Effective counting of simple closed geodesics on hyperbolic surfaces

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichm\"{u}ller geodesic flow.


Introduction
Let g ≥ 2, and let S be a compact Riemann surface of genus g. Let T (S) be the Teichmüller space of complete hyperbolic metrics on S, and let M(S) = T (S)/ Mod g be the corresponding moduli space, where Mod g is the mapping class group of S.
Let M ∈ M(S). Problems related to the asymptotic growth rate of the number of closed geodesics on M have been long studied. In particular, thanks to works of Delsart, Huber, and Selberg we have the following: There exists some δ = δ(M ) > 0 so that the number of closed geodesics of length at most L on M equals (1) Li(e L ) + O M (e L−δ ), where Li(x) = x 2 dt log t ; see [Bus] and references there. More generally, the growth rate of the number of closed geodesics on a negatively curved compact manifold was studied by Margulis, [Mar]. His proof, which is different from the above mentioned works, is based on the mixing property of the Margulis measure for the geodesic flow. In the constant negative curvature case, Margulis' method combined with an exponential mixing rate for the geodesic flow, also provides an estimate like (1) -albeit with a weaker power saving δ, see e.g. [MMO].
1.1. Simple closed geodesics. The aforementioned fundamental results do not provide any estimates for the number of simple closed geodesics on M . Indeed, very few closed geodesics on M are simple, [BS2], and it is hard to discern them in π 1 (M ), [BS1]. More explicitly, it was shown in [Ri] that the number of simple closed geodesics of length at most L on M is bounded above and below by O M (L 6g−6 ).
A.E. acknowledges support by the NSF and the Simons Foundation. A.M. acknowledges support by the NSF and Alfred P. Sloan Research Fellowship. In her PhD thesis, [Mir1] and [Mir2], Mirzakhani proved an asymptotic growth rate for the number of simple closed geodesics of a given topological type on a hyperbolic surface Mrecall that two simple closed geodesics γ and γ ′ on M are of the same topological type if there exists some g ∈ Mod g so that γ ′ = gγ.
Let X be a compact surface equipped with a Riemannian metric of negative curvature. We emphasize that the curvature is not assumed to be constant; indeed, elements in M(S) will be denoted by M to minimize the confusion. By a multi-geodesic γ on X we mean γ = d i=1 a i γ i where γ i 's are disjoint, essential, simple closed geodesics, and a i > 0 for all 1 ≤ i ≤ d. In this case, we define ℓ X (γ) := a i ℓ X (γ), where ℓ X denotes the length function on X. The multi-geodesic γ will be called integral (resp. rational) if a i ∈ Z (resp. a i ∈ Q).
Mirzakhani, [Mir2,Thm. 1.1], proved the following estimate when M is a hyperbolic surface: where n γ 0 : M(S) → R + (the Mirzakhani function) is a continuous proper function; geometric informations carried by n γ 0 are also studied in [Mir2].
In this paper we obtain a quatitative version of (2); moreover, our approach allows us to prove such a result in the more general setting of variable negative curvature.
Theorem 1.1. There exists some κ = κ(g) > 0 so that the following holds. Let X be a compact surface of genus g equipped with a Riemannian metric of negative curvature. Let γ 0 be a rational multi-geodesic on X. Then s X (γ 0 , L) = n γ 0 (X)L 6g−6 + O γ 0 ,X (L 6g−6−κ ) where n γ 0 (X) is a positive constant which depends on γ 0 and X.
The proof of Theorem 1.1 is based on the study of a related counting problem in the space of geodesic measured laminations on S,à la Mirzakhani. The space of measured laminations on S, which we denote by ML(S), is a piecewise linear integral manifold homeomorphic to R 6g−6 ; but it does not have a natural differentiable structure, [Th1]. Train tracks were introduced by Thurston as a powerful technical device for understanding measured laminations. Roughly speaking, train tracks are induced by squeezing almost parallel strands of a very long simple closed geodesic to simple arcs on a surface; they provide linear charts for ML(S).
The mapping class group Mod g of S acts naturally on ML(S). Moreover, there is a natural Mod g -invariant locally finite measure on ML(S), the Thurston measure µ Th , given by the piecewise linear integral structure on ML(S), [Th1]. For any open subset U ⊂ ML(S) and any t > 0, we have µ Th (tU ) = t 6g−6 µ Th (U ).
On the other hand, any metric of negative curvature X on S induces the length function λ → ℓ X (λ) on ML(S), which satisfies ℓ X (tλ) = tℓ X (λ) for all t > 0. It is proved in [Mir1,App. A] that ℓ M is a convex function on ML(S) when M is a hyperbolic surface. This fact remains valid in the more general setting of variable negative curvature, see §5.5.
The source of the polynomially effective error term in Theorem 1.1 is the exponential mixing property of the Teichmüller geodesic flow proved by Avila, Gouëzel, and Yoccoz, [AGY, AR, AG]. We combine this estimate with ideas developed by Margulis in his PhD thesis, [Mar], to prove the following theorem which is of independent interest -see Theorem 7.1 for a more general statement.
Let τ be a train track and let U (τ ) be the corresponding train track chart. For every λ ∈ U (τ ) we let λ τ denote the sum of the weights of λ in U (τ ), see §5.
It is worth noting that in view of Theorem 1.2, the asymptotic behavior of the number of points in one Mod g -orbit in the cone {λ : λ τ ≤ L} and that of the number of integral points in this cone agree up to multiplicative constant.
Theorem 1.2, in the more general form Theorem 7.1, plays a crucial role in our analysis. Indeed, using the aforementioned convexity of the length function, we will prove Theorem 1.1 using Theorem 7.1 in §8.
It is an intriguing problem to investigate the asymptotic behavior of functions similar to and different from s X (γ 0 , L) or the complexity considered in Theorem 1.2. For instance, for a suitable formulation of a combinatorial length -using intersection numbers -the count is exactly a polynomial, see [FLP]. We also refer the reader to [CMP] where a related problem is studied for thrice punctured sphere.
1.2. Outline of the paper. In §2 we collect some preliminary results. In §3 we prove an equidistribution result with an error term, Proposition 3.2, which may be of independent interest; see, e.g. [KM, LMir]. The proof of this proposition is based on the exponential mixing rate for the Teichmüller geodesic flow, [AGY], and the so called thickening technique, see [Mar, EMc]. In §4 we prove Proposition 4.1; this proposition is one of the main ingredients in the proof, and could be compared to arguments in [Mar,Chap. 6]. We will recall some basic facts about ML(S), and study the relation between the linear structures on ML(S) and the space of quadratic differentials in §5 and §6. The orbital counting in sectors of ML(S) is studied in §7; the main result here is Theorem 7.1. We prove Theorem 1.1 in §8.
1.3. Acknowledgement. This project originated in fall of 2015 when the authors were members of the Institute for Advanced Study (IAS), we thank the IAS for its hospitality. We thank C. McMullen, K. Rafi, and A. Zorich for helpful discussions. We also thank F. Arana-Herrera, H. Oh, and A. Wright for their comments on an earlier version of this paper. We are in debt to G. Margulis and F. Arana-Herrera for drawing our attention to the case of variable negative curvature, and to K. Rafi for providing the proof of Theorem 5.1.
Last, but not least, we thank the anonymous referee for their careful reading and several helpful comments.

Preliminaries and notation
Let Q(S) denote the moduli space of quadratic differentials on S, and let Q 1 (S) be the moduli space of quadratic differentials with area one on S. For any α = (α 1 , . . . , α k , ς) with α i = 4g − 4 and ς ∈ {±1}, define Q 1 (α) to be (a connected component) of the stratum of quadratic differentials consisting of pairs (M, q) where M ∈ M(S) and q is a unit area quadratic differential on M whose zeros have multiplicities α 1 , . . . , α k and ς = 1 if q is the quare of an abelian differential and −1 otherwise. Then Put Q(α) := {tq : t ∈ R, q ∈ Q 1 (α)}. Let Σ ⊂ S be a set of k distinct marked points. Let Q 1 T (α) denote the space of quadratic differentials (M, q) equipped with an equivalence class of homeomorphisms f : S → M that send the marked points to the zeros of q. The equivalence relation is isotopy rel marked points. Let π : Q 1 T (α) → Q 1 (α) be the forgetful map which forgets the marking f ; this is an infinite degree branched covering.
Similarly, let Ω(S) denote the moduli space of Abelian differentials on S, and let Ω 1 (S) be the moduli space of area one Abelian differentials. For any α = (α 1 , . . . , α k ), we let H(α) denote the corresponding stratum, and let H 1 (α) denote the area one abelian differentials.
Note that passing to a branched double coverM of M , we may realize Q 1 (α) as an affine invariant submanifold in H 1 (α) corresponding to odd cohomology classes onM , see §2.1. However, even if q belongs to a compact subset of Q 1 (S), the complex structure onM may have very short closed curves in the hyperbolic metric, e.g. a short saddle connection between two distinct zeros on (M, q) could lift to a short loop inM . Note however that if (M , ω) is the aforementioned double cover of (M, q), then the length of the shortest saddle connection in ω is bounded by the length of the shortest saddle connection in q, i.e. compact subsets of Q 1 (α) lift to compact subsets of H 1 (α).
2.1. Period coordinates. Let x = (M, ω) ∈ H(α), and let Σ ⊂ M be the set of zeros of ω. Passing to a finite cover, which we continue to denote by H(α), we assume there are no orbifold points in H(α). Define the period map Φ : H(α) → H 1 (M, Σ, C).
Let us recall that Φ can be defined as follows. Let #Σ = k. Fix a triangulation T of the surface by saddle connections of x, that is: 2g + k − 1 directed edges δ 1 , . . . , δ 2g+k−1 which form a basis for H 1 (M, Σ, Z). Define Note that this map depends on the triangulation T . If T ′ is any other triangulation, and Φ ′ is the corresponding period map, then Φ ′ • Φ −1 is linear. For any x ∈ H(α), there is a neighborhood B(x) of x so that the restriction of Φ to B(x) is a homeomorphism onto Φ(B(x)), see §2.9. We always choose B(x) small enough so that, using the Gauss-Manin connection, the triangulation at y ∈ B(x) can be identified with the triangulation at x.
We define the period coordinates at x = (M, q) ∈ Q(α) as follows. If ς = 1, then q is a square of an abelian differential, and we may define period coordinates as above. If ς = −1, we use the orienting double cover H(α) to define the period coordinates: in this case there is a canonical injection from Q(α) into H(α). Any Riemann surface in the image of this map is equipped with an involution. This way we get the period map from Q(α) to H 1 odd (M, Σ, C) -the anti-invariant subspace of the cohomology for the involution.
2.2. SL(2, R)-action on H 1 (α). Let x ∈ H 1 (α), we write Φ(x) as a 2×n matrix. The action of g = a b c d ∈ SL(2, R) in these coordinates is linear. We choose a fundamental domain for the action of the mapping class group and think of the dynamics on the fundamental domain. Then, the SL(2, R)-action becomes where A(g, x) ∈ Sp(2g, Z) ⋉ Z k−1 is the Kontsevich-Zorich cocycle. That is: A(g, x) is the change of basis one needs to perform to return the point gx to the fundamental domain. It can be interpreted as the monodromy of the Gauss-Manin connection restricted to the orbit of SL(2, R).
We have the following.
More generally, for any affine invariant manifold, M ⊂ H 1 (α), we let µ denote the SL(2, R)invariant affine measure on M. In particular, all the strata in Q 1 (S) are equipped with such invariant measures.
2.3. Mapping class group action. We denote elements in Mod g using bold letters, e.g., g denotes an element in Mod g . The action of Mod g on Q 1 T (α) commutes with the action of SL(2, R), we will however denote both these actions as left action and write, e.g. g ·x or simply denoted by gx.
2.4. The constants. In the sequel we will use κ • and N • , • = 1, 2, . . . to denote various constants. Unless it is explicitly mentioned otherwise, these constants are allowed only to depend on the genus. The constants κ • are meant to indicate small positive numbers while N • are used for constants which are expected to be > 1.
We will also use the notation A ≪ B. This expression means: there exists a constant c > 0 so that A ≤ cB; the implicit constant c is permitted to depend on the genus, but (unless otherwise noted) not on anything else. We write A ≍ B if A ≪ B ≪ A. If a constant (implicit or explicit) depends on another parameter others than the genus, we will make this clear by writing, e.g. ≪ ǫ , C(x), etc.
We also adopt the following ⋆-notation.
We have a natural map r : H 1 (M, R) → H M which sends a cohomology class c ∈ H 1 (M, R) to the holomorphic 1-form r(c) ∈ H M such that the real part of r(c) (which is a harmonic 1-form) represents c. We can thus define the Hodge inner product on H 1 (M, R) by c 1 , c 2 = r(c 1 ), r(c 2 ) . Then where * denotes the Hodge star operator and we choose harmonic representatives of c 1 and * c 2 to evaluate the integral. We denote the associated norm by · M . This is the Hodge norm, see [FK].
If x = (M, ω) ∈ H 1 (α), we will often write · H,x to denote the Hodge norm · M on H 1 (M, R). Since · H,x depends only on M , we have c H,kx = c H,x for all c ∈ H 1 (M, R) and all k ∈ SO(2).
is often referred to as the standard space. We let denote the natural projection; p defines an isomorphism between E(x) and p(E(x)) ⊂ H 1 (M, R).
For our applications in the sequel (and in order to account for the loss of hyperbolicity in the thin part of the moduli space) we need to consider a modification of the Hodge norm.
The classes c α and * c α . Let α be a homology class in H 1 (M, R). We let * c α ∈ H 1 (M, R) be the cohomology class so that where i(·, ·) denotes the algebraic intersection number. Let * denote the Hodge star operator, and let c α = * −1 ( * c α ). Then, for any ω ∈ H 1 (M, R) we have where ·, · is the Hodge inner product. We note that * c α is a purely topological construction which depends only on α, but c α depends also on the complex structure of M .
Fix ǫ * > 0 (the Margulis constant) so that any two geodesics of hyperbolic length less than ǫ * must be disjoint.
Let σ denote the hyperbolic metric in the conformal class of M . For any closed curve α on M , let ℓ M (α) denote the length of the geodesic representative of α in the metric σ.
We recall the following.
Theorem 2.2. [ABEM,Thm. 3.1] For any constant L > 1 there exists a constant c > 1, such that for any simple closed curve α with ℓ M (α) < L, we have Furthermore, if ℓ M (α) < ǫ * and β is the shortest simple closed curve crossing α, then Short bases. Suppose (M, ω) ∈ H 1 (α). Fix ǫ 1 < ǫ * and let α 1 , . . . , α k be the curves with hyperbolic length less than ǫ 1 on M . For every 1 ≤ i ≤ k, let β i be the shortest curve in the flat metric defined by ω with i(α i , β i ) = 1. We can pick simple closed curves γ r , 1 ≤ r ≤ 2g − 2k on M so that the hyperbolic length of each γ r is bounded by a constant L depending only on the genus, and so that the α j , β j and γ j form a symplectic basis S for H 1 (M, R). We will call such a basis short. A short basis is not unique, and in the following we fix some measurable choice of a short basis at each point of H 1 (α).
We recall the definition of a modified Hodge norm from [EMM]; this is similar (but not the same) to the one defined in [ABEM]. The modified norm is defined on the tangent space to the space of pairs (M, ω) where M is a Riemann surface and ω is a holomorphic 1-form on M . Unlike the Hodge norm, the modified Hodge norm will depend not only on the complex structure on M but also on the choice of a holomorphic 1-form ω on M . Let {α i , β i , γ r } 1≤i≤k,1≤r≤2g−2k be a short basis for x = (M, ω).
We can write any θ ∈ H 1 (M, R) as We then define Note that · ′′ depends on the choice of a short basis; however, switching to a different short basis can change · ′′ by at most a fixed multiplicative constant depending only on the genus.
From (6) we have: for 1 ≤ i ≤ k, Remark. From the construction, we see that the modified Hodge norm is greater than the Hodge norm. Also, if the flat length of shortest curve in the flat metric defined by ω is greater than ǫ 1 , then for any cohomology class c, for some N depending on ǫ 1 and the genus, i.e., the modified Hodge norm is within a multiplicative constant of the Hodge norm.
Note however that for a fixed absolute cohomology class c, c ′′ x is not a continuous function of x, as x varies in a Teichmüller disk; this is due to the dependence on the choice of a short basis. To remedy this, we pick a positive, continuous, SO(2)-bi-invariant function φ on SL(2, R) which is supported on a neighborhood of the identity with SL(2, R) It follows from [EMM,Lemma 7.4] that for a fixed c, log c ′ x is uniformly continuous as x varies in a Teichmüller disk. In fact, there is a constant m 0 such that for all x ∈ H 1 (α), all c ∈ H 1 (M, R) and all t > 0, x is uniformly continuous as long as x varies in a Teichmüller disk, it may be only measurable in general (because of the choice of short basis).
where γ z,z ′ is any path connecting the zeroes z and z ′ of ω. Since c − p x (c) represents the zero class in absolute cohomology, the integral does not depend on the choice of γ z,z ′ . Note that the · ′ norm on H 1 (M, Σ, R) is invariant under the action of SO(2).
As above, we pick a positive continuous SO(2)-bi-invariant function φ on SL(2, R) supported on a neighborhood of the identity such that SL(2,R) φ(g) dg = 1, and define Then, the · x norm on H 1 (M, Σ, R) is also invariant under the action of SO(2).
By [EMM,Lemma 7.5] there exists some N 1 so that 2.7. The AGY-norm. Let · AGY,x denote the norm defined in [AGY,§2.2.2]. We recall the definition: let x = (M, ω) ∈ H 1 (α). For any c ∈ H 1 (M, Σ, C), define where the supremum is taken over all saddle connections of ω. This defines a norm and the corresponding Finsler metric is complete, see [AGY].
We note that x and AGY,x are commensurable to each other on compact subsets of H 1 (α).
For every x = (M, q) ∈ Q 1 (α), we define the norms x and AGY,x using the branched double coverM .

This observation and
The lower bound follows similarly.
2.9. Period box. Letx = (M, q) ∈ Q 1 T (α). For every r > 0 define The open subset B r (x) will be called a period box of radius r centered atx. Thanks to [AG,Prop. 5.3], B r (x) is well defined for all 0 < r ≤ 1/2 and allx ∈ Q 1 T (α). We also have the following.
Lemma 2.6. There exists some N 3 so that for all x ∈ Q 1 (α) and every 0 < r ≤ u(x) −N 3 the following hold. Letx ∈ Q 1 T (α) be a lift of x. Then (1) The restriction of the covering map π to B r (x) is injective.
Proof. The argument is similar to the one used in the proof of [EMM,Lemma 8.2].
For part (2) we will need the following two facts: d T ((a t u s ) ±1 z, (a t u s ) ±1 z ′ ) ≤ 16e 2t for all t ≥ 0 and s ∈ [−1, 1] where d T denotes the Tichmüller distance. Moreover, there exist a constant C ≥ 1 so that is the compact set introduced in Theorem 2.5.
We now turn to the proof of the lemma. For every x ∈ K ′ α , there exists 0 < r(x) ≤ 1/2 so that B r(x) (x) is embedded in the sense that the projection from the Teichmüller space Decreasing r 0 if necessary, we assume that for all x ∈ K ′ α and allx 1 ,x 2 ∈ B r 0 (x), the Teichmüller distance betweenx 1 andx 2 is at most 1.
Let N ≥ 1 be so that where N 2 is as in Theorem 2.5.
We will show that N 3 = N satisfies the claims in the lemma. First note that in view of [AG,Prop. 5 is well defined for all x ∈ Q 1 (α) and all the liftsx ∈ Q 1 T (α). Suppose now that there exists x ∈ Q 1 (α) andx 1 ,x 2 ∈ B(x) such thatx 2 = gx 1 for some g in the mapping class group. Writẽ Then, in view of (15) we have ≤ r 0 where for the last estimate we used (17) and the fact that u(x) ≥ 2. However,x ′ 2 = gx ′ 1 , thus, both x ′ 1 and x ′ 2 belong to the projection of B r 0 (x ′ ); this contradicts the fact that B r 0 (x ′ ) is embedded.
This contradiction shows that B u(x) −N (x) is embedded, establishing part (1).
We now turn to part (2). We use the above notation. Letx 1 ,x 2 ∈ B u(x) −N (x), and define x ′ i = a τ u s x i ∈ K ′ α andx ′ i = a τ u sxi as above. Then (18) implies that where we used (17) and u(x) ≥ 2 in the last inequality. The proof is complete.
For every x ∈ Q 1 (α) we put For every 0 < r ≤ r(x), we let B r (x) denotes π(B r (x)) wherex ∈ Q 1 T (α) is an arbitrary lift of x. We refer to B r (x) as the ball of radius r centered at x.

Horospherical foliation. Given a point
where v(x) with v(x) AGY,x = 1 determines the direction of the Teichmüller geodesic flow, The subspaces E u,s (x) depend smoothly on x, moreover, they are integrable. We denote the corresponding leaves by W u (x) and W s (x), respectively. Also put Let µ u x and µ s x denote the leafwise measures of the natural measure µ along W u (x) and W s (x), respectively. Then y → µ u,s y is constant along W u,s (x), respectively, and we have (21) (a t ) * µ u x = e −ht µ u atx and (a t ) * µ s x = e ht µ s atx ; see also [AG,§4] where these measures are defined using volume forms.
If B r (x) is a period box centered at x, then µ| Br(x) has a product structure as dLeb × dµ s × dµ u , see e.g. [AG,Prop. 4.1]. Given x ∈ Q 1 (α) and a period box B r (x) with center x and 0 ≤ r ≤ r(x), we let r (x) for • = cu, cs similarly. We also denote functions which are supported on the leaves W u , W cu , etc. using the same superscript, e.g., φ u denotes a function which is supported on a leaf W u (x).
We use the norm · AGY,x to induce a metric d W u,s (x) on B u,s r (x) for 0 < r < r(x). Hence notions such as diam etc. refer to this metric. [AG,Lemma 5.2]. Moreover, we have the following uniform hyperbolicity estimate.
Proposition 2.7. Let K ⊂ Q 1 (α) be a compact subset. There exist some κ 2 (K) and some t 0 = t 0 (K) with the following property. Let t ≥ t 0 ; suppose that x, a t x ∈ K, moreover, assume that

Proof. Let
ABEM,x denote the modified Hodge norm defined in [ABEM,§3]. Let C be a constant so that In view of [ABEM,Thm. 3.15], there exists some κ 3 (K) so that under our assumptions in this proposition we have We now compute The claim thus holds with κ 2 = κ 3 /2 and t 0 = 4 log C κ 3 .
Lemma 2.8. Let K ′ α be as in Theorem 2.5. There is a positive constant N 4 and for every 0 < θ < 1 there exists κ 4 (θ), and a compact subset K α (θ) ⊃ K ′ α with the following properties. Let x ∈ Q 1 (α), 0 < r ≤ r(x), and let B r (x) be a period box centered at x. Put Then for every t ≥ N 4 log u(x), we have Proof. See [AG,Prop. 6.1].
2.11. Smooth structure on affine manifolds. As it is done in [AG,§5.2], we use the affine structure to define a smooth structure on Q 1 T (α) and Q 1 (α). Let us recall the definition of a C k -norm from [AG], see also [AGY].
where the supremum is taken over x in the domain of ϕ and v 1 , . . . , v k ∈ T x W with AGYnorm at most 1. Define the C k -norm of ϕ as ϕ C k = k j=0 c j (ϕ). By a C k function we mean a function whose C k -norm is finite. The space of compactly supported C k functions on W will be denoted by C k c (W ), similarly, we define C ∞ c (W ). In the sequel we will only need C 1 -norm of functions. To avoid confusion between this norm and other relevant norms which will be used, and also since we often use the letter C to denote various constants, define C 1 (ϕ) := ϕ C 1 . for any C 1 function ϕ.
In the sequel we will need to replace the characteristic functions of certain sets with their smooth approximations. The following lemmas will provide such approximations.
Lemma 2.10 (Cf. [AG], Prop. 5.8). There exists N 5 so that the following holds. Let x ∈ Q 1 (α). Let D ⊂ W u (x) be a compact set, and let ǫ ≤ 0.1r(D), see (19). There exists a finite collection {ϕ i } of C ∞ functions on W u (x) with the following properties: ϕ i ≤ 1, and the equality holds on a neighborhood of D.
Proof. This is proved in [AG,Prop. 5.8]. It is worth mentioning that [AG,Prop. 5.8] is stated for balls of size ≍ 1, to get our claim here, one needs to apply the argument there not to the AGY norm, but to the AGY norm scaled by 1/ǫ.
Let W be one of the following: Q 1 (α), W u,s (x), or W cu,cs (x), for some x ∈ Q 1 (α). Let E ⊂ W be a compact subset. For any 0 < ǫ < 0.1r(E) define note that E W +,ǫ is an open subset of W which contains E. Let r > 0 and L > 1. Let S W (E, r, L) denote the class of Borel functions 0 ≤ f ≤ 1 supported and defined everywhere in E with the following properties: If W is clear from the context, we denote S W (E, r, L) and E W +,ǫ simply by S(E, r, L) and E +,ǫ , respectively.
Lemma 2.11. There exists some L depending only on α so that for all 0 < r ≤ r(x), Proof. We will show the claims hold if we choose L > 2N 5 , see Lemma 2.10, large enough. Apply Lemma 2.10 with ǫ and D = B u r−2ǫ , and denote by {ϕ i,− } the functions obtained from that lemma. For a second time, apply Lemma 2.10 with ǫ and D = B u r (x), and denote by {ϕ i,+ } the functions thus obtained. Put These functions satisfy (S-1) thanks to Lemma 2.10(1) and (5). Moreover, they satisfy (S-2) thanks to Lemma 2.10(1)-(4) and the fact that L > 2N 5 .
To see (S-3), first note that µ u x B u r (x) − B u r−2ǫ ≪ ǫ where the implied constant depends only on α. The claim in (S-3) thus holds true in view of Lemma 2.10(5) if we choose L large enough, depending on α.
The second claim follows from the first claim, using the product structure of B r (x) and of the measure µ.
We fix once and for all some L so that Lemma 2.11 holds true and drop L from the notation. In particular, S(E, r, L) will be denoted by S(E, r).
Abusing the notation we will write S(x, r) for S(E, r) if the compact subset E is not relevant except for the fact that it is a compact subset containing the point x.

Translates of horospheres
In this section we will use a fundamental result of Avila, Gouëzel, and Yoccoz, [AGY, AG] together with Margulis' thickening technique, [Mar, EMc, KM], to study translations of pieces of the horospherical foliations along the geodesic flow.
We remark that combining [AGY, AR, AG] and [Rn], the C 1 norm in Theorem 3.1 may be replaced by the p-Hölder norm for any p > 0. However, if we use the p-Hölder norm, the constant κ will, in general, depend on p; in particular, κ tends to 0 as p tends to 0, see [Rn,Thm. 1] and [AGY,Thm. 2.14].
It is also worth mentioning that the C 1 norm in Theorem 3.1 may be taken to include derivatives only in the direction of SO(2) ⊂ SL(2, R), see [CHH] and [Rn,Thm. 1] and references there. Our choice, C 1 , is more restrictive; this is tailored to our applications later, e.g., we will use the estimate that Proposition 3.2. There exists some κ 5 , depending on α, with the following property. Let We need some notation; we discuss the case ς = 1, the case ς = −1 is similar. Let Φ(x) = a + ib; recall from (20) that These spaces can alternatively be described as follows. Recall that subspace For every 0 < δ < 0.1r and every y ∈ B r (x), let be the projection along unstable leaves. Then 0.5 ≤ Jac(p cs y ) ≤ 2, moreover we have We now begin the proof of the Proposition 3.2.
Proof. The idea is to relate the integral W u (x) φ(a t y)ψ u (y) dµ u x (y) to correlations of the function a −t φ with a thickening of ψ u in the direction of W cs (x). Then we may use Theorem 3.1 to conclude the proof.
To that end, let 0 < ǫ < 0.01r(x) be a parameter which will be fixed later. In particular, it will be taken to be of the form e −κt . Let ψ cs be a smooth function supported in D cs ǫ (x) so that W sc (x) ψ cs = 1. We can choose such a function so that it moreover satisfies C 1 (ψ s ) ≪ ǫ −N 6 where N 6 and the implied constant depend on α. (30). We need the following lemma.
Lemma. There exists κ 6 depending only on α so that where the implied constant depends only on α.
Let us assume the lemma and finish the proof of the proposition. Optimizing the choice of ǫ to be of size e −κt for some small 0 < κ < 1, the proposition follows from (27) and Theorem 3.1 applied with Ψ 1 = φ and Ψ 2 = Ψ -recall again that µ(Ψ) = µ u x (ψ u ).
Proof of the Lemma. Since Ψ is supported in FB r (x), we need to estimate Thus using the definition of C 1 (φ), we have where κ 6 and the implied constant depend only on α.
In consequence, we may replace φ(a t z) by φ(a t z u ) in (28), and use the bound · ∞ ≤ C 1 (·), to conclude the following where the implied constant depends on α.
Recall the definition of Ψ from (26), in particular recall the normalizing factor λ y u . This and the product structure of µ yield the following We now combine the estimates in (29) and (30), and get the following.
where the implied constant is absolute.
Remark 3.3. It is worth mentioning that Proposition 3.2 and its proof hold for any affine invariant manifold, (M, µ). In the sequel, however, we will only need this result for Q 1 (α); and even more specifically, in our application to counting problems, we will need this result for the principal stratum Q 1 (1, . . . , 1). The main result in [AGY] was generalized to Q 1 (α) in [AR].

A counting function
Let x, z ∈ Q 1 (α). Let ψ u be a function which is supported and defined everywhere in , and let φ cs be a function which is supported and defined everywhere in where the sum is taken over all y ∈ B u 0.1r(x) (x) so that a t y ∈ B cs 0.1r(z) (z) -note that the sum is indeed over all y ∈ supp(ψ u ) so that a t y ∈ supp(φ cs ).
Alternatively, the sum is taken over connected components of a t supp(ψ u )∩supp(φ cs ) (indeed the subscript nc stands for the number of connected components); this point will be made more explicit later in this section, see e.g. Lemma 4.2 below and recall that W u and W cs are complementary foliations.
The function N nc may be thought of as a bisector counting function where one studies the asymptotic behavior of the number of translates of a piece of W u by Mod g which intersect a cone in the Teichmüller space.
The following proposition is the main result of this section and provides an asymptotic behavior for N nc . This proposition plays a prime role in the proof of Theorem 1.2 in §7.
Proposition 4.1. There exist κ 9 and N 8 with the following property. Let x, z ∈ Q 1 (α), The proof of this proposition is based on Lemma 4.5 which in turn relies on Proposition 3.2.
In particular, the main term is given by Proposition 3.2. However, we need to control the contribution of two types of exceptional points as we now describe.
Similar to Lemma 2.8, given a compact subset K ⊃ K α , define : a τ y ∈ K}| ≥ t/2 . The first (and more difficult to control) type of exceptional points are y ∈ B u r (x) so that a t y ∈ B r ′ (z), however, y ∈ H u t (x, K). The contribution coming from these points is controlled using [EMR,Thm. 1.7], see Theorem 4.4 below.
We also need to control the contribution of points y ∈ B u r (x) which are exponentially close to the boundary of B u r (x). This set has a controlled geometry, and we use a covering argument and Proposition 3.2 to control this contribution. The argument here is standard and will be presented after we establish an essential estimate in (42).
Let us begin with some preliminary statements which are essentially consequences of the fact thatW u andW cs are complimentary foliations in the spaces marked surfaces Q 1 T (α).
As was discussed above, there are two types of exceptional points. The first type will be controlled using the following theorem.
Proof. Let us write r = 0.1r(x) and r ′ = 0.1r(z). For a compact subset K ⊃ K α , put 2r (x) and B r ′ (z) for the sets B u 2r (x) and B cs r ′ (z), respectively. For every element y ∈ B u 2r (x) we fix a liftỹ ∈ B u 2r (x). Then for every y ∈ E t (x, K) there exists some g y ∈ Mod g and somez y ∈ B cs r ′ (z) so that a tỹ = g yzy . Recall from Lemma 2.6 that the diameter of B r(q) (q) in the Teichmüller metric is at most 1 for allq. Hence, for every y ∈ E t (x, K) we have (1)ỹ is within Teichmüller distance 1 fromx and a tỹ = g yzy is within Teichmüller distance 1 of g yz , and (2) |{τ ∈ [0, t] : π(a τỹ ) ∈ K)}| < t/2.
It is shown in [EMR,Thm. 1.7], see also [EMir], that there exists some K 0 so that if K ⊃ K 0 , then the number of {gz} for which such aỹ exists is where the implied constant is absolute -indeed apply with δ = 0.1 and θ = 0.9 and observe that the function G in [EMR,Thm. 1.7] is dominated by our function u here.
We now claim that there exists some C which depends on α and K so that the following holds: the map y → g yz from E t (x, K) to {gz : g ∈ Mod g } is at most C-to-one.
First note that the above discussion together with the claim implies that as we wanted to show.
To see the claim, let y 1 , y 2 ∈ E t (x, K). Then there exists g 1 , g 2 ∈ Mod g so that g i · a tỹi ∈ B cs r ′ (z). Therefore, by Corollary 4.3, applied withx i =x and b = 2r, we have • either g 1 · W u (x) = g 2 · W u (x) which in particular implies that g 1 = g 2 , • or g 1 · B u 2r (x) ∩ g 2 · B u 2r (x) = ∅ which implies g −1 1 g 2 belongs to a fixed finite subset of Mod g .
The claim thus follows and the proof is complete.
The following lemma will play a crucial role in the proof of Proposition 4.1.
Put φ(y) := φ cs (p cs y u (y))φ u (y u ), see §3. Define Proof. We will compute in terms of N ′ nc . The claim will then follow from Proposition 3.2. Let us write r = 0.1r(x) and r ′ = 0.1r(z). First note that where the diameter, diam, is measured with respect to z ′ ,AGY for all z ′ ∈ B r ′ (z), see [AG,Prop. 5.3].
LetK α be given by Theorem 4.4 and put H u t (x) := H u t (x,K α ), see (32) for the notation. Since K α ⊂K α , it follows from Lemma 2.8 that It is more convenient for the proof to treat points in H u t (x) which are too close to the boundary of B u r (x) separately. Define } where κ 11 := κ 2 (K α )/2, see Proposition 2.7 for the definition of κ 2 . The precise radius which is used in the definition of H u t,int is motivated by estimates for uniform hyperbolicity of the Teichmüller geodesic flow, see Claim 4.6 below.

Using (36) and the definition of H
) for some κ 12 depending onK α . The estimate in (37) implies the following: We now compute the term H u t,int φ(a t y)ψ u (y) dµ u x (y) appearing in (38).
For every y ∈ H u t,int so that a t y ∈ B r (z), there is an open neighborhood C y of y such that a t C y is a connected component of a t B u r (x) ∩ B r ′ (z) containing a t y. We note that C = {C y } is a disjoint collection of open subsets in B u r (x). Further, in view of (21) we have (39) µ u aty (φ) = e ht µ u y a −t φ = e ht µ u x a −t φ ; recall that a −t φ(y ′ ) = φ(a t y ′ ).
Claim 4.6. Let y ∈ H u t (x), then C y ⊂ B u 10e −κ 11 t (y). If we further assume that y ∈ H u t,int , Proof of the claim. Let y ′ ∈ C y . It follows from the definition of C y that a t y ′ ∈ W u (a t y) ∩ B r ′ (z). Let us write a t y ′ = Φ −1 (Φ(a t y) + w). Then, by (35) we have This, in view of Corollary 2.9, implies that w AGY,y ≤ e −κ 2 t w AGY,aty ≪ e −κ 2 t r ′ where the implied constant depends only on α. The claim follows from this estimate if we assume t is large enough so that the above estimate implies w AGY,y < e −κ 11 t ; recall that κ 11 = κ 2 /2. The final claim follows from the definition of H u t,int .
Claim 4.6 in particular implies that where the implied constant depends only on α.
Returning to (38), we get from (39) and (40) that Combining (38), (41), and Proposition 3.2 we get the following for some κ 13 depending on α. Thus, in order to get the conclusion, we need to control the difference between N ′ nc (t, ψ u , φ) and the summation appearing on the left side of (42). That is: the contribution of points y / ∈ H u t,int .

Contribution from points in
be so that a t y ∈ B r (z). We note that C y is not necessarily contained in B u r (x); however, in view Claim 4.6, we have C y is contained in B 10e −κ 11 t (y).
The following is a consequence of the definition.
where the implicit multiplicative constant depends only on α.
Let 0 <κ < κ 11 be a small constant which will be optimized later, and let t ≥ 2N 3 log u(x) κ . We can cover G(x) with period balls {B(y i ) : 1 ≤ i ≤ I} centered at y i and of radius e −κt with multiplicity bounded by ≪ e N 6κ t , see [Hör,Lemma 1.4.9] and also §2.11. We have (43) I ≪ e Nκt for some N depending only on α.
For every i, letB(y i ) denote the the period ball with the same center y i and with radius 0.04e −κt . Note that sinceκ < κ 11 = κ 2 /2 we have Therefore, ∪ iB (y i ) covers a set G ′ (x) ⊃ G(x) so that µ u x (G ′ (x)) ≪ e −κt . Let 0 ≤ψ u i ≤ 1 be a smooth function which is supported inB u (y i ) which equals 1 on B u 2e −κt (y i ) so that where N 6 ≥ N 5 is chosen to account for the multiplicative constant in Lemma 2.10.
Let I i be the contribution coming from B(y i ) to N nc (t, ψ u , φ). Then arguing as above and using Proposition 3.2, the choice ofψ u implies that Summing (45) over all 1 ≤ i ≤ I and using (44), (43), and ψ u dµ u x ≪ e −hκt we get Therefore, we can chooseκ so that the above upper bound yields for some κ 14 depending only on α.

Contribution from points in
The proposition now follows from (42) in view of (46) and (47).
Proof of Proposition 4.1. Let ̺ = e −κt and let ǫ = ̺ N , for two constants κ, N > 0 which will be optimized later.
We end this section with the following corollary.
The second claim follows from the first claim by optimizing the choice δ = e −⋆t .

The space of measured laminations
In this section we recall some basic facts about the space of geodesic measured laminations and train track charts. The basic references for these results are [Th1] and [HP].
The space of geodesic measured laminations on S is denoted by ML(S); it is a piecewise linear manifold homeomorphic to R 6g−6 , but it does not have a natural differentiable structure [Th1]. Train tracks were introduced by Thurston as a powerful technical device for understanding measured laminations. Roughly speaking train tracks are induced by squeezing almost parallel strands of a very long simple closed geodesic to simple arcs on a hyperbolic surface. A train track τ on a surface S is a finite closed 1 complex τ ⊂ S with vertices (switches) which is -embedded on S, -away from its switches, it is C 1 , -it has tangent vectors at every point, and -for each component R of S − τ , the double of R along the interiors of the edges of ∂(R) has negative Euler characteristic.
The vertices (or switches), V , of a train track are the points where 3 or more smooth arcs come together. Each edge of τ is a smooth path with a well defined tangent vector. That is: all edges at a given vertex are tangent. The inward pointing tangent of an edge divides the branches that are incident to a vertex into incoming and outgoing branches.
A train track τ is called maximal (or generic) if at each vertex there are two incoming edges and one outgoing edge.

Train track charts.
A lamination λ on S is carried by a train track τ if there is a differentiable map f : S → S so that -f is homotopic to the identity, -the restriction of df to a tangent line of λ is nonsingular, and -f maps λ onto τ .
Every geodesic lamination is carried by some train track. Let λ be a measured lamination with invariant measure µ. If λ is carried by the train track τ , then the carrying map defines a counting measure µ(b) to each branch line b: µ(b) is just the transverse measure of the leaves of λ collapsed to a point on b. At a switch, the sum of the entering numbers equals the sum of the exiting numbers.
The piecewise linear integral structure on ML(S) is induced by train tracks as follows. Let V(τ ) be the set of measures on a train track τ ; more precisely, u ∈ V(τ ) is an assignment of positive real numbers to the edges of the train track satisfying the switch condition: incoming e i u(e i ) = outgoing e j u(e j ).
Also, let W(τ ) be the vector space of all real weight systems on edges of τ satisfying the switch condition, i.e., u(e i ) need not be positive for u ∈ W(τ ). Then V(τ ) is a cone on a finite-sided polyhedron where the faces are of the form V(σ) ⊂ V(τ ) where σ is a sub train track of τ .

Thurston symplectic form on ML(S).
We can identify W(τ ) with the tangent space of ML(S) at a point u ∈ V(τ ), see [HP].
For any train track τ , the integral points in V(τ ) are in one to one correspondence with the set of integral multicurves in U (τ ) ⊂ ML(S). The natural volume form on V(τ ) defines a mapping class group invariant volume form µ Th in the Lebesgue measure class on ML(S).
In fact, the volume form on ML(S) is induced by a mapping class group invariant 2-form ω as follows. Suppose τ is maximal, for u 1 , u 2 ∈ W(τ ) the symplectic pairing is defined as follows.
(58) ω(u 1 , u 2 ) = 1 2 u 1 (e 1 ) u 2 (e 2 ) − u 1 (e 2 ) u 2 (e 1 ) , the sum is over all vertices v of the train track where e 1 and e 2 are the two incoming branches at v such that e 1 is on the right side of the common tangent vector.
This form defines an antisymmetric bilinear form on W(τ ).
Lemma 5.2. Let τ be maximal. The Thurston form ω, defined in (58), is non-degenerate. Therefore it gives rise to a symplectic form on the piecewise linear manifold ML(S).
See [HP,§3] for a proof and also the relationship between the intersection pairing of H 1 (S, R) and Thurston intersection pairing.

Combinatorial type of measured laminations and train tracks.
Each component of S − λ is a region bounded by closed geodesics and infinite geodesics; further, any such region can be doubled along its boundary to give a complete hyperbolic surface which has finite area.
We say a filling measured lamination λ is of type a = (a 1 , ....a k ) if and only if S−λ consists of ideal polygons with a 1 , . . . , a k sides. By extending the measured lamination λ to a foliation with isolated singularities on the complement, we see that k i=1 a i = 4g − 4 + 2k, see [Th1] and [Le].
Similarly, each component of the complement of a filling train track τ is a non-punctured or once-punctured cusped polygon of negative Euler index. We say a train track τ is of type a = (a 1 , . . . , a k ), if and only if S − τ consists of k polygons with a 1 , . . . , a k sides. Every measured lamination of type a = (a 1 , . . . , a k ) can be carried by a train track of type a. Lemma 5.3. For any filling train track τ of type a = (a 1 , . . . , a k ) we have More generally, a measured lamination λ is said to be of type a if there exists a quadratic differential q ∈ Q(a 1 − 2, . . . , a k − 2) such that λ = R(q). It is easy to check that if λ is filling, the above can happen only if S − λ consists of ideal polygons with a 1 , . . . , a k sides.
In general, see [HP,§3], we have: Proposition 5.4. Given a measured lamination λ of type a, there exists a birecurrent train track of type a such that λ is an interior point of U (τ ).
For every a = (a 1 , . . . , a k ) so that k i=1 a i = 4g −4+2k, we can fix a collection τ a,1 , . . . , τ a,ca of train tracks with the following property. Every λ which can be carried by a train track of type a can be carried by at least one τ a,i for some i. Theorem 5.5 (Hubbard-Masur, Gardiner). The mapP is a Mod g equivariant homeomorphism.
This gives rise to an equivariant homeomorphism from QT (S) onto ML(S) × ML(S) − ∆ which we continue to denote byP, see [Th1] and [Le].
Recall that PML(S) denotes the space of projective measured lamination. The mapP also gives rise to an equivariant homeomorphism Recall that π is the natural projection from Q 1 T (S) to Q 1 (S), then we have the map (59) π •P −1 1 : PML(S) × ML(S) − ∆ → Q 1 (S).

5.5.
Convexity of the hyperbolic length function. Let λ 1 , λ 2 ∈ U (τ ) = ι τ (V(τ )), see §5.1 for the definition of ι τ . The sum ) could depend on τ . However, it is proved in [Mir1,App. A] that given a closed curve γ, i(γ, .) : U (τ ) → R + defines a convex function from which convexity of the hyperbolic length function is drawn in [Mir1,Thm. A.1]. The following is an extension of [Mir1,Thm. A.1] to the case of variable negative curvature. We are grateful to K. Rafi for providing the proof of this theorem.
Theorem 5.1. Let X be a compact surface equipped with a Riemannian metric of negative curvature, and let τ be a train track. Let ℓ X : U (τ ) → R + denote the length function. For every pair of measured laminations λ 1 , λ 2 ∈ ML(S) carried by τ if µ = λ 1 ⊕ τ λ 2 , then In particular, ℓ X is convex.

The following lemma is well known
Lemma 5.6. Let τ a train-track, and let λ 1 and λ 2 be multi-curves carried by τ . Then, there exists a multi-curve µ carried by τ such that µ = λ 1 + λ 2 in coordinates given by τ . Furthermore, µ can be obtained from λ 1 and λ 2 by a sequence of surgeries.
We now turn to the proof of the Theorem 5.1.
Proof of Theorem 5.1. Let C be the space of geodesic currents on X, that is the space of π 1 (X)-invariant Radon measures on the space of geodesics in X. Recall that the space of measured laminations can be topologically embedded into the space of geodesic currents, therefore, we can think of any λ ∈ ML(S) as a geodesic current, namely, an element of C. Also recall from [Bon1] that there is a continuous intersection pairing Furthermore, there is a geodesic current L X ∈ C such that for every for all λ ∈ ML(S), see [Ot]. The set of simple closed curves with rational weights are dense in ML(S). Therefore, in view of the continuity of intersection pairing i, it is sufficient to check the statement of the theorem for rationally weighted simple closed curves only. Since, length is homogeneous, we can in fact assume the weights are integers or λ 1 , λ 2 , and µ are multi-curves with the possibility of some curve appearing more than once.
Proof of the claim. Note that λ 1 and λ 2 have unique geodesic representatives in M . Let p be an intersection point of λ 1 and λ 2 where the surgery takes place. Then the free homotopy class of β can be represented by a traversing λ 1 first (starting from p) then λ 2 . Which means β has a representative whose length is ℓ X (λ 1 ) + ℓ X (λ 2 ). This proves the claim.
Further, we note that, µ = λ 1 ⊕ τ λ 2 can be obtained from λ 1 and λ 2 by a sequence of surgery maps, see Lemma 5.6. This proves the theorem.
Let C ⊂ R n be a cone and f : C → R be a convex function. Let K be a closed and bounded set contained in the relative interior of the domain of f . Then f is Lipschitz continuous on K. That is: there exists a constant L = L(K) such that for all x, y ∈ K we have Therefore, we have the following.
Corollary 5.2. Let X be a compact surface equipped with a Riemannian metric of negative curvature. Then ℓ X : ML(S) → R + is locally Lipschitz. In other words, and in view of the fact that ℓ X (t ·) = tℓ X (·) for all t > 0, we can cover ML(S) with finitely many cones such that ℓ X is Lipschitz in each cone.
The Lipschitz constant depends on X. See also [LS].
6. Linear structure of ML(S) and QT (S) Our arguments are based on relating the counting problems in ML(S) to dynamical results in Q 1 (1, . . . , 1). To that end, we need to compare the linear structure on Q 1 (1, . . . , 1), arising from period coordinates, with the piecewise linear structure on ML(S), which arises from train track charts. This section establishes required results in this direction.
From this point to the end of the paper, we will be concerned with the principal stratum, i.e., Q 1 (1, . . . , 1). Also a = (3, . . . , 3) for the rest of the discussion.
Fix once and for all a collection τ 1 , . . . , τ c of train tracks so that every λ can be carried by at least one τ i for some i, see §5.3.
Let x = (M, q) ∈ Q 1 (1, . . . , 1). We denote by R(q 1/2 ) (resp. I(q 1/2 )) the real (resp. imaginary) foliation induced by q; abusing the notation we will often simply denote these foliations by R(q) and I(q). Note that W u,s (x), which we sometimes also denote by W u,s (q), may alternatively be defined as follows. Similarly, we will write B r (q) and B • r (q) for B r (x) and B • r (x), respectively.
Let τ be a maximal train track, i.e., a train track of type (3, . . . , 3), and let U (τ ) be a train track chart, i.e., the set of weights on τ satisfying the switch conditions. Recall from §5.1 that U (τ ) has a linear structure, indeed U (τ ) is a cone on a finite-sided polyhedron. We use the L 1 -norm on W(τ ) to define a norm on U (τ ). That is: for every measured lamination λ ∈ U (τ ), we define λ τ to be the sum of the weights of λ. Let us define For every λ ∈ U (τ ), defineλ τ := 1 λ τ λ ∈ P (τ ); if τ is fixed and clear from the context, we sometimes drop the subscript and the superscript τ and simply write λ andλ for λ τ andλ τ , respectively.
By a polyhedron U ⊂ U (τ ), we mean a polyhedron of dimension dim U (τ ) − 1 where the angles are bounded below and the number of facets are bounded, both by constants depending only on the genus. We will mainly be concerned with dim U (τ ) − 1 dimensional cubes in the sequel.
Lemma 6.2. There is some N 12 ≥ N 3 so that the following holds, see (19) for the definition of N 3 . Let q ∈ Q 1 (1, . . . , 1). There exists a 1-complex T ⊂ S with the following properties.
(1) Every edge of T is a saddle connection of q.
(3) S − T is a union of triangles.
Proof. We will find such a T with |I(e)| ≥ 0.1ℓ q (e), the proof of the fact that such a T exists with |R(e)| > 0.1ℓ q (e) is similar, by replacing a t u s with a −tūs in the following argument.
Let K be the compact set given by Theorem 2.5; let r 0 = inf{r(x) : x ∈ K}, see (19). For every q ′ ∈ K, there is a graph T ′ ⊂ S of saddle connections of q ′ so that • the q ′ length of each of these saddle connections is bounded by L 0 = L 0 (K), and • S − T ′ is a union of triangles.
We will always assume that L 0 > 2. Increasing L 0 , if necessary, we will also assume that L 0 bounds the lengths of saddle connections obtained by parallel transporting T ′ to q ′′ ∈ B r 0 (q ′ ) for all q ′ ∈ K.
Apply Theorem 2.5 with t 0 = L 0 log f (q). There exists some and some s ∈ [0, 1] so that q ′ = a t u s q ∈ K.
Let now T ′ be a graph of saddle connections for q ′ defined as above. We claim that for every e ∈ T ′ , we have e ∈ a t u s R q . To see the claim, first note that for every γ ∈ R q we have Hence a t u s γ is not contained in T ′ . In consequence, T = u −s a −t T ′ satisfies (1), (2), (3), and (4). Note that for every e ∈ T , we have u(q) −⋆ ≪ ℓ q (e) ≪ u(q) ⋆ where the implied constants depend only on the genus.
We now turn to the proof of part (5). First note that there is Let us write 0 < r = u(q) −N 12 , then 0 < r ≤ r(q), recall that N 12 ≥ N 3 . For every Let t ≤ max{2L 0 f (q), N 2 log u(q)} and 0 ≤ s ≤ 1 be so that q ′ = a t u s q ∈ K; see the preceding discussion. Note that in view the choice of t and N , we have (64) ≤ e 2 · u(q) N −N 12 v AGY,q ≤ u(q) −N 12 by the choice of r ≤ e 2 · 2 N −N 12 ≤ r 0 /2 since u(q) ≥ 2 and using (63).
Hence a t u s B r (q) ⊂ B r 0 (q ′ ) which gives the claim in view of the definitions of T and T ′ .
Increasing N 12 if necessary part (4) also holds for this exponent.
(2) The restriction ofP 1 to B r (q) is a homeomorphism.
(4) The linear structure on U I (q) := {I(p) :p ∈ B r (q), R(p) = R(q)} as a subset of U (σ) agrees with the linear structure on U I (q) which is induced by the restriction of Moreover, the radius r of B r (q) can be taken to be uniform on compact subsets of Q 1 (1, . . . , 1).
Proof. Let T be a triangulation of q given by Lemma 6.2. In particular, (i) every edge of T is a saddle connection, (ii) |I(e)| ≥ 0.1ℓ q (e) for any e ∈ T , (iii) S − T is a union of triangles, and (iv) A q ≤ ℓ q (e) ≤ A −1 q for every edge e ∈ T where A q = u(q) −N 12 Our construction of the train track σ will depend on T .
Recall that r = 0.01A 2 q . Then the balls B r (q) and B r (q) satisfy (1) and (2) in the lemma by Lemma 6.2(5).
Let σ ′ be the null-gon dual graph to T , in particular, there is one triangle of σ ′ in each component of S − T . Let σ be the train track obtained from σ ′ as follows. If ∆ is a triangle in T with edges e ∆ 1 , e ∆ 2 , e ∆ 3 , then there is a permutation put σ := σ ′ − {the edge corresponding to e ∆ i 1 in σ ′ }. We claim the lemma holds with σ. To see the claim, first note that σ is a maximal train track. Assign the weight |I(e b )| to each branch b ∈ σ where e b ∈ T is the edge which intersects b. In view of (65) and the fact that |I(γ)| = i(γ, R(q)) for every saddle connection γ, we get that λ = I(q) is carried by σ.

Counting integral points in ML(S)
Let the notation be as in §6. In particular, τ is a maximal train track. Also recall that P (τ ) denotes the finite-sided polyhedron in U (τ ) corresponding to laminations with λ τ = 1.
The smallest t so that a lamination λ ∈ U (τ ) lies in [0, e t ]P (τ ) = {λ ′ ∈ U (τ ) : λ ′ τ ≤ e t } can be thought of as a measure of complexity (or length) for the lamination λ. In this section we obtain an effective counting result with respect to this complexity. In §8 we will use the convexity of the hyperbolic length function in U (τ ) to relate the counting problem in Theorem 1.1 to this counting problem.
Let U ⊂ P (τ ) be a cube. For every t ≥ 0, define The following strengthening of Theorem 1.2 is the main result of this section.
Theorem 7.1. There exist κ 19 and κ 20 so that the following holds. Let t ≥ 1, and let U ⊂ P (τ ) be a cube of size ≥ e −κ 19 t . Then where v(γ 0 ) is defined as in (69) and h = 6g − 6.
The basic tool in the proof of Theorem 7.1 is Proposition 4.1. We relate the counting problem in Theorem 7.1 to a counting problem for translations of W u (q 0 ) in Lemma 7.2. Proposition 4.1 studies a more local version of this latter counting problem. That is: one works with translations of a small region in W u (q 0 ). Using Corollary 4.3, we will reduce to this local analysis. The main step in the proof of Theorem 7.1 is Lemma 7.6 below.
Let us begin with some preparation. Recall that ML(S) does not have a natural differentiable structure, in particular,P 1 is only a homeomorphism. The situation however drastically improves so long as we restrict to one train track chart and fix a transversal lamination. Therefore, we fix a maximal lamination η which is carried by τ for the rest of the discussion.
Lemma 7.2. Let δ > 0, and let U ⊂ P (τ ) be a cube of size ≥ δ. Let λ denote the center of U . For all ǫ ≤ δ and all large enough t ≥ 0 we have: Proof. Since τ is fixed throughout, we drop it from the subscript and superscript for the norm and the normalization.
7.1. Strebel differentials. Problems related to the existence and uniqueness of Jenkins-Strebel differentials have been extensively studied.
Theorem 7.3 (Cf. [Str], Thm. 20.3). Let γ = ⊔ d i=1 γ i be a rational multi-geodesic on M , and let r 1 , . . . , r d be positive real numbers. Then there exists a unique holomorphic quadratic differential q on M (Jenkins-Strebel differential) with the following properties.
(1) If Γ is the critical graph 1 of q, where Ω i is either empty or a cylinder whose core curve is γ i .
(2) If Ω i is not empty, it is swept out by trajectories whose q length is r i .
The following lemma will be used in the sequel.
Proof. We first show that W u (q) is a properly immersed submanifold of Q 1 (1, . . . , 1). This is equivalent to showing the following two statements.
Let γ be as in the statement. Write γ = i a i γ i where each γ i is a simple closed curve and a i ∈ Q. By Theorem 7.3 we have: the locus W u (q) ∩ Q 1 (1, . . . , 1) is identified with a linear subspace W = {(x ij ) : j x ij = r i , x ij > 0} in the period coordinates, where r 1 , . . . , r d are positive real numbers. Moreover, the measure ν is the pull back of the Lebesgue measure from W to W u (q). This finishes the proof of (1).
1 Recall that the critical graph of a quadratic differential is the union of the compact leaves of the measured foliation induced by q which contain a singularity of q. Using Theorem 7.3, we have W u (q) ∩ K(ǫ) ∁ ⊂ Φ −1 (W(ǫ)). The claims in part (2) now follow from Lemma 2.10. Indeed apply Lemma 2.10 with D = D(2ǫ) − D(ǫ/2), and let {ϕ i } be the collection of functions obtained by that lemma. Define This function satisfies the claims.
Lemma 7.5. For every b there exists some N (b) ≪ b −N 14 so that the following holds. There exists a collection of functions {ψ u i : 0 ≤ i ≤ N (b)} with the following properties: where N 15 is an absolute constant and N 16 is allowed to depend on q 0 .
Proof. This follows from Lemma 2.10 applied with D = D b and Lemma 7.4.
Let us also fix a fundamental domainD ⊂W u (q 0 ) which projects to W u (q 0 ). For each i ≥ 1, we letỹ i ∈D be a lift of y i , see Lemma 7.5. Let N ′ (b) be so that

Counting in linear sectors in ML(S).
Recall from the beginning of this section that U ⊂ P (τ ) is a box of size ≥ δ. Let λ be the center of U , and let ǫ ≤ δ. Let η ∈ ML(S) be fixed as in the beginning of this section. We always assume 0 < δ < 1/2 and η are so thatP −1 1 is a homeomorphism on {[η]} × {e r U : |r| < δ}. Recall also our notatioñ W cs Abusing the notation, we denote by µ Th (U ) the measure induced from µ Th on P (τ ). The following lemma is a crucial step in the proof of Theorem 7.1.
Lemma 7.6. There exist κ 22 and κ 23 so that the following holds. Let t ≥ 0 and in the above notation, define We will prove Lemma 7.6 using Proposition 4.1, more precisely Corollary 4.7. In order to use those results we need to control the geometry ofW cs U ,ǫ . Lemma 7.7. The characteristic function of Proof. Apply Lemma 6.1 with τ and let K = K(τ ) be defined as in (61). Then Let {B rp (p) : p ∈ K} be the covering of K by period boxes given by Lemma 6.3. Let B · (q 1 ), . . . , B · (q b ′ ) be a finite subcover of this covering. Consider all lifts of B(q j ) to period boxes based at liftsq j of q j in our fixed (weak) fundamental domain. Denote these lifts by B r 1 (q 1 ), . . . , B r b (q b ) -note that we only fixed a weak fundamental domain, hence there might be more than one lift, however, there is a universal bound on the number of lifts.
For every 1 ≤ j ≤ b, let σ j be a train track obtained by applying Lemma 6.3 to B r j (q j ). Assume ǫ is smaller than the radius of B r j (q j ) for all j. Write U = ∪Û i wherê By Lemma 5.1 eachÛ i is a piecewise linear subset of U i . The claim now follows from Lemma 6.3(4) if we ignore thoseÛ i 's which have size less than ǫ N for some N > 1 depending only on the dimension.
Recall from §2 that µ denotes the SL(2, R)-invariant probability measure on Q 1 (1, . . . , 1) which is in the Lebesgue measure class. The measures µ u x and µ s x are the conditional measures of µ along W u (x) and W s (x); µ cs x and µ cu x are defined accordingly.
For simplicity in notation, let us writeW cs =W cs U ,ǫ and put N = N (q 0 , t, U , ǫ).
, see Lemma 7.5 and the paragraph following that lemma; there exists some g ′ ∈ Mod g so that g ′ ·W u (q 0 ) =W u (q 0 ) and some 0 ≤ i ≤ N (b) so that (71). We claim that the following holds: where the implied constants depend on the genus.
Let us assume (74) and finish the proof. Let where the outer summation is over all N ′ (b) < i ≤ N (b) and the inner summation is over all y ∈ B u b (y i ) so that a t y ∈ π(W cs ). To see the claim, first note that by the definition of N ′ , if g ·W u (q 0 ) ∈ N ′ , then (73) holds with some N ′ (b) < i ≤ N (b). Let now g 1 , g 2 ∈ Mod g and N ′ (b) ≤ i 1 , i 2 ≤ N (b) be so that W cs ∩ gg j · a t B u b (ỹ i j ) = ∅. Then g jW u (q 0 ) =W u (q 0 ) for j = 1, 2, see the discussion preceding (73); hence by Corollary 4.3 we haveW where the outer summation is over all N ′ (b) < i ≤ N (b) and the inner summation is over all y ∈ B u b (y i ) so that a t y ∈ π(W cs ). Moreover, in view of the fact that B u b (ỹ i ) ∩ B u b (ỹ 0 ) = ∅ for all i ≥ N ′ (b) and using Lemma 7.5(2) and (4), we have i y where the implied constant depends on α. The claim in (75) thus follows in view of the estimate in (74).
Let us now investigate i y ψ u i (y). Using the definition of N nc in (34), we have where the summations are over all y ∈ B u b (y i ) so that a t y ∈ π(W cs ) = supp(φ cs ). Now apply Corollary 4.7, see in particular (53), with ψ u i and φ cs , and get that In view of (72) and the estimate C 1 (ψ u i ) ≤ N 16 b −N 15 , see (70), we get the following from (76).
Summing up (77) over all N ′ (b) ≤ i ≤ N (b) and using the fact that N (b) ≪ b −⋆ , we get that We now compare i µ u q 0 (ψ u i ) and v(γ 0 ). Indeed, using Lemma 7.4, see also (69), and the relationship between ν and µ u q 0 we get the following: The estimate in (79) implies that We now use these estimates to get an estimate for #N ′ . First note that where the implied constant depends only on the genus. This estimate and (80) imply that Putting this estimate and (78) together we get that We now choose ǫ and b of size e −⋆t so that (74) is < e (h−⋆)t and so that N 16 ǫ −⋆ b −⋆ e −κ 18 t on the right side of (81) is < (1 − e −hǫ )e −⋆t . The lemma follows from this in view of (74).

Moreover, by Proposition 3.2 we have
Combining these two estimates and using the fact that in view of the estimates in (82) we have µ(φ 1 )/µ u p (B ̺ (p)) ≪ 1 we conclude that In view of (70) we have . If we now choose κ small enough, (74) follows from (83) and the proof of complete.
This last statement is proved in Lemma 7.6.
Proof of Theorem 7.1. Let ǫ ≥ e −κ 24 t , and for every n ≥ 0 define t n := t − nǫ. Then (84) applied with t = t n implies that This implies the proposition -note that by basic lattice point count in Euclidean spaces 2 , we have the number of integral points γ ∈ U (τ ) so that γ ≤ e h−1 h t is ≪ e (h−1)t .
2 As we remarked in the introduction, the point here is that we are counting the number of point in one Modg-orbit.

Proof of Theorem 1.1
We are now in the position to prove Theorem 1.1. The proof relies on Theorem 7.1. We cover ML(S) with finitely many train track charts U (τ 1 ), . . . , U (τ c ). Using the convexity of the hyperbolic length function, we can reduce the counting problem in Theorem 1.1 to an orbital counting in sectors on U (τ i ), with respect to linear structure, where the length function ℓ X is well approximated by the τ i . Theorem 7.1 is then brought to bear in the study of the latter counting problem.
Let X be a compact surface equipped with a Riemannian metric of negative curvature. Recall that ℓ X : ML(S) → ML(S) denotes the length function. It satisfies ℓ X (tλ) = tℓ X (λ) for any t > 0.
Let τ be a maximal train track. By Corollary 5.2, ℓ X is Lipschitz in U (τ ). Let L τ be the Lipschitz constant, hence Recall that U (τ ) is a cone on the polyhedron P (τ ).
Proof of Theorem 1.1. Let X be as above. Let τ 1 , . . . , τ c be finitely many maximal train tracks with the following properties.
Let L = max L i ; increasing L if necessary we will also assume that the conclusion of Lemma 8.1 holds with L.
Let us fix some 1 ≤ i ≤ c and write τ = τ i ; when there is no confusion we drop τ from the notation for the norm and normalization in U (τ ). We will first consider the contribution coming from U (τ ) and then will combine contributions of different τ i for 1 ≤ i ≤ c.
In the following we will use the following upper bound estimate for the number of integral point in a Euclidean region: the number of lattice points in a Euclidean region is ≪ the volume of the 1-neighborhood of the region.
Cover P (τ ) with cubes of size δ with disjoint interior. Let {U j : j ∈ J δ } be the subcollection of these cubes so that U j ∩ P ≥δ (τ ) = ∅ For every j, let λ j ∈ U j be the center of U j . The number of U j 's required to cover P (τ ) is ≪ δ −N 17 for some N 17 depending on τ .
For each j, let U j,− denote the cube which has the same center λ j as U j , but has size δ −δ N 18 where N 18 = N 17 + 1.
This conclude the contribution arising from a single train track chart U (τ ).
Recall now that the regions in U (τ i ) which are carried by other U (τ i ′ ) are finite sided polyhedra, see Lemma 5.1. We may thus find disjoint finite sided polyhedra U i ⊂ P (τ i ) to the ∪R + . U i = ML(S). Repeating the above argument for each U i , the theorem follows from the estimate in (94).