Asymptotic formulas on flat surfaces

Alex Eskin^{∗} and Howard Masur^{†}

October 20, 2000

1 Introduction.

This paper concerns asymptotic formulas for the number of closed trajectories and saddle connections on a flat surface. It is inspired by some recent work of W. Veech, especially the recent paper [Ve5]. In particular, the analogy noted in [Ve5] between the problem discussed in this paper and the quantitative version of the Oppenheim Conjecture [EMM] allowed us to use some of the ideas developed by G.A. Margulis in the context of the Oppenheim conjecture to simplify and improve the quadratic upper bound of [Ma2]. This, together with an ergodic theorem proved for this purpose by A. Nevo [Nevo], allowed us to find asymptotic quadratic growth rates for the number of closed trajectories on a generic surface. This sharpens results of [Ve5] where these growth rates were found in the mean.

Flat metrics and quadratic differentials. Given a Riemann surface structure on a closed surface of genus g > 1, recall that a holomorphic quadratic differential ϕ is a tensor of the form ϕ(z)dz^{2} in local coordinates with ϕ holomorphic. Away from the zeroes, a coordinate ζ can be chosen so that ϕ = dζ^{2}, which determines a Euclidean metric |dζ^{2}| in that chart. The change of coordinates away from the zeroes of ϕ are of the form ζ → ±ζ + c, which preserves the Euclidean metric. Consequently, quadratic differentials are sometimes referred to as translation surfaces or flat structures with parallel line fields. A general flat structure or equivalently a surface with zero Gaussian curvature would allow for rotational components in the transition functions and therefore would not have a parallel line field. If one can always take the + sign in the change of coordinates, the quadratic differential is the square of an Abelian differential and the parallel line field is a vector field. We will henceforth adopt the notation S to refer to the structure of a quadratic differential. A local coordinate ζ can be chosen in the neighborhood of a zero of order k ≥ 1 so that quadratic differential is given by ζ^{k}dζ^{2}. The metric is |ζ^{k}||dζ^{2}|. The total angle around the zero is (k + 2)π, so we say that S has a cone angle (k + 2)π singularity. We will denote the set of singularities of S (i.e the zeroes of ϕ) by Z_{S}. Note that the curvature of the metric is 0, except for δ-function singularities at the points of Z_{S}.

∗Research partially supported by NSF grant DMS-9704845, the Sloan Foundation and the Packard Foundation

†Research partially supported by NSF grant 9803497

The number of zeroes counting multiplicity is 4g − 4. Let α be a partition of 4g − 4 (i.e. a representation of 4g − 4 as a sum of positive integers), and let ϵ denote a sign (+ or −). We say a quadratic differential ϕ determines (α, ϵ) if the orders of its zeroes are the partition α, and ϵ is + if ϕ is the square of an Abelian differential; −, if ϕ is not. An obvious necessary, but not sufficient condition for ϕ to be a square of an Abelian differential is to have all zeroes with even order.

Let QD(g, α, ϵ) denote the space of pairs (M, ϕ) where M is a Riemann surface of genus g, and ϕ is a holomorphic quadratic differential which determines (α, ϵ). This space is called a stratum; this term is justified by the fact that the space of all quadratic differentials on Riemann surfaces of genus g is stratified by the spaces QD(g, α, ±), as α varies over the partitions of 4g −4. These spaces have been studied by various authors, e.g. [Ma1], [MaSm], [Ve2], [Ve4]. The strata QD(g, α, ±) have projection maps to the moduli space R_{g} of Riemann surfaces of genus g. We note that QD(g, α, ϵ) are not always connected, but the number of connected components is always finite: see [KZ] for a classification. Let j be a label for a connected component, and let P denote the triple (α, ϵ, j). Thus QD(P) refers to a connected component of a stratum.

One way to visualize a point S ∈ QD(P) = QD(α, ϵ, j) where α = {k_{1}, … k_{n}} is as a finite union of polygons glued along parallel (or antiparallel) sides, such that each side is glued to exactly one other, and with special vertices v_{1}, … v_{n} such that at v_{i} there is a conical singularity of excess angle πk_{i}.

There is an SL(2, ℝ) action on QD(P). In each coordinate patch ζ ∈ ℝ^{2}, and for A ∈ SL(2, ℝ), the action is the linear action ζ → Aζ. The fact that the change of coordinates is given by ζ → ±ζ + c says that this is well-defined. If one visualizes S ∈ QD(P) as a union of polygons R_{1} ∪ ⋯ ∪ R_{n}, then gS = gR_{1} ∪ ⋯ ∪ gR_{n}. The orbits of this action (in Teichmuller space) are called Teichmuller disks, and have been studied extensively: see e.g. [Kra], [Ve3], [EG], [GuJ]. The SL(2, ℝ) action preserves an ergodic probability measure μ_{0} on QD(P) ([Ma1], [Ve1]). This measure is absolutely continuous with respect to the orbitfold structure and may be interpreted as the Liouville measure. One can also construct other invariant measures: those which come from certain invariant suborbitfolds of QD(P). The authors are not aware of any other constructions.

Cylinders of closed geodesics and saddle connections. A saddle connection of S is a geodesic segment joining two zeroes of S which has no zeroes in its interior. We note that the metric on the surface has non-positive curvature. Every homotopy class contains a geodesic; however this geodesic typically consists of a union of several saddle connections. If we give the saddle connection an orientation then it determines a vector in ℝ^{2} since in each coordinate chart it is geodesic with respect to the Euclidean metric. This vector can be thought of as the holonomy along the path given by the oriented saddle connection. The SL(2, ℝ) action on surfaces maps saddle connections to saddle connections. Given a (oriented) saddle connection β we can decompose the associated vector (which we also denote by β) as a sum of its horizontal part h_{β} and its vertical part v_{β}. The action on these components is the linear action

[ \begin{matrix}h_{β} \\ v_{β} \end{matrix} ] ↦ A · [ \begin{matrix}h_{β} \\ v_{β} \end{matrix} ] .

A closed geodesic that does not pass through any zeroes determines a cylinder of parallel freely homotopic closed geodesics of the same length. The cylinder is metric in that it induces an isometry from a right-angled flat Euclidean cylinder into the surface. Each boundary component of the cylinder consists of one or more parallel saddle connections. A cylinder of closed geodesics is called regular if both its boundaries consist of a single saddle connection.

Pointwise asymptotic formulas. One goal of this paper is to prove the following asymptotic formulas for the number of (oriented) closed geodesics and saddle connections on translation surfaces.

Theorem 1.1. For a quadratic differential S and T > 0, let N_{1}(S, T) denote the number of cylinders of closed geodesics on S such that the norm of the associated vector is at most T. For each P there exists a constant c_{P} > 0 such that for μ_{0} almost all S ∈ QD(P),

N_{1}(S, T) ∼ c_{P} T^{2}

Theorem 1.2. For a quadratic differential S and T > 0 let N_{2}(S, T) denote the number of saddle connections on S such that the norm of the associated vector is at most T. Then, for any P there exists a constant s_{P} > 0 such that for μ_{0} almost all S ∈ QD(P),

N_{2}(S, T) ∼ s_{P} T^{2}

Versions of the above theorems where convergence a.e. replaced by convergence in L^{1} are proved in [Ve5]. We also note some related results all of which will follow from our general result, Theorem 2.1:

Theorem 1.3. For a quadratic differential S and T > 0 let N_{3}(S, T) denote the number of irregular cylinders of closed geodesics on S such that the norm of the associated vector is at most T. Then, for any P, for μ_{0} almost all S ∈ QD(P),

N_{3}(S, T) = o(T^{2}).

Note that in Theorem 1.3, μ_{0} cannot be replaced by any ergodic SL(2, ℝ) invariant measure μ; for example on a Veech surface, one may have N_{3} ∼ cT^{2} for some c > 0, see [Ve3].

Some results on the value of the constants c_{P} and s_{P} are known, see [EO], [EMZ]. In this paper, for two related problem, we prove the asymptotic formula and also write down the constants explicitly. The expression for these constants is much simpler then the expressions for the constants c_{P} and s_{P} in the known cases.

Let Y_{1} → QD(P) the fiber bundle over QD(P) where the fiber over each point in QD(P) is the corresponding surface with flat structure S. Similarly, let Y_{2} → QD(P) be the bundle whose fiber is S × S. Thus, a point in Y_{1} is a pair (S, p), where S is a surface with a flat structure, and p is a point in S. A point in Y_{2} is a triple (S, p_{1}, p_{2}) where S is a surface with a flat structure, and p_{1}, p_{2} are points in S. The fiber bundle Y_{i} inherits a natural SL(2, ℝ) invariant probability measure μ_{i}, which is the measure μ_{0} on the base QD(P) and is just Euclidean measure induced by the flat structure (product of flat structures) on the fiber.

Theorem 1.4.

(1) Given a flat structure S and p ∈ S\Z_{S} let N_{4}(S, p, T) be the number of segments of length at most T joining p to a zero of the quadratic differential. Then for μ_{1} a.e. (S, p) ∈ Y_{1},

N_{4}(S, p, T) ∼ πkT^{2}/2,

where kπ is the sum of the cone angle at the zeroes, Z_{S}.

(2) For (p_{1}, p_{2}) ∈ (S \ Z_{S}) × (S \ Z_{S}) let N_{5}(S, p_{1}, p_{2}, T) denote the number of segments joining p_{1} and p_{2} of length at most T. Then for μ_{2} a.e (S, p_{1}, p_{2}) in Y_{2},

N_{5}(S, p_{1}, p_{2}, T) ∼ πT^{2}.

The lack of connection to billiards. We note that there is a well known construction which associates a surface with a flat structure to each rational polygon (see [ZK], [Gut], [KMS]). In particular counting families of periodic trajectories of the billiard in the polygon is equivalent to counting cylinders of closed geodesics on the surface constructed from the polygon. However, the set of surfaces which are constructed from polygons has μ_{0} measure 0 in the stratum (see however [EZ]). Thus, Theorem 1.1 cannot be used to obtain information about billiards in rational polygons. This is in contrast with the quadratic upper bound [Ma2] and lower bound [Ma3] theorems, and the asymptotic formulas for Veech surfaces [Ve3]. However, the techniques of this paper, together with Ratner’s theorem [Rat] can also be used to obtain asymptotic formulas for billiards in some special polygons which do not give rise to Veech surfaces; see [EMaS]. We also note that we give another proof of the upper bound of [Ma2] in §5.

Spherical averages of K-finite functions. Let r_{θ} = ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ). Let

K = {r_{θ} : θ ∈ [0, 2π]} denote the standard SO(2) ⊂ SL(2, ℝ). Let M be a space on which SL(2, ℝ) acts. A function f on M is called K-finite if the linear span of the functions {f ∘ r_{θ} : θ ∈ [0, 2π]} is finite dimensional. (Thus f is K-finite if and only if there exists an integer m such that for almost all x ∈ SO(2)\M, p(θ) = f(r_{θ}x) is a trigonometric polynomial of degree at most m.) The main ergodic tool we use is the following general ergodic theorem:

Theorem 1.5 (Nevo). Let a_{t} = ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} ), and let K denote the standard SO(2) ⊂

SL(2, ℝ). Let M be a space on which SL(2, ℝ) acts and let μ be a ergodic SL(2, ℝ) invariant probability measure on M. Suppose ϕ ∈ C_{0}^{∞}(ℝ) is any nonnegative function, ϵ > 0 and suppose f ∈ L^{1+ϵ}(M, μ) is a K-finite function. Then, for μ-almost every x in M,

lim_{τ→∞} ∫_{−∞}^{∞} ϕ(τ − t) (\frac{1}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}x) dθ) dt = ∫_{−∞}^{∞} ϕ(t) dt ∫_{M} f(y) dμ(y).

Note in this theorem, ϕ is a smoothing function. Basically the assertion of the theorem is a smoothed version of the assertion

lim_{t→∞} \frac{1}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}x) dθ = ∫_{M} f(y) dμ(y).

Readers Guide. The paper is organized as follows: in §2 we formulate the general case of the counting problem considered in this paper. We then state our main theorem Theorem 2.1. The constants in that formula are given by the Siegel-Veech formula Theorem 2.2. As a consequence of these two theorems we prove Theorem 1.1, Theorem 1.2, and Theorem 1.3 in §2. The rest of the paper is devoted to the proof of Theorem 2.1. In §3, following [Ve5], we relate the counting problem to consideration of certain integrals over orbits of SL(2, ℝ). In §4 we use Nevo’s Theorem 1.5 to obtain asymptotic results almost everywhere for the integrals in §3.

The results of §3-§4 are conditional on the assumptions (A), (B) and (C_{μ}) formulated in §2. For the key case of counting saddle connections, (A) is immediate, but (B) and (C_{μ}) require certain a priori upper bounds. These upper bounds are verified in §5, based on two results of independent interest, Theorem 5.1 and Theorem 5.2. We also show in §5 that Theorem 5.1 and Theorem 5.2 imply the quadratic upper bound result of [Ma2].

In §6-§8, which are more technical then the rest of the paper, we prove Theorem 5.1 and Theorem 5.2. In §9, we compute the constants in the asymptotic formulas of Theorem 1.4. Acknowledgements. We would like to thank W. Veech, who besides inspiring this project also corrected an error in the initial version of this paper. We would also like to thank the referee for many valuable comments.

2 The counting problems

The space M and the measure μ. It is convenient to axiomatize the properties we will need. Let M denote a space on which SL(2, ℝ) acts. For most of the paper, M will denote QD(P), but in §9, M will denote the fiber bundles Y_{i}. Let μ be a probability measure on M invariant under the SL(2, ℝ) action. Let B(T, 0) denote the Euclidean ball of center (0, 0) and radius T, and let | · | denote the cardinality of a finite set.

In the rest of the paper V (S) will denote a subset of ℝ^{2}−{(0, 0)} with multiplicity; that is, a set of vectors assigned positive weights. They typically will be integer weights, but in some applications, they need not be. Suppose to every point S ∈ M we associate such a set so that the following hold:

(A) V varies linearly under SL(2, ℝ) action. More precisely, for g ∈ SL(2, ℝ), V (gS) = gV (S).

(B) For any S ∈ M, there exists a constant c = c(S) > 0, such that for any T > 1 the cardinality of V (S) ∩ B(T, 0) is at most cT^{2}. The constant c can be chosen uniformly as S varies over any compact subset of M.

(C_{μ}) For each T > 0 let N_{V} (T, S) denote the cardinality of V (S) ∩ B(T, 0). Then there exist some R > 0 and ϵ > 0 such that as a function of S ∈ M, N_{V} (R, S) ∈ L^{1+ϵ}(M, μ).

We note that if V (·) denotes the holonomy vectors associated to oriented closed geodesics or to saddle connections, then assumption (A) is clearly satisfied. The fact that assumption (B) is satisfied is a theorem of H. Masur [Ma2] which we will reprove as Theorem 5.4 in §5. We will also verify that condition (C_{μ}) is also satisfied for every invariant probability measure μ. These statements together with the following theorem will allow us to prove our asymptotic formulas.

Theorem 2.1. Let μ be an ergodic invariant probability measure on M. Suppose V (·) satisfies assumptions (A), (B) and (C_{μ}). Then there exists a constant c_{V,μ} = c(V, μ, M) such that for μ-almost all S ∈ M,

lim_{T →∞} \frac{1}{T^{2}} N_{V} (T, S) = πc_{V,μ}

The constant c_{V,μ} is the constant in a “Siegel-Veech formula”; see Theorem 2.2 below.

We note that we do not know the value of c_{V,μ} in many cases. For some results, see [EO], [EMZ].

Remark. Conditions (A), (B) and (C_{μ}) are not sufficient to ensure that c_{V,μ} > 0; indeed, Theorem 1.3 is an example when c_{V,μ} = 0. See the statement of Theorem 2.2 for a necessary and sufficient condition to ensure that c_{V,μ} > 0 (and thus that the asymptotic formula is quadratic).

2.1 The Siegel-Veech formula

In this subsection we recall some constructions and results from [Ve5]. For completeness, and since our assumptions differ somewhat from those of [Ve5], we reproduce the proofs below.

The functions f and f̂. For f ∈ C_{0}^{∞}(ℝ^{2}), let f̂: QD(P) → ℝ be defined by

f̂(S) = ∑_{v∈V (S)} f(v).

The terms in the sum are counted with multiplicity. By assumption (B), the sum is finite, and also f̂ is a bounded function on compact subsets of QD(P)

For technical reasons one needs f to be a smooth function, but if one takes f to be the characteristic function of a ball of radius T centered at the origin, then f̂(S) will count the number of elements of V (S) which lie in a ball of radius T.

A generalization of the following theorem is proved in [Ve5]:

Theorem 2.2 (Siegel-Veech formula). Suppose V satisfies assumptions (A), (B), and (C_{μ}). Let μ be any ergodic invariant probability measure on M. Then there exists a constant c_{V,μ} such that for any f ∈ C_{0}^{∞}(ℝ^{2}),

∫_{M} f̂(S) dμ(S) = c_{V,μ} ∫_{ℝ^{2}} f(x, y) dxdy.

The constant c_{V,μ} is always non-negative; it is zero if and only if f̂ = 0 μ-almost everywhere.

Proof. We present the idea of the proof from [Ve5] for completeness. By assumption (C_{μ}) and since μ is a probability measure, f̂ ∈ L^{1+ϵ}(M, μ) ⊂ L^{1}(M, μ). Consider the linear functional

Φ(f) = ∫_{M} f̂(S) dμ(S).  (1)

Since μ is invariant under the SL(2, ℝ) action on M, Φ is invariant under the SL(2, ℝ) action on continuous functions on ℝ^{2}. Thus it follows from the general form of such functionals that for all f ∈ C_{0}^{∞}(ℝ^{2}),

Φ(f) = af(0, 0) + b ∫_{ℝ^{2}} f.  (2)

where a and b are independent of f. Pick a function ϕ ∈ C_{0}^{∞}(ℝ^{2}) so that ϕ(0, 0) = 1, and ϕ ≤ 1, and let f_{j}(x, y) = ϕ(jx, jy)f(x, y). Then, f_{j}(0, 0) = f(0, 0), and ∫_{ℝ^{2}} f_{j} → 0 by dominated convergence. Note that for every fixed S ∈ M, since f̂(S) < ∞ and (0, 0) ∉ V (S), lim_{j→∞} f̂_{j}(S) = 0. Hence, using (1) and the fact that f̂ ∈ L^{1}, we get (by dominated convergence) Φ(f_{j}) → 0. Substituting f_{j} instead of f into (2) and taking the limit j → ∞, we get a = 0. Now set c_{V,μ} = b. The final assertion is immediate.

Proof of Theorem 1.2 This is immediate from Theorem 2.1 and Theorem 2.2 since every surface has a saddle connection of bounded length so that for some f, f̂(S) ≠ 0 for all S.

Proof of Theorem 1.3 Now we need to show that f̂(S) = 0 for μ_{0} a.e. S. In other words we have to show that a generic surface does not have an irregular cylinder. We first consider the case of differentials that are squares of Abelian differentials. Note that all the periodic trajectories comprising a cylinder are all homotopic to the same curve on the surface; we call this curve the “core curve” of the cylinder. The holonomy along the core curve is the height of the cylinder, and is thus nonzero. Since the holonomy along any curve γ is equal to the integral of the Abelian differential along γ, this implies that the core curve of the cylinder is not homologous to 0. Thus the saddle connections on the boundary of an irregular cylinder are linearly independent in H_{1}(S, Z_{S}) (the homology relative to the zeroes). Local coordinates on the stratum are given by the holonomy vectors of a basis for H_{1}(S, Z_{S}). Thus the fact that these linearly independent holonomy vectors are real multiples of each other means that the set of flat structures defining an irregular cylinder near the given one is a positive codimension set, and hence has μ_{0} measure 0. The result follows by taking the union over the at most countable set of homotopy classes.

If the stratum is not defined by Abelian differentials, the proof is the same if the cylinder is not homologous to 0. If it is, we pass to a double cover S′, ramified over the zeroes. The quadratic differential lifts to an Abelian differential and the cylinder lifts to a pair of cylinders, each of which is not homologous to 0; although they are homologous to each other, and are interchanged by the involution on the double cover. Now local coordinates on the original space QD(P) are given by the holonomy vectors of a basis for the part of the relative homology group H_{1}(S′, Z_{S′}) which is odd with respect to the involution. Hence, again the set of surfaces with an irregular metric cylinder are a positive codimension subset.

Proof of Theorem 1.1 We again apply Theorem 2.1 and Theorem 2.2. We need to show that the constant c_{P} given in Theorem 2.2 is positive. Let S_{0} be any surface with a metric cylinder. Such surfaces are easy to construct. For example, one can take a right angled flat Euclidean cylinder and make certain isometric gluings of the sides on the top and bottom to form a surface. Such a cylinder is not regular but it is immediate that every nearby surface also has a cylinder whose core curve is in the same homotopy class. Now by Theorem 1.3 the set of nearby surfaces such that the cylinder is irregular has positive codimension. Thus there must be a nearby surface with a regular cylinder and that surface has an open neighborhood of surfaces with a regular cylinder. Thus for an appropriate f, the function f̂ is positive on an open set and we are done.

3 Counting and integration over M

Let a_{t} = ( \begin{matrix}e^{t} & 0 \\ 0 & e^{−t} \end{matrix} ), and for θ ∈ [0, 2π], let r_{θ} = ( \begin{matrix}cos θ & sin θ \\ − sin θ & cos θ \end{matrix} ). Let ℝ_{+}^{2} denote

the upper half-plane.

Definition 3.1 (Generic Points). Suppose μ is an ergodic invariant probability measure on M. A point S ∈ M is called μ-generic if for any f ∈ C_{0}^{∞}(ℝ_{+}^{2}) and all nonnegative ϕ ∈ C_{0}^{∞}(ℝ),

lim_{τ→∞} \frac{1}{2π} ∫_{−∞}^{∞} ϕ(τ − t) (∫_{0}^{2π} f̂(a_{t}r_{θ}S) dθ) dt = ∫_{−∞}^{∞} ϕ(t) dt ∫_{M} f̂(S_{1}) dμ(S_{1}). (3)

Note that the integral on the left is smoothed version of \frac{1}{2π} ∫_{0}^{2π} f̂(a_{t}r_{θ}S) dθ. The proof of Theorem 2.1 breaks up naturally into two parts, which we state as Proposition 3.2 and Proposition 3.3 below.

Proposition 3.2. Suppose μ is an ergodic invariant probability measure on M, V (·) satisfies conditions (A),(B) and (C_{μ}) and S is μ-generic. Then,

lim_{T →∞} \frac{1}{T^{2}} N_{V} (T, S) = πc_{V,μ},

where c_{V,μ} is as in Theorem 2.2.

Proposition 3.3. Suppose μ is a ergodic invariant probability measure on M, and V (·) satisfies conditions (A) (B) and (C_{μ}). Then, for μ-almost all S ∈ M, S is μ-generic.

Proposition 3.3 will be derived from Theorem 1.5. In this section, we will prove Proposition 3.2. We will also prove an elementary result, Proposition 3.5, which will be used in the proofs of the upper bounds (B).

The results in this section are essentially contained in §5 of [Ve5] (which were in turn inspired in part by [EMM] following an analogy noted in [Ve5]). In fact, our treatment, in this section and the next, is an almost verbatim translation of §3 of [EMM].

Lemma 3.4. (cf. [EMM, Lemma 3.6]) For f ∈ C_{0}^{∞}(ℝ_{+}^{2}), let

J_{f}(y) = \frac{1}{2πy} ∫_{−∞}^{∞} f(x, y) dx

Then for every ϵ > 0 there exists T_{0} > 0 such that for every t with e^{t} > T_{0}, and every v ∈ ℝ^{2} with ∥v∥ > T_{0},

|\frac{e^{2t}}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ − J_{f}(∥v∥e^{−t})| < ϵ.  (4)

Heuristic argument for Lemma 3.4. (See figure ??) Suppose f is the characteristic function of the rectangle R = [−b, b] × [c, d] (where b > 0, and 0 < c < d). Then, by definition, J_{f}(∥v∥e^{−t}) = 2b/(2π∥v∥e^{−t}) if ce^{t} ≤ ∥v∥ ≤ de^{t} and is zero otherwise. Also, a_{t}^{−1}R is the rectangle [−be^{−t}, be^{−t}] × [e^{t}c, e^{t}d]. Since the integrand in (4) is zero unless r_{θ}v ∈ a_{t}^{−1}R, \frac{1}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ is \frac{1}{2π∥v∥} times the length of the intersection of the circle of radius ∥v∥ with the rectangle a_{t}^{−1}R (the factor of 1/(2π∥v∥) arises because the length of an arc of angle θ on a circle of radius ∥v∥ is ∥v∥θ). Now, the intersection of the circle with a_{t}^{−1}R is empty if ∥v∥ is outside of [ce^{t}, de^{t}] and is essentially the horizontal segment [−be^{−t}, be^{−t}] × {∥v∥} otherwise. Thus, \frac{1}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ ≈ \frac{2}{2π}(be^{−t})/∥v∥ = e^{−2t}J_{f}(∥v∥e^{−t}).

Proof of Lemma 3.4. Clearly, the left hand side of (4) depends on v only via the norm ∥v∥. Thus, we may assume that v = (0, ∥v∥). Then,

\frac{1}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ = \frac{1}{2π} ∫_{0}^{2π} f(e^{t}∥v∥ sin θ, e^{−t}∥v∥ cos θ) dθ  (5)

Since t and ∥v∥ are large, and f has compact support, the integrand is zero unless sin θ is near 0. Hence θ must be near 0 or π. Since f is supported on the upper half plane ℝ_{+}^{2}, θ must be near 0. From the form of (5) it is clear that the integrand is 0 unless ∥v∥e^{−t} is bounded between c and c^{−1}, and unless θ = O(e^{−t}∥v∥^{−1}) = O(e^{−2t}), where the c > 1 and the implied constants depend only on the support of f.

Let δ(ρ) = sup{|f(x, y_{1}) − f(x, y_{2})| such that |y_{1} − y_{2}| < ρ}. By the uniform continuity of f, δ(ρ) → 0 as ρ → 0.

We now make the change of variable u = e^{t}∥v∥ sin θ in (5); the integral becomes:

\frac{e^{−t}}{2π∥v∥} ∫_{−∞}^{∞} f (u, e^{−t}∥v∥\sqrt{1 − (e^{−t}u/∥v∥)^{2}}) (1 − (e^{−t}u/∥v∥)^{2})^{−1/2} du =  (6)
\frac{e^{−t}}{2π∥v∥} ∫_{−∞}^{∞} f(u, e^{−t}∥v∥) du + E = e^{−2t}J_{f}(e^{−t}∥v∥) + E,

where the error term E is bounded by:

|E| = O ((e^{−t}/∥v∥)δ((e^{−t}/∥v∥)^{2})) + O ((e^{−t}/∥v∥)∥f∥_{∞}(e^{−t}/∥v∥)^{2})) = O(e^{−2t}δ(c^{2}e^{−4t})).

After multiplying both sides by e^{2t}, the lemma follows.

As an illustration of the basic technique, we first prove the following elementary result:

Proposition 3.5. Suppose V (·) satisfies condition (A) and the intersection of V (·) with any compact set is finite. Then, there exists an absolute constant c < ∞ such that for any T > 0 and any S ∈ M,

N_{V} (2T, S) − N_{V} (T, S) ≤ cT^{2} ∫_{0}^{2π} N_{V} (4, a_{t}r_{θ}S) dθ

where t = log T.

Proof of Proposition 3.5. Let ℝ_{+} denote the positive reals. Note that any function in C_{0}^{∞}(ℝ_{+}) can be written as J_{f} for some f ∈ C_{0}^{∞}(ℝ_{+}^{2}). Let χ denote the characteristic function of the interval [1, 2], and let ϵ > 0 be arbitrary. Choose f ∈ C_{0}^{∞}(ℝ_{+}^{2}) so that J_{f} ≥ (1 + ϵ)χ, and the support of f is contained in [−2.5, 2.5] × [0, 2.5]. Let T_{0} be as in Lemma 3.4, and suppose e^{t} > T_{0}. By Lemma 3.4, if e^{t} ≤ ∥v∥ ≤ 2e^{t}, then

1 ≤ J_{f}(e^{−t}∥v∥) − ϵ ≤ \frac{e^{2t}}{2π} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ.  (7)

Let S be a point in M. Summing the outermost inequality in (7) over those v in V (S) that belong to the annulus e^{t} ≤ ∥v∥ ≤ 2e^{t}, we get

N_{V} (2e^{t}, S) − N_{V} (e^{t}, S) = ∑_{\begin{matrix}v∈V (S) \\ e^{t}≤∥v∥≤2e^{t} \end{matrix}} 1 ≤ \frac{e^{2t}}{2π} ∑_{\begin{matrix}v∈V (S) \\ e^{t}≤∥v∥≤2e^{t} \end{matrix}} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ ≤  (8)

≤ \frac{e^{2t}}{2π} ∑_{v∈V (S)} ∫_{0}^{2π} f(a_{t}r_{θ}v) dθ = \frac{e^{2t}}{2π} ∫_{0}^{2π} f̂(a_{t}r_{θ}S) dθ (9)

Set c = sup f. The function f is bounded by c times the characteristic function of the rectangle [−2.5, 2.5] × [0.2.5]. This rectangle is a subset of the disk B(4, 0) of radius 4 centered at the origin, so f is majorized by cg where g is the characteristic function of the disk. Hence f̂(S) is majorized by cĝ(S). But ĝ(S) = N_{V} (4, S) and so (8) becomes

N_{V} (2T, S) − N_{V} (T, S) < \frac{cT^{2}}{2π} ∫_{0}^{2π} N_{V} (4, a_{t}r_{θ}S) dθ

where we set T = e^{t}. This gives the Lemma (for a new constant c).

Proposition 3.5 will be used in §5 and a variation, Lemma 8.1 will be used in §8.

Proof of Proposition 3.2. Let χ be the characteristic function of the interval [1, 2], and let ϵ > 0 be arbitrary. Let c be as in condition (B) for the generic surface S. Choose h_{+} ∈ C_{0}^{∞}(ℝ_{+}) h_{−} ∈ C_{0}^{∞}(ℝ_{+}) so that h_{−} ≤ χ ≤ h_{+} and ∫_{0}^{∞} y(h_{+}(y) − h_{−}(y)) dy ≤ ϵ/c. Choose f_{±} ∈ C_{0}^{∞}(ℝ_{+}^{2}) so that J_{f_{±}} = h_{±}. Choose ϕ ∈ C_{0}^{∞}(ℝ) be non-negative with ∫_{−∞}^{∞} ϕ = 1, and with the support of ϕ so close to 0 that J_{ϕ∗f_{−}} ≤ χ ≤ J_{ϕ∗f_{+}}, where we use the notation (ϕ ∗ f)(v) = ∫_{−∞}^{∞} ϕ(t)f(a_{−t}v)dt. Then, for any v ∈ ℝ^{2} and any τ > 0,

J_{ϕ∗f_{−}}(∥v∥e^{−τ}) ≤ χ(∥v∥e^{−τ}) ≤ J_{ϕ∗f_{+}}(∥v∥e^{−τ}).

Using Lemma 3.4, for sufficiently large τ we get

\frac{e^{2τ}}{2π} ∫_{0}^{2π}(ϕ ∗ f_{−})(a_{τ}r_{θ}v) dθ − ϵ/c ≤ χ(∥v∥e^{−τ}) ≤ \frac{e^{2τ}}{2π} ∫_{0}^{2π}(ϕ ∗ f_{+})(a_{τ}r_{θ}v) dθ + ϵ/c. (10)

Summing over v ∈ V (S) and noting that by condition (B), the number of terms with χ(∥v∥e^{−τ}) ≠ 0 is at most ce^{2τ}, we get

\frac{e^{2τ}}{2π} ∫_{0}^{2π}(ϕ ∗ f̂_{−})(a_{τ}r_{θ}S) dθ − ϵe^{2τ} ≤ N_{V} (2e^{τ}, S) − N_{V} (e^{τ}, S) ≤

≤ \frac{e^{2τ}}{2π} ∫_{0}^{2π}(ϕ ∗ f̂_{+})(a_{τ}r_{θ}S) dθ + ϵe^{2τ} (11)

Since S is μ-generic and ∫_{−∞}^{∞} ϕ = 1,

lim_{τ→∞} \frac{1}{2π} ∫_{0}^{2π}(ϕ ∗ f̂_{±})(a_{τ}r_{θ}S) dθ = ∫_{M} f̂_{±} dμ = c_{V,μ} ∫_{ℝ^{2}} f_{±},  (12)

where in the last identity we used the Siegel-Veech formula Theorem 2.2. But, by construction,

∫_{ℝ^{2}} f_{±}(x, y) dx dy = 2π ∫_{0}^{∞} yJ_{f_{±}}(y) dy = 2π(3/2 + E)  (13)

where |E| < ϵ/c. Hence, after dividing (11) by e^{2τ}, taking the limit as τ → ∞, and using (12) and (13) we get

3πc_{V,μ} − (2πc_{V,μ}/c + 1)ϵ ≤ lim inf_{τ→∞} e^{−2τ}(N_{V} (2e^{τ}, S) − N_{V} (e^{τ}, S))
≤ lim sup_{τ→∞} e^{−2τ}(N_{V} (2e^{τ}, S) − N_{V} (e^{τ}, S))
≤ 3πc_{V,μ} + (2πc_{V,μ}/c + 1)ϵ.

Since ϵ > 0 is arbitrary, this implies

lim_{T →∞} \frac{1}{T^{2}} (N_{V} (2T, S) − N_{V} (T, S)) = 3πc_{V,μ}

where we let T = e^{τ}. From this, it is easy to conclude that

lim_{T →∞} \frac{1}{T^{2}} N_{V} (T, S) = πc_{V,μ}

as required.

4 Approximation by K-finite functions, Proof of Proposition 3.3.

Lemma 4.1. Let C be a compact set in ℝ^{2} − {0}. There exists a countable family of (complex valued) functions F ∈ C_{0}^{∞}(ℝ^{2} − {0}) with the following properties:

(1) For each f ∈ F there exists m ∈ ℤ such that for all v ∈ ℝ^{2} and all θ ∈ [0, 2π], f(r_{θ}v) = e^{imθ}f(v).

(2) For every q ∈ C_{0}(ℝ^{2} − {0}) such that the support of q is contained in C and for every ϵ > 0 there exist f_{1} and f_{2} finite linear combinations of functions in F such that the f_{i} are real-valued,

f_{1} ≤ q ≤ f_{2}

and

∫_{ℝ^{2}}(f_{2} − f_{1}) < ϵ  (14)

Also there exists a countable family Φ of functions in C_{0}^{∞}(ℝ) such that for every ϕ ∈ C_{0}^{∞}(ℝ) there exist ϕ_{1}, ϕ_{2} finite linear combinations of functions in Φ such that ϕ_{1} ≤ ϕ ≤ ϕ_{2} and ∫_{ℝ}(ϕ_{2} − ϕ_{1}) < ϵ.

Proof. We may fix a radially symmetric function h ∈ C_{0}^{∞}(ℝ^{2} − {0}) such that h = 1 on C. Let F = {hf_{m,n}(r, θ) : m ∈ ℕ, n ∈ ℤ}, where f_{m,n}(r, θ) = r^{m}e^{inθ}. Then F satisfies the conditions of the lemma by e.g. the Stone-Weierstrass theorem. For Φ one may use a similar construction.

Proof of Proposition 3.3. We note that if f is K-finite, then so is f̂. By Theorem 1.5 there exists a set E ⊂ M with μ(E) = 1 such that for any S ∈ E any f ∈ F, and any ϕ ∈ Φ, (3) holds. Let ϵ > 0 be arbitrary, and let ϕ_{1}, ϕ_{2}, f_{1}, f_{2} be as in Lemma 4.1. Then f̂_{1} ≤ f̂ ≤ f̂_{2}, hence

∫_{−∞}^{∞} ϕ_{1}(τ − t) (∫_{0}^{2π} f̂_{1}(a_{t}r_{θ}S) dθ) dt ≤ ∫_{−∞}^{∞} ϕ(τ − t) (∫_{0}^{2π} f̂(a_{t}r_{θ}S) dθ) dt

≤ ∫_{−∞}^{∞} ϕ_{2}(τ − t) (∫_{0}^{2π} f̂_{2}(a_{t}r_{θ}S) dθ) dt (15)

Suppose S ∈ E. Since f_{i} and ϕ_{i} are finite linear combinations of functions in F and Φ,

lim_{τ→∞} ∫_{−∞}^{∞} ϕ_{i}(τ − t) (∫_{0}^{2π} f̂_{i}(a_{t}r_{θ}S) dθ) dt = ∫_{ℝ} ϕ_{i} ∫_{M} f̂_{i} dμ = c_{V,μ} ∫_{ℝ} ϕ_{i} ∫_{ℝ^{2}} f_{i} (16)

where in the last equality we used the Siegel-Veech formula. Since ϵ > 0 is arbitrary,

(3) follows from (15), (16) and (14).

5 Upper bounds

5.1 Statement of Results

Letting V (·) denote the set of vectors (with multiplicity) associated to saddle connections, then in this section we will prove that conditions (B) and (C_{μ}) hold for every μ. This allows us to apply Theorem 2.1. Conditions (B) and (C_{μ}) will follow from the following two theorems:

Theorem 5.1. Suppose V (·) denotes the set of vectors associated to saddle connections. Then, for any stratum QD(P) there exists κ > 0 such that (a) For any δ > 0 any R < κ and any S ∈ QD(P),

|V (S) ∩ B(R, 0)| ≤ C (\frac{R}{ℓ(S)})^{1+δ} ,  (17)

where C = C(δ, P) and ℓ(S) denotes the length of the shortest saddle connection in S.

(b) For any δ > 0 any R < ∞ and any S ∈ QD(P), (17) holds with C = C(δ, R, P).

Theorem 5.2. For any S ∈ QD(P) and any β < 2,

sup_{t>0} ∫_{0}^{2π} ℓ(a_{t}r_{θ}S)^{−β} dθ < ∞

and the upper bound can be chosen uniformly as S varies over compact subsets of QD(P).

Corollary 5.3. Let K = {r_{θ} : 0 ≤ θ ≤ 2π}, so that K ≅ SO(2). Let S any flat structure and let ν denote the K-invariant measure supported on the set KS. Then the set of measures a_{t}ν is compact (in the weak-* topology),

The proofs of Theorem 5.1 and Theorem 5.2, which combines the ideas of [Ma2] with the ideas of G. Margulis, will be given in §6-§8. In the rest of this section we will show how Theorem 5.1 and Theorem 5.2 can be used to derive condition (B) for the saddle connections) and also to verify that condition (C_{μ}) holds for the saddle connections for every invariant measure μ.

We begin, however by showing how Corollary 5.3 follows from Theorem 5.2.

Proof of Corollary 5.3 For any ϵ > 0, let QD_{ϵ}(P) = {S ∈ QD(P) : ℓ(S) < ϵ}. Then QD(P) \ QD_{ϵ}(P) is compact. Taking β = 1 in Theorem 5.2, there is a constant M such that for all ϵ and t, a_{t}ν(QD_{ϵ}(P)) ≤ Mϵ. Since the set of probability measures on a compact space is weak-* compact, we are done.

Theorem 5.4. Suppose V (S) denotes the set associated to saddle connections on S. Then, V (·) satisfies condition (B), i.e. for any S ∈ QD(P), there exists a constant c = c(S) such that for all T > 0,

N_{V} (T, S) ≤ cT^{2}

The constant c can be chosen uniformly as S varies over compact subsets of QD(P).

This theorem was originally proved in [Ma2]. Of course Theorem 5.4 implies the analogous result for the number of cylinders of closed geodesics since the number of such cylinders on a surface is bounded trivially by the number of saddle connections.

Proof of Theorem 5.4. By Proposition 3.5, Theorem 5.1 and Theorem 5.2, there exist R > 0 and β < 2 such that we have

N_{V} (2T, S) − N_{V} (T, S) ≤ cT^{2} ∫_{0}^{2π} N_{V} (R, a_{t}r_{θ}S) dθ ≤ c′T^{2} ∫_{0}^{2π} \frac{dθ}{ℓ(a_{t}r_{θ}S)^{β}} ≤ c′′T^{2}

Now the theorem follows from an easy geometric series argument.

Lemma 5.5. (cf. [Ve5, Corollary 2.8]) Let μ be any ergodic invariant probability measure on QD(P). Then for any 1 ≤ β < 2, ℓ(·)^{−1} ∈ L^{β}(QD(P), μ).

Proof. If μ = μ_{0}, it is possible to prove this directly. For general μ, we derive this by an indirect argument from Theorem 5.2 and the ergodic Theorem 1.5. In fact, we will only need the K-fixed version of Theorem 1.5, see [NS]. It suffices to prove that for any β < 2, ℓ^{−β} ∈ L^{1}(QD(P), μ). Assume this is false.

Let χ be the characteristic function of [1, ∞] and for j ∈ ℕ let h_{j}(S) = ℓ(S)^{−β}χ(jℓ(S)). Then h_{j} are bounded with compact support, since the set of quadratic differentials whose shortest saddle is bounded below is compact. As j → ∞, h_{j} ↑ ℓ^{−β}. Then by monotone convergence, ∫_{QD(P)} h_{j}(S) dμ(S) → ∞.

Fix a positive ϕ ∈ C_{0}^{∞}(ℝ), and consider the integrals

I_{j}(τ) = ∫_{−∞}^{∞} ϕ(τ − t) (∫_{0}^{2π} h_{j}(a_{t}r_{θ}S) dθ) dt.

Since h_{j} is K-fixed (i.e. h_{j}(r_{θ}S) = h_{j}(S) for all θ ∈ [0, 2π] and all S), by Theorem 1.5, for almost all S and all j, lim_{τ→∞} I_{j}(τ) = ∫_{QD(P)} h_{j}dμ(S) ∫_{−∞}^{∞} ϕ. Since for all j, ℓ^{−β} ≥ h_{j} we get, for all j,

lim inf_{τ→∞} ∫_{−∞}^{∞} ϕ(τ − t) (∫_{0}^{2π} ℓ(a_{t}r_{θ}S)^{−β} dθ) dt ≥ lim_{τ→∞} I_{j}(τ) = ∫_{QD(P)} h_{j}dμ(S) ∫_{−∞}^{∞} ϕ.

Taking j → ∞ we get a contradiction since the right-hand side tends to infinity, and the left hand side remains bounded by Theorem 5.2.

Proof of condition (C_{μ}) for the saddle connections. This follows immediately from Theorem 5.1 and Lemma 5.5.

6 Combinatorial Constructions

Saddle connections, p_{0}, complexes and domains. We say that two saddle connections are disjoint if the only common points (if any) are zeroes. A union of disjoint saddle connections Γ is called a system. By |Γ| we mean the maximum length of any member of Γ. By a complex K we mean a closed subset of S, whose boundary, denoted ∂K, is a system of saddle connections. Here ∂K refers to the subset of K consisting of those points such that every neighborhood in S of the point intersects the complement of K. Thus if K consists of a single saddle connection then ∂K = K. Note that a complex need not be connected and is not assumed to be triangulated. However we will assume that if ∂K contains three saddle connections that bound a triangle, then the triangle is included in K.

We now show that there is a uniform bound p_{0} to the number of disjoint saddle connections. Indeed, given any set of disjoint saddle connections, we may further add disjoint saddle connections so that all complimentary regions are discs and then further subdivide the discs into triangles. This gives a triangulation of the surface with edges which are saddle connections and whose vertices are the m zeroes. Let e the number of edges and t the number of triangles. Then Euler’s formula gives m − e + t = χ, the Euler characteristic of the surface. Each edge is contained in 2 triangles and each triangle has 3 edges; thus 3t = 2e. This implies that the maximum number p_{0} of disjoint saddle connections is at most 3(m − χ). By a domain Y we will mean an open set in S with boundary a system Γ of saddle connections. A domain is not considered to be triangulated.

Area, complexity, and domain-complexity. The area of a complex K is denoted ||K||. Note that if K is a complex, then K \ ∂K is a (possibly empty) domain. The complexity of a complex is the number of saddle connections needed to triangulate the complex, including the boundary. The complexity of a domain is the number of saddle connections needed to triangulate it, not counting the number on the boundary.

Cross Products. Given a saddle connection β considered as a vector we may decompose β as a sum of its horizontal component β_{h} and its vertical component β_{v}. Given ω and β saddle connections we can form the length of the vector product

|ω × β| = |β_{h}ω_{v} − β_{v}ω_{h}|.

This number is invariant under the SL(2, ℝ) actions on flat structures and saddle connections. For example if S has area 1 and ω and β are two sides of a triangle, then since the triangle has area at most 1, and the length of the vector product measures the area of the corresponding parallelogram, we must have |ω × β| ≤ 2.

Proposition 6.1 (Cross Products). Suppose γ_{1}, γ_{2}, β are saddle connections. Then

(1) |γ_{1} × γ_{2}| ≤ \frac{|γ_{1}×β||γ_{2}|}{|β|} + \frac{|γ_{2}×β||γ_{1}|}{|β|}. (Hence if |γ_{1}| and |γ_{2}| are bounded in terms of |β|, then |γ_{1} × γ_{2}| can be bounded in terms of |γ_{1} × β| and |γ_{2} × β|).

(2) If γ_{1}, γ_{2}, β are three sides of a triangle and γ is another saddle connection, then |β × γ| ≤ ∑_{i=1}^{2} |γ_{i} × γ|.

Proof. To prove (1), rotate the coordinate system so that β is vertical; |γ_{i,h}| = \frac{|γ_{i}×β|}{|β|}.

Since |γ_{i,v}| ≤ |γ_{i}|, (1) follows. To prove (2), rotate so γ is vertical so |γ_{i,h}| = \frac{|γ_{i}×γ|}{|γ|}. Then (2) follows from the fact that |β_{h}| ≤ |γ_{1,h}| + |γ_{2,h}| and |β × γ| = |β_{h}||γ|.

In the next section we will need to combine or merge intersecting complexes to form bigger complexes. The main combinatorial lemma is the following.

Proposition 6.2 (Combining complexes). Suppose K is a complex of complexity i with nonempty boundary and there is a saddle connection σ which is either disjoint from K or crosses ∂K. Then there is a complex K′ containing K of complexity j > i with the following properties:

(i) |∂K′| ≤ |∂K| + max(|∂K|, |σ|).

(ii) ||K′|| ≤ ||K|| + |∂K| max(|σ|, |∂K|).

(iii) For any ω′ ⊂ K′ \ K and any γ we have |ω′ × γ| ≤ max_{ω∈∂K}(|ω × γ| + max(|ω × γ|, |σ × γ|)).

(iv) If β is a saddle connection disjoint from both K and σ, then β is disjoint from K′ or β ⊂ K′.

Proof. Heuristically, to get K′, we add σ to K and retriangulate the boundary, if necessary. More carefully, the construction can be done as follows: f σ is disjoint from K, we may just add it to K and all the estimates hold. Thus we may assume that σ crosses ∂K. Let Q be the point where σ crosses ∂K on a segment ω with endpoints A, B. Suppose in following σ from Q into the exterior of K σ hits a singularity P in the exterior of K before again crossing ∂K or hits a singularity on ∂K. Let σ_{1} be the subsegment of σ joining Q and P. (see figure ??) Consider the path AQ ∗ σ_{1} joining A to P, where ∗ denotes juxtaposition of paths. There is a unique geodesic l_{A} joining A to P in the homotopy class of that path. If not all saddle connections in l_{A} are already in K, we add those that are not in K giving us a larger complex K′. Since |l_{A}| ≤ |ω| + |σ| and ||Δ|| ≤ |σ||ω|,(recall that |∂K| is not the length of ∂K, but rather the length of the longest saddle connection in ∂K), the length and area bounds (i) and (ii) are immediate. The second statement of Proposition 6.1 implies that for any segment ω′ of l_{A} and any γ,

|ω′ × γ| ≤ |σ_{1} × γ| + |ω × γ| ≤ |σ × γ| + max_{ω∈∂K} |ω × γ|

giving (iii). Since l_{A}, σ_{1} and AQ bound a simply connected domain which contains no singularities in its interior, it is clear that if β is disjoint from K and σ, it is either one of the segments of l_{A} or disjoint from K′ so we have (iv).

If every saddle connection of l_{A} is already a subset of K, then we apply the same analysis to l_{B}. Now l_{A}, l_{B} and ω bound a simply connected domain Δ. If Δ is a triangle, then it cannot be the case that the single segment l_{B} is also already in K, for then by assumption, Δ would already be in K and σ would not cross ω into the exterior of K. In this case can add l_{B} to form K′. If Δ is not a triangle and every segment of l_{B} is already in K, then we may add a saddle connection in Δ that goes from an endpoint of ω, say A to a singularity on l_{B}. The same estimates still hold.

Thus we are reduced to the case that before next hitting a singularity, σ crosses ∂K at Q′ an interior point of a segment ω′ with endpoints C, D. (see figure ??) It may be the case that A = D or B = C or both. Let σ_{1} be the subsegment of σ joining Q to Q′. There are two cases. The first case is if σ_{1} is homotopically nontrivial rel ∂K; in that case move σ_{1} parallel to itself with one endpoint moving along ω toward one endpoint, say A, the other along ω′, towards say D. We can choose the direction so that the lengths of the parallel segments do not increase. This can be done until for a first time there is a singularity P and a pair of segments, σ_{2}, one of which goes from ω to P, the other from ω′ to P. (If P is either A or D, then σ_{2} is a single segment.) The assumption that σ_{1} is homotopically nontrivial guarantees that σ_{2} does not lie in ∂K. It is clear that if β does not intersect σ, ω or ω′, then it does not intersect σ_{2}. We proceed exactly as in the previous paragraph. We can use P to build a bigger complex whose boundary satisfies the same estimates on lengths and cross products as before.

The second possibility is that σ_{1} is homotopically trivial rel ∂K. (see figure ??) In that case σ_{1} together with segments of ω and ω′ and possibly other segments of ∂K bound a simply connected domain in the complement of K. This implies that there is a geodesic γ in the complement of K joining an endpoint of ω and an endpoint of ω′ such that γ, ω, ω′ and the same other segments of ∂K bound a simply connected domain in the complement of K. Then we may add the segments of γ to K. The above estimates hold. It is also the case that if β is disjoint from σ and ∂K, then either it is disjoint from γ or coincides with a segment of γ.

Corollary 6.3 (Complex generated by short saddle connections). There are constants c = c(p_{0}) < ∞ and c′ = c′(p_{0}) such that for any a > 0, there is a complex Y containing all saddle connections of S of length at most a with |∂Y | ≤ c′a and ||Y || ≤ ca^{2}.

Proof. We start with a complex consisting of a maximal set of disjoint saddle connections of length at most a. We fill in any triangles. Each has area at most a^{2}. If there are no additional segments of length at most a, we are done, with c the number of triangles. Suppose otherwise and suppose inductively we have built a complex Y_{j} of complexity j, with boundary length at most c′_{j}a and area at most c_{j}a^{2}. If any saddle connection of length at most a crosses ∂Y_{j} by the above Proposition we can find a complex Y_{j+1} containing Y_{j} whose boundary has length has length at most 2c′_{j}a + a = c′_{j+1}a and area at most c_{j}a^{2} + c′_{j}a(a + c′_{j}a) = c_{j+1}a^{2}. Since the maximum complexity of any complex is p_{0}, the induction ends after at most p_{0} steps.

Definition 6.4. For Y satisfying the conclusions of Corollary 6.3 we say Y is generated by saddle connections of length a.

The number ϵ. We now fix a number ϵ = ϵ(p_{0}) so that cϵ^{2} < 1 where c is as in Corollary 6.3. This implies that if ||Y || = 1, any complex K ⊂ Y generated by saddle connections of length ϵ must in fact be a proper subset of Y.

Corollary 6.5 (Intersections of complexes).

(a) Suppose Y is a complex and Ω is a set of saddle connections contained in Y. Then there is a complex Z ⊂ Y containing every saddle connection in Ω.

(b) Suppose Y_{1}, Y_{2} are two complexes. Then there exists a complex Z ⊂ Y_{1} ∩ Y_{2} which contains every saddle connection in Y_{1} ∩ Y_{2}.

Proof. To prove (a), as in the proof of Corollary 6.3 we build inductively Z out of the saddle connections in Ω. The only thing to check is that at each stage the resulting complex is a subset of Y. So suppose K is a complex which is a subset of Y and σ ∈ Ω crosses ∂K. Then either σ ∈ ∂Y or every segment in ∂Y must be either contained in ∂K or disjoint from both σ and ∂K. Hence, by Proposition 6.2 (iv), ∂Y cannot cross the enlarged K′. This implies that K′ ⊂ Y. This completes the inductive step in the proof of (a). To prove (b), apply (a) with Ω equal to the set of saddle connections in Y_{1} ∩ Y_{2}.

Definition 6.6 (Filled in Complex). Given a complex Y with boundary ∂Y and a complex K which does not contain ∂Y, we define the filled in complex K_{f} to be the set of points which are separated from ∂Y by K. Equivalently. K_{f} is the union of K with all the components of Y \ K whose closure does not intersect ∂Y.

Definition 6.7 (Separate). A system Ω separates two disjoint saddle connection β and γ from each other if every arc from β to γ intersects Ω. We allow Ω to coincide with one of the saddle connections.

Definition 6.8 (Accessible, Weakly Accessible). Suppose Y is a domain, γ ⊂ ∂Y, and let M > 1 and C > 0. We say that a saddle connection β ⊂ Y is M accessible from γ if for any system Ω that separates β from γ

(1) |Ω| > |γ|/M.

(In particular, since we are allowing Ω to coincide with β, we must have |β| > |γ|/M).

We say β is (M, C) weakly accessible from γ if for any separating system Ω either (1) holds or for some saddle connection ω ⊂ Ω,

(2) |ω × γ| > C||Y ||.

Remarks. We will use Definition 6.8 with M ≫ 1 and C ≫ 1. Note that always, if β is M-accessible from γ, then β is (M, C) weakly accessible from γ. If ||Y || = 1, C ≥ 2, |γ| ≤ 1 and (2) holds, then |ω| ≥ 1 and so (1) holds; this means that if ||Y || = 1, |γ| ≤ 1, and C ≥ 2, then β is M-accessible from γ if and only if β is (M, C) weakly-accessible from γ.

Proposition 6.9. There exists a sequence of constants D_{j} such that if Y is a domain which can be triangulated by j saddle connections (not counting the saddle connections on the boundary of Y), ||Y || = 1, β ⊂ Y is a saddle connection weakly (p_{0}, D_{j}) accessible from γ ⊂ ∂Y then there is a chain of saddle connections β = β_{1}, β_{2}, …, β_{n} = γ such that

(1) β_{i} is disjoint from β_{i+1}

(2) |β_{i} × β_{i+1}| ≤ D_{j}.

(3) β_{i} is weakly (p_{0}, D_{j}) accessible from β_{i+1}.

The proof of Proposition 6.9 is based on the following:

Claim 6.10. Suppose C′ > 0. Let Y be a domain with ||Y || = 1, and let β ⊂ Y and γ ⊂ ∂Y be saddle connections.

For D > 2 let F_{D} denote the set of saddle connections λ ⊂ Ȳ such that |λ×γ| ≤ D and λ and β are disjoint. Let n(D) denote the maximal cardinality of a disjoint system contained in F_{D}. Then at least one of the following is true:

(a) There exists a saddle connection σ ⊂ Y with |σ| ≤ |γ|/p_{0} and |σ × γ| ≤ D.

(b) There exists a saddle connection λ ∈ F_{D} such that β is (p_{0}, C′) weakly accessible from λ.

(c) for D′ = p_{0}C′ + (\frac{1}{p_{0}} + 2)D we have n(D′) > n(D).

(d) |β × γ| < p_{0}C′ + (\frac{1}{p_{0}} + 2)D.

Proof of Claim 6.10. Suppose (a) and (b) do not hold. We note that F_{D} is not empty, since there exists a saddle connection τ which bounds a triangle with γ, so that |τ × γ| ≤ 2.

Choose a maximal system Γ of disjoint saddle connections in F_{D}, and let K be the complex that they form. Let Y_{β} be the connected component of Y \ ∂K which contains β, and choose a saddle connection λ ⊂ K ∩ ∂Y_{β}. Since |λ × γ| ≤ D and (a) does not hold,

|λ| ≥ |γ|/p_{0}.  (18)

Since (b) does not hold, β is not (p_{0}, M′) weakly accessible from λ. This means that there is some system Ω separating β from λ such that neither (1) nor (2) holds. Choose any ω ∈ Ω \ K. Since (1) and (2) do not hold

|λ| ≥ p_{0}|ω| and |ω × λ| ≤ C′.  (19)

Now from Proposition 6.1 (i) applied to the saddle connections ω, γ, λ we get

|ω × γ| ≤ |ω × λ|\frac{|γ|}{|λ|} + |γ × λ|\frac{|ω|}{|λ|} ≤ p_{0}C′ + \frac{D}{p_{0}}  (20)

We now apply Proposition 6.2 to K and ω in place of σ to get a complex K′ ⊃ K. Let ω_{1} be any saddle connection in K′ which is disjoint from K. Then it follows from Proposition 6.2 part (iii) and (20) that

|ω_{1} × γ| ≤ |ω × γ| + 2 max_{λ∈K} |λ × γ| ≤ p_{0}C′ + \frac{D}{p_{0}} + D.  (21)

Now by Proposition 6.2 (iv), either w_{1} = β or w_{1} is disjoint from β. In the former case, (d) holds in view of (21). In the latter case, Γ ∪ ω_{1} ∈ F_{D′} where D′ is the right-hand-side of (21), so (c) holds.

Proof of Proposition 6.9. The proof is by induction on the number d(Y) of saddle connections needed to triangulate Y, not counting the saddle connections on the boundary of Y. (We note that d(Y) is not exactly the complexity of the complex \overline{Y} since we do not count boundary saddle connections in d(Y)). The proposition is clearly true if d(Y) = 0, so that Y is a triangle or a disjoint union of triangles. Now suppose the proposition has been proved for all domains Z with d(Z) < j, and Y is a domain with d(Y) = j. We assume D_{i}, i < j have been chosen. Let C′ = D_{j−1}, and for D > 0, let f(D) = p_{0}C′ + (\frac{1}{p_{0}} + 2)D. Let D_{j} = f^{(p_{0}+1)}(D_{j−1}) where f^{(k)} denotes f iterated k times.

Suppose first that for some D < D_{j}, (a) of Claim 6.10 occurs. Consider the domain Y \ σ. We claim that if β is not (p_{0}, D_{j−1}) weakly accessible from γ on Y \ σ, it is not (p_{0}, D_{j}) weakly accessible from γ on Y, contrary to assumption. For adding σ to any system Ω that separates β from γ on Y \ σ gives a system Ω′ that separates β from γ on Y for which neither (1) nor (2) hold. Now, since d(Y \ σ) = j − 1, we can apply the induction hypothesis to Y \ σ to get the required chain of saddle connections.

Hence we may assume that (a) of Claim 6.10 does not occur for any D_{j−1} ≤ D ≤ D_{j}. This implies that every saddle connection on Y is (p_{0}, D_{j}) weakly accessible from β_{n} = γ. We now apply Claim 6.10 repeatedly to

D_{j−1}, f(D_{j−1}), f^{(2)}(D_{j−1}), …, f^{(p_{0}+1)}(D_{j−1}) = D_{j}.

Note that (c) of Claim 6.10 can occur at most p_{0} times. Thus, for some D < D_{j}, either (d) or (b) occurs. In the first case we set n = 2 and we have the chain β, γ. In the latter case we can set β_{n−1} = λ, and apply the induction hypothesis.

7 A system of integral inequalities

In this section, we prove Theorem 5.2, together with a variation, Proposition 7.2, which will be used in §8 in the proof of Theorem 5.1. The main idea of this section (i.e. the system of inequalities Lemma 7.5) is similar to ideas developed by G.A. Margulis for the “quantitative version of the Oppenheim conjecture” (cf. [EMM]).

For a domain Y, let

ℓ(Y) = min {|λ| : λ is a saddle connection contained in Y } .

For γ a saddle connection on S, for some angle θ, γ determines a vector [0, |γ|] on the surface r_{θ}S. The angle θ is called the vertical angle for γ.

Given a domain Y of area 1 we may use the SL(2, ℝ) action to form the domains a_{t}r_{θ}Y. Fix γ ⊂ ∂Y. Recall the constant p_{0}, which is the maximum number of disjoint saddle connections.

Definition 7.1 (Accessible saddle connection). For any θ and t, we say that a a saddle connection λ ⊂ a_{t}r_{θ}Y is accessible if r_{θ}^{−1}a_{t}^{−1}λ ⊂ Y is p_{0}-accessible from γ.

In this definition notice that the accessible condition refers to lengths on Y. Let

ℓ_{γ}(a_{t}r_{θ}Y) = min {|λ| : λ is an accessible saddle connection contained in a_{t}r_{θ}Y } .

Proposition 7.2. Let Y be a domain, γ ⊂ ∂Y and |γ| ≤ 1. By rotating let 0 be the vertical angle for γ. For each constant ρ and t, let I(t) = I(t, γ, ρ) be the interval of angles [ρe^{−t}/|γ|, ρe^{−t}/|γ|]. Then for any δ, 0 < δ < 1/2 there exists ρ and C′ (depending only on δ and ρ) such that for any t,

∫_{I(t)} \frac{dθ}{ℓ_{γ}(a_{t}r_{θ}X)^{1+δ}} ≤ C′ \frac{e^{−(1−2δ)t}}{|γ|^{1+2δ}} .

Remarks. Proposition 7.2 differs from Theorem 5.2 as follows: In Proposition 7.2 we integrate only over the exponentially small sector I(t) (instead of [0, 2π]) and consider only accessible saddle connections (instead of all saddle connections). Also Proposition 7.2 asserts that the integral is small (instead of bounded). However, the proofs of Proposition 7.2 and Theorem 5.2 are virtually identical. We will only give the proof of Proposition 7.2 in detail, and indicate how to modify it to obtain Theorem 5.2 at the end of this section. In §8, Proposition 7.2 will be used in the inductive step of the proof of Theorem 5.1.

7.1 Admissible complexes and the function α.

Definition 7.3 (Admissible Complex). Fix γ ⊂ ∂Y. For any θ, t, a complex K′ ⊂ a_{t}r_{θ}Y with boundary ∂K′ and of complexity i is admissible on a_{t}r_{θ}Y if

(1) ||K′|| ≤ 2^{i}ϵ^{2} where ϵ is the constant defined following Corollary 6.3.

(2) for any system Ω ⊂ Y of saddle connections separating r_{θ}^{−1}a_{t}^{−1}K′ ⊂ Y from γ ⊂ ∂Y, |Ω| > |γ|/p_{0}.

In particular if K′ is admissible then |∂(r_{θ}^{−1}a_{t}^{−1}K′)| ≥ |γ|/p_{0}.

The functions α_{i}. Fix δ > 0. Let Y be a domain with ||Y || = 1. We define

α_{i}(a_{t}r_{θ}Y) = max (sup_{K_{i}} \frac{1}{|∂K_{i}|^{1+δ}} , 1)

where the sup is over all admissible subcomplexes K_{i} of a_{t}r_{θ}Y of complexity i.

The operator A_{τ}. For any τ > 0 let A_{τ} denote the averaging operator which sends the function f defined on the SL(2, R) orbit of a surface S_{0} to

(A_{τ}f)(S_{0}) = \frac{1}{2π} ∫_{0}^{2π} f(a_{τ}r_{θ}S_{0}) dθ,

We will be applying A_{τ} to functions f on SL(2, ℝ) which are invariant under the action of SO(2) on the right, and can thus be considered as functions on the hyperbolic plane SO(2)\SL(2, R). For such f, (A_{τ}f)(z) is the average of f over the circle of radius τ centered at z.

7.2 A system of inequalities.

In this subsection we show that the functions α_{i} satisfy a certain system of integral inequalities (see Lemma 7.5 below). The idea is that if α_{i} is large, so that S_{0} is near infinity in a stratum, then most points on the circle of radius τ centered at S_{0} are further from infinity, which means that the average of the boundary length of K_{i} is bigger on the circle than at S_{0}. If K_{i} defines the supremum over the entire circle in the definition of α_{i}, then A_{τ}α_{i}(S_{0}) is smaller than S_{0}. If for some point on the circle, some other K′_{i} of complexity i has a shorter boundary than K_{i}, then we ”merge” K_{i} and K′_{i} as in §6 to form a complex K_{j} of complexity j > i, and bound α_{i} in terms of α_{j}.

Lemma 7.4. Fix 0 < δ < 1. Then there exists a constant c(δ) such that for any τ > 0 and any v ∈ ℝ^{2} − {(0, 0)},

\frac{1}{2π} ∫_{0}^{2π} \frac{dθ}{∥a_{τ}r_{θ}v∥^{1+δ}} ≤ \frac{c(δ)e^{−τ(1−δ)}}{∥v∥^{1+δ}}  (22)

Proof. By rescaling and rotating may assume that v = (0, 1). Then, the left-hand-side of (22) becomes:

\frac{1}{2π} ∫_{0}^{2π}(e^{2τ} sin^{2} θ + e^{−2τ} cos^{2} θ)^{−(1+δ)/2} dθ

We decompose [0, 2π] = R_{1} ∪ R_{2}, where R_{1} = {θ : e^{2τ} sin^{2} θ ≤ e^{−2τ} cos^{2} θ} and R_{2} is the set where the opposite inequality holds. Note that there exist absolute constants 0 < c_{1} < c_{2} such that

c_{1}e^{−2τ} ≤ |R_{1}| ≤ c_{2}e^{−2τ}  (23)

On R_{1}, the integrand is bounded by a constant multiple of e^{τ(1+δ)}. Hence, in view of (23), the integral over R_{1} is O(e^{−τ(1−δ)}) as required. Now the integral over R_{2} is bounded by

e^{−τ(1+δ)} ∫_{R_{2}} | sin θ|^{−(1+δ)} dθ = O(e^{−τ(1+δ)} ∫_{R_{2}} |θ|^{−(1+δ)} dθ = O(e^{−τ(1−δ)}),

where in the last estimate we used (23).

Lemma 7.5 (System of Inequalities). For any 0 < δ < 1 and any τ > 0 there exist constants ω = ω(τ, δ) and b = b(τ, δ) such that for any i and any S_{0} = a_{t}r_{ϕ}Y

(A_{τ}α_{i})(S_{0}) ≤ ce^{−(1−δ)τ}α_{i}(S_{0}) + b + ω ∑_{j>i} α_{j}(S_{0}).

where c is the constant of Lemma 7.4

Proof. We can assume that on S_{0} the length of the boundary of the complex K_{i} that defines the supremum in α_{i} has length at most e^{−2τ}ϵ. Otherwise on the circle of radius τ centered at S_{0}, α_{i} is uniformly bounded, and thus we can absorb the left-hand-side in the constant b.

Suppose first K_{i} is the complex that defines α_{i} throughout the circle of radius τ. Then A_{τ}α_{i} is the average of the minimum of

\frac{1}{∥a_{τ}r_{θ}v∥^{1+δ}} ,

the minimum taken over v ∈ ∂K_{i}. Since the average of a minimum is smaller than the minimum of an average, by Lemma 7.4, we have the desired bound for α_{i}. Now suppose for some θ ∈ [0, 2π] some other complex K_{i}^{θ} ⊂ S_{0} with the same complexity defines α_{i}(a_{τ}r_{θ}S_{0}), i.e. |∂(a_{τ}r_{θ}K_{i}^{θ})| ≤ |∂(a_{τ}r_{θ}K_{i})|. Since lengths on a_{τ}r_{θ}S_{0} differ from lengths on S_{0} by a factor of at most e^{τ},

|∂K_{i}^{θ}| ≤ e^{τ}|∂(a_{τ}r_{θ}K_{i}^{θ})|  (24)

Here |∂K_{i}^{θ}| refers to lengths on S_{0}. Assume first that K_{i} and K_{i}^{θ} have boundaries that cross. We take σ ∈ ∂K_{i}^{θ} \ ∂K_{i} and using Proposition 6.2 we can build a complex K_{j}^{θ} of complexity j = j(θ) > i containing K_{i}. We check that K_{j}^{θ} is admissible. On the contrary, if there were a system Ω separating r_{ϕ}^{−1}a_{t}^{−1}K_{j}^{θ} from γ such that |Ω| ≤ |γ|/p_{0}, then Ω would separate r_{ϕ}^{−1}a_{t}^{−1}K_{i} from γ, contradicting that K_{i} is admissible. Thus (2) in the definition of admissible holds. Since the boundary lengths of each of K_{i} and K_{i}^{θ} are at most ϵ and the area of K_{i} is at most 2^{i}ϵ^{2}, the area bound (1) follows from Proposition 6.2.

We also have

|∂K_{j}^{θ}| ≤ |∂K_{i}| + max(|∂K_{i}|, |∂K_{i}^{θ}|) ≤ 2|∂K_{i}^{θ}|,  (25)

the first inequality by Proposition 6.2 and the second since K_{i} defines the supremum at S_{0}. Thus, in view of (25), for any θ

α_{i}(a_{τ}r_{θ}S_{0}) = \frac{1}{|∂(a_{τ}r_{θ}K_{i}^{θ})|^{1+δ}} ≤ \frac{ω}{|∂K_{j}^{θ}|^{1+δ}} ≤ ωα_{j}(S_{0}),  (26)

where ω = (2e^{τ})^{1+δ}. Hence the desired estimate also holds in this case.

If the boundaries of K_{i} and K_{i}^{θ} do not cross, then we take K_{j}^{θ} = K_{i}∪K_{i}^{θ}. Note that if Ω separates r_{ϕ}^{−1}a_{t}^{−1}K_{j}^{θ} from γ, then Ω separates r_{ϕ}^{−1}a_{t}^{−1}K_{i} from γ, a contradiction. This shows that (2) in the definition of admissible holds. We have ∥K_{j}^{θ}∥ ≤ ∥K_{i}∥ + ∥K_{i}^{θ}∥ ≤ 2^{i+1}ϵ^{2} ≤ 2^{j}ϵ^{2}, so K_{j}^{θ} is admissible. Clearly, (25) and hence (26) hold in this case as well. This completes the proof of the lemma.

7.3 Averages over arcs.

In this subsection we will deduce Theorem 5.2 and Proposition 7.2 from Lemma 7.5.

We identify the right cosets SO(2)\SL(2, ℝ) with the hyperbolic disk D^{2}; thus a function on SL(2, ℝ) such that f(r_{θ}g) = f(g) for all θ ∈ [0, 2π], g ∈ SL(2, ℝ) may be considered to be a function on D^{2}. Let o denote the origin of D^{2} (corresponding to the identity coset). For θ ∈ S^{1} and t > 0 the point z_{0} = o · a_{t}r_{θ} in the hyperbolic disk D^{2} is on the circle of hyperbolic radius t centered at the origin and at angle θ. Now as ϕ varies over S^{1}, consider the points o · a_{τ}r_{ϕ}a_{t}r_{θ}. These points form the circle of radius τ centered at z_{0}. In polar coordinates we write

o · a_{τ}r_{ϕ}a_{t}r_{θ} = o · a_{s}r_{θ+ψ}

where s = s(t, τ, ϕ) and ψ = ψ(t, τ, ϕ). Thus o · a_{τ}r_{ϕ}a_{t}r_{θ} is at hyperbolic distance s(t, τ, ϕ) from the origin and is visible at the angle θ + ψ(t, τ, ϕ).

Recall the intervals I(t, γ, ρ) defined in Proposition 7.2. For each t > 0, κ > 0 and θ ∈ I(t + τ) ⊂ I(t) let U_{θ}(κ, t, τ) ⊂ S^{1} denote the set of points ϕ ∈ S^{1} such that s(t, τ, ϕ) ≥ t + τ − κ and θ + ψ(t, τ, ϕ) ∈ I(t + τ), i.e. o · a_{τ}r_{ϕ}a_{t}r_{θ} lies outside the circle of hyperbolic radius t + τ − κ and its radial projection to the circle of radius t + τ lies in I(t + τ).

Lemma 7.6. There exists c′′ > 0 so that for each κ, there is a τ and ρ such that for any θ ∈ I(t + τ), |U_{θ}| ≥ c′′|S^{1}|. The map Ψ : U_{θ} → S^{1} sending ϕ to ψ(t, τ, ϕ) is a diffeomorphism onto its image and

2πc′′ ≤ |U_{θ}| = ∫_{U_{θ}} dϕ = ∫_{Ψ(U_{θ})} |\frac{dϕ}{dψ} | dψ  (27)

Figure 1. (Lemma 7.6). The dotted circle has center o and radius t + τ − κ. The shaded circle has hyperbolic radius τ and is centered at a point z ∈ J(t). The dotted lines starting at z are hyperbolic geodesics; |U_{θ}| is the angle between them. Hyperbolic geometry and the assumptions that τ is much larger then κ and that the diameter of J(t + τ) ≥ 2τ + 2κ imply that |U_{θ}| is bounded from below.

Proof. (See figure ??.) We use the notation J(t) = {o · a_{t}r_{θ} : θ ∈ I(t)}, so J(t) ⊂ D^{2} is the part of the circle of radius t around o which is seen from the origin at angles in I(t). Given κ, for τ large enough, the assertion that o · a_{τ}r_{ϕ}a_{t}r_{θ} lies outside the circle of hyperbolic radius t + τ − κ for most angles ϕ is just an exercise in hyperbolic geometry. Now choose ρ = ρ(κ, |γ|) large enough so that for any t, J(t + τ) has hyperbolic diameter at least 2τ + 2κ (it is enough to do this for t = 0). By construction then, for any θ ∈ I(t + τ) for at least 1/2 of the angles ϕ, θ + ψ(t, τ, ϕ) ∈ I(t + τ). The second assertion follows formally from the first.

Lemma 7.7. Let f be any non-negative continuous function on SL(2, ℝ) invariant under left multiplication by SO(2); (that is, a function on D^{2}) with the property that given any σ > 1 there exists κ > 0 such that for g ∈ D^{2} of hyperbolic distance at most κ from o, and any h ∈ SL(2, R) σ^{−1}f(h) ≤ f(gh) ≤ σf(h). Choose τ, ρ such that Lemma 7.6 holds. Then

∫_{I(t+τ)} f(a_{t+τ}r_{θ}) dθ ≤ c′σ ∫_{I(t)}(A_{τ}f)(a_{t}r_{θ}) dθ

where c′ does not depend on τ.

Proof. For each θ ∈ I(t + τ),

(A_{τ}f)(a_{t}r_{θ}) = \frac{1}{2π} ∫_{0}^{2π} f(a_{τ}r_{ϕ}a_{t}r_{θ}) dϕ ≥ \frac{1}{2π} ∫_{U_{θ}} f(a_{τ}r_{ϕ}a_{t}r_{θ}) dϕ ≥  (28)
≥ \frac{σ^{−1}}{2π} ∫_{Ψ(U_{θ})} f(a_{t+τ}r_{θ+ψ})|\frac{dϕ}{dψ} | dψ

where for the last estimate we used the definition of U_{θ} and the condition on f. Now integrating (28) over θ ∈ I(t + τ) we get

∫_{I(t+τ)}(A_{τ}f)(a_{t}r_{θ}) dθ ≥ \frac{σ^{−1}}{2π} ∫_{I(t+τ)} ∫_{Ψ(U_{θ})} f(a_{t+τ}r_{θ+ψ})|\frac{dϕ}{dψ} | dψdθ

= \frac{σ^{−1}}{2π} ∫_{I(t+τ)} f(a_{t+τ}r_{ξ}) (∫_{−Ψ(U_{ξ})} |\frac{dϕ}{dψ} | dψ) dξ (29)
≥ c′′σ^{−1} ∫_{I(t+τ)} f(a_{t+τ}r_{ξ}) dξ

where in the second line we made the change of variable ξ = ψ + θ and for the last inequality we used (27) to estimate from below the inner integral. (To see the domain of the inner integral in (29) one can argue as follows: Take point on circle of radius t and on the real axis. Take the circle of radius τ centered at that point. Let p_{−}, p_{+} be the two points where that circle intersects the circle of radius t + τ − κ centered at the origin. Denote the angles of p_{−} and p_{+} as viewed from the origin by −w(t) and w(t) respectively. Then set W = [−w(t), w(t)]. The domain of the inner integral in (29) is −Ψ(U_{ξ}) since before the change of variables the domain of integration was {(θ, ψ) : θ ∈ I(t + τ) and ψ ∈ W ∩ (I(t + τ) − θ)}. After the change of variables the domain becomes {(ξ, ψ) : ξ ∈ I(t + τ) and ψ ∈ W ∩ (−I(t + τ) + ξ)} which for the inner integral translates exactly to −Ψ(U_{ξ}).)

Since I(t) ⊃ I(t + τ), (29) implies the estimate of the lemma.

Proof of Proposition 7.2. For notational convenience we set α_{i} = 1 for i > p_{0}. the maximal complexity of a complex. Fix σ > 1. Since α_{i} is invariant under left multiplication by rotations, and since elements of SL(2, ℝ) near the identity distort lengths of saddle connections by a multiplicative factor at most σ, we may choose κ so that α_{i} satisfies the hypotheses of Lemma 7.7. We may then choose τ sufficiently large so that applying Lemma 7.5 with Lemma 7.7 we get

∫_{I(t+τ)} α_{i}(a_{t+τ}r_{θ}Y) dθ ≤ cc′σe^{−(1−δ)τ} ∫_{I(t)} α_{i}(a_{t}r_{θ}Y) dθ + c′σb|I(t)| +

+ c′σω ∑_{j=i+1}^{p_{0}} ∫_{I(t)} α_{j}(a_{t}r_{θ}Y) dθ (30)

We want to prove that for all i, there is a constant C_{i}(ρ, τ, b) such that

∫_{I(t)} α_{i}(a_{t}r_{θ}Y) dθ ≤ C_{i} \frac{e^{−t}e^{2δt}}{|γ|^{1+2δ}}  (31)

This will be proved by reverse induction on i. Then, (31) is trivially true for i > p_{0} (recall that |I(t)| = 2ρe^{−t}/|γ| and |γ| ≤ 1). Now suppose (31) has already been proved for all j > i. It is clearly enough to prove (31) for i when t = nτ. Let a_{n} = ∫_{I(nτ)} α_{i}(a_{nτ}r_{θ}Y) dθ. Then, the inequality (30) can be rewritten for a new constant ω′ = ω′(ω, b, C_{j}, c′, σ) as

a_{n+1} ≤ e^{−τ}(cc′σe^{δτ})a_{n} + \frac{ω′}{|γ|^{1+2δ}} e^{−nτ}e^{2δnτ}  (32)

Iterating this equation, and taking out the common factor of e^{−nτ} we get for a new constant ω′′

a_{n} ≤ e^{−nτ} ((cc′σ)^{n}e^{nδτ}a_{0} + \frac{ω′′}{|γ|^{1+2δ}} ∑_{k=1}^{n−1}(cc′σe^{δτ})^{k}e^{2(n−k)δτ})

Note that a_{0} ≤ (p_{0}/|γ|)^{1+δ}, since by definition an admissible complex has boundary length at least |γ|/p_{0}. If we choose τ large enough (e.g. so that cc′σe^{−δτ} < 1/2), the geometric series above is bounded, up to an absolute constant, by the term with k = 1. Thus, we get for all n,

a_{n} ≤ \frac{C_{4}e^{−nτ}e^{2nδτ}}{|γ|^{1+2δ}}

This implies (31). Now Proposition 7.2 follows from the fact that α_{1}(K) = 1/ℓ_{γ}(K)^{1+δ}.

Proof of Theorem 5.2. The proof of this theorem is a simplified version of the proof of Proposition 7.2. The only differences are that one declares all complexes to be admissible, and one integrates over the entire circle [0, 2π] instead of over I(t). Then, since Lemma 7.5 holds on the entire hyperbolic plane, the proof goes through without modifications.

8 The Induction

In this section we prove Theorem 5.1. We will use the following version of Proposition 3.5:

Lemma 8.1. Let Y be a domain, and for Θ > 0 and R_{2} > R_{1} > 0 let A(R_{1}, R_{2}, Θ) denote the set of vectors in ℝ^{2} of length between R_{1} and R_{2} which make an angle of at most Θ with the vertical. Let V (·) be a (set-valued) function on the orbit SL(2, ℝ)Y which satisfies condition (A) in §2. For T > 0 and S ∈ SL(2, ℝ)Y let N_{V} (T, S) denote the cardinality of V (S) ∩ B(T, 0). Then, for any ϵ, 0 < ϵ < 1, there exists a constant c = c(ϵ) > 0 such that for any T > 1, and any Θ > 1/T,

|A(ϵT/4, ϵT/2, Θ) ∩ V (Y)| ≤ cT^{2} ∫_{−2Θ}^{2Θ} N_{V} (ϵ, a_{t}r_{θ}Y) dθ  (33)

where t = log T.

Outline of proof. Let f(x, y) be 16π/ϵ^{2} times the characteristic function of the rectangle [−ϵ/4, ϵ/4] × [ϵ/8, 3ϵ/4]. Then, it is easy to check (cf. the proof of Lemma 3.4) that for any t > 0, any Θ > e^{−t} and any v ∈ A(ϵe^{t}/4, ϵe^{t}/2, Θ),

e^{2t} ∫_{−2Θ}^{2Θ} f(a_{t}r_{θ}v) dθ ≥ 1.

The rest of the argument follows exactly the proof of Proposition 3.5.

The next theorem is the major step in the proof of Theorem 5.1. First recall the constants D_{l} given by Proposition 6.9

Theorem 8.2. For any l, the following hold:

(U_{l}) Let Y be a domain of complexity at most l and ||Y || ≤ 1. Suppose γ ∈ ∂Y, and for R > 0, let N_{1}(γ, R) denote the number of saddle connections β such that |β| ≤ R, β is (p_{0}, D_{l})-weakly accessible from γ and |β × γ| ≤ D_{l}||Y ||. Then, for each δ > 0 there is C_{1} = C_{1}(l, δ, D_{l}) such that for each R > 0

N_{1}(γ, R) ≤ C_{1} (\frac{R}{|γ|})^{1+δ}

(V_{l}) Let Y and γ be as in (U_{l}). Let N_{2}(γ, R) denote the number of saddle connections β such that |β| < R and β is (p_{0}, D_{l}) weakly- accessible from γ. Then, for each δ > 0 there is C_{2} = C_{2}(l, δ) such that for each R > 0

N_{2}(γ, R) ≤ C_{2} (\frac{R}{|γ|})^{1+δ}

(W_{l}) Let Y be as in (U_{l}), and K a possibly empty complex that does not contain ∂Y. Let Y_{K} denote the complement of the filled in complex K_{f}. Let N(Y, K, R) denote the number of saddle connections in Y of length at most R which intersect Y_{K}; that is to say, are not contained in K_{f}. Then, for each δ > 0, there is C_{3} = C_{3}(l, δ) such that for each R > 0

N(Y, K, R) ≤ C_{3} (\frac{R}{ℓ(Y, K)})^{1+δ}

where ℓ(Y, K) denotes the length of the shortest saddle connection in Y that intersects Y_{K} (i.e. not contained in K_{f}).

Remark. We note that Theorem 5.1 (a) follows easily from (W_{l}). Indeed, let ϵ > 0 be the constant defined following Corollary 6.3, and let S ∈ QD(P) be arbitrary. Form the complex Y ⊂ S containing all saddle connections of length at most ϵ. Y is a complex with boundary, and the shortest saddle connection on S is contained in Y. We take K to be the empty complex and apply (W_{l}). Then, ℓ(Y, K) = ℓ(S), and thus (17) holds.

The proof of Theorem 8.2 is by induction on l. Clearly (U_{1}), (V_{1}) and (W_{1}) are all true since only 1 saddle connection can be added to the domains. The induction is done in the following steps:

(W_{l−1}) and Proposition 7.2 imply (U_{l}). The assumption on β implies that |β|/|γ| ≥ 1/p_{0}; otherwise the curve β itself would separate itself from γ, contradicting the weak accessibility of β to γ, contrary to the assumption. By rescaling, we may assume that ∥Y ∥ = 1. Suppose first |γ| ≤ 1. This implies that the notions of p_{0} accessible and (p_{0}, D_{l}) weakly accessible coincide.

By Corollary 6.3 we form a complex K_{0} ⊂ Y which generated by the saddle connections of length at most |γ|/p_{0}c′; then |∂K_{0}| ≤ |γ|/p_{0} so any saddle connection contained in K_{0} is not p_{0}-accessible from γ. We replace K_{0} by its “filled-in” version (see §6). It is still the case that any saddle connection contained in K_{0} is not p_{0} accessible from γ.

Now, for S = a_{t}r_{θ}Y ∈ SL(2, ℝ)Y, let V (S) denote the set of saddle connections λ ⊂ S such that λ ⊄ a_{t}r_{θ}K_{0}. Then, V (·) satisfies the condition (A) of §2.

Let 0 correspond to the angle for which γ is vertical. Suppose a saddle connection β ⊂ Y contributes to N_{1}(γ, R) − N_{1}(γ, R/2). Then,

R/2 ≤ |β| ≤ R  (34)

and

|β × γ| ≤ D_{l}.  (35)

From (35) and the fact that γ is vertical, |h_{β}| ≤ D_{l}/|γ| (where as in §6, h_{β} denotes the horizontal component of β). Let ϕ denote the angle, smaller than π/2 between β and γ. Then from (34),

sin ϕ = \frac{|h_{β}|}{|β|} ≤ \frac{2D_{l}}{|γ|R}.

Choose ϵ > 0 to be as in §6 (following Corollary 6.3). Let T = \frac{2R}{ϵ} so that saddle connections satisfying (34) have length between ϵT/4 and ϵT/2. Also set e^{t} = T, and Θ = sin^{−1}(min(1, 2D_{l}/|γ|R)). Then, |ϕ| is always bounded by Θ, hence

N_{1}(γ, R) − N_{1}(γ, R/2) ≤ |A(ϵT/4, ϵT/2, Θ) ∩ V (Y)|.  (36)

Assume that T ≥ 1. Then, by Lemma 8.1,

|A(ϵT/4, ϵT/2, Θ) ∩ V (Y)| ≤ cT^{2} ∫_{−2Θ}^{2Θ} N_{V} (ϵ, a_{t}r_{θ}Y) dθ.  (37)

Note that

Θ ≤ \frac{4D_{l}}{|γ|R} = \frac{8D_{l}e^{−t}}{ϵ|γ|}  (38)

Now suppose θ ∈ [−2Θ, 2Θ]. We will estimate N_{V} (ϵ, a_{t}r_{θ}Y). By Corollary 6.3 we build the complex L ⊂ Y of complexity of at most l − 1 generated by the saddle connections of length at most ϵ on a_{t}r_{θ}Y. Thus, all the saddle connections which contribute to N_{V} (ϵ, a_{t}r_{θ}Y) are contained in L. It is our intention to count the number of saddle connections of length ϵ in L which are not contained in K_{0}, since the saddle connections in K_{0} are not accessible. By Corollary 6.5 part (b) there exists a complex K′ ⊂ L ∩ K_{0} which contains all the saddle connections in L ∩ K_{0}. By construction, if a saddle connection in L is contained in V (Y), it is not contained in the filled in K′_{f}, with respect to L. We may assume that K′ does not contain ∂L (otherwise L ⊂ K_{0} and there is nothing to count). Thus for some constant C_{3} = C_{3}(δ, ϵ),

N_{V} (ϵ, a_{t}r_{θ}Y) ≤ N(L, K′_{f}, ϵ) ≤  (39)
≤ \frac{C_{3}}{ℓ(L, K′_{f})^{1+δ/2}} = \frac{C_{3}}{ℓ(a_{t}r_{θ}Y, a_{t}r_{θ}K_{0})^{1+δ/2}} = \frac{C_{3}}{ℓ_{γ}(a_{t}r_{θ}Y)^{1+δ/2}}

where in the second estimate we have used (W_{l−1}) and the last equality uses the definition of ℓ_{γ} in §7.

Combining (36), (37) and (39) we obtain:

N_{1}(γ, R) − N_{1}(γ, R/2) ≤ C_{3}e^{2t} ∫_{−2Θ}^{2Θ} \frac{dθ}{ℓ_{γ}(a_{t}r_{θ}Y)^{1+δ/2}}

Now set ρ = \frac{16D_{l}}{ϵ} so by (38) the domain of integration in the above integral is contained in the interval [\frac{−ρe^{−t}}{|γ|} , \frac{ρe^{t}}{|γ|}]. By Proposition 7.2, for some C′′ = C′′(D_{l}, δ, ϵ), C_{1} = C_{1}(D_{l}, δ, ϵ),

N_{1}(γ, R) − N_{1}(γ, R/2) ≤ C′′e^{2t} \frac{e^{−t(1−δ)}}{|γ|^{1+δ}} = C′′ (\frac{e^{t}}{|γ|})^{1+δ} = C_{1} (\frac{R}{|γ|})^{1+δ} .

Now, an easy geometric series argument shows that

N_{1}(γ, R) ≤ C′_{1}(D_{l}, δ, ϵ) (\frac{R}{|γ|})^{1+δ} .

This completes the proof of (U_{l}) in the case that |γ| ≤ 1 and T ≥ 1.

If |γ| ≤ 1 and T ≤ 1, then R ≤ ϵ, and in view of the upper bound in (34),

N_{1}(γ, R) ≤ N_{V} (R, Y).  (40)

We do exactly the same except that we set L to be the complex generated by the saddle connections of length R ≤ ϵ on the original domain Y. Again, set K′ ⊂ L∩K_{0} to be the complex containing all the saddle connections in L∩K_{0}. Then, using (W_{l−1}) as in (39),

N_{V} (R, Y) ≤ N(L, K′_{f}, R) ≤ C_{4} (\frac{R}{ℓ(L, K′_{f})})^{1+δ} = C_{4} (\frac{R}{ℓ(Y, K_{0})})^{1+δ} ≤ C_{5} (\frac{R}{|γ|})^{1+δ} , (41)

where the constants C_{4} and C_{5} depend on l and δ, and the last estimate follows from Corollary 6.3 and the definition of K_{0}. Now (U_{l}) follows from (40) and (41). This completes the proof of (U_{l}) in the case |γ| ≤ 1.

If |γ| ≥ 1 then we can rotate γ so it is vertical and then flow by a_{t} so that its length is 1. Since rotations are isometries, a_{t} contracts a vertical saddle connection the most and cross products are preserved, it is clear that after this action by an element of SL(2, R) that any β which is (p_{0}, D_{l})- weakly accessible before is still (p_{0}, D_{l})- weakly accessible. Let | · |′ denote lengths on the deformed surface. Then we have, for some C_{6},

\frac{R}{|γ|} ≤ |β|′ ≤ \frac{2R}{|γ|} + D_{l} ≤ C_{6} \frac{R}{|γ|},

with the first inequality on the right following from |β|′ ≥ \frac{|β|}{|γ|} ≥ \frac{R}{|γ|}, the middle inequality following from the condition on the cross product, and the rightmost inequality following from the fact that R/|γ| is bounded below by \frac{1}{p_{0}} . This implies that the estimate for |γ| = 1 gives the general estimate.

(U_{l}) and Proposition 6.9 imply (V_{l}). We say a saddle connection σ has size n if e^{n−1} < |σ| ≤ e^{n}. Up to an additive constant there are log R/|γ| sizes between |γ| and R. For each saddle connection β which contributes to N_{2}(γ, R), first note that β is (p_{0}, D_{l}) weakly accessible from γ. Thus we find the chain (see Proposition 6.9) β = β_{1}, …, β_{n} = γ where |β_{i} × β_{i+1}| ≤ D_{l}, β_{i} is (p_{0}, D_{l}) weakly-accessible from β_{i+1}. This implies that |β_{i}|/|β_{i+1}| ≥ \frac{1}{p_{0}} and so up to an additive constant β_{i} has size between size(|γ|) and size(|β|). Now (U_{l}) implies that the number of β is bounded by a constant times a product

(\frac{|β_{1}|}{|β_{2}|})^{1+δ/2} × … × (\frac{|β_{n−1}|}{|β_{n}|})^{1+δ/2} = C_{3} (\frac{|β|}{|γ|})^{1+δ/2} ≤ C_{3} (\frac{R}{|γ|})^{1+δ/2} ,

where the product has to be summed over all the possible sizes of |β_{i}|. This introduces an extra factor of log(R/|γ|)^{p_{0}} into the above quantity, which is incorporated into the exponent 1 + δ.

(V_{l}) and (W_{l−1}) imply (W_{l}). Let ω be the shortest saddle connection intersecting Y_{K}; that is, not contained in the filled in K_{f}. Let K′ ⊂ Y denote the complex generated by all the saddle connections in Y of length smaller than |ω|/c′ where c′ is as in Corollary 6.3 so K′ ⊂ K_{f}. Then K′_{f} ⊂ K_{f} as well so it is sufficient to prove (W_{l}) for saddle connections intersecting Y_{K′}; that is, not contained in K′_{f}. Let γ the longest saddle connection in ∂Y \ K so |γ| ≥ |ω|. Let L be the subcomplex of Y generated by all the saddle connections of length smaller then |γ|/c′, so K′ ⊂ L and therefore K′_{f} ⊂ L_{f}. Since, γ ∉ L_{f}, L_{f} is a proper subcomplex of Y. Thus, its complexity is at most l − 1.

We wish first to count the number of saddle connections in L_{f} \ K′_{f}, if this set is nonempty. By (W_{l−1}),

N(L_{f}, K′_{f}, R) ≤ C_{3}(l, δ) (\frac{R}{ℓ(L_{f}, K′_{f})})^{1+δ} ≤ C_{4}(l, δ) (\frac{R}{|ω|})^{1+δ}

Now we wish to count the saddle connections that intersect Y_{K′} that are not contained in L_{f}. Each such saddle connection is p_{0} accessible from γ, and hence is (p_{0}, D_{l}) weakly accessible from γ. Hence, by (V_{l}) the total number N(γ, R) of such saddle connections is at most

N(γ, R) ≤ C_{2}(l, δ) (\frac{R}{|γ|})^{1+δ} ≤ C_{2}(l, δ) (\frac{R}{|ω|})^{1+δ}

This completes the proof of (W_{l}) and the inductive step.

Proof of Theorem 5.1 (b). Let V (·) denote the set of vectors, counting multiplicity associated to saddle connections. Let ϵ = ϵ(P) > 0 be as in Theorem 5.1 (a) (and §6). Let C(δ) be as in Theorem 5.1 (a).

We note that for any R > ϵ there exists an integer m = m(R) and m elements g_{1}, …, g_{m} of SL(2, ℝ) depending only on R such that B(R, 0) ⊂ ⋃_{j=1}^{m} g_{j}B(ϵ, 0). Hence, for any S ∈ QD(g, P),

|V (S) ∩ B(R, 0)| ≤ ∑_{j=1}^{m} |V (S) ∩ g_{j}B(ϵ, 0)| = ∑_{j=1}^{m} |V (g_{j}^{−1}S) ∩ B(ϵ, 0)|. (42)

By Theorem 5.1 (a), for each j,

|V (g_{j}^{−1}S) ∩ B(ϵ, 0)| ≤ C(δ) (\frac{ϵ}{ℓ(g_{j}^{−1}S)})^{1+δ} .  (43)

Since lengths on g_{j}^{−1}S differ from lengths on S by a factor depending only on g_{j} (and hence only on R), (42) and (43) imply the theorem.

Remark. An easy geometric series argument shows that Theorem 5.1 (a), and hence (B) and (C_{μ}) hold for cylinders of imprimitive closed geodesics. This implies that the pointwise asymptotic formula holds for cylinders of imprimitive closed geodesics on almost every surface.

We now relate the constants in the asymptotic formulas for primitive and imprimitive geodesics. Let V (·) denote the primitive closed geodesics, and let W (·) denote the imprimitive closed geodesics. For a a bounded function f on ℝ^{2} and a surface S let

f̂(S) = ∑_{v∈V (S)} f(v) and f̃(S) = ∑_{v∈W (S)} f(v)

We claim that for any ergodic invariant probability measure μ, the Siegel-Veech constants are related by c_{V,μ} = c_{W,μ}/ζ(2). Choose any R > 0, and let f denote the characteristic function of the ball of radius R. Let f_{k} denote the characteristic function of the annulus {v : R/(k + 1) < ∥v∥ ≤ R/k}. Then, for any S ∈ QD(P),

f̃(S) = ∑_{k=1}^{∞} kf̂_{k}(S)  (44)

From the Siegel-Veech formula Theorem 2.2 we have,

∫_{QD(P)} f̃(S) dμ(S) = πc_{W,μ}R^{2}, and ∫_{QD(P)} f̂_{k}(S) dμ(S) = πc_{V,μ} (\frac{R^{2}}{k^{2}} − \frac{R^{2}}{(k + 1)^{2}} ) (45)

Integrating (44) over QD(P) with respect to μ and substituting (45), we get

πc_{W,μ}R^{2} = πc_{V,μ} ∑_{k=1}^{∞} k (\frac{R^{2}}{k^{2}} − \frac{R^{2}}{(k + 1)^{2}} ) = πc_{V,μ}R^{2} ∑_{k=1}^{∞} \frac{1}{k^{2}} = πc_{V,μ}R^{2}ζ(2).

Hence c_{W,μ} = ζ(2)c_{V,μ}.

9 Proof of Theorem 1.4

In this section we use the notations Y_{i}, μ_{i}, N_{4}(·), N_{5}(·) defined in §1.

Theorem 9.1. Each μ_{i} is ergodic under the SL(2, ℝ) action on Y_{i}.

Proof. We follow the Hopf proof of ergodicity of the diagonal action a_{t}. Let f ∈ C_{0}(Y_{i}) a continuous function with compact support. For each x ∈ Y_{i} we form

f^{∗}(x) = lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(a_{t}x)dt

and

f_{∗}(x) = lim_{T →∞} \frac{1}{T} ∫_{0}^{T} f(a_{−t}x)dt,

the limits existing for μ_{i} a.e. x ∈ Y_{i}. Moreover f^{∗} = f_{∗} a.e. as well. We need to show that f^{∗} = f_{∗} = c a.e. for some constant c. Now for each x, f^{∗} is constant along the vertical leaves (pairs of vertical leaves) of the flat structure (pair of flat structures) corresponding to x. If the vertical foliation of the flat structure is ergodic then f^{∗} is constant a.e. on each (product) fiber. Since μ_{i} a.e. fibers have ergodic vertical foliations, this implies that f^{∗} is constant on fibers. This reduces the proof to that of [Ma1].

Lemma 9.2. Let ϵ > 0 be as in §6.

(1) For any δ > 0 there exists C_{5} = C_{5}(P, δ) such that for any S ∈ QD(P) and any p ∈ S, N_{4}(S, P, p, ϵ) ≤ C_{5}/ℓ(S)^{1+δ}.

(2) For any δ > 0 there exists C_{6} such that for any p, q, N_{5}(S, P, p, q, ϵ) ≤ C_{6}/ℓ(S)^{1+δ}.

Proof. Recall that Z_{S} denotes the set of singularities of S. To prove (1), let Z′_{S} = Z_{S}∪p. Let S′ be the same surface as S except that on S′ we define a saddle connection to be a geodesic segment which begins and ends at a point of Z′_{S}, and does not contain a point of Z′_{S} in its interior.

By Corollary 6.3, there exists a complex Y′ ⊂ S′ with boundary such that every saddle connection on S′ of length at most ϵ is contained in Y′. Note that any saddle connection on S′ of length shorter then ℓ(S) must begin or end at p. Let b = ℓ(S)/3, and form the complex K′ generated by saddle connections of length at most b. Now we claim K′ consists of at most a single segment from p to a vertex v. Suppose on the contrary there are two segments joining p to vertices v_{1}, v_{2} or a closed loop from p to itself with length at most b. In the first case there would be geodesic joining v_{1} to v_{2} of length at most 2b < ℓ(S), contradicting that the shortest segment on S has length at most ℓ(S). In the second case, in the homotopy class of the closed loop would be a segment of length at most b, again a contradiction.

Since K′ consists of at most single segment which is not a closed loop, it cannot contain the entire boundary of Y′. Thus by Theorem 8.2 (W_{l}),

N(Y′, K′, ϵ) ≤ \frac{C_{3}}{ℓ(Y′, K′)^{1+δ}} = \frac{C_{5}}{ℓ(S)^{1+δ}} .

The proof of (2) is similar. This time set b = ℓ(S)/4. We add both p, q to the vertex set Z_{S} and again form Y′ and K′. Now K′ can consist of at most 3 segments; no two of which can be both be from p (resp. q) to vertices or from p to q. Nor there can there be a segment from p (resp. q) to itself. Thus the only possible closed loop must be homotopically trivial consisting of segments joining a vertex v to p and q and a segment joining p to q. This closed loop bounds a triangle Δ. Now K′ can be the entire boundary of Y′ only if Y′ = Δ, in which case the bound in the lemma is trivial. Otherwise we apply (W_{l}) again.

Proof of Theorem 1.4. We first verify that Theorem 2.1 applies. Let M be one the fiber bundles Y_{i}. If i = 1, let V (S, p) ⊂ ℝ^{2} denote the set of segments on S joining p ∈ S to a vertex. If i = 2, let V (S, p, q) ⊂ ℝ^{2} denote the set of segments on S joining p ∈ S to q ∈ S. The condition (A) is clearly satisfied. Also (C_{μ_{i}}) is clearly satisfied in view of Lemma 9.2 and Lemma 5.5. Also, (B) is satisfied in view of Lemma 9.2 and Theorem 5.2 (cf. the proof of Theorem 5.4 in §5). Thus, to prove Theorem 1.4, it remains to evaluate the Siegel-Veech constants.

Fix a small ϵ and let f ∈ C_{0}^{∞}(R^{2}) be a positive function which is 1 on the disc of radius ϵ, is 0 outside the disc of radius 2ϵ and has integral πϵ^{2} + o(ϵ^{2}). To prove (1), for p ∈ S \Z_{S}, let V = V (S, P, p) denote the set of segments joining p to a singularity.

By the Siegel-Veech formula

∫_{Y_{1}} f̂dμ_{1} = c_{V,μ_{1}} ∫_{ℝ^{2}} f = c_{V,μ_{1}}πϵ^{2} + o(ϵ^{2}).  (46)

We will show that

c_{V,μ_{1}} = \frac{k}{2}  (47)

We will compute c_{V,μ_{1}} by computing the left hand side of (46), dividing both sides by ϵ and let ϵ → 0.

Let QD_{ϵ}(P) denote the set of flat structures S such that the shortest saddle connection on S has length at most 4ϵ. For each S ∉ QD_{ϵ}(P) and v ∈ P let D_{ϵ}(v) denote the ball of radius ϵ centered at v. For each such S,

∫_{S} f̂(S, p) dλ_{S}(p) = ∑_{v∈P} D_{ϵ}(v) + o(ϵ^{2}) = kπϵ^{2}/2 + o(ϵ^{2}).  (48)

where λ_{S} denotes the Euclidean measure on S.

Let Y_{1,ϵ} be the fiber space over QD_{ϵ}(P). It is enough to show that

∫_{Y_{1,ϵ}} f̂dμ_{1} = o(ϵ^{2})  (49)

for then adding the contribution of (48) integrated over QD(P) to (49), substituting into (46), dividing both sides by ϵ and let ϵ → 0, we get (47).

Now by Lemma 9.2, for each S ∈ QD_{ϵ}(P),

f̂(S) ≤ \frac{C_{5}}{ℓ(S)^{1+δ}} .

Since f̂ is supported on a set of measure O(ϵ^{2}) on S,

∫_{S} f̂(S, p) dλ_{S}(p) = O (\frac{ϵ^{2}}{ℓ(S)^{1+δ}} ) .

Hence, since by Lemma 5.5 the function \frac{1}{ℓ(S)^{1+δ}} ∈ L^{1}(QD(P)), and the measure of QD_{ϵ}(P) → 0 as ϵ → 0, (49) holds. This concludes the proof of (1).

For (2), let V = V_{S,P,p_{1},p_{2}} denote the set of segments joining p_{1}, p_{2} and again apply the Siegel-Veech formula to f. This time we will show

c_{V,μ_{2}} = 1 and then use Theorem 2.1.

Again we need to compute the left side of the Siegel-Veech formula, divide by ϵ and let ϵ → 0. Again we first restrict to fibers over QD(P) \ QD_{ϵ}(P). In each fiber S ×S, let W_{ϵ} be the set of pairs (p_{1}, p_{2}) where either p_{1} or p_{2} is in the 2ϵ neighborhood of a singularity. For (p_{1}, p_{2}) ∈ S × S \ W_{ϵ}, f̂ = 1 if p_{1}, p_{2} belong to the same ϵ ball. Thus

∫_{S×S\W_{ϵ}} f̂ = πϵ^{2} + o(ϵ^{2}).

If (p_{1}, p_{2}) ∈ W_{ϵ}, say, p_{1} lies in a 2ϵ neighborhood of a singularity, then p_{2} must lie within a 4ϵ neighborhood of the singularity; otherwise f̂ = 0. The integral of f̂ over the set of pairs (p_{1}, p_{2}) such that both p_{i} lie in a 4ϵ neighborhood of a singularity is O(ϵ^{4}) = o(ϵ^{2}). Thus the integral of f̂ on the fiber space over QD(P) \ QD_{ϵ}(P) is πϵ^{2} + o(ϵ^{2}).

Now using the second part of Lemma 9.2, on each fiber S over QD_{ϵ}(P), f̂ is O(\frac{1}{ℓ(S)^{1+δ}}) and since f̂ is supported on a set of measure O(ϵ^{2}), the integral of f̂ is O(\frac{ϵ^{2}}{ℓ(S)^{1+δ}}) on each such fiber. Exactly as before this implies that the total contribution to ∫ f̂ is o(ϵ^{2}) on the fibers over QD_{ϵ}(P).

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