Every flat surface is Birkhoff and Oseledets generic in almost every direction

We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of large numbers, and on recent rigidity results for the action of the upper triangular subgroup of SL(2,R) on the moduli space of flat surfaces. Most of the results also use a theorem about continuity of splittings of the Kontsevich-Zorich cocycle recently proved by S. Filip.


Introduction
Flat surfaces and strata. Suppose g ≥ 1, and let α = (α 1 , . . . , α n ) be a partition of 2g − 2, and let H(α) be a stratum of Abelian differentials, i.e. the space of pairs (M, ω) where M is a Riemann surface and ω is a holomorphic 1-form on M whose zeroes have multiplicities α 1 . . . α n . The form ω defines a canonical flat metric on M with conical singularities at the zeros of ω. Thus we refer to points of H(α) as flat surfaces or translation surfaces. For an introduction to this subject, see the survey [Zo2].
Affine measures and manifolds. Let H 1 (α) ⊂ H(α) denote the subset of surfaces of (flat) area 1. An affine invariant manifold is a closed subset of H 1 (α) which is invariant under the SL(2, R) action and which in period coordinates (see [Zo2, Chapter 3]) looks like an affine subspace. Each affine invariant manifold M is the support of an ergodic SL(2, R) invariant probability measure ν M . Locally, in period coordinates, this measure is (up to normalization) the restriction of Lebesgue measure to the subspace M, see [EM] for the precise definitions. It is proved in [EMM] that the closure of any SL(2, R) orbit is an affine invariant manifold.
The most importatant case of an affine invariant manifold is a connected component a stratum H 1 (α). In this case, the associated affine measure is called the Masur-Veech or Lebesgue measure [Mas1], [Ve1].
The element r θ ∈ SL(2, R) acts by (M, ω) → (M, e iθ ω). This has the effect of rotating the flat surface by the angle θ. The action of g t is called the Teichmüller geodesic flow. The orbits of SL(2, R) are called Teichmüller disks.
A variant of the Birkhoff ergodic theorem. We use the notation C c (X) to denote the space of continuous compactly supported functions on a space X. One of our main results is the following: Theorem 1.1. Suppose x ∈ H 1 (α). Let M = SL(2, R)x be the smallest affine invariant manifold containing x. Then, for any φ ∈ C c (H 1 (α)), for almost all θ ∈ [0, 2π), we have (1.1) lim II. For almost all θ, the limit lim t→∞ (A * (g t , r θ x)A(g t , r θ x)) 1 2t ≡ Λ(x, θ) exists. Moreover, the eigenvalues of the matrix Λ(x, θ), taken with their multiplicities, coincide with the numbers e λ i . Furthermore, III. Let α 1 < · · · < α s denote the distinct Lyapunov exponents λ i . Let U i (x, θ) ⊂ H 1 (M, R) denote the corresponding eigenspaces of Λ(x, θ). We set V 0 (x, θ) = {0} and V i (x, θ) = U 1 (x, θ) ⊕ · · · ⊕ U i (x, θ). Then, for almost all θ, and for Remark. The fact that the conclusions of Theorem 1.2 hold for almost all x with respect to the affine measure ν M (or in particular with respect to the Masur-Veech measure) is just the classical Osceledets multiplicative ergodic theorem. The main point of Theorem 1.2 is that the conclusion holds for all x ∈ H 1 (α). This has some applications which partly motivated this paper, in particular in connection to the wind-tree model [DHL], [FU] and earlier results on IETs [Zo], [MMY]. By the arguments in [DHL] Theorem 1.2 strengthens [DHL, Theorem 1 part 2] to apply to all obstacles. It is likely that the arguments [FU] extend [FU,Theorem 1.2] to apply for all obstacles with the input of Theorem 1.1 and 1.2. Theorems 1.1 and 1.2 extend [MMY] to apply to a full measure subset of the one parameter family of IETs coming first return to a transversal on any flat surface. Theorem 1.1 implies condition (a). In particular, an open set in the space of flat surfaces (in a fixed affine invariant submainfold) can be related to the IETs coming from it having γ (n) taking all possible names in fixed bounded time. Condition (b) holds by [Fo]. Condition (c) holds by Theorem 1.2. Theorem 1.2 extends [Zo] to apply to a full measure subset of the one parameter family of IETs coming first return to a transversal on any flat surface. The question of whether Theorem 1.2 is true was raised in [Fo2].
It is well known that parts II and III of Theorem 1.2 follow from part I by an argument which does not involve any ergodic theory (see [GM], from which our statement of the multiplicative ergodic theorem was taken). It is thus enough to show that part I holds for all x and almost all θ. Our proof of I is based on the same ideas as the proof of Theorem 1.1, namely the results of [EM], [EMM] and the strong law of large numbers. However, we also need another important input: the theorem of Filip stated as Theorem 1.6 below. This complication can be traced back to the fact that the Kingman and Osceledets ergodic theorems can fail to hold at some points even for uniquely ergodic systems (see [Fu] and references therein). Definition 1.3 (ν-measurable almost invariant splitting). Let X be a space on which G = SL(2, R) acts, preserving a measure ν. Suppose V is a real vector space, and suppose A : G × X → SL(V ) is a cocycle. We say that A has an almost invariant splitting if there exists n > 1 and for a.e x there exist nontrivial subspaces for 1 ≤ i < j ≤ n and also for a.e g ∈ G and ν-a.e. x ∈ X, for some 1 ≤ j ≤ n .
The map x → {W 1 (x), . . . , W n (x)} is required to be ν-measurable. In this paper, we prove the following: Theorem 1.5. Fix x ∈ H 1 (α), and let M = SL(2, R)x be the smallest affine invariant manifold containing x. Let V be SL(2, R) invariant subbundle of (some exterior power of ) the Hodge bundle which is defined and is continuous on M. Let A V : SL(2, R) × M → V denote the restriction of (some exterior power of ) the Kontsevich-Zorich cocycle to V , and suppose that A V is weakly irreducible with respect to the affine measure ν M whose support is M. Then, for almost all θ ∈ [0, 2π), exists and coincides with the top Lyapunov exponent of A V .
The main additional input needed for the proof of Theorem 1.2 is the following: Theorem 1.6 ( [Fi]). Let A(·, ·) denote (some exterior power of ) the Kontsevich-Zorich cocycle restricted to an affine invariant submanifold M. Let ν M be the affine measure whose support is M, and suppose A has a ν M -measurable almost-invariant non-degenerate splitting. Then, the subspaces W i (x) in Definition 1.3 can be taken to depend continuously on x ∈ M.
Proof of Theorem 1.2 from Theorem 1.5 and Theorem 1.6. Let A(·, ·) denote the Kontsevich-Zorich cocycle restricted to an affine invariant submanifold M. Then by [EM,Theorem A.6] A(·, ·) is semisimple, in the sense that (after passing to some finite cover) for ν M -almost all x ∈ M there is a ν M -measurable direct sum decomposition where all the subbundles V i are ν M -measurable, SL(2, R)-invariant and strongly irreducible. This remains true when passing to any exterior power, see [Fi]. We now combine all the V i with the same top Lyapunov exponent to obtain a direct sum decomposition where each subbundle W i is weakly irreducible (see Definition 1.4). By Theorem 1.6, the W i (x) can be taken to depend continuously on x. Then, by Theorem 1.5 it follows that the top Lyapunov exponent on each W i is defined for almost all θ. (To connect the conclusion of Theorem 1.5 with (2.4), note that the top eigenvalue of To get that the rest of the Lyapunov exponents are defined for almost all θ it suffices to repeat the argument for the cocycle acting on the exterior powers of the Hodge bundle. (Note that the norm of A V (g t , r θ x) acting on d (V ) is the product of the top d eigenvalues of [A V (g t , r θ x) * A V (g t r θ x)] 1/2 acting on V ). This proves statement I of Theorem 1.2, and then statements II and III of Theorem 1.2 follow as in [GM].
Remark 1.7. For the case of a two-dimensional continuous subbundle V of the Hodge bundle, Theorem 1.2 follows from Theorem 1.5 (without the need for Theorem 1.6). Indeed, by [AEM,Theorem 1.4] any SL(2, R)-invariant measurable subbundle of the Hodge bundle is symplectic, and thus even dimensional. Thus, the restriction of the cocycle to a two-dimensional subbunde is automatically strongly irreducible. (This is the case which arises in [DHL], [FU]).

Random walks
To provide intuition, we first prove versions of Theorem 1.1 and Theorem 1.5 for random walks. We use the following setup. Let µ be an SO(2)-bi-invariant compactly supported measure on SL(2, R) which is absolutely continuous with respect to Haar measure. We consider the random walk on SL(2, R) whose transition probabilities are given by µ. This also defines a random walk on H 1 (α), via the SL(2, R) action. (The trajectories of this random walk stay in Teichmüller disks). Letḡ = (g 1 , . . . , g 2 , . . . , ) denote an element of SL(2, R) N . Let µ N denote the product measure on SL(2, R) N . It follows from the Osceledets multiplicative ergodic theorem that for µ N -almost-allḡ, the trajectory g 1 , g 2 g 1 , . . . , g n−1 . . . g 1 , g n g n−1 . . . g 1 tracks, up to sublinear error, a geodesic of the form {g t r θ : t ∈ R} with respect the the right-invariant metric on SL(2, R). (This will be made more precise in §4). The angle θ depends onḡ, but as we show in §4, the distribution of θ's induced by µ N is uniform. Thus, we expect to have analogues of Theorem 1.1 and Theorem 1.2 (and Theorem 1.5) in the random walk setup, where the clause "for almost all θ" is replaced by the clause "for almost allḡ". This is indeed the case, and we find the proofs of the random walk versions, namely Theorem 2.1 and Theorem 2.6 a bit cleaner and easier to follow. Also we will see below that Theorem 1.5 follows formally from its random walk version Theorem 2.6.
2.1. A Birkhoff type theorem for the random walk.
where ν M is the affine measure whose support is M.
where ν M is the affine measure whose support is M.
Our proof of Theorem 2.1 follows [BQ]. Let x, M and ν M be as in Theorem 2.1. We begin with the following: Lemma 2.3. For almost everyḡ ∈ SL(2, R) N , ifν is a weak-* limit point of Proof. It suffices to check a countable subset of C c (M), so it suffices to have the result for each fixed function in C c (M). We follow [BQ,Lemma 3 By definition f 1 ∈A 1 ,...,f n−1 ∈An−1 f n (x,ḡ)dµ N = 0 for any n ∈ N and any subsets A 1 , ..., A n−1 of R. Additionally, f n ∞ ≤ 2 φ ∞ . So by the strong law of large numbers Thusν is µ-stationary almost everywhere.
We also use the following (which is the main technical result of [EMM]): Proposition 2.4 (see [EMM,Proposition 2.13], [EMM,Lemma 3.2]). Let N ⊂ H 1 (α) be an affine submanifold. (In this proposition N = ∅ is allowed). Then there exists an SO (2) There exists b > 0 (depending on N ) and for every 0 < c < 1 there exists n 0 > 0 (depending on N and c) such that for all x ∈ H 1 (α) and all n > n 0 , Here µ (n) denotes the convolution µ * · · · * µ (n times). (c) There exists σ > 1 such that for all g ∈ SL(2, R) with g ≤ 1 and all x Lemma 2.5 ( [BQ,Proposition 3.9]). Suppose f N is a function satisfying the conditions of Proposition 2.4. Then, for any 0 < c < 1 any M > 0 and µ N -almost-all g ∈ SL(2, R) N , we have, for all sufficiently large n, where C depends only on the constants n 0 , b and σ of Proposition 2.4.
By [EM,Theorem 1.4], any ergodic SL(2, R)-invariant measure is affine. Therefore, sinceν is supported on M,ν has can be decomposed into ergodic components as where a N ∈ [0, 1] and the sum is over the affine invariant submanifolds N contained in M. (Here N = M is allowed). By [EMM,Proposition 2.16] this is a countable sum. By applying (2.2) for the case N = ∅ we get that for µ N -almost allḡ,ν is a probability measure. Then, by applying (2.2) again with N any affine invariant submanifold properly contained in M, we see that for µ N -almost-allḡ,ν(N ) = 0. Thus a N = 0 for N properly contained in M. Sinceν is a probability measure, this forcesν = ν M , completing the proof of Theorem 2.1.

An Osceledets type theorem for the random walk.
Theorem 2.6. Fix x ∈ H 1 (α), and let M = SL(2, R)x be the smallest affine invariant manifold containing x. Let V be SL(2, R) invariant subbundle of (some exterior power of ) the Hodge bundle which is defined and is continuous on M. Let A V : SL(2, R) × M → V denote the restriction of (some exterior power of ) the Kontsevich-Zorich cocycle to V , and suppose that A V is weakly irreducible (see Definition 1.4) with respect to the affine measure ν M whose support is M. Then, for µ N -almost-allḡ = (g 1 , . . . , g n , . . . ), where λ 1 is the top Lyapunov exponent of A V restricted to M (and depends only on µ, V and M).
Let m = dim(V ). We recall the statement of the Oseledets multiplicative ergodic theorem from e.g. [GM] in this setting: Theorem 2.7. For ν M -almost all y ∈ M and µ N -almost-allḡ ∈ SL(2, R) N , the following hold: I. Let ψ 1 (n,ḡ, y) ≤ · · · ≤ ψ m (n,ḡ, y) denote the eigenvalues of the matrix Here the numbers λ 1 ≥ · · · ≥ λ m depend only on ν M and V . They are the Lyapunov exponents of the cocycle A V on M. II. The limit lim n→∞ (A * V (g n . . . g 1 , y)A V (g n . . . g 1 , y)) 1 2n ≡ Λ(y,ḡ) exists. Moreover, the eigenvalues of the matrix Λ(y,ḡ), taken with their multiplicities, coincide with the numbers e λ i . Furthermore, denote the corresponding eigenspaces of Λ(y,ḡ). We set V 0 (y,ḡ) = {0} and V i (y,ḡ) = U 1 (y,ḡ) ⊕ · · · ⊕ U i (y,ḡ). Then, for almost all y,ḡ, and for any v ∈ V i (y,ḡ) \ V i−1 (y,ḡ), we have Remark 2.8. As was done §1, one can use the theorem of Filip Theorem 1.6 and Theorem 2.6 to show that the conclusions of Theorem 2.7 hold for all y (and almost all g) provided M is the smallest affine invariant manifold containing y (or equivalently M = SL(2, R)y).
The set E good (ǫ, L). Suppose ǫ > 0, L ∈ N. Let E good (ǫ, L) denote the set of y ∈ M such that for each v ∈ V there exists a subset H(v) ⊂ SL(2, R) L so that and for all (h 1 , . . . , h L ) ∈ H(v), The following Lemma is a key step in our proof. We may write where η x is a measure on Gr s (V ). For Lemma 2.10 ([EM, Lemma C.9(i)]). Suppose the cocycle A V is strongly irreducible with respect to ν M . Then for almost all y ∈ M, for any v y ∈ V , η y (I(v y )) = 0.
Proof. The proof is given in [EM, Appendix C]. The essential idea is that if conclusion of Lemma 2.10 is false, then the cocycle would have to permute some finite collection of subspaces, contradicting the strong irreducibility assumption.
Proof of Corollary 2.11. Since V is weakly irreducible, we may write where each W i (x) is strongly irreducible, and all the W i have the same top Lyapunov exponent. Then, s−1 is the analogue of V s−1 for W i in place of V . Now, (2.8) follows from the corresponding statement for each W i . Thus, without loss of generality, we may assume that V is strongly irreducible. For and letν denote the measure on the bundle M × Gr s−1 (V ) given by Then,ν is a stationary measure for the random walk. Let Z = {y ∈ M :ν y (I(w)) > 0 for some w ∈ P 1 (V )}, Suppose ν M (Z) > 0. Let r ≥ 1 be maximal so that for a positive measure subset Z r ⊂ M and all y ∈ Z r , there exists an r-dimensional subspace U y withν y (I(U y )) > 0.
Here I(U y ) = {W ∈ Gr s−1 (V ) : W ⊃ U y }. Since r is maximal, for almost all y, the distinct U y for whichν y (I(U y )) > 0 are disjoint. Thus, the set of U y for whichν y (I(U y )) is maximal is finite, and by ergodicity, the cardinality of this set is constant almost everywhere. Then, for each x ∈ Z r we can measurably choose U x ∈ P 1 (V ) such thatν x (I(U x )) > 0. Then, We now measurably choose w x ∈ U x . Then, (2.9)ν x∈Zx {x} × I(w x ) > 0. Therefore, (2.9) holds for some ergodic component ofν. However, this contradicts Lemma 2.10, since the action of the cocycle on V is strongly irreducible. Thus, ν(Z) = 0 and ν(Z c ) = 1. By definition, for all y ∈ Z c and all w y ∈ V , (2.8) holds.
Let U i (n, y,ḡ) denote the direct sum of the eigenspaces of A * V (g n . . . g 1 , y)A V (g n . . . g 1 , y) which correspond to those eigenvalues which will converge as n → ∞ to 2α i . Let V i (n, y,ḡ) = U 1 (n, y,ḡ)⊕· · ·⊕U i (n, y,ḡ). Then, it follows from part II of Theorem 2.7 that for almost all y and almost allḡ, The set F good (ǫ, σ, L). Suppose ǫ > 0, σ > 0, and L ∈ N. Let F good (ǫ, σ, L) denote the set of y ∈ M such that for any v y ∈ V Since the cocycle A V is continuous and both (2.12) and (2.13) depend onḡ only via g 1 , . . . , g L , the set F good (ǫ, σ, L) is open.
We also use the following trivial result: Lemma 2.14. For any σ > 0 there is a constant c(σ) > 0 with the following property: Let A ∈ GL(V ) be a linear map, and let V ⊂ V denote the subspace spanned by the eigenspaces of all but the top eigenvalue of A * A. Then, for any v with v = 1 and Proof of Lemma 2.9. Suppose ǫ > 0 and δ > 0 are given, and let σ > 0 and L 0 > 0 be as in Lemma 2.13. Choose L > L 0 such that (λ 1 − ǫ/2) L c(σ) > (λ − ǫ) L , where c(σ) is as in Lemma 2.14. Pick v y ∈ V . Then, in view of Lemma 2.14, for all g satisfying (2.12) and (2.13),

2.2.2.
Proof of Theorem 2.6. In view of Lemma 2.9, we choose L so that ν M (E good (ǫ, L)) > 1 − ǫ. Pick an arbitrary v 0 ∈ V , and let is as in the definition of E good (ǫ, L). By Corollary 2.2 and (2.6), for almost allḡ, the lower density ofJ(ḡ) is at least 1 − 3ǫ. For almost every suchḡ we can find a subset I(ḡ) ⊂ J(ḡ) so that where the intervals [i, i + L] are disjoint for i ∈ I(ḡ), g i+1 , ..., g i+L ∈ H(v i ) (by the strong law of large numbers) and the upper density of K is at most 4ǫ. Now suppose n ≫ L. Then, Let C be such that for all g in the support of µ and all y ∈ M, A(g, y) ≤ C. Then, |S 3 | ≤ L log C. Also, since the upper density of K is at most 3ǫ, |S 2 | ≤ 3ǫn log C. However, by (2.7), Thus, for almost allḡ and any n ≫ L, Since ǫ > 0 is arbitrary, we get that for almost allḡ, = S 1 +S 2 +S 3 .
Since ǫ > 0 is arbitrary this completes the proof of Theorem 1.5.
3. Proof of Theorem 1.1 3.1. An analogue of Lemma 2.3. Let η T,θ denote the measure on SL(2, R) given by In this subsection we prove the following: Proposition 3.1. Fix x ∈ M. For almost every θ ∈ [0, 2π], if ν θ is any weakstar limit point (as T → ∞) of η T,θ * δ x , then then ν θ is invariant under P , where The proof of Proposition 3.1 is based on the strong law of large numbers. In fact, Proposition 3.1 holds for arbitrary measure-preserving SL(2, R) actions.
It is clear from the definition, that for any θ, any weak-* limit point ν θ is invariant under g t . Let Hence it is enough to show that ν θ is invariant underū α for every α. Fix 0 < α < 1.
Let J 1 be the contribution of the first term in parethesis in (3.7) to (3.5) and let J 2 be the contribution of the second term. We have, using (3.1) and (3.2), tan αt a αt − I dt = 0, by (3.6) and α t = O(e −t ). Also J 2 = 0 by Proposition 3.2. Thus (3.5) holds. This shows that for any fixed 0 < α < 1 for almost all θ, the measures ν θ of Proposition 3.1 are invariant under u α (as well as g t for all t). We now repeat the proof with two different α's linearly independent over Q. We get that for almost all θ, any limit point of η T,θ * δ x is invariant under a dense subgroup of P , hence invariant under all of P . This completes the proof of Proposition 3.1.
3.2. Proof of Lemma 3.4. We recall the following basic facts: First, because f t (θ) is an 2M-Lipshitz function of t for each θ it suffices to show that for any ǫ and almost every θ we have: We will show that (3.12) follows from (3.9), the Borel-Cantelli lemma, and Chebyshev's inequality. To see this, observe that ( n i=1 f ǫi (θ)) 2 = 3.3. Completion of the proof of Theorem 1.1.
Proposition 3.7 ([EMM, Proposition 2.13]). Let N be any affine submanifold. Then there exists an SO(2) invariant function f N : Also {x : f N (x) ≤ N} is compact for any N.
(2) There exists b > 0 (depending on N ) and for every 0 < c < 1 there exists t 0 > 0 (depending on N and c) such that for all x ∈ H 1 (α) and all t > t 0 ,  (2) and (3) of Proposition 3.7, we have that for any 0 < β < 1 there exist M < ∞ and γ < 1 such that for every x we have λ ({θ : f (g t r θ x) > M for at least β-fraction of t ∈ [0, T ]}) < γ T for all large enough T .
Proof of Theorem 1.1. Let ν θ be any weak-star limit point of the measures η T,θ * δ x . By Proposition 3.1, for almost all θ, ν θ is P -invariant.
By [EM,Theorem 1.4], any ergodic P -invariant measure is SL(2, R)-invariant and affine. Therefore, since ν θ is supported on M, it has can be decomposed into ergodic components as where a N (θ) ∈ [0, 1] and the sum is over the affine invariant submanifolds N contained in M. (Here N = M is allowed). By [EMM,Proposition 2.16] this is a countable sum. By applying Theorem 3.8 for the case N = ∅ we get that for almost all θ, ν θ is a probability measure. Then, by applying Theorem 3.8 again with N any affine invariant submanifold properly contained in M, we see that for almost all θ, ν θ (N ) = 0. Thus, for almost all θ, a N (θ) = 0 for any N properly contained in M.
Since ν θ is a probability measure, this forces ν θ = ν M for almost all θ, completing the proof of Theorem 1.1.

Proof of Theorem 1.5
Let µ be as in §2. The following lemma expresses the well known fact that a typical random walk trajectory tracks a geodesic (up to sublinear error).
By [Fo], there exists C > 0 and N < ∞ so that for all g ∈ SL(2, R) and all x ∈ H 1 (α), we have (4.4) A V (g, x) ≤ C g N .