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Weakly Mixing Polygonal Billiards

We prove that there exists a residual set of (non-rational) polygons such the billiard flow is weakly mixing with respect to the Liouville measure (on the unit tangent bundle to the billiard). This follows, via a Baire category argument, from showing that for any translation surface the product of the flows in almost every pair of directions is ergodic with respect to Lebesgue measure. This in turn is proven by showing that for every translation surface the flows in almost every pair of directions do not share non-trivial common eigenvalues.


Introduction
A basic question in ergodic theory is the dynamical properties of the billiard flow in various (planar) domains (which we will call "tables") [Kat05], [MT], [Tab], [CheMar], [Gut]. Frequently, the table is assumed to have piecewise C 1 boundary and the billiard is assumed to be a massless point traveling without friction on the interior of the table which experiences elastic collision when it hits the boundary of the table. We will be concerned with such a system where the table is a polygon. The main result of this paper is, Theorem 1.1. There exists a weakly mixing billiard flow in a polygon.
In fact, we answer a conjecture stated by E. Gutkin and A. Katok (see [GutKa], §1) that the set of polygonal tables with a weakly mixing billiard flow is a dense G δ subset of the appropriate space of all polygonal tables with a fixed number of vertices.
Let P denote a polygon. The billiard flow is a flow F t : (P × S 1 )/ ∼ → (P × S 1 )/ ∼, where (p, θ) ∼ (q, ψ) if p = q ∈ ∂P and the angle between θ and the side of P at p is π minus the angle between ψ and the side of P at p. This flow is defined for any orbit that does not orbit into the vertices of P . This flow preserves the Lebesgue measure on P × S 1 , defined as the product of the Lebesgue measure on P times the Lebesgue measure on S 1 (pushed forward by ∼).
A natural dynamical consequence of this result is that there exists a polygon P of area 1 so that for any rectangles R, R ′ ⊂ P and for any intervals I, I ′ ⊂ S 1 , for almost every p, p ′ ∈ P and θ, ψ ∈ S 1 we have S. Kerckhoff, H. Masur and J. Smillie proved that there were ergodic billiard flows in polygons [KMS]. The significant general results about billiard flows in polygons are C. Boldrighini, M. Keane and F. Marchetti's result that they have at most a countable set of directions containing periodic trajectories [BKM] and A. Katok's result that the billiard flow has zero entropy [Kat87,Section 3]. By a result of A. Katok [Kat87, Section 3] the zero entropy property implies a subexponential bound on the growth of complexity, in particular for the counting function of generalized diagonals and periodic orbits. Recently, D. Scheglov [Sch13], [Sch19+] improved this bound in the case of almost all triangles to "weakly exponential". We recall that H. Masur [Mas90] showed the counting function of generalized diagonals grows quadratically for rational tables, and it has been conjectured that polynomial bounds should hold for general typical polygons ( [Kat87], §4). The most significant recent results on the ergodic properties of billiards in polygons, all of which are in different contexts, are the results of A. Avila and V. Delecroix [AvDel] and D. Aulicino, A. Avila and V. Delecroix [AuAvDel], which prove weak mixing in almost all directions in certain rational polygons and the proof by A. Málaga and S. Troubetzkoy [MaTr] of the weak mixing property in almost all directions for rational billiards in generic polygons with vertical -horizontal sides. In these cases the unit tangent bundle splits into invariant surfaces, and the weak mixing property is proved with respect to the two dimensional measure on the generic invariant surface, as opposed to our setting in which we prove weak mixing with respect to the (3dimensional) Liouville measure on the unit tangent bundle. Another recent result is the proof by J. Bobok and S. Troubetzkoy [BoTr] of topological weak mixing for the billiard map (that is, for the first return map to the sides of the polygon, which is a Z-action, as opposed to the billiard flows we consider) in the generic polygon.
Similar to the proof of existence of ergodic billiards [KMS], and of the directional weak mixing results of [GutKa], our result is obtained from a result about every translation surface via a Baire category argument. A translation surface is a pair (X, ω) where X is a Riemann surface and ω is an Abelian differential. From ω we obtain an S 1 family of vector fields on X, which in turn give flows F t r θ ω on X and a Lebesgue measure on X. We denote this normalized Lebesgue measure λ 2 , regardless of the surface, which is preserved by these flows.
P. Hubert and the first named author [ChHu] previously showed that for almost every surface (with respect to any SL(2, R)-invariant measure) the product of the flow in almost every direction is uniquely ergodic. Our methods for establishing Theorem 1.2 are spectral, as we show the following, which because the straight line flow on any translation surface is λ 2 ergodic in almost every direction [KMS], is well known to imply Theorem 1.2 (see for instance [KT], Prop. 4.2): Theorem 1.3. For every α = 0 and every translation surface ω we have |{θ ∈ S 1 : F t r θ ω has eigenvalue e 2πıα }| = 0 .
A. Avila and the second named author [AvFo] showed that for almost every translation surface (in genus at least 2) the flow in almost every direction is weakly mixing, which implies the above result for almost every surface. In fact, by an announcement of D. Aulicino, A. Avila and V. Delecroix [AuAvDel], Theorem 1.3 holds for almost every surface with respect to every SL(2, R) invariant measure. (They show that for any SL(2, R) ergodic measure, which is not supported on branched covers of tori, the vertical flow on almost every surface is weakly mixing. This gives Theorem 1.2 for these measures. By an argument based on rigidity sequences, one can show that Theorem 1.3 holds for any branched cover of a torus. In fact, one can prove, see for instance [FH19], that for any α ∈ R \ {0} and for the flow in almost every direction on the branched cover of the torus, there exists a rigidity sequence t j so that e 2πıαt j does not converge to 1). Note, it is well known that single cylinder surfaces give many examples of surfaces (even with full orbit closure) where every direction has a non-constant eigenfunction.
1.1. Organization of the paper. In Section 1.2 below we gather some fundamental open questions in the ergodic theory of billiards in polygons. In Section 2 we recall some basic material about the Teichmüller flow and the renormalization cocycle for translation flows, the Kontsevich-Zorich cocycle over the Teichmüller flow on the Hodge bundle. We conclude the section, in Section 2.1, with an outline of our argument. In Section 3 we derive several consequences of the work of [EskMir], [EskMirMo] and [ChEs], including results on the growth of vectors under the action of the Kontsevich-Zorich cocycle and on averages along horocycle arcs of the pushforward of "height" functions (functions as in Theorem 3.8) for the moduli space by the geodesic flow. Section 4 contains, in Section 4.1, preparatory results on the growth of curves in the Hodge bundle, which are derived from results of the previous section, and, in Section 4.2, some standard large deviation results, which are included for convenience of the reader. Section 5 contains the core of the argument, that is, the key proposition on the controlled growth of curves under the Kontsevich-Zorich cocycle (Prop. 5.1). In Section 6 after recalling Veech's criterion for weak mixing, we prove our main result, Theorem 1.3. Finally, in Section 7 we derive our result on the existence of weakly mixing polygonal billiards.

Open questions.
Question 1.4. Does every translation surface ω where the flow is not weakly mixing in almost every direction have the property that the line R[ℑ(r θ ω)] spanned by the cohomology class [ℑ(r θ ω)] of the imaginary part of the its rotation r θ ω intersects an integer translate of the stable subspace of the SL(2, R) subbundle in a set of directions θ ∈ S 1 of positive measure? Question 1.5. For every translation surface is the flow in almost every pair of directions uniquely ergodic? Spectrally singular (modulo constants)?
There are also many natural questions about the flow of billiards in irrational polygons.
Question 1.6. Is there a polygon with a mixing billiard flow? Topologically mixing? Minimal? Is the billiard flow in every irrational polygon ergodic? Weak mixing? Mixing?
Note that numerical experiments [CasPros] suggest that every billiard in an irrational triangle is mixing.
1.3. Acknowledgments. The first named author thanks DMS-1452762, a Poincaré chair, a Sloan fellowship and a Warnock chair for support. The first named author also thanks the University of Maryland for its hospitality. The second named author was supported by the NSF grant DMS 1600687 and by a Research Chair of the Fondation Sciences Mathématiques de Paris (FSMP). He would also like to thank the University of Utah for its hospitality. The authors would like to thank CIRM, IHP, University of Zurich, ETH Zurich and Oberwolfach for their hospitality during work on this project.

Background
As stated earlier, a translation surface is a pair (X, ω) where X is a finite type Riemann surface and ω is an Abelian differential. The space of translation surfaces is stratified by the order and number of the zeros of the Abelian differential. Let H(α) be the moduli space of translation surfaces (X, ω) such that ω has k zeros of orders α := (α 1 , . . . , α k ), which we call a stratum. The (real) Hodge bundle of the stratum is a bundle over each stratum whose fiber at (X, ω) is H 1 (X; R). Given a translation surface (X, ω), integrating ω provides charts for X \Σ where Σ is the set of zeros of ω. From its action on these charts, the group SL(2, R) of real matrices with determinant one acts on each translation surface, preserving the stratum it's in.
The group SL(2, R) also acts on the Hodge bundle since it acts on the base H(α) of the bundle and this action can be lifted to the bundle by parallel transport of cohomology classes with respect to a natural flat connection.
The Kontsevich-Zorich cocycle is given by the action of SL(2, R) on the Hodge bundle. It is a cocycle over the action of the group SL(2, R) on the stratum H(α). See the lecture notes of the second named author and Carlos Matheus [FoMa] for a detailed description of the above material. Let denote, respectively, the Teichmüller horocycle flows and the Teichmüller geodesic flow. Also let denote the group of rotations. Let I ⊂ R be any interval. Given a horocycle arc {h s ω|s ∈ I} at ω ∈ H(α), a horocycle section at ω is a map φ : Here H 1 (M, R) denotes the Hodge bundle over the moduli space of Abelian differentials and H 1 ω (M, R) denotes the fiber at ω ∈ H(α).
For every s ∈ I, let π ω (s) : H 1 hsω (M, R) → H 1 ω (M, R) denote the linear map given by the parallel transport along the horocycle arc joining h s ω to ω, that is, {h s−σ ω|σ ∈ [0, s]}. Given a horocycle section φ : Since the Kontsevich-Zorich cocycle is defined by parallel transport, the following commutation relation holds: for It follows that for any horocycle section φ : s) is a horocycle section at g t ω, hence in order to compute the evolution of horocycle sections under the Teichmüller flow it is enough to compute the evolution of curves φ ω : I → H 1 ω (M, R) under the maps φ ω (s) → KZ(g t , ω)φ ω (e −2t s) .
Let | · | denote the Hodge norm. We recall that the Hodge norm of a cohomology class γ ∈ H 1 (X, C) is given by where * denotes the Hodge star operator. This norm depends on the Riemann surface (but not the Abelian differential), though we usually suppress this dependance. See the survey [FoMaZo] for a detailed description. For every g ∈ SL(2, R) let KZ(g, ω) denote the operator norm with respect to the Hodge norms on H 1 ω (M, R) and H 1 gω (M, R). If F ⊂ H 1 (M, R) is a subbundle of the (real) Hodge bundle, let KZ(g, ω) F denote the operator norm with respect to the Hodge norm restricted to F . Lemma 2.1. Let dist denote the hyperbolic distance on SL(2, R)/SO(2). For any Abelian differential ω ∈ H(α) and for all g ∈ SL(2, R), we have KZ(g, ω) ≤ e dist(g,Id) and moreover |KZ(g, ω)v| ≥ e −dist(g,Id) |v|.
Proof. For g = g t a diagonal element of SL(2, R), the upper bound holds for the Hodge norm by the first variation formulas [Fo,§2] or [FoMa,§3.5]. The lower bounds also follows since KZ(g t , ω) is invertible with inverse K(g −t , g t ω), hence for t ≥ 0 For a general g ∈ SL(2, R), the result follows from the KAK decomposition of SL(2, R), since the action on the Hodge bundle of the group K = SO(2) ⊂ SL(2, R) of rotations is isometric.
Lemma 2.2. For every c > 0 there is a compact set K and t 0 > 0 so that if ω ∈ K then |{−1 ≤ s ≤ 1 : g t h s ω ∈ K}| > 2 − c for all t > t 0 .
Let ν be an SL(2, R)-invariant measure on a stratum of translation surfaces. We say F is a ν-almost everywhere invariant subbundle if for ν-almost every ω and any g ∈ SL(2, R) we have that g sends F ω to F gω , that is, KZ(g, ω)F ω = F gω .
Following [ChEs,Def. 1.3] we say that the KZ cocycle has a νmeasurable almost invariant splitting F if there exists a finite set of proper subbundles F 1 , . . . , F n ⊂ F such that F i ∩ F j = {0} ν-almost everywhere, for all 1 ≤ i, j ≤ n, and for ν-almost all ω and almost all g, the linear map KZ(g, ω) sends the set {F 1,ω , . . . , F n,ω } to the set {F 1,gω , . . . , F n,gω }. Following [ChEs,Def. 1.4], we say that the cocycle acts strongly irreducibly on F with respect to the measure ν if it does not admit any measurable almost invariant splitting.
The span of ℜ(ω), ℑ(ω) (respectively the real and imaginary part of the Abelian differential ω ∈ H(α)) defines a smooth invariant, symplectic subbundle of the Hodge bundle, which is then ν-invariant for any SL(2, R) invariant measure ν on H(α). We call this the SL(2, R) subbundle. LetF denote its symplectic complement, which is also a ν-almost everywhere invariant subbundle.
2.1. Outline of proof. To prove Theorem 1.3, most of our work is to rule out the 'weak stable space' (in the terminology of [AvFo]). This is different from the approach of [AvDel] and our understanding of the approach of [AuAvDel], where they rule it out for 'structural' reasons. It also differs from the approach in [AvFo], because it is centered more on moduli space. As in the previous approaches we apply the Veech criterion [Ve84] (Lemma 6.1 of this paper) which morally says that it suffices to show that for a.e. α there exists c > 0 and a compact set K so that for arbitrarily large t we have (1) KZ(g t , r θ ω)ℑ(r θ ω) Z > c and g t r θ ω ∈ K (we actually need two additional appropriate times in K). We show that for every ω this one parameter family of classes are transverse to any integer translate of the stable SL(2, R)-bundle. This allows us to apply Lemma 4.6 which makes the assignment after rescaling the segment by the geodesic flow which exponentially expands it, closer to a constant curve (for typical θ). This lets us apply Proposition 3.1 (a modification of [ChEs]) to have that KZ(g t , r θ ω)ℑ(r θ ω) typically grows in t. The key Proposition 5.1 shows that, under appropriate assumptions, there exists a 0 < ρ < 1 so that for an appropriately chosen segment of angles J we have that for most θ ∈ J there is an s > 0 so that KZ(g t+s , r θ ω)ℑ(r θ ω) Z > ( KZ(g t , r θ ω)ℑ(r θ ω) Z ) ρ and g t+s r θ ω ∈ K. A key step in the proof (Lemma 5.3) is a new large deviations estimate for the measure of the set of directions, on any translation surface, where the Kontsevich-Zorich cocycle grows slower than expected. This complements [AAEKMU], Theorem 1.5, which proves a similar result for when the growth of the cocycle is larger than expected. Iterating Proposition 5.1, we can avoid the issues of 'descendants' (contrary to [AvFo], §3 especially (3.6), although this terminology is not used there). Another difference with [AvFo]'s approach is we treat the "weak stable" and stable subbundles at the same time. We also need to treat the unstable-SL(2, R) subbundle and subbundles where the Kontsevich-Zorich cocycle acts isometrically, which we do by straightforward or standard arguments.
Throughout this proof we treat horocycles instead of circles because they behave better under the geodesic flow, and relate horocycles to circles in Section 6.2. Section 7 proves Theorem 1.1 via the previously mentioned, and now standard, Baire Category argument. In this section we follow the approach of [Vo97].

Making vectors grow
This section proves Propositions 3.1 and 3.2 and then develops, in a straightforward way, some machinery from [EskMirMo].
LetF denote the complementary subbundle, that is, the symplectic orthogonal, of the SL(2, R) subbundle and let ν be any SL(2, R)invariant ergodic probability measure.
Proposition 3.1. (Chaika-Eskin, [ChEs]) If F ⊂F is an equivariant subbundle, where the Kontsevich-Zorich cocycle acts strongly irreducibly and with a positive exponent, then there exists λ := λ F > 0 such that, for all δ, ǫ > 0, and for all L sufficiently large, there exist ǫ > 0 and an open setÛ :=Û (δ, L) so that ν(Û ) > 1 − δ, and for all ω ∈Û and for all parallel horocycle sections v at ω, that is, for the all the maps v : Proposition 3.1 is proved below. For any compact subset K, for any ω and t > 0, let Proposition 3.2. Let F ⊂F be an equivariant subbundle ofF. For any compact subset K there exists λ := λ F ,K ∈ [0, 1) such that Proof. Both estimates follow from the first variational formulas for the Hodge norm [Fo,§2] or [FoMa,§3.5].
We now prove Proposition 3.1.
3.1. Proof of Proposition 3.1. Let λ > 0 denote the largest element of the Lyapunov spectrum restricted to F . Let µ be an SO(2) invariant, compactly supported probability measure on SL(2, R), whose support generates a dense subgroup of SL(2, R). Let M be any SL(2, R)-invariant suborbifold. We recall that Eskin and M. Mirzakhani [EskMir] and A. Eskin, M. Mirzakhani and A. Mohammadi [EskMirMo] proved that all orbit closures of the SL(2, R) action on the moduli space are "affine" suborbifolds supporting a unique "affine" probability invariant measure.
Let E good (L, ǫ) be the set of all surfaces ω ∈ M so that for every Note that by assumption the cocycle acts strongly irreducibly on F justifying the application of [ChEs]. The continuity of the Hodge norm implies that E good (L, ǫ) is open.
Furthermore, the distribution ofs is in the measure class of Lebesgue.
By construction, for all ω ∈ U and for all parallel horocycle sections v at ω, that is for the all the maps v : Let now u ∈ V hsω be a vector such that ∠(u, v(s)) <ǫ. We can write u = av(s) + bv ⊥ (s) with v ⊥ (s) ⊥ v(s) (with respect to the Hodge inner product) and by assumption we have It follows that, ifǫ < e −L e (λ−ǫ)L /8, by applying Lemma 2.1, we have The argument is completed.

Developing.
Theorem 3.7. (Eskin-Mirzakhani-Mohammadi) Let U be an open set, ν be an SL(2, R) invariant and ergodic measure. For any ǫ > 0 there exists a finite set of invariant manifolds, Z 1 , . . . , Z n so that for any This follows from [EskMirMo,Theorem 2.7 (2) There exists b > 0 (depending on M) and for every 0 < c < 1 there exists t 0 > 0 (depending on M and c) such that for all ω ∈ H \ M and all t > t 0 , (3) There exists σ > 1 and V ⊂ SL(2, R) a neighborhood of the identity so that for all g ∈ V and all ω ∈ H, This implies a similar result for horocycles: Lemma 3.9. [AAEKMU,Lemma 3.5] Let f M be as in Theorem 3.8. Then there exists a constant b ′ > 0 so that for all 0 < a < 1 there exists t 0 =t 0 (a) such that for all t >t 0 and for all ω ∈ H \ M we have Proposition 3.10. [AAEKMU,Proposition 3.7] Let f M be as in Theorem 3.8 and let b ′ > 0 andt 0 =t 0 (a) be as in Lemma 3.9. There exist C 1 > 1 (independent of ω and a) such that for all a ∈ (0, 1), all ρ > C 1 b ′ /a, all t ≥t 0 such that e t ∈ N and all N ∈ N, By choosing a < min{ 1 2 , 1 2b ′ } we obtain: Proposition 3.12. [AAEKMU,Proposition 3.9] (cf [KKLM,Theorem 1.5]) Let C ρ be as above for a function f M as in Theorem 3.8. Let us assume that ω ∈ M. For any δ, a ∈ (0, 1) there exist ρ 0 > 1 and t 0 > 1, depending only on a, such that for all ρ ≥ ρ 0 , all t ≥ t 0 such that e t ∈ N, and all N ∈ N, the set , the measure of the set in the statement of Prop. 3.12 decays exponentially in N.
Corollary 3.13. Let U be an open set whose boundary has measure zero. For all ǫ > 0 there exist numbers ξ ∈ (0, 1), , open in supp(ν), and T 2 ∈ R + so that for any T > T 2 : This uses the following standard definition and result: Definition 3.14. Let ω ∈ H and ν be the unique, SL(2, R)-invariant and ergodic probability measure with Lemma 3.15. Let S be a set so that ν(∂S) = 0. If ω is Birkhoff generic then If U is an open set and ω is Birkhoff generic Proof of Corollary 3.13. First, (ii) follows free of charge by the fact that V ǫ is open and Lemma 3.15. Given an open set U and ǫ > 0, we then construct an open set V ǫ with properties (i), (iii) and (iv). For the given U and ǫ > 0, we apply Theorem 3.7 to obtain a finite number of manifolds Z 1 , . . . , Z n . For each of these manifolds we build a function f i as in Theorem 3.8 and let C (i) Applying Proposition 3.12 to each f i with δ < ǫ n and a < 1 2(2C 1 ) 1 δ (where C 1 is as in the proposition), there exists ρ i , t i as in the proposition, and so that ν(∩C Following the sentence after the proposition, for any σ > 1 there exist constants ζ 1 , . . . , ζ n < 1 and natural numbers N 1 , . . . , N n so that for each We define the set V ǫ to be the interior in supp(ν) of the compact set σρ i , so that property (iv) holds by the above argument. By applying Theorem 3.7 to the compact set and the given ǫ > 0, we conclude that there exists T 2 > 0 such that property (iii) holds for all T > T 2 . Finally, it follows from the def- We will of course be interested in V ǫ chosen for U =Û and this is whatV ǫ denotes for the remainder of the paper. We further assume thatV ǫ is contained in a fixed compact set.

Some preparatory results
4.1. Making curves grow. The main result of the next section is Lemma 4.8, which applies Proposition 3.1 to say that the geodesic flow image of (projectively) Lipschitz horocycle sections with small enough (projective) Lipschitz constants in a subbundle as in Proposition 3.1 typically grow. It also collects some results which says that the geodesic flow image of horocycle sections have their Lipschitz (and usually their projective Lipschitz) constants improve.
is a K-Lipschitz map with respect to the Hodge norm. We recall that the map φ ω is defined, for all s ∈ I, as a composition φ ω (s) = π ω (s) • φ(s) with the parallel transport π ω (s) : . By the commutativity of parallel transport and the KZ cocycle, we have obtained by parallel transport along the horocycle is K-Lipschitz with respect to the Hodge norm on H 1 ω (M, R) and the Hodge norm φ ω (s) ω is bounded below by K/κ > 0.
Remark 4.4. For any bounded interval [a, b] ⊂ R, there exists a constant C a,b > 0 such that the following holds. For any section φ : [a, b] → H 1 (M, R) (projectively) κ-Lipschitz at ω, and for any s ∈ [a, b], the section φ is also (projectively) C a,b κ-Lipschitz at h s ω.
The main result of this section is the following lemma. Let δ > 0 and let L > 0 be sufficiently large.
Notation: For all ǫ, δ > 0, let Proof. Let δ > 0 and L > 0 sufficiently large be fixed. By Corollary 3.13 (iii) (and the fact that ν(Û ) > 1 − δ), since g t h s 0 ω ∈V ǫ , we have that, for all T > T 2 , That is, From this we have that there exists a ∈ [0, L] so that It follows that except for a set of s ∈ [s 0 −e −2t , s 0 +e −2t ] with measure at most e −2t 2(ǫ + δ) + L T we have For t ≥ t 0 and for all s ∈ [s 0 − e −2t , s 0 + e −2t ], let We claim that In fact, for each j, let χ j denote the characteristic function of the set

It then follows that
which finally implies that as claimed.
Choosing T 3 > 0 so large that L T 3 < ǫ + δ gives a set of measure at least [1 − ( 3(ǫ + δ) + 2 √ δ)]e −2t , where we have that, whenever T > T 3 , for at least T L (1 − 3(ǫ + δ) − 2 √ δ) of our indices the cocycle grows by at least e λL(1−ǫ) . By Lemma 2.1, the cocycle reduces the Hodge norm by multiplication times a factor larger than e −L on the remaining indices, thereby giving the bound. 4.2. Some probabilistic results. We collect below some well-known probabilistic results for the convenience of the reader.

The key proposition
Proposition 5.1. Let κ 0 be as in Lemma 4.8. There exist constants σ > 0, τ > 0 so that for all large enough compact sets, K there exists γ K > 0 such that all horocycle sections φ : [−1, 1] → F ⊂F at ω, under the conditions that, for some s 0 ∈ [−1, 1] we have (a) e t large enough and γ ∈ (0, γ K ) ; such that, for each s ∈ S t,s 0 , there exists ℓ := ℓ(s) such that Note that by our assumption on ℓ(s 0 ), Conclusion (C) implies that there exists 0 < ρ < 1 so that : for all s ∈ S t,s 0 and for ℓ = ℓ(s), we have Before the beginning the proof of Proposition 5.1 we reduce lower estimates on the distance to the integer lattice to lower bounds on the norm of vectors in the cohomology bundle over a fixed compact set.
Proposition 5.1 follows from the next lemma, whose proof we defer until after the proposition's proof.
If ǫ, δ > 0 are small enough there exists σ ′ > 0 such that Remark 5.4. Let F ⊂F be an equivariant subbundle, where the Kontsevich-Zorich cocycle acts strongly irreducibly with a positive exponent, and let λ be the largest such exponent. Let v : [−1, 1] → F be a parallel horocycle section at ω. By the above lemma and Lemma 2.1 one has that for any ǫ ′ > 0 there exists c > 0 and T 1 ∈ R so that for all T > T 1 we have More generally, one can prove such a large deviation result for any Lipschitz horocycle section ψ : [−1, 1] → F . Indeed by Proposition 3.12 (applying it to a function f ∅ ) there exists a compact set K and a constant c so that the measure of the set of s ∈ [−1, 1] so that |{0 ≤ t ≤ T : g t h s ω / ∈ K}| < cT decays exponentially with T . (Indeed one can choose ρ so big that if g t ω ∈ C ρ 0 then g t+τ ω ∈ C ρ = K for all 0 ≤ τ ≤ t 0 .) By Lemma 4.6, the transported horocycle section KZ(g T )(ψ) at such an s ∈ [−1, 1] is κ T -projectively Lipschitz with a constant κ T which decays exponentially with T .
Proof of Proposition 5.1 assuming Lemma 5.3. For all c, γ > 0, let We first prove that for any c > 0, and for γ > 0 sufficiently small, the Lebesgue measure of B is polynomially small in γ > 0.
Let ψ = φ t,s 0 . By the hypotheses of the Proposition the horocycle section φ t,s 0 at g t h s 0 ω, defined in formula (14), satisfies the hypotheses of Lemma 5.3. Indeed, by the hypotheses (c) and (d) of the Proposition, the section φ t,s 0 is κ-projectively Lipschitz at h s 0 ω with κ ∈ (0, κ 0 ).
Let T 0 > 0 be sufficiently large and so that e T 0 ∈ N and let γ > 0 such that T := 1 2 | log γ| > T 0 . As in Lemma 5.3, we partition the interval [0, 1 2 | log γ|] into R := ⌊ | log γ| 2 1 T 0 ⌋ equal intervals of length T 0 > 0 and an additional interval of length at most T 0 . Let G be the set of formula (17) in the statement of the Lemma 5.3 applied to the horocycle section φ t,s 0 , and let S t,s 0 be defined as S t,s 0 := {s 0 + s ′ e −2t |s ′ ∈ G \ B} .
Let us assume that s ∈ S t,s 0 ⊂ [s 0 −e −2t , s 0 + e 2t ], hence by definition e 2t (s − s 0 ) ∈ B. By the above definition of the set B, there exists Clearly conditions (A) and (B) are satisfied for ℓ := ℓ s . It remains to verify Condition (C). Since e 2t (s − s 0 ) ∈ G, by formula (17) and Lemma 2.1 there exists µ ′ := µ − 16η − ǫ > 0 such that, for all s ∈ [−1, 1], and because the Hodge norm can change by a factor of at most e ±(c| log γ|+T 0 ) from time RT 0 = ⌊ | log γ| 2 1 T 0 ⌋T 0 to time ℓ s ≤ ( 1 2 + c)| log γ|, for c > 0 sufficiently small there exists τ > 0 such that Since, by our assumption on s, g ℓ h e 2t (s−s 0 ) g t h s 0 ω ∈ K, by (c) the horocycle section φ t,s 0 is γ-Lipschitz at h e 2t (s−s 0 ) g t h s 0 ω and by (d) by Lemma 5.2, there exists a parallel section z t,s 0 : In fact, we can apply Lemma 5.2 to the section φ t,s 0 to get a parallel integer section z t,s 0 : [−1, 1] → H 1 (M, Z) at g t h s 0 ω such that the above identity holds. By applying the above argument to the curve φ t,s 0 −z t,s 0 we therefore conclude that, for s ∈ S t,s 0 , Finally since by hypothesis the section φ t,s 0 is κγ-Lipschitz and by formula (18), we have the lower bound which completes the argument.
Proof of Lemma 5.3. Let us recall that by definition, for s ∈ [−1, 1], We say that the pair (j, s) is good if g jT 0 h s ω ∈ V ǫ . Observe that if (j, s) is good then, since by hypothesis the horocycle section φ is projectively κ-Lipschitz, by Lemma 4.8 for sufficiently large T 0 > 0, we have We now wish to use this estimate and Corollary 4.11 to complete the proof of the lemma. To satisfy the assumptions of Corollary 4.11 we define a sequence of nested partitions of the interval [−1, 1]: Let P (j) (s) be the unique element of P (j) such that s ∈ P (j) (s) and P We claim that by the remark preceding the statement of Lemma 5.3, if g jT 0 h s ω ∈ K then, for any s ′ ∈ P (j+1) (s), In fact, since the Teichmüller distance between g jT 0 h s ω and g jT 0 h s ′ ω is at most e −2T 0 , there exists g ∈ SL(2, R) at a hyperbolic distance at most e −2T 0 from the identity in SL(2, R)/SO(2) such that, for any cohomology class v ∈ H 1 h s ′ ω (M, R), we have By Lemma 2.1 it follows that Let K > 0 denote the Lipschitz constant of φ at ω. Since the section φ is κ-projectively Lipschitz at ω, there exists a constant C > 0 such that it is CK-Lipschitz and Cκ-projectively Lipschitz at h s ω. By Lemma 2.1, we then have Since KZ(g jT 0 , h s ω) ≤ e jT 0 and |s−s ′ | ≤ e −2(j+1)T 0 for s ′ ∈ P (j+1) (s), we also have so that we have derived the estimate The upper bound in formula (21) then follows from the above estimates, for κ ∈ (0, κ 0 ) and for T 0 > 0 sufficiently large. The lower bound follows by symmetry, hence the claim is proved. Now assume T 0 is large enough so that e −ǫT 0 < (1 + e −2T 0 ) −1 and let Observe that (20) and (21) give (23) Leb({s / ∈ B j :s ∈ G j and I j (s) = v}) Indeed, by definition s, s ′ ∈ P (j) k for some k implies I j (s) = I j (s ′ ) and (20) and (21) gives the measure estimate conditioned to s ∈ G j (which is equivalent to s ′ ∈ G j ).
This lower bound on the conditional probability provides the large deviations estimate via Corollary 4.11. Indeed, by applying the corollary to the sequence of random variables F j equal, for all j ∈ N, to the characteristic functions of the sets of s ∈ B j ∩ G j , we have that in the complement of a set of measure exponentially small in T T 0 , (for all large enough T > T 0 ). This is because by formula (23) the sequence of random variables {F j } defined above satisfies the assumptions of Corollary 4.11. Consider the set of s ∈ [−1, 1] such that Note that by Corollary 3.13 (iv) (which we may apply if T 0 is at least t 0 ) the complement of this set has that its measure decays exponentially with ⌊ T T 0 ⌋. The intersection of the sets given by (24) and (25) satisfies the desired conditions, hence the proof of the Proposition is complete.
6. Proof of Theorem 1.3 6.1. The Veech criterion. To prove Theorem 1.3 we need a condition to rule out the flow F t r θ ω having e 2πıα as an eigenvalue. This is Lemma 6.1 below, which is essentially due to Veech. Following Veech [Ve84,§7] we have the following criterion for a translation flow F t r θ ω to not have e 2πıα as an eigenvalue. There is a c > 0 and a sequence of transversals {J i } so that where ℓ ′ (J i ) denotes the length of the projection of J i in the direction θ.
• If T is the IET given by the first return to J i and I 1 , . . . , I d are the intervals that define T then |I a | > c|I b | for all a, b.
Let r i be the vector of return times of F t r θ ω to J i . If lim sup i→∞ r i α Z = 0 then e 2πıα is not an eigenvalue of F t r θ ω . We have the following trivial consequence of this: Lemma 6.1. (Veech) Let e 2πıα be a nontrivial eigenvalue of F t ω . For every compact set K there exists ℓ K so that, if there exist ℓ ≥ ℓ K and a diverging sequence (t i ) ⊂ R + with the property that g t i ω ∈ K, g t i −ℓ ω ∈ K and g t i +ℓ ω ∈ K, then lim i→∞ KZ(g t i , ω)(αℑ(ω)) Z = 0.
Proof. For each compact set K there exist constants a K , b K and d K > 0 so that if ω ′ ∈ K then we can write ω ′ as a zippered rectangles over an interval I of length at most a K , in the horizontal direction and one endpoint a singularity (that is, as a suspension with piece-wise constant roof function). Moreover the heights of the rectangles are at most b K . Lastly d K is the length of the shortest saddle connection on all ω ′ ∈ K.
Because there are singularities on each endpoint of the rectangles, if u is the width of a rectangle, there is a saddle connection on the surfaces with holonomy (u, r) with r < b K . So if ℓ > log( b K 2d K ) and g ℓ ω ′ ∈ K then u ≥ 1 2 d K e −ℓ . In fact, the holonomy of the same saddle connection on g ℓ ω is (e ℓ u, e −ℓ r) and for ℓ > log( 2b Since g ℓ ω ′ ∈ K its shortest saddle connection has length at least d K , hence we have e ℓ |u| ≥ d K /2, as claimed. Likewise, because one endpoint of I is a singularity, if F s ω ′ is discontinuous on I then there is a saddle connection on ω ′ with holonomy (a, v) with |v| ≤ |s| and |a| < a K . Thus if ℓ > log( a K 2d K ) and g −ℓ ω ′ ∈ K, then F s ω ′ acts continuously on I for all − 1 2 e −ℓ d K < s < 1 2 e −ℓ d K . In fact, the saddle connection on g −ℓ ω ′ has holonomy (e −ℓ a, e ℓ v). For hence e ℓ |v| ≥ d K /2 and so |s| ≥ d K /2e −ℓ , as claimed. We choose ℓ K = max{log( a K 2d K ), log( b K 2d K )}. Now, if ω ′ = g t i ω, by g −t i we can transport I back to ω and obtain an interval I ′ so that |I ′ | = ℓ ′ (I ′ ) ≤ e −t a K . Observe that the second and third bullet points of the Veech criterion still hold since they state properties which are equivariant with respect to the action of the Teichmüller flow g t . We now need to understand the return times to I ′ . Note that they are the imaginary parts of a basis for (absolute) homology of the surface and are all of bounded size when transported to g t i ω (because g t i ω = ω ′ ∈ K). Thus the return time vector is given by KZ(g t , ω)ℑ(ω) and the lemma follows.
6.2. Eliminating the stable subbundle of the SL(2, R)-bundle and isometric subbundles. Recall the decomposition of r θ given by formula (4). Since the horocycleĥ t fixes the real part of holomorphic differentials, that is, it fixes the leaves of the stable foliation of the Teichmüller flow, we have that g log cos θ h − tan θ ω belongs to the same stable manifold as r θ ω, hence h − tan θ ω and r θ ω have the same vertical foliation. The vertical flow of h − tan θ ω equals the vertical flow of r θ ω after a linear reparametrization by multiplication times cos θ. It follows that if α is a (fixed) eigenvalue for the vertical flow of of r θ ω, then α sec(θ) is an eigenvalue for the vertical flow of h − tan θ ω, thus by the Veech criterion (Lemma 6.1) we have that if K is a compact set, ℓ ≥ ℓ K , g t i h s ω, g t i −ℓ h s ω and g t i +ℓ h s ω ∈ K with t i → ∞ then We conclude that if α is a fixed eigenvalue for the vertical flow of r θ ω for a positive measure set of θ ∈ T, then α sec(arctan s) is an eigenvalue for the vertical flow of h s ω, for a positive measure set of s ∈ R, which implies that for a positive measure set of s ∈ R we have that if g t i h s ω, g t i −ℓ h s ω and g t i +ℓ h s ω ∈ K with t i → ∞ then Lemma 6.2. The section ψ has range in SL(2, R) subspace at ω and it is transverse to any integer translate of the stable subbundle of the SL(2, R) subbundle.
Thus the coefficient a ∈ R is uniquely determined (given z ∈ H 1 ω (M, Z)) by the first equation, hence the second equation has a unique solution.
Proposition 6.3. For every compact set K there exists a constant γ ′′ K > 0 such that the following holds. If there exist t 0 > 0 and an integer class z ∈ H 1 gt 0 ω (M, Z) such that • g t 0 ω ∈ K and {t > 0|g t ω ∈ K} has upper density at least 3 4 ; for the Kontsevich-Zorich cocycle, then e 2πıα is not an eigenvalue for the vertical flow F t ω of ω.
Proof of Proposition 6.3. Let γ ′′ K be so small that the minimum Hodge distance between points of the lattice H 1 ω (M, Z) for ω ∈ K is at least 10γ ′′ K . By the isometry property of the Kontsevich-Zorich cocycle on E 0 , we have that for all t ≥ t 0 , Since by hypothesis that the set {t > 0 : g t ω ∈ K} has upper density 3 4 , it can be proved (by an elementary argument left to the reader) that for any ℓ 0 > 0 there exists ℓ > ℓ 0 and a diverging sequence (t i ) such that g t i ω, g t i −ℓ ω and g t i +ℓ ω ∈ K for all i ∈ N. Hence, we may apply the Veech criterion (Lemma 6.1) and α is not an eigenvalue, as stated.
6.3. Proof of Theorem 1.3. Before completing the proof, we recall that if Kontsevich-Zorich cocycle does not act isometrically on a subbundle then Proposition 5.1 can be applied to it.
(2) φ(s) has a non-trivial projection onto a continuous SL(2, R)equivariant subbundle where the Kontsevich-Zorich cocycle has a positive exponent and acts strongly irreducibly.
Proof. By [EskMir,Theorem A.6] the splitting is semi-simple and each irreducible block is strongly irreducible (in the sense that it does not admit a measurable almost invariant splitting) and it is either isotropic or symplectic. By [EskMir,Theorem A.5] if the subbundle is isotropic then all of the exponents are zero. By [EskMir,Theorem A.4], if all the exponents are zero then the cocycle acts isometrically. So, we assume that we are considering an irreducible symplectic block where the cocycle has a non-zero exponent. Since the cocycle is symplectic, it has both a positive and a negative exponent (which are opposite of each other).
Proof of Theorem 1.3. We proceed with a proof by contradiction. Let B ⊂ [−1, 1] denote the set of points for which the conclusion of the theorem fails, and assume that the measure of B is ξ > 0. Let K be a compact set so that Lemma 2.2 and Proposition 5.1 hold with c = .999 and µ(Int(K)) > .999. Let a = max{t 0 , ℓ K } where t 0 is as in Lemma 2.2 and ℓ K is as in Lemma 6.1. We choose γ 0 > 0 small enough so that Proposition 5.1 holds with the compact set K, 1 2 | log(γ 0 )| > T 0 , where T 0 is as in §5, γ 0 ≤ γ ′′ K where γ ′′ K is as in Proposition 6.3, and one additional condition below (so that the RHS in the estimate of formula (33) below is at most 1 2 ). By Lemma 6.1 there exists t 0 > 0 so that (27) Leb({s ∈ [−1, 1] : for all t ≥ t 0 either KZ(g t , h s ω)ψ(s) Z < e −a−1 γ 0 or g t h s ω / ∈ K, or g t−a h s ω / ∈ K, or g t+a h s ω / ∈ K}) > .99ξ.
Let B ′ denote the set in the LHS of formula (27). By the Lebesgue density theorem, there exists r 0 > 0 so that Let B ′′ be the set in the LHS of formula (28). We now choose t ≥ t 0 and s ∈ B ′′ such that h s ω is Birkhoff generic, which is a full measure condition by [ChEs,Theorem 1.1], so that The first assumption is trivial. The second and third assumption can be simultaneously satisfied by Lemma 6.1 and Birkhoff genericity. (Indeed µ(K) > .999 > 3 4 and so, for any ℓ, there exist arbitrarilly large t so that g t−ℓ h s ω, g t h ω , g t+ℓ h s ω ∈ K.) By the third, and last, of the above assumptions there exists an integer class z ∈ H 1 gthsω (M, Z) such that This smallness assumption shows that, if ψ(s) − z is contained in an isometric subbundle, then for almost every θ ∈ S 1 , the flow in direction θ does not have α as an eigenvalue. Indeed, we are assuming all but the first assumption of Proposition 6.3 and that assumption holds for almost every s ∈ [−1, 1], by Birkhoff genericity. Thus, up to the translation of ψ by the integer vector z, that is, by considering the section ψ − z instead of ψ, we can assume that ψ has a non-zero projection on a strongly irreducible subbundle with a non-zero top Lyapunov exponent.
By Lemma 6.2, the section ψ is transverse to any integer translate of the stable subbundle of the SL(2, R) subbundle, so clearly the projection to the stable subspace of the SL(2, R) subbundle can be ignored.
To complete the proof we now separately consider the projection of φ to the unstable part of the SL(2, R) subbundle and to the symplectic orthogonalF of the SL(2, R) subbundle.
We show that for each of these individually, at the time t ′ > t when the norm of the section has grown to size γ 0 under the action of the cocycle, the subset of B(s, e −2t ) of points which are not in the set in formula (27) has at least half the measure of B(s, e −2t ), a contradiction which will conclude our argument. First, we consider the projection onto the unstable part of the SL(2, R)-subbundle. By Lemma 2.2, for any t ′ > t 0 + a we have that Let δ > 0 be the norm of the projection of KZ(g t , h s ω)ψ(s) onto the unstable part of the SL(2, R) subbundle. Let ℓ = log( γ 0 δ ) + 1. Because the unstable part of the SL(2, R) subbundle grows by a constant factor e t ′ at time t ′ > 0, we have our claim by applying (29) at t ′ = t + ℓ.
It now suffices to prove the analogous result forF . Let φ be the projection of ψ onF. By choosing a possibly larger t > 0, we can further assume that Indeed by Lemma 4.6, for every compact set K ′ , there exists υ > 0 so that to establish the second bullet point it suffices to show that |{0 < t ′′ < t : g t ′′ h s ′ ω ∈ K ′ }| > υ > 0 for all s ′ ∈ [s − e −2t , s + e −2t ]. We choose K ′ to be the closure of a small neighborhood of K. By the Birkhoff genericity of h s ω with respect to the Teichmüller flow, there exists t 1 so that |{0 < t ′′ < t 1 : g t ′′ h s ω ∈ K}| > υ > 0. There exists U, a small neighborhood of s, so that for every s ′ ∈ U and 0 ≤ t ′′ ≤ t 1 , s ′ ∈ [s−e −2t , s+e −2t ] so that KZ(g t+L min{j:s ′ / ∈S j }−1 (s ′ ) , h s ′ ω)φ(s ′ ) Z < γ 0 is at most (since υ = ⌈log ρ (β)⌉ and by formula (30) Clearly this bound goes to 0 with γ 0 , and so if γ 0 is small enough we are left with at least a subset of conditional measure greater than 1 2 . That is, for a set of s ′ ∈ [s − e −2t , s + e −2t ] of conditional measure at least 1 2 we have that for each s ′ in this set: • there exists τ s ′ ≥ t so that KZ(g τ s ′ , h s ′ ω)φ(s ′ ) Z ≥ γ 0 , • g τ s ′ h s ′ ω ∈ K.
By Lemma 2.2 if γ 0 is sufficiently small (so that in Proposition 5.1 ℓ(s) ≥ | log γ 0 |/2 > t 0 , hence τ s ′ −t ≥ t 0 ) the subset of s ′ ∈ [s −e −2t , s + e −2t ] such that g τ s ′ +a h s ′ ω or g τ s ′ +2a h s ′ ω are not in K has conditional measure at most 1/9. By Lemma 2.1 we have that By the above lower bound, together with the condition that g τ s ′ h s ′ ω ∈ K, g τ s ′ +a h s ′ ω ∈ K and g τ s ′ +2a h s ′ ω ∈ K, we conclude that s ′ ∈ B ′ . Since the conditional measure of such s ′ ∈ [s − e −2t , s + e −2t ] is at least 1/2 − 1/9 > 1/4, this contradicts that s ∈ B ′′ (because e −2t < r 0 ). 7. Proof of Theorem 1.1 7.1. Setup. Let P be a polygon and T 1 (P ) be the phase space for its billiard flow. Let X(P ) = T 1 (P ) × T 1 (P ). The phase space T 1 (P ) is endowed with the Liouville measure for the billiard flow, hence the space X(P ) can be endowed with the product measure. All integrals below will be taken with respect to the square of the Liouville measure on X(P ).
We also need to define Lip c (X(P )) the space of c-Lipschitz functions on X(P ). For instance, we view T 1 (P ) as a quotient of P × S 1 (which is a metric space) and use the path metric to define a metric on T 1 (P ).

By Theorem 1.2 we have
Corollary 7.2. For every ǫ > 0 there exists N ǫ such that the following holds. If P is a rational polygon with the property that the group of reflections about its sides (translated to the origin) contains a rotation by 2π M with M ≥ N ǫ , then there exists T 0 such that for all T ≥ T 0 and f ∈ Lip 1 (X(P )) we have Proof. Consider the flat surface M P obtained by unfolding P . By Theorem 1.2 we have that for almost every pair of directions the product flow is ergodic on M P × M P . Considering these flows on T 1 (P ), by our assumption on the group of reflections, they equidistribute in the product of the table cross M ≥ N ǫ evenly spaced copies of a discrete set in S 1 . The corollary follows.
Proof of Theorem 1.1 assuming Lemma 7.1. Let n be given and, for all k ∈ N, let U k be the set of polygons P with n ≥ 3 sides so that there exists T > 0 such that, for all f ∈ Lip 1 (X(P )), By Lemma 7.1, U k is open for all k ∈ N as union (over T > 0) of a family of open sets. By Corollary 7.2 it is dense for all k ∈ N. So by the Baire Category theorem there exists P ∈ ∩ k U k . Because 1-Lipschitz functions have dense span in the set of continuous functions, the product of the billiard flow with itself on X(P ) is ergodic. In fact, if P ∈ ∩ k U k we have that, for every k ∈ N, there exists T k > 0 such that for all f ∈ Lip 1 (X(P )), Ergodicity then follows (for instance) from the ergodic theorem. The ergodicity of the product billiard flow implies that the billiard flow is weakly mixing.
Definition 7.3. (Y. Vorobets [Vo97], Definition 2.1) We say a polygon Q is a δ-perturbation of P if there exists a homeomorphism φ P,Q : P → Q that establishes a one-to one correspondence between the vertices of the two polygons and so that the distance between the corresponding vertices is at most δ.
For every polygon P there exists δ P > 0 such that for δ < δ P , any δ-perturbation Q of P has a triangulation whose triangles are in a bijective correspondence with triangles of a triangulation of P . For instance, one can triangulate P by diagonals in an arbitrary manner. Then for δ < d(P )/2, with d(P ) the minimum of the smallest nonzero distance between vertices, for each two vertices of P which can be joined by a diagonal inside P , the corresponding vertices of Q can also be joined by a diagonal inside P . Then there exists a triangulation of Q corresponding to the above triangulation of P by diagonals.
In the next lemma we assume the homeomorphism φ P,Q is as above, affine on the triangles in a (fixed) triangulation of P and takes them to the corresponding triangles of Q.
Proof. Indeed Lip 1 (X(P )) is compact in · sup and so there exists ǫ ′′ so that for all f ∈ Lip 1 (X(P )).
Proposition 7.6. (Vorobets [Vo97, Proposition 2.3]) Let Q be a δperturbation of P . For each t > 0 there exists B ⊂ P × S 1 dependent on the polygons P, Q, the map φ := φ P,Q and t and of measure at most C 3 (C 1 t + C 2 ) 3 δ such that for each (x, θ) ∈ P × S 1 \ B and for each τ , 0 ≤ τ ≤ t, at least one of the following two possibilities hold: (1) The distance between π (1) P F t p (x, v) and φ −1 (π (1) is at most C 4 (C 1 t + C 2 ) 2 δ and the angle between the directions of π (2) F t P (x, v) and π (2) F t Q (φ(x), v) is at most C 5 (C 1 t + C 2 )δ.
(2) The points π (1) P F t p (x, v) and π (1) Q F t Q (φ(x), v) lie at a distance at most C 6 (C 1 t + C 2 ) 2 δ from the boundaries of P and Q respectively.