1 . One can verify, using the unfolding process first described in [10], that the flow (ϕ θ t ) t∈R on the table T (h, a, λ) is isomorphic to the directional flow ( φ θ t ) t∈R on a non-compact translation surface ( M , ω) which is obtained gluing four copies of T (h, a, λ) along the segments of the same name, as shown in Figure 2. Moreover, the surface ( M , ω) can be represented as gluing two Z-periodic polygons (obtained by gluing pairs of copies of T (h, a, λ) along b and b ′ ) as shown in the Figure 3 (here R n = r n ∪ r ′ n and L n = l n ∪ l ′ n ). Next, let us cut these polygons into rectangles P n , P ′ n along the segments U n ,  The group GL(2, R) acts naturally on Q(M, Σ, κ) and M(M, Σ, κ), by postcomposition with the charts defined by local primitives of the holomorphic 1-form. The Abelian differential obtained acting by g ∈ GL(2, R) on ω will be denoted by g · ω. The Teichmüller flow (G t ) t∈R is the restriction of this action to the diagonal subgroup (diag(e t , e −t )) t∈R of GL(2, R) on Q(M, Σ, κ) and M(M, Σ, κ).

2.4.
Ergodicity for Z-periodic surfaces. In this section we formulate a result from [9] which provides an effective method to prove the ergodicity of translation flows on recurrent Z-covers of compact translation surfaces and will be exploited to prove our main results.
Let (M, ω) be a compact connected translation surface. Let ( M γ , ω γ ) be one of its recurrent Z-covers. Suppose that C ⊂ M is a cylinder. Let δ(C) ∈ H 1 (M, Z) be the homology class of any core curve of the cylinder C. We will use the following notation (introduced in [9]): Definition 1 (see [9]). A direction θ ∈ S 1 is well approximated by strips of the The following result follows directly from the proof of Theorem 1 in [9] (more precisely from Claim 12).
In order to prove ergodicity of the translation flow, which is required to apply the above Theorem, the following result from [11], know as Masur's criterion, will be helpful.
Theorem 2.2 (Masur's criterion [11]). Let (M, ω) ∈ M(κ) be a compact translation surface. Let g ∈ SL(2, R) be an element that maps the direction θ to the vertical direction. Suppose that there exists a bounded subset B ⊂ M(κ) and a sequence t n → +∞ such that G tn (g · (M, ω)) ∈ B for all n ∈ N. Then the directional flow (φ θ t ) t∈R on (M, ω) is uniquely ergodic.

Construction of ergodic directions
In this section we describe a procedure to construct strips on the translation surface ( M γ0 , ω γ0 ) (see the end of Section 2.2) which will allow us to apply Theorem 2.1. We will study the behavior of the SL(2, Z) orbit of γ 0 ∈ H 1 (M, Z) for the SL(2, Z)-action induced on H 1 (M, Z). For simplicity, we always assume that a = 1 = 2h, then λ = |L|/2 = |R|/2 ∈ (0, 1/2).
Let T 2 0 denote the set be the translation surface drawn in Figure 5. We will distinguish between the singular points (• and in Figure 5) and z = (x, y) will denote the position of the singular point while −z will be the position of the singular point •. Let L ⊂ M(1, 1) be the locus containing all the surfaces  Figure 5.
The group GL(2, R) acts naturally on R 2 by matrix multiplication. We will denote the image of z = (x, y) ∈ R 2 under g ∈ GL(2, R) by g(x, y) or gz. This action induces also a natural action GL(2, Z) ⊂ GL(2, R) on the torus T 2 0 . We will also denote by gz ∈ T 2 0 the image of z ∈ T 2 0 by the automorphism of T 2 0 induced by g. One can show that for every g ∈ GL(2, Z): In order to verify (2), it suffices to check it on generators of GL(2, Z) and this is obtained as a byproduct of the proof of Lemma 3.3. Since Aut(M (z)) = {id, τ }, there exist exactly two affine maps ζ g , τ •ζ g : M (z) → g ·M (z) whose derivatives are equal to g. Thus, if we denote by ζ * : Representing this action in standard basis we obtain a matrix, denoted by g * (z), which is an element of P GL(2, Z). Finally note that for all g 1 , g 2 ∈ GL(2, Z) we have 3.1. Construction of strips and an ergodicity criterion. We now present the procedure of construction of infinite family of infinite strips required by Theorem 2.1. For any z = (x, y) ∈ T 2 0 let us consider the cylinder which is shaded in Figure 5. The holonomy vector of the core δ( For any g ∈ SL(2, Z) let z g = (x g , y g ) := g −1 (z) ∈ T 2 0 and ζ g : M (z g ) → M (z) be an affine transformation whose derivative is g ∈ SL(2, Z).
In summary, the above procedure allows to construct strips on M (z) β satisfying (1) whenever Combining this with Theorem 2.2 (Masur's criterion) and Theorem 2.1 (Hubert-Weiss criterion) yields the following criterion for the ergodicity of directional flows on the surface M (z) β .
. .], 0 < a < b < 1/2 and there exists an increasing sequence of even numbers (k n ) n≥1 such that Then the directional flow in direction (1, α) on the Z-cover M (z 0 ) β given by β is ergodic.
Proof. First we will show that the directional flow in direction (1, α) on the surface M (z 0 ) is ergodic. Let (p n /q n ) n≥0 stand for the sequence of convergents of the continued fraction of α. Then it is clear that σ maps the direction of the vector (1, α) to the vertical direction. We will prove that the sequence is bounded in the moduli space. Indeed, by assumption, Moreover, |q kn (q kn α − p kn )| < |q kn (q kn−1 α − p kn−1 )| < 1 and, since k n is by construction even, so that p kn /q kn < α, we have Therefore, all σ n belong to the subset G 0 ⊂ SL(2, R) of matrices whose coefficients belong to [−1, 1]. Of course, the set G 0 is compact. Let us consider the set which is compact in the moduli space M (1, 1). 1) is a compact subset in the moduli space M(1, 1) as well. Since σ n · M (z n ) ∈ G 0 · B 0 for every natural n, in view of Theorem 2.2, the directional flow in direction (1, α) on the surface M (z 0 ) is ergodic.
In the rest of the proof, Theorem 2.1 combined with the construction described before Theorem 3.1 will give the ergodicity of directional flow in direction (1, α) on the Z-cover M (z 0 ) β .
By assumption, for every n ≥ 1 there exists an affine transformation us consider the cylinder C n := ζ n (C zn ) ⊂ M (z 0 ) for which the homology class of the core is δ(C n ) = (ζ n ) * (δ(C zn )). Then Therefore each cylinder C n ⊂ M (z 0 ) is lifted to an infinite strip C n ⊂ M (z 0 ) β . Using this sequence of strips we can show that the direction (1, α) is well approximated by strips. Indeed, (1, α) .
The ergodicity of the directional flow in direction (1, α) on the surface M (z 0 ) β hence follows directly from Theorem 2.1.
3.2. SL + (2, Z)-action induced on homologies. Denote by SL + (2, Z) the semigroup of non-negative matrices in SL(2, R). In this section we establishes some principal rules of the SL + (2, Z)-action induced on homologies. These rules will help us to find elements g ∈ SL + (2, Z) satisfying (4), first for irrational λ in Section 4 and then for rational λ in Section 5. Set Then SL + (2, Z) is generated by h + and h − and one can check that: Let us consider two involutions −id and ϑ in GL(2, Z). They generate two involutions (denoted also by −id and ϑ) acting on the locus L . Geometrically, • −id rotates the squares of M (z) by angle π, therefore its induced action on H 1 (M, Z) exchanges the basis elements α and β. These two involutions (symmetries) will help us to describe the action of SL + (2, Z) on the homology level. For this purpose, we will use the following two lemmas. ϑ * (z) = ϑ and (−id) * (z) = id whenever z ∈ E.
Proof. In Figures 6 and 7 we present the surface h + · M (z) for z ∈ S and z / ∈ S respectively and using cutting and pasting we show how to represent h + · M (z) as M (h + z). The cut and paste in Figure 6 is straightforward. Let us explain Figure 7: after the linear action of h + and a first cut and paste, we consider the shaded areas labeled by A, A ′ , B and B ′ in Figure 7. Recalling the gluings between slits, one can verify that A and A ′ can be exchanged and similarly B and B ′ . The surface that we obtain after this operation is one of the canonical representatives of the locus L , more precisely it is M (z ′ ), where z ′ = h + z.
Let us denote by ζ = ζ h + : M (z) → M (h + z) the affine map obtained combining the linear action of h + with the cut and paste operations. By construction, we have Dζ h + = h + . In order to describe its induced action ζ h + * on homology, in Figures 6  and 7 we draw the images of the homology classes of α, β under ζ h + . By changing representatives as shown in Figures 6 and 7, one can verify that: In view of (6) and (8), for any z ∈ T 2 Since S is symmetric with respect to the involution ϑ, z ∈ S if and only if ϑ −1 z = ϑz ∈ S. By (6) and (8) In view of Lemma 3.3, for every z ∈ T 2 0 and n ∈ Z there exists m ± n (z) ∈ Z such that Lemma 3.4. If z = (x, y) ∈ T 2 0 with irrational x and y then (i) m ± n (z) → +∞ as n → +∞; (ii) the sequences m ± n (z) n≥1 take all natural values.
Proof. We will give the proof for the matrix h + . The case of the matrix h − is similar. By Lemma 3.3, for every z ′ ∈ T 2 0 we have where 1 S is the indicator function of the subset S. In view of (3), it follows that Since m + 0 (z) = 0 and m + n+1 (z) = m + n (z) ± 1, the condition (i) implies (ii).

Irrational λ
The aim of this section is to give the proof of the following result on the existence of ergodic directions on M (0, λ) β whenever λ ∈ (0, 1/2) is irrational. This result follows from Theorem 3.1 and the following auxiliary lemma.
Lemma 4.2. Let λ ∈ (0, 1/2) be an irrational number and let J ⊂ (0, 1/2) be a closed interval. Suppose that z = (x, y) = g −1 (0, λ) for some g ∈ SL + (2, Z) so that y > 0. Then there exist natural numbers a, b, c, d such that if Moreover, the number a can be chosen arbitrary large. Remark 4.3. Note that the irrationality of λ implies the irrationality of y. If additionally the matrix g has positive entries then x is also irrational.
In view of the proof of Lemma 4.2, we have a freedom of choice of d n for fixed a n , b n , c n . It follows that the set of ergodic directions is uncountable. Let α = [0; e 1 , e 2 , . . .]. In view of (14), we have Moreover, y n ∈ [1/6, 1/3] and e 8n+1 = a n+1 ≥ 6 ≥ 2 1 − 2y n .
Finally Theorem 3.1 applied to the sequence (k n ) n≥1 , k n = 8n yields the ergodicity of the directional flow along the vector (1, α).

Rational λ
The aim of this section is to describe more precisely a subset of ergodic directions on M (0, λ) β whenever λ ∈ (0, 1/2) is rational. Such precise description will help us to show that the Hausdorff dimension of the set of ergodic directions is greater than 1/2.
Notation. For every λ = p/2q with 0 < p < q relatively prime natural numbers set where a is the unique natural number satisfying 0 < a ≤ q and ap = −1 mod q. The estimate from below of the Hausdorff dimension in Theorem 5.1 follows immediately form the following result, whose proof is fairly standard, but is included for the convenience of the reader. In view of (16), ψ a,l (E u ) ⊂ E u for every l ∈ B u and the intervals ψ a,l (E u ), l ∈ B u are pairwise disjoint.
From now on, we will deal with square tiled translation surfaces M (z) for which z = (r/2q, s/2q) ∈ T 2 0 with r, s, q ∈ Z, |r|, |s| < q, s is non-zero and coprime with q. Lemma 5.3. Suppose that at least one number s or r is odd. Let a, b be natural numbers such that (17) 0 < a, b ≤ 2q and r + as = −q mod 2q, bs + s − q = r mod 2q.
Then setting we have g z · M (z) = M (z) and (g z ) * (z) = id.
Notation. Let z = (r/2q, s/2q) ∈ T 2 0 , where r, s, q are integer numbers with |r|, |s| < q and such that s = 0 is coprime with q. Suppose that The following result is a more general version of Theorem 5.1.