Periodic geodesics on generic translation surfaces

Yaroslav Vorobets

Abstract The objective of the paper is to study properties of periodic geodesics on translation surfaces that hold for generic elements of the moduli space of translation surfaces.

2000 Mathematics Subject Classification: 37C27.

1 Introduction

Let M be a compact connected oriented surface endowed with a flat metric that has a finite number of conical singularities. The flat structure on M can be determined by an atlas of coordinate charts such that all transition functions are (affine) rotations or translations in ℝ^{2} and chart domains cover the whole surface M except for the singularities. Suppose that the atlas can be chosen so that all transition functions are translations. Then this atlas endows M with a translation structure and M is called a translation surface. Translation surfaces are closely related to Abelian differentials on compact Riemann surfaces. If x is a conical singularity of a translation surface M, then the total cone angle at this point is of the form 2πm, where m is an integer. m is called the multiplicity of the singular point x.

A translation structure on M endows the surface punctured at its singular points with a smooth structure, a flat Riemannian metric, and an area element. Moreover, it allows us to identify the tangent space at any nonsingular point with Euclidean space ℝ^{2} so that velocity is an invariant of the geodesic flow. In particular, each oriented geodesic is assigned a unique direction v ∈ S^{1} = {x ∈ ℝ^{2} : |x| = 1}. The study of the geodesic flow reduces to the study of the family of directional flows F_{v}, v ∈ S^{1}, on the surface M.

A geodesic on a translation surface cannot have self-intersections. Therefore a geodesic joining a nonsingular point to itself is necessarily closed (or periodic). We regard periodic geodesics as simple closed curves. The flat structure implies that any periodic geodesic belongs to a family of freely homotopic parallel periodic geodesics of the same length. The geodesics of the family fill either the whole surface or a cylindrical subset. We call this subset a cylinder of periodic geodesics (or a periodic cylinder). A periodic cylinder is bounded by geodesic segments whose endpoints are singular points. Such segments are called saddle connections. As a default, periodic geodesics are assumed to be unoriented but we shall consider oriented periodic geodesics and cylinders as well.

The fundamental properties of periodic geodesics on translation surfaces were established by Masur.

Theorem 1.1 ([M1], [M2], [M3]) Let M be a translation surface.

(a) There exists a periodic geodesic on M of length at most c\sqrt{a}, where a is the area of M and c > 0 is a constant depending only on the genus of M.

(b) The directions of periodic geodesics on M are dense in S^{1}.

(c) Let N_{1}(M, R) denote the number of periodic cylinders of M of length at most R > 0. Then there exist 0 < c_{1}(M) < c_{2}(M) < ∞ such that

c_{1}(M) ≤ N_{1}(M, R)/R^{2} ≤ c_{2}(M)

for R sufficiently large.

Theorem 1.1 can be generalized as follows.

Theorem 1.2 ([Vo]) Let M be a translation surface of area a, m ≥ 1 be the sum of multiplicities of singular points of M, and s be the length of the shortest saddle connection of M. Then

(a) M has a periodic cylinder of length at most 2^{2^{4m}} \sqrt{a} and of area at least a/m;

(b) for almost every x ∈ M directions of periodic geodesics passing through the point x are dense in S^{1};

(c) for any R ≥ 2^{2^{4m}} \sqrt{a},

((600m)^{(2m)^{2m}})^{−1}a^{−2}s^{2}R^{2} ≤ N_{1}(M, R) ≤ (400m)^{(2m)^{2m}}s^{−2}R^{2}.

Two translation surfaces M and M′ are called isomorphic if there exists an orientation-preserving homeomorphism f : M → M′ such that f maps the set of singular points of M onto the set of singular points of M′ and f is a translation with respect to translation structures of M and M′. For any integers p, n ≥ 1 let MQ(p, n) denote the set of equivalence classes of isomorphic translation surfaces of genus p with n singular points (of arbitrary multiplicity). MQ(p, n) is called the moduli space of such surfaces. An element of MQ(p, n) is a translation surface considered up to isomorphism. The moduli space MQ(p, n) is endowed with the structure of an affine orbifold of dimension 2(2p + n − 1) and with a canonical volume element (see Section 6). MQ(p, n) need not be connected but the number of its connected components is finite. By MQ_{1}(p, n) denote the subset of MQ(p, n) corresponding to translation surfaces of area 1. MQ_{1}(p, n) is a real analytic suborbifold of MQ(p, n) of codimension 1. The volume element on MQ(p, n) induces a canonical volume element on MQ_{1}(p, n) such that the volume of MQ_{1}(p, n) is finite (see [V2], [MS]). Every connected component of MQ_{1}(p, n) is of the form C ∩ MQ_{1}(p, n), where C is a connected component of MQ(p, n).

Suppose P is a property of translation surfaces such that P simultaneously holds or does not hold for any pair of isomorphic translation surfaces. Let C be a nonempty open subset of MQ(p, n) or MQ_{1}(p, n). We say that the property P is generic for translation surfaces in C or that P holds for a generic M ∈ C if P holds for almost all M ∈ C with respect to the volume element on MQ(p, n) (resp. MQ_{1}(p, n)).

The simplest property of periodic geodesics on generic translation surfaces that is not enjoyed by all translation surfaces concerns regularity of periodic cylinders. A periodic cylinder is called regular if it is bounded by saddle connections of the same length as geodesics in the cylinder.

Proposition 1.3 ([EM]) Generic translation surfaces admit only regular periodic cylinders.

Two nonintersecting periodic geodesics are called homologous if they break the translation surface into two components. Homologous geodesics are parallel and of the same length. Two periodic geodesics in the same cylinder are obviously homologous but the converse is not true. As shown by Eskin, Masur, and Zorich [EMZ], for any positive integer k a generic translation surface of sufficiently high genus admits k distinct cylinders of homologous periodic geodesics.

Proposition 1.4 (a) For a generic translation surface any two nonhomologous periodic geodesics are of different length and direction.

(b) For a generic translation surface M ∈ MQ(p, n), (p, n) ≠ (1, 1), all periodic cylinders are of different area.

The group SL(2, ℝ) acts on the set of translation surfaces of genus p with n singular points by postcomposition of the chart maps with linear transformations from SL(2, ℝ). This action descends to an action on the moduli space MQ(p, n), which is affine and leaves invariant the subspace MQ_{1}(p, n). The SL(2, ℝ) action on MQ_{1}(p, n) is volume preserving and ergodic on each connected component of MQ_{1}(p, n) (see [V1]). This fact is basic in establishing genericity of many properties of translation surfaces, in particular, the properties stated below.

For any translation surface M and any R > 0 let N_{1}(M, R) denote the number of periodic cylinders of M of length at most R. By N_{2}(M, R) denote the sum of areas of these cylinders. Further, for any x ∈ M let N_{3}(M, x, R) denote the number of periodic geodesics on M of length at most R that pass through the point x. For any σ ≥ 0 let N_{4}(M, σ, R) denote the number of periodic cylinders of M of length at most R and of area greater than σ.

Let C be a connected component of the space MQ_{1}(p, n).

Theorem 1.5 ([EM]) For a generic translation surface M ∈ C,

lim_{R→∞} N_{1}(M, R)/R^{2} = c_{1}(C),

where c_{1}(C) > 0 depends only on the component C.

Theorem 1.5 was proved by Eskin and Masur [EM] who sharpened results of Veech [V3]. The constants c_{1}(C) were found by Eskin, Masur, and Zorich [EMZ]. It turns out that explicit evaluation of these constants involves a lot of calculation.

Theorem 1.6 (a) For a generic translation surface M ∈ C,

lim_{R→∞} N_{2}(M, R)/R^{2} = c_{2}(C),

where c_{2}(C) > 0 depends only on the component C.

(b) c_{2}(C) = c_{1}(C)/m_{C}, where m_{C} = 2p − 2 + n is the sum of multiplicities of singular points for translation surfaces in C.

Theorem 1.7 For a generic translation surface M ∈ C,

lim_{R→∞} N_{3}(M, x, R)/R^{2} = c_{2}(C)

for almost every x ∈ M.

The following theorem answers, in particular, Question 13.4 of the paper [V3].

Theorem 1.8 For a generic translation surface M ∈ C,

lim_{R→∞} N_{4}(M, σ, R)/R^{2} = (1 − σ)^{m_{C}−1}c_{1}(C)

for all σ ∈ [0, 1).

For every translation surface M of area 1 we define three families α_{M,R}, δ_{M,R}, and D_{M,R} of measures depending on the parameter R > 0. α_{M,R} is a measure on [0, 1]; for any K ⊂ [0, 1] let α_{M,R}(K) be the number of periodic cylinders of M of length at most R and of area in the set K. δ_{M,R} is a measure on the unit circle S^{1}; for any U ⊂ S^{1} let δ_{M,R}(U) be the number of oriented periodic cylinders of length at most R with directions in the set U. D_{M,R} is a measure on S^{1} × [0, 1]; for any U ⊂ S^{1} × [0, 1] let D_{M,R}(U) be the number of oriented periodic cylinders of length at most R with direction v and area a such that (v, a) ∈ U. Let R_{M} be the length of the shortest periodic geodesic on M. For any R ≥ R_{M} the measures α_{M,R}, δ_{M,R}, and D_{M,R} are nonzero so we define probability measures α̃_{M,R} = (α_{M,R}([0, 1]))^{−1}α_{M,R}, δ̃_{M,R} = (δ_{M,R}(S^{1}))^{−1}δ_{M,R}, and D̃_{M,R} = (D_{M,R}(S^{1} × [0, 1]))^{−1}D_{M,R}. The measures α̃_{M,R} describe the distribution of areas of periodic cylinders on M. The measures δ̃_{M,R} describe the distribution of their directions. The measures D̃_{M,R} describe the joint distribution of directions and areas.

For any integer m ≥ 1 let λ_{m} denote a unique Borel measure on [0, 1] such that λ_{m}([σ, 1]) = (1 − σ)^{m−1} for all σ ∈ [0, 1). Let 𝔪_{1} denote Lebesgue measure on S^{1} normalized so that 𝔪_{1}(S^{1}) = 1.

Theorem 1.9 For a generic translation surface M ∈ C we have the following weak convergence of measures:

lim_{R→∞} α̃_{M,R} = λ_{m_{C}},

lim_{R→∞} δ̃_{M,R} = 𝔪_{1},

lim_{R→∞} D̃_{M,R} = 𝔪_{1} × λ_{m_{C}}.

Theorem 1.9 means that for a generic translation surface M ∈ C the directions of periodic geodesics on M are uniformly distributed in S^{1} while the areas of periodic cylinders are distributed according to the measure λ_{m_{C}}. Moreover, the distributions of directions and areas of periodic cylinders are independent. In view of Theorems 1.5 and 1.6(a), the ratio c_{2}(C)/c_{1}(C) may be regarded as the mean area of periodic cylinders on a generic translation surface M ∈ C. Indeed, c_{2}(C)/c_{1}(C) = 1/m_{C} is the expectation of a random variable taking values in [0, 1] with probabilities given by the measure λ_{m_{C}}.

For any translation surface M and any x ∈ M we define a family δ_{M,x,R} of measures on S^{1} depending on R > 0. For any U ∈ S^{1} let δ_{M,x,R}(U) be the number of oriented periodic geodesics on M of length at most R passing through the point x in directions from U. If the measure δ_{M,x,R} is nonzero then we define a probability measure δ̃_{M,x,R} = (δ_{M,x,R}(S^{1}))^{−1}δ_{M,x,R}.

Theorem 1.10 For a generic translation surface M ∈ C the weak convergence of measures

lim_{R→∞} δ̃_{M,x,R} = 𝔪_{1}

takes place for almost all x ∈ M.

Thus the directions of periodic geodesics passing through a generic point on a generic translation surface are uniformly distributed in S^{1}.

The paper is organized as follows. In Section 2 we review Veech’s theory of Siegel measures. In Section 3 this theory is applied to a general counting problem related to the growth of the number of periodic geodesics. The tools to treat limit distributions are developed in Section 4. Section 5 contains preliminaries on translation surfaces and Delaunay partitions. In Section 6 we consider moduli spaces of translation surfaces. Section 7 is devoted to proofs of main results of the paper. Some proofs rely on estimates of volumes of certain subsets in the moduli space of translation surfaces. These estimates are obtained in Section 8. Acknowledgements. This paper was inspired by the papers of Veech [V3], Eskin and Masur [EM], Masur and Smillie [MS], and Eskin, Masur, and Zorich [EMZ]. I am grateful to all these authors for the inspiration. I would like to thank Anton Zorich for several helpful discussions.

Part of the paper was written while the author was visiting the Max Planck Institute of Mathematics in Bonn, whose hospitality is gratefully acknowledged.

2 Siegel measures

Let 𝔐 denote the set of locally finite Borel measures on ℝ^{2}. Given a bounded, compactly supported Borel function ψ on ℝ^{2}, set

ψ̂(ν) = ∫_{ℝ^{2}} ψ(x) dν(x)

for any ν ∈ 𝔐. Let C_{c}(ℝ^{2}) denote the space of continuous, compactly supported functions on ℝ^{2}. Endow 𝔐 with the C_{c}(ℝ^{2}) weak-∗ topology. By definition, this is the smallest topology such that ψ̂ is continuous on 𝔐 when ψ ∈ C_{c}(ℝ^{2}).

For any R > 0 let B(R) denote the disk of radius R in ℝ^{2} centered at the origin. For any ν ∈ 𝔐 and any R > 0 set N_{ν}(R) = ν(B(R)). The function N_{ν} is called the growth function of the measure ν. By 𝔐_{2} denote the set of measures ν ∈ 𝔐 such that M(ν) < ∞, where

M(ν) = sup_{R>0} \frac{N_{ν}(R)}{R^{2}}.

As a topological space, 𝔐_{2} is a countable union of metrizable compacta. Namely, for any c > 0 the subspace 𝔐_{2}(c) = {ν ∈ 𝔐_{2} | M(ν) ≤ c} is compact and metrizable. A measure ν ∈ 𝔐 is called even if ν(U) = ν(−U) for any Borel set U ⊂ ℝ^{2}. The set 𝔐^{e} of all even measures is a closed subset of 𝔐. By 𝔪 denote Lebesgue measure on ℝ^{2}. Clearly, 𝔪 ∈ 𝔐_{2}(π) ⊂ 𝔐_{2} and 𝔪 ∈ 𝔐^{e}.

Let g ∈ SL(2, ℝ). We regard g as a linear transformation of ℝ^{2}. For any ν ∈ 𝔐 define a measure gν by gν(U) = ν(g^{−1}(U)), U ⊂ ℝ^{2} a Borel set. The map SL(2, ℝ) × 𝔐 ∋ (g, ν) ↦ gν defines a continuous action of the group SL(2, ℝ) on 𝔐. The sets 𝔐_{2} and 𝔐^{e} are invariant under this action.

For any R > 0 define a transformation T_{R} : 𝔐 → 𝔐 by T_{R}ν(U) = R^{−2}ν(RU), ν ∈ 𝔐, U ⊂ ℝ^{2} a Borel set. The family {T_{R}}_{R>0} defines a continuous action of the group ℝ^{+} on 𝔐. The sets 𝔐_{2}, 𝔐^{e}, and each of the subsets 𝔐_{2}(c), c > 0, are invariant under this action. Besides, the action of ℝ^{+} on 𝔐 commutes with the action of SL(2, ℝ).

Denote by P(𝔐_{2}) the set of Borel probability measures on 𝔐_{2}. An element μ ∈ P(𝔐_{2}) is called a Siegel measure if the SL(2, ℝ) action on 𝔐_{2} leaves μ invariant and is ergodic with respect to μ. Siegel measures were introduced and studied by Veech [V3].

Theorem 2.1 ([V3]) Assume μ ∈ P(𝔐_{2}) is a Siegel measure. Then there exists c(μ) ≥ 0 such that for any compactly supported bounded Borel function ψ on ℝ^{2}, the function ψ̂ belongs to L^{1}(𝔐_{2}, μ) and

∫_{𝔐_{2}} ψ̂(ν) dμ(ν) = c(μ) ∫_{ℝ^{2}} ψ(x) d𝔪(x).

The number c(μ) is called the Siegel-Veech constant of the measure μ.

Theorem 2.2 ([V3]) Assume μ ∈ P(𝔐_{2}) is a Siegel measure. Then

lim_{R→∞} ∫_{𝔐_{2}} |R^{−2}N_{ν}(R) − c(μ)π| dμ(ν) = 0.

As a consequence, there exists a sequence R_{n} → ∞ such that for μ-a.e. ν ∈ 𝔐_{2},

lim_{n→∞} \frac{N_{ν}(R_{n})}{R_{n}^{2}} = c(μ)π.

Let 𝔐_{2}^{e} = 𝔐_{2}∩𝔐^{e}. Denote by P(𝔐_{2}^{e}) the set of measures μ ∈ P(𝔐_{2}) supported on 𝔐_{2}^{e}.

Theorem 2.3 ([V3]) Assume μ ∈ P(𝔐_{2}^{e}) is a Siegel measure. Then there exists a sequence R_{n} → ∞ such that for μ-a.e. ν ∈ 𝔐_{2}^{e},

lim_{n→∞} T_{R_{n}}ν = c(μ)𝔪,

where convergence is in the C_{c}(ℝ^{2}) weak-∗ topology.

Theorem 2.3 is derived from Theorem 2.2 by applying the following criterion.

Theorem 2.4 ([V3]) Let {ν_{α} | a ∈ A} be a net of even, locally finite Borel measures on ℝ^{2}. Assume there exist c < ∞ and a dense set F ⊂ SL(2, ℝ) × ℝ^{+} such that for any (g, t) ∈ F,

lim_{α∈A} ν_{α}(tgB(1)) = πt^{2}c.

Then

lim_{α∈A} ∫_{ℝ^{2}} ψ(x) dν_{α}(x) = c ∫_{ℝ^{2}} ψ(x) d𝔪(x)

for any ψ ∈ C_{c}(ℝ^{2}), i.e., lim_{α∈A} ν_{α} = c𝔪 in 𝔐.

Let μ ∈ P(𝔐_{2}) be a Siegel measure. Set L^{1+}(𝔐_{2}, μ) = ⋃_{ϵ>0} L^{1+ϵ}(𝔐_{2}, μ). The measure μ is called regular if ψ̂ ∈ L^{1+}(𝔐_{2}, μ) for any ψ ∈ C_{c}(ℝ^{2}).

Theorem 2.5 If μ ∈ P(𝔐_{2}) is a regular Siegel measure, then for μ-a.e. ν ∈ 𝔐_{2},

lim_{R→∞} \frac{N_{ν}(R)}{R^{2}} = c(μ)π.

A closely related result, which is reproduced below as Part II of Theorem 3.2, was proved by Eskin and Masur [EM]. The proof of Theorem 2.5 is almost the same and will be omitted.

Theorem 2.6 If μ ∈ P(𝔐_{2}^{e}) is a regular Siegel measure, then for μ-a.e. ν ∈ 𝔐_{2}^{e},

lim_{R→∞} T_{R}ν = c(μ)𝔪.

Proof. Let c(μ) be the Siegel-Veech constant of the measure μ. By Theorem 2.5, there exists a Borel set U ⊂ 𝔐_{2}^{e} such that μ(U) = 1 and for all ν ∈ U,

lim_{R→∞} R^{−2}N_{ν}(R) = c(μ)π.

Suppose G is a countable dense subset of the group SL(2, ℝ). Set U_{0} = ⋂_{g∈G} gU. Clearly, U_{0} is a Borel subset of 𝔐_{2}^{e} and μ(U_{0}) = 1. Take a measure ν ∈ U_{0}. For any g ∈ G and any t > 0 we have

T_{R}ν(tgB(1)) = R^{−2}ν(gB(Rt)) = R^{−2}(g^{−1}ν)(B(Rt)) = R^{−2}N_{g^{−1}ν}(Rt).

Since g^{−1}ν ∈ U, it follows that lim_{R→∞} T_{R}ν(tgB(1)) = t^{2}c(μ)π. By Theorem 2.4, lim_{R→∞} T_{R}ν = c(μ)𝔪 in 𝔐.

3 Counting problem

Let V denote a sequence v_{1}, v_{2}, … of vectors in ℝ^{2} equipped with a sequence w_{1}, w_{2}, … of positive reals. The number w_{k} is called the weight of the vector v_{k}. It is assumed that the sequence of vectors tends to infinity or is finite (possibly empty). By V denote the set of all such sequences of vectors with weights. Two elements V_{1}, V_{2} ∈ V are considered to be equal if one of them can be obtained from the other by rearranging its vectors along with the corresponding rearrangement of weights. The group SL(2, ℝ) acts on the set V by the natural action on vectors and the trivial action on weights. To each V ∈ V we assign a linear functional Φ[V] on the space of compactly supported functions on ℝ^{2}; the functional is defined by the relation Φ[V](ψ) = ∑_{k=1}^{∞} w_{k}ψ(v_{k}). Furthermore, for any R > 0 set N_{V} (R) = Φ[V](χ_{B(R)}) = ∑_{k:|v_{k}|≤R} w_{k}. The function N_{V} is called the growth function of V. If all weights of V are equal to 1, then N_{V} (R) counts the number of vectors of V in the disk B(R).

Given V ∈ V, let v_{1}, v_{2}, … be the sequence of vectors of V and w_{1}, w_{2}, … be the sequence of weights. For any Borel set U ⊂ ℝ^{2} let ν_{V} (U) = ∑_{k:v_{k}∈U} w_{k}. It is easy to see that ν_{V} is a Borel measure on ℝ^{2}. Since the sequence v_{1}, v_{2}, … either tends to infinity or is finite, the measure ν_{V} is locally finite, i.e., ν_{V} ∈ 𝔐. The growth function of V coincides with the growth function of the measure ν_{V}. For any compactly supported Borel function ψ on ℝ^{2},

Φ[V](ψ) = ∫_{ℝ^{2}} ψ(x) dν_{V} (x).

Let M be a locally compact metric space endowed with a finite nonzero Borel measure μ. Suppose the group SL(2, ℝ) acts on the space M by homeomorphisms. We assume that the measure μ is invariant under this action and the action is ergodic, that is, any measurable subset of M invariant under the action is of zero or full measure. In what follows we consider maps V : M → V satisfying all or at least some of the following conditions:

(0) for any ψ ∈ C_{c}(ℝ^{2}) the function M ∋ ω ↦ Φ[V (ω)](ψ) is Borel;

(A) the map V intertwines the actions of the group SL(2, ℝ) on the spaces M and V, that is, V (gω) = gV (ω) for any g ∈ SL(2, ℝ) and any ω ∈ M;

(B) for any ω ∈ M there exists c(ω) < ∞ such that N_{V (ω)}(R) ≤ c(ω)R^{2} for all R > 0;

(B′) for any compact subset K ⊂ M there exists c(K) < ∞ such that N_{V (ω)}(R) ≤ c(ω)R^{2} for all ω ∈ K and R > 0;

(C) there exists R_{0} > 0 such that the function ω ↦ N_{V (ω)}(R_{0}) belongs to the space L^{1+}(M, μ) = ⋃_{ϵ>0} L^{1+ϵ}(M, μ);

(C′) for any R > 0 the function ω ↦ N_{V (ω)}(R) belongs to L^{1+}(M, μ); (C′′) for any ψ ∈ C_{c}(ℝ^{2}) the function ω ↦ Φ[V (ω)](ψ) belongs to L^{1+}(M, μ);

(E) V (ω) = −V (ω) for any ω ∈ M.

Note that conditions (C), (C′), and (C′′) depend on the measure μ.

Lemma 3.1 Suppose a map V : M → V satisfies condition (0). Then conditions (A) and (C) imply condition (C′), while conditions (C′) and (C′′) are equivalent.

Proof. Given a bounded, compactly supported Borel function ψ on ℝ^{2}, set ψ̃(ω) = Φ[V (ω)](ψ) for any ω ∈ M. For any R > 0 there exists a sequence ψ_{1}, ψ_{2}, … of functions in C_{c}(ℝ^{2}) such that ψ_{1} ≥ ψ_{2} ≥ … and ψ_{n} → χ_{B(R)} pointwise as n → ∞. It follows that ψ̃_{1} ≥ ψ̃_{2} ≥ … and ψ̃_{n} → χ̃_{B(R)} pointwise as n → ∞. Condition (0) implies ψ̃_{1}, ψ̃_{2}, … are Borel functions on M. Then χ̃_{B(R)} is a Borel function as well.

Suppose conditions (A) and (C) hold for the map V. Condition (C) means that χ̃_{B(R_{0})} ∈ L^{1+}(M, μ) for some R_{0} > 0. Condition (A) implies χ̃_{gB(R_{0})}(ω) = χ̃_{B(R_{0})}(g^{−1}ω) for all g ∈ SL(2, ℝ) and ω ∈ M. It follows that χ̃_{gB(R_{0})} ∈ L^{1+}(M, μ) for any g ∈ SL(2, ℝ). Since SL(2, ℝ) acts transitively on ℝ^{2} \ {(0, 0)}, for any x ∈ ℝ^{2} there exist a neighborhood U_{x} of x and an operator g_{x} ∈ SL(2, ℝ) such that g_{x}U_{x} ⊂ B(R_{0}). Given R > 0, the disk B(R) is covered by finitely many neighborhoods U_{x_{1}}, U_{x_{2}}, …, U_{x_{k}}. Then B(R) ⊂ ⋃_{i=1}^{k} h_{i}B(R_{0}), where h_{i} = g_{x_{i}}^{−1}. It follows that χ̃_{B(R)} ≤ ∑_{i=1}^{k} χ̃_{h_{i}B(R_{0})}. By the above the functions χ̃_{h_{1}B(R_{0})}, …, χ̃_{h_{k}B(R_{0})} belongs to L^{1+}(M, μ). Since χ̃_{B(R)} is a nonnegative Borel function, it belongs to L^{1+}(M, μ) as well. Thus, χ̃_{B(R)} ∈ L^{1+}(M, μ) for any R > 0, i.e., condition (C′) holds.

For any ψ ∈ C_{c}(ℝ^{2}) there exist c, R > 0 such that |ψ| ≤ cχ_{B(R)}. Then |ψ̃| ≤ cχ̃_{B(R)}. It follows that ψ̃ ∈ L^{1+}(M, μ) whenever χ̃_{B(R)} ∈ L^{1+}(M, μ). Thus condition (C′) implies condition (C′′). On the other hand, for any R > 0 there exists ψ ∈ C_{c}(ℝ^{2}) such that χ_{B(R)} ≤ ψ. Then 0 ≤ χ̃_{B(R)} ≤ ψ̃. By the above χ̃_{B(R)} is a Borel function, hence χ̃_{B(R)} ∈ L^{1+}(M, μ) whenever ψ̃ ∈ L^{1+}(M, μ). Thus condition (C′′) implies condition (C′).

Theorem 3.2 Let V : M → V be a map satisfying condition (0).

I. Suppose the map V satisfies conditions (A) and (B). Then there exists c_{V,μ} ≥ 0 such that for any compactly supported bounded Borel function ψ on ℝ^{2}, the function ω ↦ Φ[V (ω)](ψ) belongs to L^{1}(M, μ) and

\frac{1}{μ(M)} ∫_{M} Φ[V (ω)](ψ) dμ(ω) = c_{V,μ} ∫_{ℝ^{2}} ψ(x) d𝔪(x).

II. If V satisfies conditions (A), (B), and (C), then for μ-a.e. ω ∈ M,

lim_{R→∞} \frac{N_{V (ω)}(R)}{R^{2}} = πc_{V,μ}.

III. If V satisfies conditions (A), (B), (C), and (E), then for μ-a.e. ω ∈ M,

lim_{R→∞} ∫_{ℝ^{2}} \frac{1}{R^{2}} ψ(\frac{x}{R}) dν_{V (ω)}(x) = c_{V,μ} ∫_{ℝ^{2}} ψ(x) d𝔪(x)

for each compactly supported, Riemann integrable function ψ : ℝ^{2} → ℝ. In particu-

lar, lim_{R→∞} T_{R}ν_{V (ω)} = c_{V,μ}𝔪 in 𝔐 for μ-a.e. ω ∈ M.

The constant c_{V,μ} is called the Siegel-Veech constant of the pair (V, μ).

Part I of Theorem 3.2 was proved by Veech [V3] in some particular cases. The proof in the general case requires no changes. Parts I and II was proved by Eskin and Masur [EM] assuming conditions (A), (B′), (C), and, implicitly, condition (0) hold. In fact, the weaker condition (B) was used in the proof instead of condition (B′). Although condition (C) is not necessary for Part I to be true, it simplifies the proof significantly. Besides, condition (C) allows one to prove Part I without using ergodicity of the SL(2, ℝ) action on M. Proof of Theorem 3.2. We shall derive Parts I–III of the theorem from Theorems 2.1, 2.5, and 2.6, respectively.

Define a map F : M → 𝔐 by F (ω) = ν_{V (ω)}, ω ∈ M. It is easy to observe that F is a Borel map if and only if condition (0) holds for the map V. Further, F (M) ⊂ 𝔐_{2} if and only if V satisfies condition (B). If condition (A) holds, then F (gω) = gF (ω) for all g ∈ SL(2, ℝ) and ω ∈ M. If condition (E) holds, then F (M) ⊂ 𝔐^{e}.

Suppose the map V satisfies conditions (0), (A), and (B). For any Borel subset U ⊂ 𝔐_{2} let μ_{V} (U) = μ(F^{−1}(U))/μ(M). Condition (0) implies μ_{V} is a well-defined finite Borel measure on 𝔐_{2}. Condition (B) implies μ_{V} is a probability measure. Since μ is invariant under the SL(2, ℝ) action on M, condition (A) implies μ_{V} is invariant under the SL(2, ℝ) action on 𝔐_{2}. Ergodicity of the action of SL(2, ℝ) on M with respect to μ implies ergodicity of the action on 𝔐_{2} with respect to μ_{V}. Thus, μ_{V} is a Siegel measure. By Theorem 2.1, for any compactly supported bounded Borel function ψ : ℝ^{2} → ℝ the function ψ̂ belongs to L^{1}(𝔐_{2}, μ_{V}) and

∫_{𝔐_{2}} ψ̂(ν) dμ_{V} (ν) = c_{V,μ} ∫_{ℝ^{2}} ψ(x) d𝔪(x),

where c_{V,μ} is the Siegel-Veech constant of the measure μ_{V}. Then ψ̂ ∘ F ∈ L^{1}(M, μ) and

\frac{1}{μ(M)} ∫_{M} ψ̂(F (ω)) dμ(ω) = ∫_{𝔐_{2}} ψ̂(ν) dμ_{V} (ν).

As ψ̂(F (ω)) = Φ[V (ω)](ψ) for any ω ∈ M, Part I of the theorem follows.

Now suppose V satisfies conditions (0), (A), (B), and (C). By Lemma 3.1, condition (C′′) also holds for V. Condition (C′′) means that ψ̂ ∘ F ∈ L^{1+}(M, μ) for any ψ ∈ C_{c}(ℝ^{2}). It follows that ψ̂ ∈ L^{1+}(𝔐_{2}, μ_{V}) for any ψ ∈ C_{c}(ℝ^{2}). Therefore μ_{V} is a regular Siegel measure. By Theorem 2.5,

lim_{R→∞} R^{−2}N_{ν}(R) = πc_{V,μ}

for μ_{V} -a.e. ν ∈ 𝔐_{2}, or, equivalently,

lim_{R→∞} R^{−2}N_{F (ω)}(R) = πc_{V,μ}

for μ-a.e. ω ∈ M. As for any ω ∈ M the growth function N_{F (ω)} of the measure F (ω) = ν_{V (ω)} coincides with the growth function N_{V (ω)}, Part II of the theorem follows.

Now suppose V satisfies conditions (0), (A), (B), (C), and (E). By the above μ_{V} is a regular Siegel measure. Condition (E) implies μ_{V} ∈ P(𝔐_{2}^{e}). By Theorem 2.6,

lim_{R→∞} T_{R}ν = c_{V,μ}𝔪

for μ_{V} -a.e. ν ∈ 𝔐_{2}^{e}. It follows that

lim_{R→∞} T_{R}ν_{V (ω)} = c_{V,μ}𝔪

for ω ∈ U, where U ⊂ M is a Borel set such that μ(U) = μ(M). Given any compactly supported, Riemann integrable function ψ : ℝ^{2} → ℝ, there exist two sequences ψ_{1}^{+}, ψ_{2}^{+}, … and ψ_{1}^{−}, ψ_{2}^{−}, … of functions in C_{c}(ℝ^{2}) such that ψ_{n}^{−} ≤ ψ ≤ ψ_{n}^{+} and ∫ (ψ_{n}^{+} − ψ_{n}^{−}) d𝔪 < 1/n for n = 1, 2, . . .. Obviously,

∫_{ℝ^{2}} R^{−2}ψ_{n}^{−}(x/R) dν_{V (ω)} ≤ ∫_{ℝ^{2}} R^{−2}ψ(x/R) dν_{V (ω)} ≤ ∫_{ℝ^{2}} R^{−2}ψ_{n}^{+}(x/R) dν_{V (ω)}

for any ω ∈ M, and

lim_{n→∞} ∫_{ℝ^{2}} ψ_{n}^{+} d𝔪 = lim_{n→∞} ∫_{ℝ^{2}} ψ_{n}^{−} d𝔪 = ∫_{ℝ^{2}} ψ d𝔪.

If ω ∈ U, then

∫_{ℝ^{2}} R^{−2}ψ_{n}^{+}(x/R) dν_{V (ω)} = c_{V,μ} ∫_{ℝ^{2}} ψ_{n}^{+} d𝔪,
∫_{ℝ^{2}} R^{−2}ψ_{n}^{−}(x/R) dν_{V (ω)} = c_{V,μ} ∫_{ℝ^{2}} ψ_{n}^{−} d𝔪

for n = 1, 2, . . .. It follows that

∫_{ℝ^{2}} R^{−2}ψ(x/R) dν_{V (ω)} = c_{V,μ} ∫_{ℝ^{2}} ψ d𝔪

for any ω ∈ U. Part III of the theorem is proved.

Proposition 3.3 Let V : M → V be a map satisfying conditions (0), (A), and (B). Let c_{V,μ} denote the Siegel-Veech constant of the pair (V, μ). Then c_{V,μ} = 0 if and only if the sequence V (ω) is empty for μ-a.e. ω ∈ M.

Proof. By Part I of Theorem 3.2,

\frac{1}{μ(M)} ∫_{M} Φ[V (ω)](χ_{B(R)}) dμ(ω) = πc_{V,μ}R^{2}

for any R > 0. Observe that Φ[V (ω)](χ_{B(R)}) ≥ 0, and Φ[V (ω)](χ_{B(R)}) = 0 if and only if the sequence V (ω) contains no vectors of length at most R. Clearly, if V (ω) is empty for μ-a.e. ω ∈ M, then c_{V,μ} = 0. Conversely, if c_{V,μ} = 0, then for any R > 0 we have Φ[V (ω)](χ_{B(R)}) = 0 for μ-a.e. ω ∈ M. Hence there exists a Borel set U ⊂ M such that μ(U) = μ(M) and Φ[V (ω)](χ_{B(1)}) = Φ[V (ω)](χ_{B(2)}) = … = 0 for ω ∈ U. Then for any ω ∈ U the sequence V (ω) is empty.

4 Limit distributions

Denote by V_{1} the subset of V consisting of sequences with all weights equal to 1. For any K ⊂ ℝ^{+} = (0, ∞) define a map W_{K} : V → V_{1} as follows. Given V ∈ V, the sequence W_{K}V is obtained from V by removing all vectors with weights outside K and then changing weights of the remaining vectors to 1.

Let V ∈ V, v_{1}, v_{2}, … be the sequence of vectors of V, and w_{1}, w_{2}, … be the sequence of weights. For any compactly supported function ψ on ℝ^{2} and any function f on ℝ^{+} set Ψ[V](f, ψ) = ∑_{k≥1} f(w_{k})ψ(v_{k}). Clearly, Φ[V](ψ) = Ψ[V](id, ψ) and Φ[W_{K}V](ψ) = Ψ[V](χ_{K}, ψ) for any K ⊂ ℝ^{+}. By C_{c}(ℝ^{+}) denote the space of continuous, compactly supported functions on ℝ^{+}. Note that any function f ∈ C_{c}(ℝ^{+}) vanishes in a neighborhood of 0. Ψ[V] is a bilinear functional on C_{c}(ℝ^{+}) × C_{c}(ℝ^{2}).

Furthermore, we associate to V three families α_{V,R}, δ_{V,R}, and D_{V,R} of measures depending on the parameter R > 0. α_{V,R} is a measure on ℝ^{+}; for any K ⊂ ℝ^{+} let α_{V,R}(K) be the number of indices k such that |v_{k}| ≤ R and w_{k} ∈ K. δ_{V,R} is a measure on the unit circle S^{1} = {v ∈ ℝ^{2} : |v| = 1}; for any U ⊂ S^{1} let δ_{V,R}(U) be the number of indices k such that 0 < |v_{k}| ≤ R and v_{k}/|v_{k}| ∈ U. Finally, D_{V,R} is a measure on S^{1} × ℝ^{+}; for any U ⊂ S^{1} × ℝ^{+} let D_{V,R}(U) be the number of indices k such that 0 < |v_{k}| ≤ R and (v_{k}/|v_{k}|, w_{k}) ∈ U. The measures α_{V,R}, R > 0, describe the distribution of weights of vectors in V. The measures δ_{V,R}, R > 0, describe the distribution of their directions. The measures D_{V,R}, R > 0, describe the joint distribution of directions and weights.

Suppose V contains nonzero vectors and denote by R_{0} the length of the shortest nonzero vector in V. For any R ≥ R_{0} the measures α_{V,R}, δ_{V,R}, and D_{V,R} are nonzero so we can define probability measures α̃_{V,R} = (α_{V,R}(ℝ^{+}))^{−1}α_{V,R}, δ̃_{V,R} = (δ_{V,R}(S^{1}))^{−1}δ_{V,R}, and D̃_{V,R} = (D_{V,R}(S^{1} × ℝ^{+}))^{−1}D_{V,R}.

Now let V be a map of the space M introduced in Section 3 to V. We add two more conditions to the list started in Section 3:

(0′) for any a > 0 the map M ∋ ω ↦ W_{(a,∞)}V (ω) satisfies condition (0);

(0′′) for any f ∈ C_{c}(ℝ^{+}) and ψ ∈ C_{c}(ℝ^{2}) the function M ∋ ω ↦ Ψ[V (ω)](f, ψ) is Borel.

Lemma 4.1 Suppose a map V : M → V satisfies condition (0′). Then for any Borel set K ⊂ ℝ^{+} the map W_{K}V satisfies condition (0).

Proof. Let K denote the set of subsets K ⊂ ℝ^{+} such that the map W_{K}V satisfies condition (0). We shall verify the following properties of K:

(i) if A, B ∈ K and A ∩ B = ∅, then A ∪ B ∈ K;

(ii) if A, B ∈ K and B ⊂ A, then A \ B ∈ K;

(iii) if A_{1}, A_{2}, … ∈ K and lim_{n→∞} A_{n} exists, then lim_{n→∞} A_{n} ∈ K.

Let A, B ⊂ ℝ^{+}. If A ∩ B = ∅, then Φ[W_{A∪B}V (ω)] = Φ[W_{A}V (ω)] + Φ[W_{B}V (ω)] for all ω ∈ M. This implies property (i). If B ⊂ A, then Φ[W_{A\B}V (ω)] = Φ[W_{A}V (ω)] − Φ[W_{B}V (ω)] for all ω ∈ M. This implies property (ii). Now suppose A_{1}, A_{2}, … are subsets of ℝ^{+} such that the set A = lim_{n→∞} A_{n} exists. Then χ_{A}(t) = lim_{n→∞} χ_{A_{n}}(t) for all t > 0. It follows that Φ[W_{A}V (ω)](ψ) = lim_{n→∞} Φ[W_{A_{n}}V (ω)](ψ) for any ψ ∈ C_{c}(ℝ^{2}) and ω ∈ M. This implies property (iii).

Condition (0′) means that (a, ∞) ∈ K for any a > 0. Property (iii) implies ℝ^{+} = ⋃_{n=1}^{∞}(n^{−1}, ∞) ∈ K and ∅ = ⋂_{n=1}^{∞}(n, ∞) ∈ K. For any a, b, 0 ≤ a < b, we have (a, b] = (a, ∞) \ (b, ∞) ∈ K. Further, (a, b) = ⋃_{n=1}^{∞}(a, b − n^{−1}] ∈ K. Since any open subset of ℝ^{+} is a disjoint union of intervals, properties (i) and (iii) imply K contains all open subsets of ℝ^{+}. Then it follows from the properties (i), (ii), and (iii) that K contains all Borel subsets of ℝ^{+}.

Lemma 4.2 Conditions (0′) and (0′′) are equivalent. Condition (0′′) implies condition (0).

Proof. Let F denote the set of functions f on ℝ^{+} such that the function M ∋ ω ↦ Ψ[V (ω)](f, ψ) is Borel for any ψ ∈ C_{c}(ℝ^{2}). F have the following properties:

(i) if f_{1}, f_{2} ∈ F, then a_{1}f_{1} + a_{2}f_{2} ∈ F for any a_{1}, a_{2} ∈ ℝ;

(ii) if f_{1}, f_{2}, … ∈ F and f = lim_{n→∞} f_{n} pointwise, then f ∈ F.

The first property follows from the fact that Ψ[V (ω)](f, ψ) depends linearly on f. To verify the second property observe that Ψ[V (ω)](f, ψ) = lim_{n→∞} Ψ[V (ω)](f_{n}, ψ) for any ψ ∈ C_{c}(ℝ^{2}) and ω ∈ M if f(t) = lim_{n→∞} f_{n}(t) for all t > 0.

Suppose the map V satisfies condition (0′). Then Lemma 4.1 implies χ_{K} ∈ F for all Borel sets K ⊂ ℝ^{+}. Given f ∈ C_{c}(ℝ^{+}), for any ϵ > 0 there exist Borel sets K_{1}, …, K_{n} and constants a_{1}, …, a_{n} such that sup |f −(a_{1}χ_{K_{1}} +⋯+a_{n}χ_{K_{n}})| < ϵ. It follows from properties (i) and (ii) that f ∈ F. Thus C_{c}(ℝ^{+}) ⊂ F, i.e., condition (0′′) holds. Conversely, if the map V satisfies condition (0′′), that is, C_{c}(ℝ^{+}) ⊂ F, then it easily follows from property (ii) that χ_{(a,∞)} ∈ F for any a > 0 and id_{(0,∞)} ∈ F. This means that conditions (0′) and (0) hold for V.

Proposition 4.3 Let a map V : M → V satisfy conditions (0′), (A), and (B). Then (a) for any Borel set K ⊂ ℝ^{+} the map W_{K}V satisfies conditions (0) and (A);

(b) for any K ⊂ (ϵ, ∞), ϵ > 0, the map W_{K}V satisfies condition (B);

(c) there exists a unique Borel measure λ_{V,μ} on ℝ^{+} with the following property: for any Borel set K ⊂ ℝ^{+} such that W_{K}V satisfies conditions (0), (A), and (B) the Siegel-Veech constant of the pair (W_{K}V, μ) is equal to λ_{V,μ}(K);

(d) if c_{V,μ} denotes the Siegel-Veech constant of the pair (V, μ), then

c_{V,μ} = ∫_{0}^{∞} t dλ_{V,μ}(t);

(e) if condition (B) holds for the map W_{(0,∞)}V, then the measure λ_{V,μ} is finite and condition (B) holds for W_{K}V for any K ⊂ ℝ^{+}.

The measure λ_{V,μ} is called the Siegel-Veech measure of the pair (V, μ). Proof of Proposition 4.3. By Lemma 4.1, the map W_{K}V satisfies condition (0) for any Borel set K ⊂ ℝ^{+}. For any K ⊂ ℝ^{+} condition (A) holds for W_{K}V since the SL(2, ℝ) action on V does not affect weights. If K ⊂ (ϵ, ∞), ϵ > 0, then N_{W_{K}V (ω)}(R) ≤ ϵ^{−1}N_{V (ω)}(R) for all ω ∈ M and R > 0, hence condition (B) holds for W_{K}V. If condition (B) holds for the map W_{(0,∞)}V, then this condition holds for W_{K}V for any K ⊂ ℝ^{+} since N_{W_{K}V (ω)}(R) ≤ N_{W_{(0,∞)}V (ω)}(R) for all ω ∈ M and R > 0.

Let K ⊂ ℝ^{+} be a Borel set. By the above for any ψ ∈ C_{c}(ℝ^{2}) the function M ∋ ω ↦ Φ[W_{K}V (ω)](ψ) is Borel. Then it follows that for any R > 0 the function ω ↦ Φ[W_{K}V (ω)](χ_{B(R)}) = N_{W_{K}V (ω)}(R) is Borel (cf. the proof of Lemma 3.1). Set

λ_{V,μ}(K) = \frac{1}{π · μ(M)} ∫_{M} N_{W_{K}V (ω)}(1) dμ(ω).

Obviously, 0 ≤ λ_{V,μ}(K) ≤ ∞. Suppose K_{1}, K_{2}, … are disjoint Borel subsets of ℝ^{+} and denote K = ⋃_{n=1}^{∞} K_{n}. Then N_{W_{K}V (ω)}(1) = ∑_{n=1}^{∞} N_{W_{Kn}V (ω)}(1) for any ω ∈ M. It follows that λ_{V,μ}(K) = ∑_{n=1}^{∞} λ_{V,μ}(K_{n}). Thus λ_{V,μ} is a measure. If the map W_{K}V satisfies conditions (0), (A), and (B) for some Borel set K ⊂ ℝ^{+}, then the Siegel-Veech constant of (W_{K}V, μ) is equal to λ_{V,μ}(K) due to Part I of Theorem 3.2. If condition (B) holds for the map W_{(0,∞)}V, then λ_{V,μ}(ℝ^{+}) is the Siegel-Veech constant of (W_{(0,∞)}V, μ), in particular, λ_{V,μ}(ℝ^{+}) < ∞.

Suppose λ̃ is a Borel measure on ℝ^{+} having the property required in the statement (c). Take any Borel set K ⊂ ℝ^{+}. For each ϵ > 0 let K_{ϵ} = K ∩ (ϵ, ∞). By the above conditions (0), (A), and (B) hold for the map W_{K_{ϵ}}V, hence λ̃(K_{ϵ}) is the Siegel-Veech constant of (W_{K_{ϵ}}V, μ), in particular, λ̃(K_{ϵ}) = λ_{V,μ}(K_{ϵ}). Then λ̃(K) = lim_{ϵ→0} λ̃(K_{ϵ}) = lim_{ϵ→0} λ_{V,μ}(K_{ϵ}) = λ_{V,μ}(K). Thus λ̃ = λ_{V,μ}.

Let f_{1}, f_{2}, … be the functions on ℝ^{+} such that f_{n}(t) = 2^{−n}k for 2^{−n}k < t ≤ 2^{−n}(k + 1), k = 0, 1, . . .. Then 0 ≤ f_{1} ≤ f_{2} ≤ … and lim_{n→∞} f_{n}(t) = t for all t > 0. It follows that

lim_{n→∞} ∫_{0}^{∞} f_{n}(t) dλ_{V,μ}(t) = ∫_{0}^{∞} t dλ_{V,μ}(t),

lim_{n→∞} ∫_{M} Ψ[V (ω)](f_{n}, χ_{B(1)}) dμ(ω) = ∫_{M} N_{V (ω)}(1) dμ(ω).

For any integer n ≥ 1 we have

∫_{M} Ψ[V (ω)](f_{n}, χ_{B(1)}) dμ(ω) = ∑_{k=1}^{∞} 2^{−n}k ∫_{M} N_{W_{(2^{−n}k,2^{−n}(k+1)]}V (ω)}(1) dμ(ω) =

π · μ(M) ∑_{k=1}^{∞} 2^{−n}k · λ_{V,μ}((2^{−n}k, 2^{−n}(k + 1)]) = π · μ(M) ∫_{0}^{∞} f_{n}(t) dλ_{V,μ}(t),

hence

∫_{0}^{∞} t dλ_{V,μ}(t) = \frac{1}{π · μ(M)} ∫_{M} N_{V (ω)}(1) dμ(ω).

By Part I of Theorem 3.2, the right-hand side of the last equality is equal to the Siegel-Veech constant of the pair (V, μ).

Proposition 4.4 Suppose a map V : M → V satisfies conditions (0′), (A), (B), and (C). Then for μ-a.e. ω ∈ M,

lim_{R→∞}(πR^{2})^{−1}α_{V (ω),R} = λ_{V,μ}

in the following sense:

lim_{R→∞} \frac{1}{πR^{2}} ∫_{0}^{∞} f(t) dα_{V (ω),R}(t) = ∫_{0}^{∞} f(t) dλ_{V,μ}(t)  (1)

for any bounded function f ∈ C(ℝ^{+}) vanishing in a neighborhood of 0.

If, in addition, condition (B) holds for the map W_{(0,∞)}V and the Siegel-Veech constant of the pair (V, μ) is nonzero, then for μ-a.e. ω ∈ M,

lim_{R→∞} α̃_{V (ω),R} = (λ_{V,μ}(ℝ^{+}))^{−1}λ_{V,μ}

in the following sense:

lim_{R→∞} ∫_{0}^{∞} f(t) dα̃_{V (ω),R}(t) = \frac{1}{λ_{V,μ}(ℝ^{+})} ∫_{0}^{∞} f(t) dλ_{V,μ}(t)  (2)

for any bounded function f ∈ C(ℝ^{+}).

Proof. Let K ⊂ ℝ^{+} be a Borel set such that K ⊂ (ϵ, ∞) for some ϵ > 0. By Proposition 4.3, the map W_{K}V satisfies conditions (0), (A), and (B). Condition (C) also holds for W_{K}V since N_{W_{K}V (ω)}(R) ≤ ϵ^{−1}N_{V (ω)}(R) for any ω ∈ M and R > 0. By Part II of Theorem 3.2, for μ-a.e. ω ∈ M we have

lim_{R→∞} R^{−2}N_{W_{K}V (ω)}(R) = πλ_{V,μ}(K).  (3)

Let K denote the countable collection of sets consisting of intervals (r_{1}, r_{2}] such that r_{1}, r_{2} are rational numbers and 0 < r_{1} < r_{2}. By the above there exists a Borel set U ⊂ M, μ(U) = μ(M), such that for any ω ∈ U and any K ∈ K relation (3) holds.

Take any ω ∈ U. For each function f ∈ L^{1}(ℝ^{+}, λ_{V,μ}) set

F_{ω}(f) = lim sup_{R→∞} |\frac{1}{πR^{2}} ∫_{0}^{∞} f(t) dα_{V (ω),R}(t) − ∫_{0}^{∞} f(t) dλ_{V,μ}(t)|.

Suppose f ∈ C_{c}(ℝ^{+}) and choose a > 1 such that f = 0 outside the segment [1/a, a]. For any ϵ > 0 there exist disjoint sets K_{1}, …, K_{n} ∈ K and constants b_{1}, …, b_{n} such that the function f_{ϵ} = b_{1}χ_{K_{1}} + ⋯ + b_{n}χ_{K_{n}} satisfies the inequality sup |f − f_{ϵ}| < ϵ. Observe that

∫_{0}^{∞} f_{ϵ}(t) dα_{V (ω),R}(t) = ∑_{i=1}^{n} b_{i}N_{W_{K_{i}}V (ω)}(R),

∫_{0}^{∞} f_{ϵ}(t) dλ_{V,μ}(t) = ∑_{i=1}^{n} b_{i}λ_{V,μ}(K_{i}).

Since ω ∈ U, it follows that F_{ω}(f_{ϵ}) = 0. Further,

∫_{0}^{∞} |f(t) − f_{ϵ}(t)| dα_{V (ω),R}(t) ≤ ϵN_{W_{[1/a,a]}V (ω)}(R),
∫_{0}^{∞} |f(t) − f_{ϵ}(t)| dλ_{V,μ}(t) ≤ ϵλ_{V,μ}([a^{−1}, a]).

It follows that F_{ω}(f) ≤ ϵM(a, ω), where M(a, ω) = sup_{R>0} R^{−2}N_{W_{[1/a,a]}V (ω)}(R) + λ_{V,μ}([1/a, a]). Note that M(a, ω) < ∞ while ϵ can be chosen arbitrarily small. Thus F_{ω}(f) = 0.

Now let f be a bounded continuous function on ℝ^{+} vanishing in a neighborhood of 0. For any a > 0 there exists f̃_{a} ∈ C_{c}(ℝ^{+}) such that f̃_{a} = f on (0, a) and sup |f̃_{a} − f| ≤ sup |f|. We have

∫_{0}^{∞} |f(t) − f̃_{a}(t)| dα_{V (ω),R}(t) ≤ N_{W_{[a,∞)}V (ω)}(R) sup |f|,
∫_{0}^{∞} |f(t) − f̃_{a}(t)| dλ_{V,μ}(t) ≤ λ_{V,μ}([a, ∞)) sup |f|.

By the above F_{ω}(f̃_{a}) = 0, hence F_{ω}(f) ≤ M_{∞}(a, ω) sup |f|, where M_{∞}(a, ω) = sup_{R>0} R^{−2}N_{W_{[a,∞)}V (ω)}(R) + λ_{V,μ}([a, ∞)). Since N_{W_{[a,∞)}V (ω)}(R) ≤ a^{−1}N_{V (ω)}(R), it follows that M_{∞}(a, ω) → 0 as a → ∞. Thus F_{ω}(f) = 0, i.e., equality (1) holds.

Now suppose the map W_{(0,∞)}V satisfies condition (B). Then λ_{V,μ} is a finite measure. Let K_{0} denote the collection of intervals (r_{1}, r_{2}] such that r_{1}, r_{2} are rational numbers and 0 ≤ r_{1} < r_{2}. Then there exists a Borel set U_{0} ⊂ U, μ(U_{0}) = μ(M), such that for any ω ∈ U_{0} and any K ∈ K_{0} relation (3) holds. Take any ω ∈ U_{0}. Let f be a bounded continuous function on ℝ^{+}. For any ϵ > 0 there exists a bounded function f̂_{ϵ} ∈ C(ℝ^{+}) such that f̂_{ϵ} vanishes in a neighborhood of 0, f̂_{ϵ} = f on (ϵ, ∞), and sup |f̂_{ϵ} − f| ≤ sup |f|. We have

∫_{0}^{∞} |f(t) − f̂_{ϵ}(t)| dα_{V (ω),R}(t) ≤ N_{W_{(0,ϵ]}V (ω)}(R) sup |f|,
∫_{0}^{∞} |f(t) − f̂_{ϵ}(t)| dλ_{V,μ}(t) ≤ λ_{V,μ}((0, ϵ]) sup |f|.

By the above F_{ω}(f̂_{ϵ}) = 0, therefore F_{ω}(f) ≤ M_{0}(ϵ, ω) sup |f|, where M_{0}(ϵ, ω) = lim sup_{R>0}(πR^{2})^{−1}N_{W_{(0,ϵ]}V (ω)}(R) + λ_{V,μ}((0, ϵ]). Since ω ∈ U_{0}, we have M_{0}(ϵ, ω) ≤ λ_{V,μ}((0, r]) + λ_{V,μ}((0, ϵ]) for any rational r ≥ ϵ. Since λ_{V,μ}(ℝ^{+}) < ∞, it follows that M_{0}(ϵ, ω) ≤ 2λ_{V,μ}((0, ϵ]) and M_{0}(ϵ, ω) → 0 as ϵ → 0. Thus F_{ω}(f) = 0, i.e., equality (1) holds for f. In particular, equality (1) holds for f = 1 so we get

lim_{R→∞}(πR^{2})^{−1}α_{V (ω),R}(ℝ^{+}) = λ_{V,μ}(ℝ^{+}).

If the Siegel-Veech constant of (V, μ) is nonzero, then λ_{V,μ}(ℝ^{+}) > 0 and equality (2) follows.

Proposition 4.5 Suppose a map V : M → V_{1} satisfies conditions (0), (A), (B), (C), (E), and the Siegel-Veech constant of the pair (V, μ) is nonzero. Then for μ-a.e. ω ∈ M,

lim_{R→∞} δ̃_{V (ω),R} = 𝔪_{1},

where 𝔪_{1} is Lebesgue measure on S^{1} normalized so that 𝔪_{1}(S^{1}) = 1. The convergence means that

lim_{R→∞} ∫_{S^{1}} ϕ(θ) dδ̃_{V (ω),R}(θ) = ∫_{S^{1}} ϕ(θ) d𝔪_{1}(θ)  (4)

for any Riemann integrable function ϕ on S^{1}.

Proof. By Part III of Theorem 3.2, there exists a Borel set U ⊂ M, μ(U) = μ(M), such that for any ω ∈ U,

lim_{R→∞} ∫_{ℝ^{2}} \frac{1}{R^{2}} ψ(\frac{x}{R}) dν_{V (ω)}(x) = c_{V,μ} ∫_{ℝ^{2}} ψ(x) d𝔪(x)  (5)

for each compactly supported, Riemann integrable function ψ on ℝ^{2}. Here c_{V,μ} is the Siegel-Veech constant of (V, μ).

Take any ω ∈ U. Let ϕ be an arbitrary Riemann integrable function on S^{1}. For any x ∈ ℝ^{2} set ϕ_{1}(x) = ϕ(x/|x|) if 0 < |x| ≤ 1, and ϕ_{1}(x) = 0 otherwise. Then ϕ_{1} is a Riemann integrable function on ℝ^{2}, therefore equality (5) holds for ψ = ϕ_{1}. Since V (M) ⊂ V_{1}, we have

∫_{ℝ^{2}} ϕ_{1}(x/R) dν_{V (ω)}(x) = ∫_{S^{1}} ϕ(θ) dδ_{V (ω),R}(θ)

for all R > 0. Besides,

∫_{ℝ^{2}} ϕ_{1}(x) d𝔪(x) = π ∫_{S^{1}} ϕ(θ) d𝔪_{1}(θ).

Hence

lim_{R→∞} R^{−2} ∫_{S^{1}} ϕ(θ) dδ_{V (ω),R}(θ) = πc_{V,μ} ∫_{S^{1}} ϕ(θ) d𝔪_{1}(θ).

In particular, the latter relation holds for ϕ = 1 so we get

lim_{R→∞} R^{−2}δ_{V (ω),R}(S^{1}) = πc_{V,μ}.

Since c_{V,μ} > 0, equality (4) follows.

Proposition 4.6 Suppose a map V : M → V satisfies conditions (0′), (A), (B), (C), and (E). Then for μ-a.e. ω ∈ M,

lim_{R→∞}(πR^{2})^{−1}D_{V (ω),R} = 𝔪_{1} × λ_{V,μ}

in the following sense:

lim_{R→∞} \frac{1}{πR^{2}} ∫_{S^{1}×ℝ^{+}} f(Θ) dD_{V (ω),R}(Θ) = ∫_{S^{1}} ∫_{0}^{∞} f(θ, t) d𝔪_{1}(θ) dλ_{V,μ}(t)  (6)

for any bounded function f ∈ C(S^{1} × ℝ^{+}) vanishing in a neighborhood of S^{1} × {0}.

If, in addition, condition (B) holds for the map W_{(0,∞)}V and the Siegel-Veech constant of the pair (V, μ) is nonzero, then for μ-a.e. ω ∈ M,

lim_{R→∞} D̃_{V (ω),R} = (λ_{V,μ}(ℝ^{+}))^{−1}𝔪_{1} × λ_{V,μ}

in the following sense:

lim_{R→∞} ∫_{S^{1}×ℝ^{+}} f(Θ) dD̃_{V (ω),R}(Θ) = \frac{1}{λ_{V,μ}(ℝ^{+})} ∫_{S^{1}} ∫_{0}^{∞} f(θ, t) d𝔪_{1}(θ) dλ_{V,μ}(t)

for any bounded function f ∈ C(S^{1} × ℝ^{+}).

Proof. Let K ⊂ ℝ^{+} be a Borel set such that K ⊂ (ϵ, ∞) for some ϵ > 0. Then the map W_{K}V satisfies conditions (0), (A), (B), (C), and (E) (cf. the proof of Proposition 4.4). As shown in the proof of Proposition 4.5, for μ-a.e. ω ∈ M we have

lim_{R→∞} R^{−2} ∫_{S^{1}} ϕ(θ) dδ_{W_{K}V (ω),R}(θ) = πλ_{V,μ}(K) ∫_{S^{1}} ϕ(θ) d𝔪_{1}(θ)  (7)

for any Riemann integrable function ϕ on S^{1}. Let K denote the collection of intervals (r_{1}, r_{2}] such that r_{1}, r_{2} are rational numbers and 0 < r_{1} < r_{2}. By the above there exists a Borel set U ⊂ M, μ(U) = μ(M), such that for any ω ∈ U, any K ∈ K, and any function ϕ ∈ C(S^{1}) relation (7) holds.

Take any ω ∈ U. For each function f ∈ L^{1}(S^{1} × ℝ^{+}, 𝔪_{1} × λ_{V,μ}) set

F_{ω}(f) = lim sup_{R→∞} |\frac{1}{πR^{2}} ∫_{S^{1}×ℝ^{+}} f(Θ) dD_{V (ω),R}(Θ) − ∫_{S^{1}} ∫_{0}^{∞} f(θ, t) d𝔪_{1}(θ) dλ_{V,μ}(t)|.

Suppose f ∈ C_{c}(S^{1}×ℝ^{+}), i.e., f is a continuous function on S^{1}×ℝ^{+} vanishing outside S^{1} × [1/a, a] for some a > 1. For any ϵ > 0 there exist disjoint sets K_{1}, …, K_{n} ∈ K and functions ϕ_{1}, …, ϕ_{n} ∈ C(S^{1}) such that the function f_{ϵ} defined by f_{ϵ}(θ, t) = ϕ_{1}(θ)χ_{K_{1}}(t)+⋯+ϕ_{n}(θ)χ_{K_{n}}(t), θ ∈ S^{1}, t ∈ ℝ^{+}, satisfies the inequality sup |f −f_{ϵ}| < ϵ. Observe that

∫_{S^{1}×ℝ^{+}} f_{ϵ}(Θ) dD_{V (ω),R}(Θ) = ∑_{i=1}^{n} ∫_{S^{1}} ϕ_{i}(θ) dδ_{W_{K_{i}}V (ω),R}(θ),

∫_{S^{1}} ∫_{0}^{∞} f_{ϵ}(θ, t) d𝔪_{1}(θ) dλ_{V,μ}(t) = ∑_{i=1}^{n} λ_{V,μ}(K_{i}) ∫_{S^{1}} ϕ_{i}(θ) d𝔪_{1}(θ).

Since ω ∈ U, it follows that F_{ω}(f_{ϵ}) = 0. Further,

∫_{S^{1}×ℝ^{+}} |f(Θ) − f_{ϵ}(Θ)| dD_{V (ω),R}(Θ) ≤ ϵN_{W_{[1/a,a]}V (ω)}(R),

∫_{S^{1}} ∫_{0}^{∞} |f(θ, t) − f_{ϵ}(θ, t)| d𝔪_{1}(θ) dλ_{V,μ}(t) ≤ ϵλ_{V,μ}([a^{−1}, a]).

It follows that F_{ω}(f) ≤ ϵM(a, ω), where M(a, ω) = sup_{R>0} R^{−2}N_{W_{[1/a,a]}V (ω)}(R) + λ_{V,μ}([1/a, a]). Since M(a, ω) < ∞ while ϵ can be chosen arbitrarily small, we have F_{ω}(f) = 0, i.e., equality (6) holds.

The remaining part of the proof is completely analogous to the corresponding part of the proof of Proposition 4.4 and we omit it.

5 Translation surfaces. The Delaunay partitions

Let M be a compact connected oriented surface. A translation structure on M is an atlas of coordinate charts ω = {(U_{α}, f_{α})}_{α∈A}, where U_{α} is a domain in M and f_{α} is an orientation-preserving homeomorphism of U_{α} onto a domain in ℝ^{2}, such that:

(i) all transition functions are translations in ℝ^{2};

(ii) chart domains U_{α}, α ∈ A, cover all surface M except for finitely many points (called singular points);

(iii) the atlas ω is maximal relative to the conditions (i) and (ii);

(iv) a punctured neighborhood of any singular point covers a punctured neighborhood of a point in ℝ^{2} via an m-to-1 map which is a translation in coordinates of the atlas ω; the number m is called the multiplicity of the singular point.

A translation surface is a compact connected oriented surface equipped with a translation structure. In what follows we assume that each translation surface has at least one singular point.

Let M be a translation surface and ω be the translation structure of M. Each translation of the plane ℝ^{2} is a smooth map preserving Euclidean metric and Lebesgue measure on ℝ^{2}. Hence the translation structure ω induces a smooth structure, a flat Riemannian metric, and a finite Borel measure on the surface M punctured at the singular points of ω. Each singular point of ω is a cone type singularity of the metric. The cone angle is equal to 2πm, where m is the multiplicity of the singular point. A singular point of multiplicity 1 is called removable as the flat metric can be continuously extended to this point. Any geodesic of the metric is a straight line in coordinates of the atlas ω. The translation structure ω allows us to identify the tangent space at any nonsingular point x ∈ M with the Euclidean space ℝ^{2} so that velocity be an integral of the geodesic flow with respect to this identification. Thus each oriented geodesic is assigned a direction v ∈ S^{1}.

Let M be a compact connected oriented surface. Suppose M is endowed with a complex structure given by an atlas of charts (U_{α}, z_{α}), α ∈ A, and let q be a nonzero Abelian differential (holomorphic 1-form) on M. A chart (U, z) is called a natural parameter of q if q = dz in U. Let ω denote the atlas of all natural parameters of q. By identifying ℂ with ℝ^{2} we can assume that charts of ω range in ℝ^{2}. Then ω becomes a translation structure on M. The singular points of ω are the zeroes of the differential q, namely, a zero of order n is a singular point of multiplicity n+1. Every translation structure on M without removable singular points can be obtained this way.

A simple and effective way to construct translation surfaces is to glue them from polygons. Let P_{1}, …, P_{n} be disjoint plane polygons. The natural orientation of ℝ^{2} induces an orientation of the boundary of every polygon. Suppose all sides of the polygons P_{1}, …, P_{n} are distributed into pairs such that two sides in each pair are of the same length and direction, and of opposite orientations. Glue the sides in each pair by translation. Then the union of the polygons P_{1}, …, P_{n} becomes a surface M. By construction, the surface M is compact and oriented. Suppose M is connected (if it is not, then we should apply the construction to a smaller set of polygons). The restrictions of the identity map on ℝ^{2} to the interiors of the polygons P_{1}, …, P_{n} can be regarded as charts of M. This finite collection of charts extends to a translation structure ω on M. The translation structure ω is uniquely determined if we require that the set of singular points of ω be the set of points corresponding to vertices of the polygons P_{1}, …, P_{n}.

Let M be a translation surface. A saddle connection of M is a geodesic segment joining two singular points or a singular point to itself and having no singular points in its interior. Two saddle connections of M are said to be disjoint if they have no common interior points (common endpoints are allowed). A domain U ⊂ M containing no singular points is called a triangle (a polygon, an n-gon) if there is a map f : U → ℝ^{2} such that the chart (U, f) belongs to the translation structure of M and f(U) is the interior of a triangle (resp. a polygon, an n-gon) in the plane ℝ^{2}. Suppose we have a finite collection of pairwise disjoint saddle connections dividing the surface M into finitely many domains such that each domain is a polygon and, moreover, each n-gon is bounded by n saddle connections (in general, an n-gon on M may be bounded by more than n saddle connections). Then M can be obtained by gluing together plane polygons as described above so that the saddle connections correspond to sides of glued polygons. The following well-known proposition shows, in particular, that any translation surface can be obtained this way.

Proposition 5.1 (a) Any collection of pairwise disjoint saddle connections of a translation surface M can be extended to a maximal collection.

(b) Any maximal collection of pairwise disjoint saddle connections partitions the surface M into triangles, each triangle being bounded by three saddle connections.

(c) For any maximal collection, the number of saddle connections is equal to 3m and the number of triangles in the corresponding triangulation is equal to 2m, where m is the sum of multiplicities of singular points.

(d) m = 2p − 2 + k, where p is the genus of M and k is the number of singular points of M.

Every translation surface has a useful Delaunay partition. We shall define this partition following the paper of Masur and Smillie [MS].

Let Z be the set of singular points of a translation surface M. For any x ∈ M \Z let d(x, Z) denote the distance from x to the set Z. A geodesic segment joining x to a singular point is called a length-minimizing path if its length is equal to d(x, Z). Any point x ∈ M \ Z admits at least one length-minimizing path and the number of all such paths is at most finite (see [MS]).

For any R > 0 let B(R) denote the disk of radius R in ℝ^{2} centered at the origin. Given a point x ∈ M \ Z, there is a unique map ı_{x} : B(d(x, Z)) → M \ Z such that ı_{x}(0) = x and ı_{x} is a translation with respect to the translation structure of M. The map ı_{x} is uniquely extended to a continuous map of the closure of B(d(x, Z)) to M. Let Z′_{x} be the set of points in ∂B(d(x, Z)) that map to points in Z. Z′_{x} is a nonempty finite set. Each segment joining the origin to a point in Z′_{x} is mapped by ı_{x} to a length-minimizing path. The cardinality of Z′_{x} is equal to the number of length-minimizing paths starting at x. Let H_{x} be the convex hull of Z′_{x}. If Z′_{x} consists of two points then H_{x} is a segment and its image under ı_{x} is a saddle connection called a Delaunay edge. If Z′_{x} consists of more than two points then H_{x} is a polygon inscribed in the circle ∂B(d(x, Z)). It is shown in [MS] that the restriction of the map ı_{x} to the interior of H_{x} is injective. The image under ı_{x} of the interior of H_{x} is a polygon on M called a Delaunay cell.

Proposition 5.2 ([MS]) Distinct Delaunay edges are disjoint saddle connections. Distinct Delaunay cells are disjoint domains. Any Delaunay cell is bounded by Delaunay edges. Any Delaunay edge separates two Delaunay cells or a Delaunay cell from itself.

Proposition 5.2 implies that Delaunay edges partition the surface M into Delaunay cells. This partition is called the Delaunay partition of the translation surface M. By Proposition 5.1, the number of Delaunay edges and Delaunay cells is finite.

Proposition 5.3 Suppose τ is a triangulation of a translation surface M by a maximal set of disjoint saddle connections. Then τ is the Delaunay partition of M if and only if for any edge L of τ two angles θ_{1}, θ_{2} opposite L in two triangles of τ bounded by L satisfy the inequality θ_{1} + θ_{2} < π.

Proof. By Proposition 5.2, the triangulation τ is the Delaunay partition of the translation surface M if and only if each triangle of τ is a Delaunay cell.

Denote by ω the translation structure of M. For any triangle T of τ there is a map f_{T} : T → ℝ^{2} such that (T, f_{T}) ∈ ω. The map f_{T} is determined up to translation. We can assume without loss of generality that the triangle f_{T} (T) ⊂ ℝ^{2} is inscribed in a circle centered at the origin. By R_{T} denote the radius of the circle.

Let L be an edge of τ and T_{1}, T_{2} be the triangles of τ bounded by L. Let θ_{1} and θ_{2} be the angles of T_{1} and T_{2} opposite L. By U denote the union of T_{1}, T_{2}, and the interior of the edge L. Then U is a polygon. Let f : U → ℝ^{2} be a map such that (U, f) ∈ ω and f = f_{T_{1}} on T_{1}. Suppose T_{1} is a Delaunay cell. By Z denote the set of singular points of M. Then there exists a point x ∈ M \ Z such that d(x, Z) = R_{T_{1}} and f(ı_{x}(x′)) = x′ for any x′ ∈ f(T_{1}). It is easy to see that f(ı_{x}(x′)) = x′ for any x′ ∈ f(U) ∩ B(R_{T_{1}}). The only points in the closure of B(R_{T_{1}}) mapped by ı_{x} to singular points are vertices of the triangle f(T_{1}). Therefore the vertex of the triangle f(T_{2}) opposite its side f(L) lies outside the circle ∂B(R_{T_{1}}). The latter condition is equivalent to the inequality θ_{1} + θ_{2} < π.

Now suppose that for any edge L of τ the sum of two angles opposite L is less than π. Let T be a triangle of τ, x_{0} ∈ T, and I be a geodesic segment joining x_{0} to a singular point. Further, let x′_{0} = f_{T} (x_{0}) and y′ be a point in ℝ^{2} such that the vector y′ − x′_{0} is of the same length and direction as I. We claim that y′ lies outside the circle ∂B(R_{T}) unless it is a vertex of the triangle f_{T} (T). It easily follows from the claim that T is a Delaunay cell. The claim is proved by induction on the number n of times the segment I intersects edges of the triangulation τ. In the case n = 0 the segment I does not leave the triangle T hence the point y′ is a vertex of f_{T} (T). Now let n > 0 and assume the claim holds for all smaller numbers of intersections. Let L be the edge of τ first intersected by I and T_{1} be the triangle of τ separated from T by L. Let U denote the union of triangles T, T_{1}, and the interior of the edge L. There exists a map f : U → ℝ^{2} such that (U, f) ∈ ω and f = f_{T} on T. Let z′ be the center of the circle circumscribed around the triangle f(T_{1}). Then f(x) = z′ +f_{T_{1}}(x) for all x ∈ T_{1}. Take a point x_{1} ∈ T_{1} ∩ I such that the subsegment of I joining x_{0} to x_{1} intersects L only once and does not intersect the other edges of τ. By I_{1} denote the subsegment of I joining x_{1} to a singular point. Then the point x′_{1} = f(x_{1}) lies on the segment joining x′_{0} to y′ and the vector y′ − x′_{1} is of the same length and direction as I_{1}. By construction, I_{1} has n − 1 intersections with edges of τ. By the inductive assumption the point y′ lies outside the circle z′ + ∂B(R_{T_{1}}) circumscribed around f(T_{1}) unless y′ is the vertex of f(T_{1}) opposite the side f(L). Since the sum of two angles opposite f(L) in triangles f(T) and f(T_{1}) is less than π, it follows that the disk z′ + B(R_{T_{1}}) contains the part of the disk B(R_{T}) separated from the triangle f(T) by f(L). Hence y′ lies outside the circle ∂B(R_{T}). The claim is proved and so is the proposition.

Lemma 5.4 Suppose M is a translation surface of area at most 1, x ∈ M, and d is the distance from x to the closest singular point of M. If d ≥ \sqrt{2/π} then x belongs to a periodic cylinder of length at most d^{−1}.

Proof. There is a map ı : B(d) → M such that ı(0) = x and ı is a translation with respect to the translation structure of M. Let r > 0 be the maximal number such that ı is injective on B(r). Then πr^{2} ≤ 1 since the area of M is at most 1. Assuming d ≥ \sqrt{2/π}, one has r < d. By the choice of r, there are distinct points x′_{1}, x′_{2} ∈ ∂B(r) such that ı(x′_{1}) = ı(x′_{2}). It is easy to observe that the segment joining x′_{1} to x′_{2} is a diameter of the disk B(r) and ı maps the segment to a periodic geodesic passing through x. Hence x belongs to a periodic cylinder C of length 2r. Let w denote the width of C. The area of the cylinder is equal to 2rw, therefore 2rw ≤ 1. The Pythagorean theorem implies the distance from x to the closest singular point on the boundary of C is at most \sqrt{(w/2)^{2} + r^{2}}. Hence d^{2} ≤ (w/2)^{2} + r^{2}. Since d ≥ \sqrt{2/π}, we have r^{2} ≤ d^{2}/2, then d^{2}/2 ≤ (w/2)^{2}. In particular, 2r ≤ w^{−1} ≤ d^{−1}.

6 Moduli spaces of translation surfaces

Let M, M′ be translation surfaces and ω, ω′ be their translation structures. An orientation-preserving homeomorphism f : M → M′ is called an isomorphism of the translation surfaces if f maps the set of singular points of M onto the set of singular points of M′ and f is a translation in local coordinates of the atlases ω and ω′. The translation structures ω and ω′ are called isomorphic if there is an isomorphism f : M → M′. If M = M′ and ω = ω′ then the isomorphism f is called an automorphism of ω. Automorphisms of the translation structure ω form a group Aut(ω), which is finite.

Given positive integers p and n, let M_{p} be a compact connected oriented surface of genus p and Z_{n} be a subset of M_{p} of cardinality n. Denote by Ω(p, n) the set of translation structures on M_{p} such that Z_{n} is the set of singular points. Let ω = {(U_{α}, ϕ_{α})}_{α∈A} be a translation structure on M_{p}. Given an orientationpreserving homeomorphism f : M_{p} → M_{p}, the atlas ωf = {(f^{−1}(U_{α}), ϕ_{α}f)}_{α∈A} is a translation structure on M_{p} isomorphic to ω. Let H(p, n) denote the group of orientation-preserving homeomorphisms of the surface M_{p} leaving invariant the set Z_{n}. By H_{0}(p, n) denote the subgroup of H(p, n) consisting of homeomorphisms isotopic to the identity. For any ω ∈ Ω(p, n) and f ∈ H(p, n) the translation structure ωf belongs to Ω(p, n). Isomorphic translation structures ω, ω′ ∈ Ω(p, n) are called isotopic if ω′ = ωf_{0} for some f_{0} ∈ H_{0}(p, n). Let Q(p, n) denote the set of equivalence classes of isotopic translation structures in Ω(p, n) and MQ(p, n) denote the set of equivalence classes of isomorphic translation structures in Ω(p, n). The map H(p, n) × Ω(p, n) ∋ (f, ω) ↦ ωf^{−1} defines an action of the group H(p, n) on the set Ω(p, n). By definition, Q(p, n) = Ω(p, n)/H_{0}(p, n) and MQ(p, n) = Ω(p, n)/H(p, n). The mapping class group Mod(p, n) = H(p, n)/H_{0}(p, n) acts naturally on the set Q(p, n) and MQ(p, n) = Q(p, n)/ Mod(p, n).

Any translation structure on a surface of genus p with n singular points is isomorphic to a translation structure in Ω(p, n). Thus MQ(p, n) is the moduli space of translation surfaces of genus p with n singular points.

As mentioned in Section 5, there is a one-to-one correspondence between translation structures in Ω(p, n) and pairs (X, q) such that X is a complex structure on M_{p} and q is an Abelian differential of X whose zeroes are contained in Z_{n}. This allows one to regard MQ(p, n) as the moduli space of Abelian differentials on Riemann surfaces of genus p with at most n zeroes.

The set Q(p, n) has the natural structure of an affine manifold while the moduli space MQ(p, n) has the structure of an affine orbifold. We shall describe these structures following the paper [MS].

Let γ : [0, 1] → M_{p} be a continuous path. For any ω ∈ Ω(p, n) there exists a continuous path γ_{ω} : [0, 1] → ℝ^{2} such that γ is a translation of γ_{ω} in coordinates of the atlas ω. The path γ_{ω} is determined up to translation. The vector γ_{ω}(1)−γ_{ω}(0) is called the holonomy vector of γ with respect to ω and is denoted by hol_{ω}(γ). If γ is a geodesic segment of ω, then the vector hol_{ω}(γ) is of the same length and direction as γ. Suppose q is an Abelian differential associated to the translation structure ω (see Section 5). If the path γ is piecewise smooth then, up to the natural identification of ℂ with ℝ^{2},

hol_{ω}(γ) = ∫_{γ} q.

The holonomy vector hol_{ω}(γ) does not change when we replace the path γ by a homologous one. If the path γ is closed or its endpoints belong to the set Z_{n}, then hol_{ω}(γ) does not change when we replace the translation structure ω by an isotopic one. The map γ ↦ hol_{ω}(γ) gives rise to a map dev(ω) : H_{1}(M_{p}, Z_{n}; ℤ) → ℝ^{2} that is an element of the relative cohomology group H^{1}(M_{p}, Z_{n}; ℝ^{2}). As the relative cohomology class dev(ω) depends only on the isotopy class of the translation structure ω, we have a well-defined map dev : Q(p, n) → H^{1}(M_{p}, Z_{n}; ℝ^{2}). The group H^{1}(M_{p}, Z_{n}; ℝ^{2}) is a real vector space of dimension 2N, where N = 2p + n − 1 is the rank of the relative homology group H_{1}(M_{p}, Z_{n}; ℤ).

Let T (p, n) denote the set of pairs (ω, τ), where ω ∈ Ω(p, n) and τ is a partition of the surface M_{p} by a maximal set of pairwise disjoint saddle connections of the translation structure ω. By Proposition 5.1, cells of the partition τ are triangles with respect to ω. We call the pair (ω, τ) a triangulation. A homeomorphism f ∈ H(p, n) maps τ to a partition fτ, which is a partition by disjoint saddle connections of the translation structure ωf^{−1}. The map H(p, n) × T (p, n) ∋ (f, (ω, τ)) ↦ (ωf^{−1}, fτ) defines an action of the group H(p, n) on T (p, n). Two elements (ω_{1}, τ_{1}), (ω_{2}, τ_{2}) ∈ T (p, n) are called affine equivalent if there exists f_{0} ∈ H_{0}(p, n) such that f_{0} maps each triangle of τ_{1} to a triangle of τ_{2} and f_{0} is an affine map in local coordinates of the atlases ω_{1} and ω_{2} when restricted to any triangle of τ_{1}. By T (p, n) denote the set of affine equivalence classes of the above triangulations. T (p, n) is a countable set. The affine equivalence relation is preserved by the action of H(p, n) and, moreover, each equivalence class is invariant under the action of the subgroup H_{0}(p, n). Hence the H(p, n) action on T (p, n) gives rise to an action of the group Mod(p, n) on T (p, n). Let MT (p, n) = T (p, n)/ Mod(p, n). An element of MT (p, n) can be regarded as a pattern to glue a translation surface of genus p with n singular points from triangles. Since the number of triangles to be glued together is fixed, the set MT (p, n) is finite.

For any τ ∈ T (p, n) let N(τ) denote the set of translation structures in Ω(p, n) that admit a triangulation in the class τ. The set N(τ) is invariant under the action of the group H_{0}(p, n), therefore we shall consider N(τ) as a subset of Q(p, n).

Lemma 6.1 For any τ ∈ T (p, n) the restriction of the map dev to the set N(τ) is injective. The image dev(N(τ)) is an open set that can be determined by a system of inequalities Q_{i}(v) > 0, 1 ≤ i ≤ s, where Q_{1}, …, Q_{s} are quadratic forms on the vector space H^{1}(M_{p}, Z_{n}; ℝ^{2}). For any τ_{1}, τ_{2} ∈ T (p, n) the set dev(N(τ_{1}) ∩ N(τ_{2})) is also open.

Proof. Let (ω′_{0}, τ_{0}) ∈ T (p, n), τ ∈ T (p, n) be the affine equivalence class of (ω′_{0}, τ_{0}), and ω_{0} ∈ Q(p, n) be the isotopy class of ω′_{0}. Take any triangle T of the triangulation τ_{0}. There is a map f_{T} : T → ℝ^{2} such that (T, f_{T}) ∈ ω′_{0}. Let Λ denote the euclidean area form on ℝ^{2}. For any vectors v = (v_{1}, v_{2}) and u = (u_{1}, u_{2}), Λ(v, u) = v_{1}u_{2} −u_{1}v_{2}.

Let γ_{1T} and γ_{2T} be two sides of T oriented so that Λ(hol_{ω_{0}}(γ_{1T}), hol_{ω_{0}}(γ_{2T})) > 0. For any q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) let Q_{T} (q) = Λ(q(γ_{1T}), q(γ_{2T})). Clearly, Q_{T} is a quadratic form on H^{1}(M_{p}, Z_{n}; ℝ^{2}). Note that Q_{T} (dev(ω_{0})) is the area of the triangle T. We claim that dev(N(τ)) is the set of q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) such that Q_{T} (q) > 0 for any triangle T of τ_{0}. Given ω ∈ N(τ), for any triangle T of τ_{0} there exists a linear map b_{T} : ℝ^{2} → ℝ^{2} such that charts (T, b_{T} f_{T}) belong to a translation structure ω′ ∈ ω. Clearly, each operator b_{T} is invertible and orientation-preserving. Then

Q_{T} (dev(ω)) = Λ(b_{T} hol_{ω_{0}}(γ_{1T}), b_{T} hol_{ω_{0}}(γ_{2T})) = (det b_{T}) Q_{T} (dev(ω_{0})) > 0

for all T. If dev(ω) = dev(ω_{0}) then each b_{T} is the identity. Since charts (T, f_{T}) uniquely determine the translation structure ω′_{0}, it follows that ω = ω_{0}. Thus the map dev is injective on N(τ). Now consider any q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) such that Q_{T} (q) > 0 for all triangles T of τ_{0}. For each T let b_{T} be a linear operator in ℝ^{2} such that b_{T} hol_{ω_{0}}(γ_{1T}) = q(γ_{1T}), b_{T} hol_{ω_{0}}(γ_{2T}) = q(γ_{2T}). Since Q_{T} (q) > 0, b_{T} is invertible and orientation-preserving. If γ is a common edge of triangles T_{1} and T_{2}, then b_{T_{1}} hol_{ω_{0}}(γ) = b_{T_{2}} hol_{ω_{0}}(γ) = q(γ). Hence the set of charts (T, b_{T} f_{T}) extends to a translation structure ω′_{1} ∈ Ω(p, n). By construction (ω′_{1}, τ_{0}) ∈ τ, therefore the isotopy class ω_{1} of ω′_{1} belongs to N(τ). Also, dev(ω_{1}) = q since the two cohomologies take the same values on edges of τ_{0}.

Suppose ω_{0} ∈ N(τ_{1}) ∩ N(τ_{2}) for some τ_{1}, τ_{2} ∈ T (p, n). Let ω′_{0} be a translation structure in the isotopy class ω_{0} and τ′_{1}, τ′_{2} be triangulations such that (ω′_{0}, τ′_{1}) ∈ τ_{1}, (ω′_{0}, τ′_{2}) ∈ τ_{2}. Choose an edge γ of τ′_{2}. If γ is not an edge of τ′_{1} then edges of τ′_{1} break this saddle connection into several geodesic segments x_{0}x_{1}, x_{1}x_{2}, …, x_{k}x_{k+1}, where x_{0} and x_{k+1} are endpoints of γ and x_{1}, …, x_{k} are interior points of edges of τ′_{1}. Take any ϵ > 0. Let x′_{1}, …, x′_{k} be points of M_{p} such that each x′_{i} (1 ≤ i ≤ k) lies on the same edge of τ′_{1} as x_{i} and the length of the subsegment of the edge bounded by x′_{i} and x_{i} is less than ϵ. Assuming ϵ is small enough, there are geodesic segments x_{0}x′_{1}, x′_{1}x′_{2}, …, x′_{k}x_{k+1} that form a path homotopic to γ. By U_{ϵ}(γ) denote the set of all such piecewise geodesics. If γ is a common edge of τ′_{1} and τ′_{2}, let U_{ϵ}(γ) = {γ}. Let γ_{1}, …, γ_{l} be all edges of τ′_{2}. If ϵ is small enough then for any γ′_{j} ∈ U_{ϵ}(γ_{j}), 1 ≤ j ≤ l, the curves γ′_{1}, …, γ′_{l} are simple and disjoint except for endpoints. The partition of M_{p} by these curves can be mapped onto τ′_{2} by a homeomorphism f ∈ H_{0}(p, n). By P_{ϵ} denote the set of all such partitions. For any ω ∈ N(τ_{1}) there exists ω′ ∈ ω such that (ω′, τ′_{1}) ∈ τ_{1} and the identity map of M_{p} is affine on every triangle of τ′_{1} with respect to translation structures ω′_{0} and ω′. If dev(ω) is close to dev(ω_{0}) then linear parts of the restrictions to triangles of τ′_{1} are close to 1. It follows that if dev(ω) is close enough to dev(ω_{0}) then each of the sets U_{ϵ}(γ_{1}), …, U_{ϵ}(γ_{l}) contains a saddle connection of the translation structure ω′. Then P_{ϵ} contains a partition τ′ such that (ω′, τ′) ∈ τ_{2}, hence ω ∈ N(τ_{2}). Thus dev(ω_{0}) is an interior point of dev(N(τ_{1}) ∩ N(τ_{2})).

By Proposition 5.1, the sets N(τ), τ ∈ T (p, n), cover Q(p, n). We use this covering and the map dev to endow Q(p, n) with a topology, smooth and affine structures, and a measure. By Lemma 6.1, there exists a unique topology on Q(p, n) such that any N(τ) is an open set and the restriction of dev to N(τ) is a homeomorphism onto dev(N(τ)). The topological space Q(p, n) is a 2N-dimensional manifold. Further, there is a unique affine (smooth, real analytic) structure on Q(p, n) such that the restriction of the map dev to any N(τ), τ ∈ T (p, n), is an affine (resp. smooth, analytic) map. The euclidean volume form on the vector space H^{1}(M_{p}, Z_{n}; ℝ^{2}) is made canonical by the requirement that the lattice H^{1}(M_{p}, Z_{n}; ℤ^{2}) have covolume 1. The volume element on Q(p, n) is obtained from the pull-back, this volume form by the map dev.

Lemma 6.2 Let ω ∈ Ω(p, n). Then each set of disjoint saddle connections of ω that do not divide the surface M_{p} can be extended to a maximal set of this kind. Each maximal set consists of N = 2p + n − 1 saddle connections whose homology classes comprise a basis for H_{1}(M_{p}, Z_{n}; ℤ).

Proof. Let γ_{1}, …, γ_{k} be disjoint saddle connections of ω that do not divide the surface M_{p}. Let m = 2p − 2 + n. By Proposition 5.1, k ≤ 3m and there exist γ_{k+1}, …, γ_{3m} such that γ_{1}, …, γ_{3m} are disjoint saddle connections of ω that form a triangulation τ of M_{p}. There are 2m triangles in the triangulation τ. We can arrange them in a sequence T_{0}, T_{1}, …, T_{2m−1} so that there exist a sequence of domains U_{0} = T_{0} ⊂ U_{1} ⊂ … ⊂ U_{2m−1} ⊂ M_{p} and a sequence of edges e_{1}, …, e_{2m−1} of τ such that for any j, 1 ≤ j ≤ 2m − 1, the triangle T_{j} is disjoint from U_{j−1}, e_{j} separates T_{j} from U_{j−1}, and U_{j} is the union of U_{j−1}, T_{j}, and the interior of e_{j}. The complement of U_{2m−1} is the union of the set of singular points of ω and N = m + 1 edges of τ. Since saddle connections γ_{1}, …, γ_{k} do not divide the surface M_{p}, it can be assumed without loss of generality that all of them are in the complement of U_{2m−1}.

It is easy to observe that the group H_{1}(M_{p}, Z_{n}; ℤ) is generated by the homology classes of edges of τ. For any j, 1 ≤ j ≤ 2m − 1, two sides of the triangle T_{j} different from e_{j} are disjoint from U_{j}. It follows that H_{1}(M_{p}, Z_{n}; ℤ) is generated by the homology classes of N edges of τ disjoint from U_{2m−1}. Since H_{1}(M_{p}, Z_{n}; ℤ) is a free Abelian group of rank N, the latter classes constitute its basis.

Suppose Γ = (γ_{1}, …, γ_{N}) is an ordered basis of cycles for H_{1}(M_{p}, Z_{n}; ℤ). Define a map f_{Γ} : H^{1}(M_{p}, Z_{n}; ℝ^{2}) → (ℝ^{2})^{N} ≈ ℝ^{2N} by f_{Γ}(q) = (q(γ_{1}), …, q(γ_{N})). f_{Γ} is an isomorphism of vector spaces. Since f_{Γ}(H^{1}(M_{p}, Z_{n}; ℤ^{2})) = ℤ^{2N}, f_{Γ} preserves volumes. Now define a map F_{Γ} : Q(p, n) → (ℝ^{2})^{N} ≈ ℝ^{2N} by F_{Γ}(ω) = (hol_{ω}(γ_{1}), …, hol_{ω}(γ_{N})). Clearly, F_{Γ}(ω) = f_{Γ}(dev(ω)). The map F_{Γ} is a local homeomorphism. The volume element on Q(p, n) is the pull-back of the canonical volume form on ℝ^{2N} by the map F_{Γ}.

Lemma 6.3 The group Mod(p, n) acts on Q(p, n) by affine homeomorphisms preserving volume element. The action is properly discontinuous.

Proof. Every mapping class ϕ ∈ Mod(p, n) induces an automorphism ϕ_{∗} of the group H_{1}(M_{p}, Z_{n}; ℤ). Define a linear operator ϕ^{∗} on H^{1}(M_{p}, Z_{n}; ℝ^{2}) by (ϕ^{∗}q)(γ) = q(ϕ_{∗}γ), γ ∈ H_{1}(M_{p}, Z_{n}; ℤ). ϕ^{∗} is invertible and preserves the lattice H^{1}(M_{p}, Z_{n}; ℤ^{2}), hence it is volume preserving. Observe that dev(ωϕ^{−1}) = ϕ^{∗}(dev(ω)) for all ω ∈ Q(p, n). Besides, if ω ∈ N(τ) for some τ ∈ T (p, n) then ωϕ^{−1} ∈ N(ϕτ). It follows that the Mod(p, n) action on Q(p, n) is affine and preserves the volume element.

Let ω_{0} ∈ N(τ), τ ∈ T (p, n). Choose a triangulation (ω′_{0}, τ′_{0}) ∈ τ such that ω′_{0} is in the isotopy class ω_{0}. There exists a neighborhood U ⊂ N(τ) of ω_{0} such that for every translation structure ω′ in an isotopy class ω ∈ U any saddle connection of ω′ of length l is homotopic to a piecewise geodesic path of ω′_{0} of length at most 2l, while any saddle connection of ω′_{0} of length l_{0} is homotopic to a piecewise geodesic path of ω′ of length at most 2l_{0}. Let us show that Uϕ^{−1} ∩ U ≠ ∅ for only finitely many ϕ ∈ Mod(p, n). Denote by K the set of relative homotopy classes of edges of τ′_{0}. Let R be the maximal length of edges of τ′_{0}. By K′_{1} denote the set of paths γ on M_{p} with endpoints in Z_{n} such that γ is a piecewise geodesic path of length at most 4R with respect to ω′_{0}. For any γ ∈ K′_{1} there is a homotopic path γ′ ∈ K′_{1} that is a saddle connection or the sum of several saddle connections of ω′_{0}. By K_{1} denote the set of relative homotopy classes of paths in K′_{1}. Clearly, K ⊂ K_{1}. Since the translation structure ω′_{0} has only finitely many saddle connections of length at most 4R, the set K_{1} is finite. Suppose Uϕ^{−1} ∩ U ≠ ∅ for some ϕ ∈ Mod(p, n), i.e., ω_{1}ϕ^{−1} = ω_{2} for some ω_{1}, ω_{2} ∈ U. Pick translation structures ω′_{1} ∈ ω_{1} and ω′_{2} ∈ ω_{2}. Let τ′_{1} be a partition such that (ω′_{1}, τ′_{1}) ∈ τ. By the choice of U, all edges of τ′_{1} are of length at most 2R. There is a homeomorphism ϕ_{0} ∈ ϕ that maps each edge of τ′_{1} onto a saddle connection of ω′_{2} of the same length. By the choice of U, the mapping class ϕ sends each homotopy class from K to a homotopy class in K_{1}. Since ϕ is uniquely determined by its action on elements of K and K_{1} is a finite set, there are only finitely many such ϕ. Furthermore, there exists a neighborhood U_{0} ⊂ U of ω_{0} such that U_{0}ϕ^{−1} ∩ U_{0} ≠ ∅ for a ϕ ∈ Mod(p, n) if and only if ϕω_{0} = ω_{0}, i.e., ϕ is the mapping class of an automorphism of ω′_{0}.

Since the action of Mod(p, n) on Q(p, n) is properly discontinuous and volume preserving, the quotient space MQ(p, n) inherits the structure of an affine orbifold along with a volume element.

Lemma 6.4 The moduli space MQ(p, n) has finitely many connected components.

Proof. It follows from Lemma 6.1 that any of the open sets N(τ), τ ∈ T (p, n), has only finitely many connected components. Since MT (p, n) is a finite set, there exist τ_{1}, …, τ_{k} ∈ T (p, n) such that the natural projection π_{0} : Q(p, n) → MQ(p, n) maps the union N(τ_{1}) ∪ … ∪ N(τ_{k}) onto MQ(p, n). Then the number of connected components of MQ(p, n) is at most the number of connected components of N(τ_{1})∪ … ∪ N(τ_{k}), which is finite.

Let α = (m_{1}, …, m_{n}) be a nondecreasing sequence of positive integers such that m_{1} + ⋯ + m_{n} = 2p − 2 + n. By H(α) denote the set of isomorphy classes of translation surfaces with n singular points of multiplicities m_{1}, …, m_{n}. H(α) is called a stratum. The stratum H(α) is a nonempty open subset of MQ(p, n). The moduli space MQ(p, n) is a disjoint union of strata. By Lemma 6.4, each stratum has only finitely many connected components. The complete classification of all connected components for each stratum was obtained by Kontsevich and Zorich [KZ]. In particular, any stratum has at most 3 connected components.

For any τ ∈ T (p, n) let M(τ) denote the set of translation structures ω ∈ Ω(p, n) such that the Delaunay partition of ω is a triangulation in the class τ. Let ≺ represent an ordering of the homology classes of edges of the triangulation τ. Note that edges of τ need not be oriented so ≺ is actually an ordering of pairs of opposite homology classes. By M(τ, ≺) denote the set of translation structures ω ∈ M(τ) such that for any two nonhomologous Delaunay edges e_{1} and e_{2} of ω the edge e_{1} is shorter than e_{2} if the homology class of e_{1} is less than the homology class of e_{2} in the ordering ≺. Let h be one of the least homology classes with respect to ≺ (the other is −h). By M^{h}(τ, ≺) denote the set of ω ∈ M(τ, ≺) such that the holonomy vector hol_{ω}(h) belongs to the halfplane ℝ_{+}^{2} = {(x_{1}, x_{2}) ∈ ℝ^{2} : x_{2} > 0}. The sets M(τ), M(τ, ≺), and M^{h}(τ, ≺) are invariant under the action of H_{0}(p, n) so we consider them as subsets of Q(p, n). Then M^{h}(τ, ≺) ⊂ M(τ, ≺) ⊂ M(τ) ⊂ N(τ). M(τ, ≺) is called a Delaunay triangulation piece while M^{h}(τ, ≺) is called a halfpiece. Clearly, the sets M(τ), τ ∈ T (p, n), are disjoint. Two Delaunay triangulation pieces M(τ_{1}, ≺_{1}) and M(τ_{2}, ≺_{2}) are disjoint unless τ_{1} = τ_{2} and ≺_{1}=≺_{2}. Two halfpieces M^{h_{1}}(τ_{1}, ≺_{1}) and M^{h_{2}}(τ_{2}, ≺_{2}) are disjoint unless τ_{1} = τ_{2}, ≺_{1}=≺_{2}, and h_{1} = h_{2}.

Lemma 6.5 M(τ), M(τ, ≺), and M^{h}(τ, ≺) are open subsets of Q(p, n). The natural projection π_{0} : Q(p, n) → MQ(p, n) is injective on M^{h}(τ, ≺). There exist finitely many disjoint Delaunay triangulation halfpieces M^{h_{1}}(τ_{1}, ≺_{1}), …, M^{h_{k}}(τ_{k}, ≺_{k}) such that π_{0} is injective on their union and π_{0}(M^{h_{1}}(τ_{1}, ≺_{1})∪…∪M^{h_{k}}(τ_{k}, ≺_{k})) is a subset of MQ(p, n) of full volume.

Proof. Let ω′ ∈ Ω(p, n) be a representative of an ω ∈ N(τ) and τ′ be a triangulation of M_{p} such that (ω′, τ′) ∈ τ. Let e_{0} be an edge of τ′ and T_{1}, T_{2} be triangles of τ′ bounded by e_{0}. By θ_{i} (i = 1, 2) denote the angle of T_{i} opposite e_{0}. Let e_{1}, e_{2} be sides of T_{1} different from e_{0}. Then cos θ_{1} = (2l_{1}l_{2})^{−1}(l_{1}^{2} + l_{2}^{2} − l_{0}^{2}), where l_{i} = | hol_{ω}(e_{i})|, i = 0, 1, 2. The homotopy classes of e_{0}, e_{1}, and e_{2} depend only on τ. It follows that cos θ_{1} depends continuously on ω. Likewise, cos θ_{2} is a continuous function of ω ∈ N(τ). Since 0 < θ_{1}, θ_{2} < π, we have θ_{1} + θ_{2} < π if and only if cos θ_{1} + cos θ_{2} > 0. By Proposition 5.3, M(τ) can be determined as the set of ω ∈ N(τ) satisfying inequalities f_{i}(ω) > 0, i = 1, …, k, where f_{1}, …, f_{k} are continuous functions on N(τ). Hence M(τ) is open. Now M(τ, ≺) is the intersection of M(τ) with a finite number of open sets of the form {ω ∈ Q(p, n) : | hol_{ω}(h_{1})| < | hol_{ω}(h_{2})|}, where h_{1}, h_{2} ∈ H_{1}(M_{p}, Z_{n}; ℤ). The halfpiece M^{h}(τ, ≺) is the intersection of M(τ, ≺) with the open set {ω ∈ Q(p, n) : hol_{ω}(h) ∈ ℝ_{+}^{2}}.

For any mapping class ϕ ∈ Mod(p, n) let ϕ_{∗} denote the induced automorphism of H_{1}(M_{p}, Z_{n}; ℤ). ϕ acts on Q(p, n) so that any M(τ) is mapped onto M(τ)ϕ^{−1} = M(ϕτ). A Delaunay triangulation piece M(τ, ≺) is mapped onto M(ϕτ, ≺′), where by definition e_{1} ≺′ e_{2} if and only if ϕ_{∗}^{−1}e_{1} ≺ ϕ_{∗}^{−1}e_{2}. A halfpiece M^{h}(τ, ≺) is mapped onto M^{ϕ_{∗}h}(ϕτ, ≺′). Suppose M^{h}(τ, ≺)ϕ^{−1} = M^{h}(τ, ≺). Then ϕ permutes homotopy classes of edges of τ. Moreover, ϕ_{∗}h_{0} = ±h_{0} for any homology class h_{0} of an edge. Since ϕ is orientation-preserving and homology classes of edges of τ generate H_{1}(M_{p}, Z_{n}; ℤ), it follows that ϕ_{∗} is the multiplication by 1 or −1. As ϕ_{∗}h = h, ϕ_{∗} is the identity. Let ω_{0} ∈ Ω(p, n) be a translation structure whose isotopy class is in M^{h}(τ, ≺). By the above there exists ϕ_{0} ∈ ϕ that maps each Delaunay cell of ω_{0} onto another Delaunay cell in an affine way. ϕ_{0} maps each Delaunay edge onto a homologous one, hence ϕ_{0} is an automorphism of ω_{0}. Let e_{1} and e_{2} be two Delaunay edges of ω_{0} bounding the same triangle. If ϕ_{0}(e_{1}) ≠ e_{1} then the edges e_{1} and ϕ_{0}(e_{1}) divide the surface M_{p} into two parts. It is easy to see that the edges e_{2} and ϕ_{0}(e_{2}) are contained in different parts. None of the latter divides the part it belongs to. As a consequence, e_{2} and ϕ_{0}(e_{2}) are not homologous as they should be. This contradiction implies ϕ_{0} fixes all Delaunay edges of ω_{0}. Then ϕ_{0} is the identity while ϕ is the trivial mapping class. It follows that the halfpiece M^{h}(τ, ≺) is disjoint from M^{h}(τ, ≺)ϕ^{−1} for any nontrivial ϕ ∈ Mod(p, n), i.e., π_{0} is injective on M^{h}(τ, ≺).

Let U_{1} denote the subset of Q(p, n) corresponding to translation structures whose Delaunay partitions are not triangulations. Let us show that U_{1} has zero volume. Take any τ ∈ T (p, n) and pick an edge e of τ (edges of τ are determined up to homotopy). By U_{1}(τ, e) denote the set of ω ∈ N(τ) such that each Delaunay edge is an edge of τ but there is no Delaunay edge homotopic to e. U_{1} is the union of countably many sets of the form U_{1}(τ, e), hence it is sufficient to prove that any of them has zero volume. Let ω′ be a translation structure in an isotopy class ω ∈ U_{1}(τ, e) and τ′ be a partition of M_{p} such that (ω′, τ′) ∈ τ. Let e_{0} be an edge of τ′ homotopic to e. Let T_{1} and T_{2} be triangles of τ′ bounded by e_{0}. Then the union of T_{1}, T_{2}, and the interior of e_{0} is isometric to an inscribed quadrilateral. Let e_{1} be a side of T_{1} different from e_{0} and e_{2} be a side of T_{2} different from e_{0} and not parallel to e_{1}. If e_{0}, e_{1}, and e_{2} do not divide the surface, then their homology classes are elements of a basis for H_{1}(M_{p}, Z_{n}; ℤ) by Lemma 6.2. Once holonomy vectors hol_{ω}(e_{0}) and hol_{ω}(e_{1}) are fixed, the holonomy vector hol_{ω}(e_{2}) lies on a fixed circle in ℝ^{2} passing through the origin. Hence Fubini’s theorem implies U_{1}(τ, e) has zero volume. If e_{0}, e_{1}, and e_{2} do divide the surface, it is easy to observe that e_{2} is homologous to the side of T_{1} different from e_{0} and e_{1}. Then T_{1} and T_{2} are isometric triangles and their angles opposite e_{0} are right. It follows that hol_{ω}(e_{1}) lies on a circle depending on hol_{ω}(e_{0}). Since e_{0} and e_{1} do not divide M_{p}, Lemma 6.2 and Fubini’s theorem imply again U_{1}(τ, e) has zero volume.

Let U_{2} denote the set of ω ∈ Q(p, n) such that | hol_{ω}(γ_{1})| = | hol_{ω}(γ_{2})| for some disjoint nonhomologous saddle connections γ_{1} and γ_{2} of a translation structure in the isotopy class ω. By Lemma 6.2, homology classes of γ_{1} and γ_{2} are linearly independent. By Lemma 7.1 (see Section 7 below), U_{2} has zero volume. Let U_{3} denote the set of ω ∈ Q(p, n) such that hol_{ω}(h) is horizontal for a nonzero h ∈ H_{1}(M_{p}, Z_{n}; ℤ). U_{3} is the union of countably many codimension 1 affine submanifolds of Q(p, n), hence it is of zero volume.

Since the Mod(p, n) action on T (p, n) has only finitely many orbits, there exist τ_{1}, …, τ_{k} ∈ T (p, n) such that π_{0} maps the union of M(τ_{1}), …, M(τ_{k}), and U_{1} onto MQ(p, n). For any τ ∈ T (p, n) the set M(τ) is the union of finitely many Delaunay triangulation pieces and a subset of U_{2}, while each piece M(τ, ≺) is the union of two halfpieces and a subset of U_{3}. Since U_{1} ∪ U_{2} ∪ U_{3} is a set of zero volume, there exist finitely many halfpieces P_{1}, …, P_{l} such that π_{0}(P_{1} ∪ … ∪ P_{l}) is a subset of MQ(p, n) of full volume. By the above any π_{0}(P_{i}) and π_{0}(P_{j}) either are disjoint or coincide. Therefore it is no loss to assume that sets π_{0}(P_{1}), …, π_{0}(P_{l}) are disjoint. Then P_{1}, …, P_{l} are also disjoint and π_{0} is injective on P_{1} ∪ … ∪ P_{l}.

Let ω = {(U_{α}, ϕ_{α})}_{α∈A} be a translation structure on M_{p}. For any operator g ∈ GL_{+}(2, ℝ) the atlas gω = {(U_{α}, gϕ_{α})}_{α∈A} is a translation structure on M_{p} with the same singular points of the same multiplicities as ω. Although the flat metrics on M_{p} induced by translation structures ω and gω need not coincide, they share the same geodesics. If g ∈ SL(2, ℝ) then ω and gω induce the same measure on M_{p}. The map GL_{+}(2, ℝ) × Ω(p, n) ∋ (g, ω) ↦ gω defines an action of the group GL_{+}(2, ℝ) on the set Ω(p, n). Obviously, this action commutes with the action of H(p, n). Therefore the action of GL_{+}(2, ℝ) descends to actions on the spaces Q(p, n) and MQ(p, n). The action of the group GL_{+}(2, ℝ) on Q(p, n) commutes with the action of Mod(p, n). Since hol_{gω}(γ) = g hol_{ω}(γ) for any g ∈ GL_{+}(2, ℝ), ω ∈ Ω(p, n), and any path γ, it follows that the GL_{+}(2, ℝ) action on Q(p, n) is affine and continuous. Moreover, the action of the subgroup SL(2, ℝ) preserves the volume element. The action of GL_{+}(2, ℝ) on MQ(p, n) is also continuous and the action of SL(2, ℝ) on MQ(p, n) is also volume preserving. For any τ ∈ T (p, n) the set N(τ) is invariant under the GL_{+}(2, ℝ) action on Q(p, n). The GL_{+}(2, ℝ) action on MQ(p, n) leaves invariant every stratum H(α) ⊂ MQ(p, n) and every connected component of MQ(p, n).

The group ℝ^{+} acts naturally on translation structures by scaling distances. We define actions of ℝ^{+} on the sets Ω(p, n), Q(p, n), and MQ(p, n) by regarding ℝ^{+} as a subgroup of GL_{+}(2, ℝ).

For any ω ∈ Ω(p, n), let a(ω) denote the area of the surface M_{p} with respect to the measure induced by ω. Since a(tω) = t^{2}a(ω) for any t > 0, every translation structure ω ∈ Ω(p, n) is uniquely represented as tω_{1}, where a(ω_{1}) = 1 and t ∈ ℝ^{+}. The area a(ω) does not change when we replace ω by an isomorphic translation structure. By Q_{1}(p, n) and MQ_{1}(p, n) denote the subsets of Q(p, n) and MQ(p, n), respectively, corresponding to translation structures ω such that a(ω) = 1. Let us consider a as a function on Q(p, n). Then it follows from the proof of Lemma 6.1 that a(ω) is locally a quadratic form of the vector dev(ω) ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}). Hence the set Q_{1}(p, n) = a^{−1}(1) is a real analytic submanifold of Q(p, n) of codimension 1. This submanifold is invariant under the actions of Mod(p, n) and SL(2, ℝ). The volume element on Q(p, n) along with the vector field grad a induce a volume element on Q_{1}(p, n). Let μ̃ and μ̃_{0} be the Borel measures on Q(p, n) and Q_{1}(p, n), respectively, induced by the volume elements. Then

μ̃_{0}(U) = 2N · μ̃({tω | ω ∈ U, 0 < t ≤ 1})

for any Borel set U ⊂ Q_{1}(p, n). The action of Mod(p, n) on Q_{1}(p, n) is properly discontinuous and volume preserving, therefore the quotient space MQ_{1}(p, n) inherits the structure of a real analytic orbifold along with a volume element. MQ_{1}(p, n) is a suborbifold of MQ(p, n) invariant under the action of SL(2, ℝ). Each connected component of MQ_{1}(p, n) is of the form C ∩ MQ_{1}(p, n), where C is a connected component of MQ(p, n). By μ_{0} denote the Borel measure on MQ_{1}(p, n) induced by the volume element. Let π_{0} : Q(p, n) → MQ(p, n) be the natural projection. Then μ_{0}(π_{0}(U)) = μ̃_{0}(U) for any Borel set U ⊂ Q_{1}(p, n) such that π_{0} is injective on U. The measure μ_{0} is invariant under the SL(2, ℝ) action on MQ_{1}(p, n).

Theorem 6.6 ([V2], [MS]) μ_{0}(MQ_{1}(p, n)) < ∞.

Theorem 6.7 ([V1]) The SL(2, ℝ) action on MQ_{1}(p, n) is ergodic with respect to the measure μ_{0} on each connected component of MQ_{1}(p, n).

Now we shall define the moduli space MY(p, n) of pairs (M, x) such that M is a translation surface of genus p with n singular points and x ∈ M. The maps H(p, n)× Ω(p, n) × M_{p} ∋ (f, ω, x) ↦ (ωf^{−1}, f(x)) and SL(2, ℝ) × Ω(p, n) × M_{p} ∋ (g, ω, x) ↦ (gω, x) define commuting actions of the groups H(p, n) and SL(2, ℝ) on the set Ω(p, n) × M_{p}. Let Y(p, n) = (Ω(p, n) × M_{p})/H_{0}(p, n) and MY(p, n) = (Ω(p, n) × M_{p})/H(p, n). Let p̃_{0} : Y(p, n) → Q(p, n) and p_{0} : MY(p, n) → MQ(p, n) be the natural projections. The group Mod(p, n) acts naturally on Y(p, n) and MY(p, n) = Y(p, n)/ Mod(p, n). The SL(2, ℝ) action on Ω(p, n) × M_{p} descends to actions on Y(p, n) and MY(p, n).

Let τ ∈ T (p, n). Take a translation structure ω_{0} ∈ Ω(p, n) that admits a triangulation τ_{0} in the class τ. For any ω ∈ N(τ) there exists a unique translation structure X_{ω_{0},τ_{0}}(ω) in the isotopy class ω such that the identity map of M_{p} is affine on each triangle of τ_{0} in local coordinates of the atlases ω_{0} and X_{ω_{0},τ_{0}}(ω), that is, for any triangle T of τ_{0} there are a map f : T → ℝ^{2} and an affine map b : ℝ^{2} → ℝ^{2} such that (T, f) ∈ ω_{0} and (T, bf) ∈ X_{ω_{0},τ_{0}}(ω). For any ω ∈ N(τ) and x ∈ M_{p} define Y_{ω_{0},τ_{0}}(ω, x) ∈ Y(p, n) to be the H_{0}(p, n)-orbit of the pair (X_{ω_{0},τ_{0}}(ω), x) ∈ Ω(p, n)×M_{p}. Then the map Y_{ω_{0},τ_{0}} : N(τ)×M_{p} → Y(p, n) is injective and Y_{ω_{0},τ_{0}}(N(τ)×M_{p}) = p̃_{0}^{−1}(N(τ)). The collection of maps Y_{ω_{0},τ_{0}}, (ω_{0}, τ_{0}) ∈ T (p, n), endows Y(p, n) with the structure of a fiber bundle over Q(p, n) with the fiber M_{p}.

The action of the group Mod(p, n) on Y(p, n) is properly discontinuous, therefore the quotient space MY(p, n) inherits the quotient topology. For any ω ∈ MQ(p, n) the following conditions are equivalent: (i) translation structures in the isomorphy class ω have no automorphisms different from the identity; (ii) for any ω̃ ∈ π_{0}^{−1}(ω) the restriction of the projection π_{0} to some neighborhood of ω̃ is a homeomorphism. Let U_{0} be the set of ω ∈ MQ(p, n) satisfying these conditions. U_{0} is an open dense subset of MQ(p, n) of full volume. The set p_{0}^{−1}(U_{0}) ⊂ MY(p, n) is a fiber bundle over U_{0} with the fiber M_{p}. Suppose ω ∈ MQ(p, n) \ U_{0} and ω_{0} ∈ ω; then p_{0}^{−1}(ω) is homeomorphic to M_{p}/ Aut(ω_{0}). Slightly abusing notation, we shall consider MY(p, n) as a fiber bundle over MQ(p, n) with the fiber M_{p} (even though some fibers may be not homeomorphic to M_{p}).

The group SL(2, ℝ) acts on the spaces Y(p, n) and MY(p, n) by homeomorphisms. The subspaces Y_{1}(p, n) = p̃_{0}^{−1}(Q_{1}(p, n)) and MY_{1}(p, n) = p_{0}^{−1}(MQ_{1}(p, n)) are invariant under these actions. Note that each connected component of MY_{1}(p, n) is of the form p_{0}^{−1}(C), where C is a connected component of MQ_{1}(p, n).

For any ω_{0} ∈ Ω(p, n), let ξ_{ω_{0}} be the Borel measure on M_{p} induced by the translation structure ω_{0}. Let ω be the isotopy class of ω_{0}. The map h_{ω_{0}} : M_{p} → p̃_{0}^{−1}(ω) defined by the relation (ω_{0}, x) ∈ h_{ω_{0}}(x), x ∈ M_{p}, is a homeomorphism. The measure ρ̃_{ω} = ξ_{ω_{0}}h_{ω_{0}}^{−1} on the fiber p̃_{0}^{−1}(ω) does not depend on the choice of ω_{0} ∈ ω. Likewise, for any ω ∈ MQ(p, n) the measures on M_{p} induced by translation structures in the isomorphy class ω define a Borel measure ρ_{ω} on p_{0}^{−1}(ω) (even if the fiber p_{0}^{−1}(ω) is not homeomorphic to M_{p}). The space Y_{1}(p, n), which is a fiber bundle over Q_{1}(p, n) with the fiber M_{p}, carries a natural measure μ̃_{1} that is the measure μ̃_{0} on the base Q_{1}(p, n) and is the measure ρ̃_{ω} on the fiber p̃_{0}^{−1}(ω). In other words, dμ̃_{1}(η) = dρ̃_{ω}(η) dμ̃_{0}(ω). Similarly, the space MY_{1}(p, n) carries a natural measure μ_{1} such that dμ_{1}(η) = dρ_{ω}(η) dμ_{0}(ω). Notice that μ_{1}(MY_{1}(p, n)) = μ_{0}(MQ_{1}(p, n)) < ∞. The measure μ_{1} is invariant under the action of the group SL(2, ℝ) on MY_{1}(p, n).

Theorem 6.8 ([EM]) The SL(2, ℝ) action on MY_{1}(p, n) is ergodic with respect to the measure μ_{1} on each connected component of MY_{1}(p, n).

7 Periodic geodesics

Let M be a translation surface. Any geodesic joining a nonsingular point of M to itself is periodic (or closed) since it cannot change its direction. We only consider primitive periodic geodesics, that is, a periodic geodesic is a simple closed curve. Each unoriented periodic geodesic corresponds to two oriented periodic geodesics of the same length and of opposite directions. If a geodesic starting at a point x ∈ M is periodic, then all geodesics starting at nearby points in the same direction are also periodic. Actually, each periodic geodesic belongs to a family of freely homotopic periodic geodesics of the same length and direction. If M is a torus without singular points, then this family fills the whole surface M. Otherwise the family fills a domain homeomorphic to an annulus. This domain is called a cylinder of periodic geodesics (or simply a periodic cylinder) since it is isometric to a cylinder ℝ/lℤ×(0, w), where l, w > 0. The number l is called the length or the waist of the periodic cylinder; it is equal to the length of periodic geodesics in the cylinder. w is called the width or the height of the cylinder. The cylinder is bounded by saddle connections parallel to its geodesics. The boundary of the cylinder is a union of two components. If geodesics in the cylinder are oriented then we can refer to them as the left and the right components. If M is a translation torus with one singular point then both components coincide and consist of a single saddle connection. Otherwise the left and the right components are different although they may share some saddle connections. A periodic cylinder is called regular if each component of its boundary consists of a single saddle connection or, equivalently, if bounding saddle connections are of the same length as periodic geodesics in the cylinder.

Each cylinder of unoriented periodic geodesics corresponds to two oriented periodic cylinders, i.e., cylinders of oriented periodic geodesics. By definition, the direction and the holonomy vector of an oriented periodic cylinder are the direction and the holonomy vector of an arbitrary periodic geodesic in the cylinder.

This section is devoted to the proofs of statements formulated in Section 1. We begin with Propositions 1.3 and 1.4.

Lemma 7.1 (a) Let h_{1}, h_{2} ∈ H_{1}(M_{p}, Z_{n}; ℤ) be relative homology classes that are not multiples of the same homology class. Then for almost all q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) vectors q(h_{1}), q(h_{2}) ∈ ℝ^{2} have different lengths and directions.

(b) Let h′_{1}, h′_{2} ∈ H_{1}(M_{p}, Z_{n}; ℤ) be another pair of relative homology classes. Let Λ be the euclidean area form on ℝ^{2}. Then Λ(q(h′_{1}), q(h′_{2})) ≠ Λ(q(h_{1}), q(h_{2})) for almost all q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) unless h′_{1} = b_{11}h_{1} + b_{12}h_{2} and h′_{2} = b_{21}h_{1} + b_{22}h_{2}, where b_{ij} ∈ ℚ, b_{11}b_{22} − b_{12}b_{21} = 1.

Proof. Since h_{1} and h_{2} are not multiples of the same homology class, there exist h_{3}, …, h_{N} ∈ H_{1}(M_{p}, Z_{n}; ℤ), N = 2p+n−1, such that h_{1}, …, h_{N} is a basis for a finite index subgroup of H_{1}(M_{p}, Z_{n}; ℤ). Then H^{1}(M_{p}, Z_{n}; ℝ^{2}) ∋ q ↦ (q(h_{1}), …, q(h_{N})) ∈ (ℝ^{2})^{N} is an isomorphism of vector spaces. The set of pairs (v_{1}, v_{2}) ∈ (ℝ^{2})^{2} such that v_{1} = 0 or v_{2} = 0 or vectors v_{1} and v_{2} have the same length or direction is of zero Lebesgue measure. By Fubini’s theorem, for almost all q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}) vectors q(h_{1}) and q(h_{2}) are nonzero vectors of different length and direction.

Any h′_{1}, h′_{2} ∈ H_{1}(M_{p}, Z_{n}; ℤ) are uniquely represented as h′_{1} = b_{11}h_{1} + ⋯ + b_{1N}h_{N} and h′_{2} = b_{21}h_{1}+⋯+b_{2N}h_{N}, where b_{ij} ∈ ℚ. Since Λ(q(h_{1}), q(h_{2})) and Λ(q(h′_{1}), q(h′_{2})) are quadratic forms of a vector q ∈ H^{1}(M_{p}, Z_{n}; ℝ^{2}), we have either Λ(q(h′_{1}), q(h′_{2})) ≠ Λ(q(h_{1}), q(h_{2})) for almost all q or Λ(q(h′_{1}), q(h′_{2})) = Λ(q(h_{1}), q(h_{2})) for all q. In the latter case it follows that b_{ij} = 0 for j ≥ 3, that is, h′_{1} = b_{11}h_{1} + b_{12}h_{2} and h′_{2} = b_{21}h_{1} + b_{22}h_{2}. Then Λ(q(h′_{1}), q(h′_{2})) = (b_{11}b_{22} − b_{12}b_{21})Λ(q(h_{1}), q(h_{2})) for any q. For almost all q vectors q(h_{1}) and q(h_{2}) are nonzero and of different directions, hence Λ(q(h_{1}), q(h_{2})) ≠ 0. This implies b_{11}b_{22} − b_{12}b_{21} = 1.

Proof of Propositions 1.3 and 1.4(a). Suppose U ⊂ M_{p} is a domain bounded by disjoint saddle connections of a translation structure ω ∈ Ω(p, n). Let γ_{1}, …, γ_{k} be boundary saddle connections that are not slits, i.e., not surrounded by U. The orientation of M_{p} induces orientations of γ_{1}, …, γ_{k}. Assuming these orientations, the sum of the relative homology classes of γ_{1}, …, γ_{k} in H_{1}(M_{p}, Z_{n}; ℤ) is equal to zero. In particular, hol_{ω}(γ_{1}) + ⋯ + hol_{ω}(γ_{k}) = 0. The above argument easily implies that saddle connections belonging to the same component of boundary of a periodic cylinder do not divide the surface.

Let [γ] ∈ H_{1}(M_{p}, Z_{n}; ℤ) be the relative homology class of a saddle connection γ of some ω ∈ Ω(p, n). By Lemma 6.2, [γ] is an element of a basis for H_{1}(M_{p}, Z_{n}; ℤ). In particular, if [γ] = bh for some h ∈ H_{1}(M_{p}, Z_{n}; ℤ) and b ∈ ℤ then b = ±1. It follows that homology classes of distinct saddle connections γ, γ′ of ω are multiples of the same homology class only if γ and γ′ are homologous, i.e., they are disjoint and divide the surface M_{p}. Then Lemma 7.1(a) implies that for a generic translation surface any nonhomologous saddle connections are of different length and direction. By the above the boundary of any irregular periodic cylinder contains nonhomologous saddle connections, which are parallel. Therefore generic translation surfaces admit only regular periodic cylinders. Two periodic geodesics in different regular cylinders are homologous if and only if saddle connections bounding these cylinders are homologous. Since boundary saddle connections have the same length and direction as periodic geodesics in the cylinder, it follows that for a generic translation surface any nonhomologous periodic geodesics are of different length and direction.

Proof of Proposition 1.4(b). Suppose C is a regular periodic cylinder of a translation structure ω ∈ Ω(p, n), where (p, n) ≠ (1, 1). Let γ be a saddle connection bounding C. Let γ′ be a saddle connection that crosses C from one side to another and does not leave the cylinder. By Λ denote the euclidean area form on ℝ^{2}. Then the area of C is equal to |Λ(hol_{ω}(γ), hol_{ω}(γ′))|. Clearly, γ and γ′ are disjoint and do not divide the surface. By Lemma 6.2, there exist saddle connections γ_{3}, …, γ_{N} (N = 2p + n − 1) such that γ, γ′, γ_{3}, …, γ_{N} are also disjoint and do not divide the surface. The relative homology classes [γ], [γ′], [γ_{3}], …, [γ_{N}] comprise a basis for H_{1}(M_{p}, Z_{n}; ℤ). Let C_{1} be a periodic cylinder of ω different from C. Let L be a periodic geodesic in C_{1} and L′ be a saddle connection crossing C_{1}. Then the area of C_{1} is equal to |Λ(hol_{ω}(L), hol_{ω}(L′))|. Since (p, n) ≠ (1, 1) and C is a regular cylinder, the geodesic L is not contained in the closure of C. If L is disjoint from C then its homology class [L] is a linear combination of [γ], [γ_{3}], …, [γ_{N}]. Otherwise L is freely homotopic to the sum of saddle connections L_{1}, L′_{1}, …, L_{k}, L′_{k} such that L_{1}, …, L_{k} are disjoint from C while L′_{1}, …, L′_{k} are contained in the closure of C. The relative homology classes of L_{1}, …, L_{k} are linear combinations of [γ], [γ_{3}], …, [γ_{N}] while the relative homology classes of L′_{1}, …, L′_{k} are linear combinations of [γ] and [γ′]. Note that projections of holonomy vectors hol_{ω}(L_{i}), hol_{ω}(L′_{i}), i = 1, …, k, on the direction orthogonal to γ are all nonzero and of the same sign. It follows that the sum of homology classes of L_{1}, …, L_{k} is not a multiple of [γ]. Thus [L] can be a linear combination of [γ] and [γ′] only if it is a multiple of [γ]. But then the cylinders C and C_{1} are parallel and disjoint, hence the homology class of L′ is not a linear combination of [γ] and [γ′].

By the above and Lemma 7.1(b), for a translation surface in a generic isomorphy class ω ∈ MQ(p, n), (p, n) ≠ (1, 1), all regular periodic cylinders are of different area. It remains to recall that generic translation surfaces do not contain irregular cylinders.

We proceed to the proofs of Theorems 1.5–1.10. Let M be a translation surface of genus p with n singular points (p, n ≥ 1). We assign to M a sequence V_{1}(M) of vectors in ℝ^{2}. These are the holonomy vectors of oriented periodic cylinders on M. Note that any (unoriented) periodic cylinder corresponds to two oriented cylinders with opposite holonomy vectors. Both vectors are supposed to be in the sequence. If a vector is the holonomy vector of k > 1 distinct periodic cylinders, it is to appear k times in V_{1}(M). By Theorem 1.1(c), the sequence V_{1}(M) tends to infinity. To make V_{1}(M) an element of the set V defined in Section 3, we have to equip vectors with weights. Let all weights be equal to 1. Another choice is to let the weight of the holonomy vector of a cylinder be equal to the area of the cylinder. Then we obtain a different element of V that is denoted by V_{2}(M). Further, for any x ∈ M we define V_{3}(M, x) ∈ V to be the sequence of holonomy vectors of oriented periodic geodesics on M passing through the point x. By definition, all weights of V_{3}(M, x) are equal to 1. Now the growth functions N_{1}, N_{2}, N_{3}, and N_{4} defined in Section 1 can be expressed as follows: N_{1}(M, R) = N_{V_{1}(M)}(R)/2, N_{2}(M, R) = N_{V_{2}(M)}(R)/2, N_{3}(M, x, R) = N_{V_{3}(M,x)}(R)/2, and N_{4}(M, σ, R) = N_{W_{(σ,∞)}V_{2}(M)}(R)/2 for all R > 0.

Let C be a connected component of MQ_{1}(p, n). Let p_{0} : MY(p, n) → MQ(p, n) be the natural projection. Then Y = p_{0}^{−1}(C) is a connected component of MY_{1}(p, n). The sequences V_{1}(M) and V_{2}(M) do not change when M is replaced by an isomorphic translation surface. The sequence V_{3}(M, x) does not change when (M, x) is replaced by a pair representing the same equivalence class in MY(p, n). Hence the assignments M ↦ V_{1}(M), M ↦ V_{2}(M), and (M, x) ↦ V_{3}(M, x) give rise to welldefined maps V_{1} : C → V, V_{2} : C → V, and V_{3} : Y → V. Recall that there are continuous actions of the group SL(2, ℝ) on C and Y. By Theorems 6.6 and 6.7, the Borel measure μ_{0} induced by the canonical volume element on C is finite and the SL(2, ℝ) action on C is ergodic relative to this measure. The measure μ_{1} on Y is also finite and the SL(2, ℝ) action on Y is ergodic by Theorem 6.8. Thus the results of Sections 3 and 4 apply to the maps V_{1}, V_{2}, and V_{3}. Let us check whether the conditions formulated in Sections 3 and 4 hold for these maps.

Proposition 7.2 The maps V_{1}, V_{2}, and V_{3} satisfy conditions (0), (A), (B′), (C), and (E). The map V_{2} satisfies condition (0′).

Proof. By definition of the SL(2, ℝ) actions on C, Y, and V, the maps V_{1}, V_{2}, and V_{3} satisfy condition (A). Condition (E) holds trivially.

Let S(p, n) be the set of free homotopy classes of simple closed oriented curves in M_{p} \Z_{n}. Given γ ∈ S(p, n), let U(γ) ⊂ Ω(p, n) be the set of translation structures that admit a periodic geodesic in the homotopy class γ. By U_{1}(γ) denote the set of pairs (ω, x) ∈ Ω(p, n) × M_{p} such that some periodic geodesic of the translation structure ω passing through the point x is in the homotopy class γ. Given ω ∈ U(γ), all periodic geodesics of ω that belong to the homotopy class γ comprise one periodic cylinder. Let a_{γ}(ω) denote the area of this cylinder. For any ω ∉ U(γ) let a_{γ}(ω) = 0. The sets U(γ) and U_{1}(γ) are invariant under the H_{0}(p, n) actions on Ω(p, n) and Ω(p, n) × M_{p}, respectively. Therefore we consider U(γ) as a subset of Q(p, n) and U_{1}(γ) as a subset of Y(p, n). It is easy to see that U(γ) and U_{1}(γ) are open sets. The map Q(p, n) ∋ ω ↦ hol_{ω}(γ) ∈ ℝ^{2} is well-defined and continuous. a_{γ} descends to a function on Q(p, n), which is also continuous. Indeed, let ω_{0} ∈ U(γ) and ω′_{0} be a translation structure in the isotopy class ω_{0}. Take a saddle connection γ_{1} of ω′_{0} that crosses the cylinder of periodic geodesics with homotopy γ and does not leave this cylinder. Let [γ_{1}] ∈ H_{1}(M_{p}, Z_{n}; ℤ) be the homology class of γ_{1}. Then a_{γ}(ω_{0}) = |Λ(hol_{ω_{0}}(γ), hol_{ω_{0}}([γ_{1}]))|, where Λ is the euclidean area form on ℝ^{2}. If the cylinder is regular then a_{γ}(ω) = |Λ(hol_{ω}(γ), hol_{ω}([γ_{1}]))| for all ω ∈ Q(p, n) in a neighborhood of ω_{0}. In the general case a_{γ}(ω) ≤ |Λ(hol_{ω}(γ), hol_{ω}([γ_{1}]))| but we can choose several saddle connections γ_{1}, …, γ_{k} of ω′_{0} so that

a_{γ}(ω) = min_{1≤j≤k} |Λ(hol_{ω}(γ), hol_{ω}([γ_{j}]))|

for all ω close to ω_{0}.

Let π_{0} : Q(p, n) → MQ(p, n), π_{1} : Y(p, n) → MY(p, n), and p̃_{0} : Y(p, n) → Q(p, n) be the canonical projections. Then for any ω ∈ π_{0}^{−1}(C), η ∈ π_{1}^{−1}(Y), ψ ∈ C_{c}(ℝ^{2}), and f ∈ C_{c}(ℝ^{+}) we have

Φ[V_{1}(π_{0}(ω))](ψ) = ∑_{γ∈S(p,n)} χ_{U(γ)}(ω) ψ(hol_{ω}(γ)),
Ψ[V_{2}(π_{0}(ω))](f, ψ) = ∑_{γ∈S(p,n)} f(a_{γ}(ω)) ψ(hol_{ω}(γ)),
Φ[V_{3}(π_{1}(η))](ψ) = ∑_{γ∈S(p,n)} χ_{U_{1}(γ)}(η) ψ(hol_{p̃_{0}(η)}(γ)).

All three sums are locally finite. It follows that the functions π_{0}^{−1}(C) ∋ ω ↦ Φ[V_{1}(π_{0}(ω))](ψ) and π_{1}^{−1}(Y) ∋ η ↦ Φ[V_{3}(π_{1}(η))](ψ) are Borel, while the function π_{0}^{−1}(C) ∋ ω ↦ Ψ[V_{2}(π_{0}(ω))](f, ψ) is continuous. Then the functions C ∋ ω ↦ Φ[V_{1}(ω)](ψ) and Y ∋ η ↦ Φ[V_{3}(η)](ψ) are also Borel and the function C ∋ ω ↦ Ψ[V_{2}(ω)](f, ψ) is also continuous. Thus the maps V_{1} and V_{3} satisfy condition (0) while the map V_{2} satisfies condition (0′′). By Lemma 4.2, V_{2} satisfies conditions (0) and (0′) as well.

For any ω ∈ C, let s(ω) denote the length of the shortest saddle connection of translation structures in the class ω. The function ω ↦ s(ω) is continuous and bounded on C. Therefore the upper estimate in Theorem 1.2(c) implies the map V_{1} satisfies condition (B′). To verify condition (C), we need the following theorem.

Theorem 7.3 ([EM]) (a) Given ϵ > 0, there exist C_{ϵ} > 0 and κ > 0 such that

N_{V_{1}(ω)}(R) ≤ C_{ϵ} (\frac{R}{s(ω)})^{1+ϵ}

for any R < κ and any ω ∈ C.

(b) For any β ∈ [1, 2) the function s^{−β} belongs to the space L^{1}(C, μ_{0}).

Theorem 7.3 implies that condition (C) holds for the map V_{1}. Let ω ∈ C. By definition, N_{V_{2}(ω)}(R) ≤ N_{V_{1}(ω)}(R) for any R > 0, and N_{V_{3}(η)}(R) ≤ N_{V_{1}(ω)}(R) for any η ∈ p_{0}^{−1}(ω) and any R > 0. It follows that conditions (B′) and (C) are satisfied by the maps V_{2} and V_{3} whenever these conditions are satisfied by V_{1}.

Proof of Theorems 1.5, 1.6(a), and 1.7. By Proposition 7.2, the maps V_{1}, V_{2}, and V_{3} satisfy conditions (0), (A), (B), and (C). Let c_{1}(C), c_{2}(C), and c_{3}(C) be nonnegative numbers such that 2π^{−1}c_{1}(C), 2π^{−1}c_{2}(C), and 2π^{−1}c_{3}(C) are the Siegel-Veech constants of the pairs (V_{1}, μ_{0}), (V_{2}, μ_{0}), and (V_{3}, μ_{1}), respectively. By Part II of Theorem 3.2, for μ_{0}-almost every ω ∈ C we have

lim_{R→∞} N_{V_{1}(ω)}(R)/R^{2} = 2c_{1}(C),  (8)
lim_{R→∞} N_{V_{2}(ω)}(R)/R^{2} = 2c_{2}(C),  (9)

and for μ_{1}-almost every η ∈ Y,

lim_{R→∞} N_{V_{3}(η)}(R)/R^{2} = 2c_{3}(C).  (10)

The positivity of c_{1}(C) and c_{2}(C) follows from Proposition 3.3 and Theorem 1.1(a). Let us show that c_{3}(C) = c_{2}(C). By Part I of Theorem 3.2, we have

\frac{1}{μ_{0}(C)} ∫_{C} N_{V_{2}(ω)}(1) dμ_{0}(ω) = 2c_{2}(C),
\frac{1}{μ_{1}(Y)} ∫_{Y} N_{V_{3}(η)}(1) dμ_{1}(η) = 2c_{3}(C).

For any ω ∈ C, let ρ_{ω} denote the Borel measure on the fiber p_{0}^{−1}(ω) induced by translation structures in the equivalence class ω. It is easy to observe that

N_{V_{2}(ω)}(R) = ∫_{p_{0}^{−1}(ω)} N_{V_{3}(η)}(R) dρ_{ω}(η)

for all R > 0. Then

∫_{Y} N_{V_{3}(η)}(1) dμ_{1}(η) = ∫_{C} ∫_{p_{0}^{−1}(ω)} N_{V_{3}(η)}(1) dρ_{ω}(η) dμ_{0}(ω) = ∫_{C} N_{V_{2}(ω)}(1) dμ_{0}(ω),

besides,

μ_{1}(Y) = ∫_{C} ρ_{ω}(p_{0}^{−1}(ω)) dμ_{0}(ω) = μ_{0}(C).

Hence, c_{3}(C) = c_{2}(C).

By the above there exists a Borel set U ⊂ C, μ_{0}(U) = μ_{0}(C), such that for any ω ∈ U the relations (8) and (9) hold, and, moreover, the relation (10) holds for ρ_{ω}- almost all η ∈ p_{0}^{−1}(ω). Let M be a translation surface in an isomorphy class ω ∈ U. Then N_{1}(M, R)/R^{2} → c_{1}(C) and N_{2}(M, R)/R^{2} → c_{2}(C) as R → ∞, and for almost all x ∈ M, N_{3}(M, x, R)/R^{2} → c_{2}(C) as R → ∞.

To prove Theorems 1.6(b), 1.8, and 1.9, we need the following proposition, which will be proved in Section 8.

Proposition 7.4 For any σ ∈ [0, 1) and ϵ > 0 let

b(σ, ϵ) = \frac{1}{μ_{0}(C)} ∫_{C} N_{W_{(σ,∞)}V_{2}(ω)}(ϵ) dμ_{0}(ω).  (11)

Then for any σ ∈ [0, 1),

lim_{ϵ→0} \frac{b(σ, ϵ)}{b(0, ϵ)} = (1 − σ)^{m_{C}−1},

where m_{C} = 2p − 2 + n.

Proof of Theorems 1.6(b) and 1.8. By Proposition 7.2, the map V_{2} satisfies conditions (0′), (A), and (B), hence Proposition 4.3 applies to it. Let λ be the Siegel-Veech measure of the pair (V_{2}, μ_{0}). Since the map V_{1} = W_{(0,∞)}V_{2} satisfies condition (B), λ is a finite measure on ℝ^{+}. For any a ≥ 0 the Siegel-Veech constant of (W_{(a,∞)}V_{2}, μ_{0}) is equal to λ((a, ∞)). By Part I of Theorem 3.2, b(a, ϵ) = πϵ^{2}λ((a, ∞)) for all ϵ > 0 and a ∈ [0, 1). Then it follows from Proposition 7.4 that λ((a, ∞)) = (1 − a)^{m_{C}−1}λ(ℝ^{+}) for all a ∈ [0, 1). For any ω ∈ C the sequence V_{2}(ω) contains no vectors with weights greater than 1, therefore λ((1, ∞)) = 0. Let c_{0} be the Siegel-Veech constant of (V_{2}, μ_{0}). If m_{C} = 1 then λ({1}) = λ(ℝ^{+}), hence c_{0} = λ(ℝ^{+}). If m_{C} > 1 then

c_{0} = ∫_{0}^{∞} t dλ(t) = −λ(ℝ^{+}) ∫_{0}^{1} t d(1 − t)^{m_{C}−1} = λ(ℝ^{+})/m_{C}.

By Part II of Theorem 3.2, for any a ∈ [0, 1) we have

lim_{R→∞} N_{W_{(a,∞)}V_{2}(ω)}(R)/R^{2} = π(1 − a)^{m_{C}−1}λ(ℝ^{+})  (12)

for μ_{0}-almost all ω ∈ C. Also, for μ_{0}-almost all ω ∈ C,

lim_{R→∞} N_{V_{2}(ω)}(R)/R^{2} = πm_{C}^{−1}λ(ℝ^{+}).  (13)

Let U ⊂ C be a Borel set such that μ_{0}(U) = μ_{0}(C), the relation (13) holds for any ω ∈ U, and the relation (12) holds for any ω ∈ U and any rational a ∈ [0, 1). Since N_{W_{(a,∞)}V_{2}(ω)}(R) is a nonincreasing function of a, it follows that the relation (12) holds for any ω ∈ U and any a ∈ [0, 1). Let M be a translation surface in an isomorphy class ω ∈ U. Then N_{2}(M, R)/R^{2} → πm_{C}^{−1}λ(ℝ^{+})/2 as R → ∞, and for any a ∈ [0, 1), N_{4}(M, a, R)/R^{2} → π(1 − a)^{m_{C}−1}λ(ℝ^{+})/2 as R → ∞. By Theorems 1.5 and 1.6(a), πλ(ℝ^{+})/2 = c_{1}(C) and πm_{C}^{−1}λ(ℝ^{+})/2 = c_{2}(C). It follows that c_{2}(C) = c_{1}(C)/m_{C}.

Proof of Theorems 1.9 and 1.10. By Proposition 7.2, the map V_{2} satisfies conditions (0′), (A), (B), (C), and (E). Besides, the Siegel-Veech constant of (V_{2}, μ_{0}) is nonzero and the map W_{(0,∞)}V_{2} = V_{1} satisfies condition (B). Therefore Propositions 4.4 and 4.6 apply to V_{2}. Let λ be the Siegel-Veech measure of the pair (V_{2}, μ_{0}). As shown in the previous proof, λ((a, ∞)) = (1 − a)^{m_{C}−1}λ(ℝ^{+}) for any a ∈ [0, 1) and λ((1, ∞)) = 0. In particular, we can consider λ as a Borel measure on [0, 1]. Then the normalized measure (λ(ℝ^{+}))^{−1}λ coincides with the measure λ_{m_{C}} defined in Section 1. Since for any ω ∈ C the sequence V_{2}(ω) contains no vectors with weights greater than 1, we can consider the measures α̃_{V_{2}(ω),R} and D̃_{V_{2}(ω),R} defined in Section 4 as measures on [0, 1] and S^{1} × [0, 1], respectively. Then it follows from Propositions 4.4 and 4.6 that for μ_{0}-almost all ω ∈ C we have the following weak convergence of measures:

lim_{R→∞} α̃_{V_{2}(ω),R} = λ_{m_{C}},

lim_{R→∞} D̃_{V_{2}(ω),R} = 𝔪_{1} × λ_{m_{C}}.

The maps V_{1} and V_{3} satisfy conditions (0), (A), (B), (C), and (E). The Siegel-Veech constants of the pairs (V_{1}, μ_{0}) and (V_{3}, μ_{1}) are nonzero. Then Proposition 4.5 implies that for μ_{0}-almost all ω ∈ C and for μ_{1}-almost all η ∈ Y the measures δ̃_{V_{1}(ω),R} and δ̃_{V_{3}(η),R} on S^{1} weakly converge to normalized Lebesgue measure 𝔪_{1} as R → ∞.

By the above there exists a Borel set U ⊂ C, μ_{0}(U) = μ_{0}(C), such that for any ω ∈ U the measures α̃_{V_{2}(ω),R}, δ̃_{V_{1}(ω),R}, and D̃_{V_{2}(ω),R} weakly converge to λ_{m_{C}}, 𝔪_{1}, and 𝔪_{1} × λ_{m_{C}}, respectively, as R → ∞, and for ρ_{ω}-almost all η ∈ p_{0}^{−1}(ω) the measures δ̃_{V_{3}(η),R} weakly converge to 𝔪_{1} as R → ∞. Let M be a translation surface in an isomorphy class ω ∈ U. Then the measures α̃_{M,R}, δ̃_{M,R}, and D̃_{M,R} defined in Section 1 coincide with α̃_{V_{2}(ω),R}, δ̃_{V_{1}(ω),R}, and D̃_{V_{2}(ω),R}, respectively. Further, for any x ∈ M the measure δ̃_{M,x,R} coincides with δ̃_{V_{3}(η),R}, where η ∈ p_{0}^{−1}(ω) is the equivalence class of (M, x). Theorems 1.9 and 1.10 follow.

8 Areas of periodic cylinders

For any positive integer n define sets

S_{n} = {(t_{1}, …, t_{n}) ∈ ℝ^{n} | t_{i} > 0, t_{1} + ⋯ + t_{n} ≤ 1},
S_{n}^{∗} = {(t, t_{1}, …, t_{n}) ∈ ℝ^{n+1} | t > 0, t_{i} > 0, t^{2} + t_{1} + ⋯ + t_{n} ≤ 1}.

Further, for any σ ∈ [0, 1) and i ∈ {1, …, n} define their subsets

S_{n}(σ, i) = {(t_{1}, …, t_{n}) ∈ S_{n} | t_{i} > σ(t_{1} + ⋯ + t_{n})},
S_{n}^{∗}(σ, i) = {(t, t_{1}, …, t_{n}) ∈ S_{n}^{∗} | t_{i} > σ(t^{2} + t_{1} + ⋯ + t_{n})}.

Note that S_{n}(0, i) = S_{n}, S_{n}^{∗}(0, i) = S_{n}^{∗}. Now let

J_{n,0}(σ, i) = ∫_{S_{n}(σ,i)} dt_{1} … dt_{n}

and for any positive integer k,

J_{n,k}(σ, i) = ∫_{S_{n}^{∗}(σ,i)} dt^{2k} dt_{1} … dt_{n}.

Lemma 8.1 For any integers n > 0 and k ≥ 0, any i ∈ {1, …, n}, and any σ ∈ [0, 1),

\frac{J_{n,k}(σ, i)}{J_{n,k}(0, i)} = (1 − σ)^{n+k−1}.

Proof. Since J_{n,k}(σ, i) = J_{n,k}(σ, 1) for any i ∈ {1, …, n}, we can assume without loss of generality that i = 1. The case n = 1, k = 0 is trivial as S_{1}(σ, 1) = S_{1} for any σ ∈ [0, 1). Given σ ∈ (0, 1) and b > 0, let

I_{b}(σ) = ∫_{0}^{1}(min(1 − t, t/σ − t))^{b} dt.

Then

I_{b}(σ) = ∫_{0}^{σ}(t/σ − t)^{b} dt + ∫_{σ}^{1}(1 − t)^{b} dt =

\frac{1}{b + 1}(σ^{b+1}(σ^{−1} − 1)^{b} + (1 − σ)^{b+1}) = \frac{(1 − σ)^{b}}{b + 1} .

Suppose n > 1 and σ ∈ (0, 1). Then

J_{n,0}(σ, 1) = ∫_{0}^{1} dt_{1} ∫_{min(1−t_{1},t_{1}/σ−t_{1})S_{n−1}} dt_{2} … dt_{n} = I_{n−1}(σ) ∫_{S_{n−1}} dt_{2} … dt_{n}.

Now suppose n > 0 and σ ∈ (0, 1). Then (t, t_{1}, …, t_{n}) ∈ S_{n}^{∗}(σ, 1) if and only if (t_{1}, …, t_{n}) ∈ S_{n}(σ, 1), t > 0, t ≤ (1−t_{1} −⋯−t_{n})^{1/2}, and t < (t_{1}/σ−t_{1} −⋯−t_{n})^{1/2}. Hence for any integer k > 0,

J_{n,k}(σ, 1) = ∫_{S_{n}(σ,1)}(min(1, t_{1}/σ) − t_{1} − ⋯ − t_{n})^{k} dt_{1} … dt_{n}.

In particular, J_{1,k}(σ, 1) = I_{k}(σ). If n > 1 then

J_{n,k}(σ, 1) = ∫_{0}^{1} dt_{1} ∫_{min(1−t_{1},t_{1}/σ−t_{1})S_{n−1}}(min(1, t_{1}/σ) − t_{1} − ⋯ − t_{n})^{k} dt_{2} … dt_{n} =

I_{n+k−1}(σ) ∫_{S_{n−1}}(1 − t_{2} − ⋯ − t_{n})^{k} dt_{2} … dt_{n}.

By the above for any integers n > 0 and k ≥ 0 there exists C_{n,k} > 0 such that J_{n,k}(σ, 1) = (1−σ)^{n+k−1}C_{n,k} for all σ ∈ (0, 1). Since J_{n,k}(σ, 1) → J_{n,k}(0, 1) as σ → 0, it follows that C_{n,k} = J_{n,k}(0, 1).

Let ω^{(n)} ∈ Ω(1, n) be a translation structure with n ≥ 1 singular points on the torus M_{1} such that for some v ∈ S^{1} there are n cylinders of periodic geodesics of ω^{(n)} going in direction v (in general, the number of cylinders may be less than n). Let x_{1}, …, x_{n} be singular points of ω^{(n)} and C_{1}, …, C_{n} be periodic cylinders with direction v. By L_{i} (1 ≤ i ≤ n) denote the saddle connection going out of x_{i} in direction v. We assume that the singular points and the cylinders are named so that L_{i} separates cylinders C_{i} and C_{i−1} (where by definition C_{0} = C_{n}) and, furthermore, C_{i} lies to the right of L_{i} (with respect to the direction of L_{i}). Let γ_{i} be a saddle connection that crosses the cylinder C_{i} joining x_{i} to x_{i+1} (by definition, x_{n+1} = x_{1}). Note that the holonomy vector of γ_{i} and v induce the standard orientation in ℝ^{2}. Among all saddle connections that join x_{i} to x_{i+1} by crossing C_{i}, only one is disjoint from γ_{i}. Let γ′_{i} denote this saddle connection. It is easy to see that L_{i}, γ_{i}, γ′_{i}, i = 1, …, n, is a maximal set of disjoint saddle connections of ω^{(n)}. Let τ^{(n)} ∈ T (1, n) be the affine equivalence class of the corresponding triangulation. For future references, we denote by L^{(n)} the relative homotopy class of L_{1}, which is regarded as an edge of τ^{(n)}. For any translation structure in an isotopy class ω ∈ N(τ^{(n)}) there are n cylinders of homologous periodic geodesics freely homotopic in M_{1} \ {x_{1}, …, x_{n}} to geodesics in the cylinders C_{1}, …, C_{n}. Finally, by P^{(n)} denote the set of ω ∈ N(τ^{(n)}) ⊂ Q(1, n) such that for any i, 1 ≤ i ≤ n, we have hol_{ω}(γ_{i}) = b_{i} hol_{ω}(L_{i})+v_{i}, where 0 < b_{i} < 1, v_{i} ∈ ℝ^{2} is orthogonal to hol_{ω}(L_{i}), and the pair of vectors v_{i} and hol_{ω}(L_{i}) induces the standard orientation in ℝ^{2}.

For any translation structure ω′ on a surface M let a(ω′) denote the area of M with respect to ω′. For any element ω of Q(p_{0}, n_{0}) or MQ(p_{0}, n_{0}) (p_{0}, n_{0} ≥ 1) let a(ω) denote the area of the surface M_{p_{0}} with respect to translation structures in the equivalence class ω. Further, for any translation structure ω′ in an isotopy class ω ∈ N(τ^{(n)}) and any integer i, 1 ≤ i ≤ n, let a_{i}(ω′) denote the area of the periodic cylinder of ω′ homotopic to C_{i}. The area a_{i}(ω′) does not depend on the choice of ω′ ∈ ω and we let a_{i}(ω) = a_{i}(ω′). Obviously, a_{1}(ω) + ⋯ + a_{n}(ω) = a(ω).

Given U ⊂ ℝ^{2}, let P^{(n)}(U) be the set of ω ∈ P^{(n)} such that hol_{ω}(L_{1}) ∈ U and a(ω) ≤ 1. For any i ∈ {1, …, n} and any σ ∈ [0, 1) let P_{σ,i}^{(n)}(U) be the set of ω ∈ P^{(n)}(U) such that a_{i}(ω) > σa(ω). Given sets U_{j} ⊂ Q_{1}(p_{j}, n_{j}), 1 ≤ j ≤ k, let P^{(n)}(U; U_{1}, …, U_{k}) denote the set of (ω, ω_{1}, …, ω_{k}) ∈ Q(1, n) × Q(p_{1}, n_{1}) × … × Q(p_{k}, n_{k}) such that ω ∈ P^{(n)}(U), each ω_{j} is represented as t_{j}ω′_{j}, where ω′_{j} ∈ U_{j} and 0 < t_{j} ≤ 1, and a(ω) + a(ω_{1}) + ⋯ + a(ω_{k}) ≤ 1. For any i ∈ {1, …, n} and σ ∈ [0, 1) let P_{σ,i}^{(n)}(U; U_{1}, …, U_{k}) be the set of (ω, ω_{1}, …, ω_{k}) ∈ P^{(n)}(U; U_{1}, …, U_{k}) such that a_{i}(ω) > σ(a(ω) + a(ω_{1}) + ⋯ + a(ω_{k})).

Lemma 8.2 (a) Suppose U ⊂ ℝ^{2} is a nonempty open bounded set. Then for any σ ∈ [0, 1) and i ∈ {1, …, n},

\frac{μ_{0}(P_{σ,i}^{(n)}(U))}{μ_{0}(P^{(n)}(U))} = (1 − σ)^{n−1},

where μ_{0} denotes the canonical measure on Q(1, n).

(b) Suppose U ⊂ ℝ^{2} and U_{j} ⊂ Q_{1}(p_{j}, n_{j}), 1 ≤ j ≤ k, are nonempty open sets of finite measure. Then for any σ ∈ [0, 1) and i ∈ {1, …, n},

\frac{μ(P_{σ,i}^{(n)}(U; U_{1}, …, U_{k}))}{μ(P^{(n)}(U; U_{1}, …, U_{k}))} = (1 − σ)^{K−2},

where K = n + 1 + ∑_{j=1}^{k}(2p_{j} + n_{j} − 1) is half of the dimension of Q(1, n) × Q(p_{1}, n_{1}) × … × Q(p_{k}, n_{k}) and μ denotes the product of the canonical measures on Q(1, n), Q(p_{1}, n_{1}), …, Q(p_{k}, n_{k}).

Proof. The homology classes of saddle connections L_{1}, γ_{1}, …, γ_{n} form a basis for H_{1}(M_{1}, {x_{1}, …, x_{n}}; ℤ). The map F_{1} : N(τ^{(n)}) → (ℝ^{2})^{n+1} defined by F_{1}(ω) = (hol_{ω}(L_{1}), hol_{ω}(γ_{1}), …, hol_{ω}(γ_{n})) is a volume preserving homeomorphism of N(τ^{(n)}) onto its image. Given a nonzero v ∈ ℝ^{2}, let g_{v} denote a unique element of SL(2, ℝ) such that g_{v}v = (0, 1) and g_{v}u = (1, 0) for some u orthogonal to v. Define a transformation F_{2} of the set (ℝ^{2} \ {(0, 0)}) × (ℝ^{2})^{n} by F_{2}(v, v_{1}, …, v_{n}) = (v, g_{v}v_{1}, …, g_{v}v_{n}). F_{2} is a homeomorphism preserving Lebesgue measure. Suppose (v, v_{1}, …, v_{n}) = F_{2}(F_{1}(ω)) for some ω ∈ P^{(n)}. Then the first coordinate of the vector v_{i} ∈ ℝ^{2}, 1 ≤ i ≤ n, is equal to a_{i}(ω) while the second coordinate lies between 0 and 1. Finally, define a map F_{3} : (ℝ^{2})^{n+1} → ℝ^{2}×ℝ^{n}×ℝ^{n} by F_{3}((v_{01}, v_{02}), (v_{11}, v_{12}), …, (v_{n1}, v_{n2})) = ((v_{01}, v_{02}), (v_{11}, …, v_{n1}), (v_{12}, …, v_{n2})). F_{3} is a linear map preserving Lebesgue measure. It is easy to observe that F_{3} ∘ F_{2} ∘ F_{1}(P^{(n)}(U)) = (U \ {(0, 0)}) × S_{n} × (0, 1)^{n} while F_{3}∘F_{2}∘F_{1}(P_{σ,i}^{(n)}(U)) = (U \{(0, 0)})×S_{n}(σ, i)×(0, 1)^{n} for all σ and i. Therefore

\frac{μ_{0}(P_{σ,i}^{(n)}(U))}{μ_{0}(P^{(n)}(U))} = \frac{𝔪(U)J_{n,0}(σ, i)}{𝔪(U)J_{n,0}(0, i)} = (1 − σ)^{n−1}

by Lemma 8.1.

For any t ∈ (0, 1] let U_{t}^{∗} denote the set of (ω_{1}, …, ω_{k}) ∈ Q(p_{1}, n_{1})×…×Q(p_{k}, n_{k}) such that each ω_{j} is represented as t_{j}ω′_{j}, where ω′_{j} ∈ U_{j} and 0 < t_{j} ≤ 1, and a(ω_{1}) + ⋯ + a(ω_{k}) ≤ t^{2}. Since a(tω′) = t^{2}a(ω′) for any translation structure ω′, it follows that U_{t}^{∗} = tU_{1}^{∗}, where by definition t(ω_{1}, …, ω_{k}) = (tω_{1}, …, tω_{k}) for all ω_{j} ∈ Q(p_{j}, n_{j}), 1 ≤ j ≤ k. Denote by μ^{∗} the product of the canonical measures on Q(p_{1}, n_{1}), …, Q(p_{k}, n_{k}). Then 0 < μ^{∗}(U_{1}^{∗}) < ∞ and μ^{∗}(U_{t}^{∗}) = t^{2K^{∗}}μ^{∗}(U_{1}^{∗}), where K^{∗} = ∑_{j=1}^{k}(2p_{j} + n_{j} − 1) is half of the dimension of Q(p_{1}, n_{1}) × … × Q(p_{k}, n_{k}).

Obviously, P^{(n)}(U; U_{1}, …, U_{k}) ⊂ P^{(n)}(U) × U_{1}^{∗}. Suppose ω ∈ P^{(n)}(U) and ω^{∗} ∈ U_{1}^{∗}. Then (ω, ω^{∗}) ∈ P_{σ,i}^{(n)}(U; U_{1}, …, U_{k}) if and only if ω ∈ P_{σ,i}^{(n)}(U) and ω^{∗} ∈ tU_{1}^{∗} for some t > 0 such that a(ω) + t^{2} ≤ 1, a_{i}(ω) > σ(a(ω) + t^{2}). It follows that

μ(P_{σ,i}^{(n)}(U; U_{1}, …, U_{k})) = 𝔪(U)μ^{∗}(U_{1}^{∗})J_{n,K^{∗}}(σ, i)

for all σ and i. Since P_{0,i}^{(n)}(U; U_{1}, …, U_{k}) = P^{(n)}(U; U_{1}, …, U_{k}), statement (b) of the lemma follows from Lemma 8.1.

Now we shall define 4 operations on translation surfaces and their equivalence classes: cutting along parallel saddle connections, gluing along parallel saddle connections, collapsing a short saddle connection, and inserting a short saddle connection. In what follows two saddle connections of a translation surface are called homologous if they are disjoint and break the surface into two parts. Note that homologous saddle connections are parallel and of the same length.

The cutting operation is defined on any translation surface X with a distinguished oriented saddle connection γ. If there is no saddle connection homologous to γ then the operation does nothing to X. Otherwise let γ_{1} = γ and γ_{2}, …, γ_{k} be saddle connections homologous to γ. Orient γ_{2}, …, γ_{k} so that they are of the same direction as γ. The surface X is divided by γ_{1}, …, γ_{k} into k domains C_{1}, …, C_{k}, each domain being bounded by two saddle connections. We assume γ_{2}, …, γ_{k} and C_{1}, …, C_{k} are named so that each C_{i} is bounded by γ_{i} and γ_{i+1} (by definition, γ_{k+1} = γ_{1}) and, moreover, C_{i} lies to the right of γ_{i} (with respect to the direction of γ_{i}). Choose a partition τ of X by a maximal set of disjoint saddle connections containing γ_{1}, …, γ_{k}. As described in Section 5, the translation surface X can be obtained by gluing together plane triangles so that these triangles become cells of τ and edges of τ correspond to glued sides of the triangles. Reverse this construction and decompose X into disjoint plane triangles. For any i, 1 ≤ i ≤ k, let T_{i}^{+} denote the triangle corresponding to the cell of τ bounded by γ_{i} and contained in C_{i}. By T_{i}^{−} denote the triangle corresponding to the other cell of τ bounded by γ_{i}. Now glue the triangles in a different way. Namely, the side of T_{i}^{+} parallel to γ is glued to a side of T_{i+1}^{−} (instead of T_{i}^{−}), where by definition T_{k+1}^{−} = T_{1}^{−}, while all other sides of triangles are glued together as before. The new gluing gives k distinct translation surfaces X_{1}, …, X_{k}. Namely, each X_{i} is obtained by gluing together triangles corresponding to cells of τ contained in C_{i}. By construction X_{i} is equipped with a triangulation τ_{i} by disjoint saddle connections. Also, we have a distinguished oriented edge γ′_{i} of τ_{i} that corresponds to glued sides of T_{i}^{+} and T_{i+1}^{−}. X_{i} contains no saddle connection homologous to γ′_{i}. The sum of areas of X_{1}, …, X_{k} is equal to the area of X.

Now suppose X = M_{p} and the translation structure ω of X belongs to Ω(p, n) (p, n ≥ 1). Let τ̃ ∈ T (p, n) be the affine equivalence class of τ and ω̃ ∈ N(τ̃) be the isotopy class of ω. Let γ̃ be the oriented edge of τ̃ corresponding to γ (note that γ̃ is actually a homotopy class). Then the translation structure of any X_{i} (1 ≤ i ≤ k) is isomorphic to some ω_{i} ∈ Ω(p_{i}, n_{i}), where p_{i} and n_{i} are determined by τ̃ and γ̃. The triangulation τ_{i} corresponds to a triangulation in a class τ̃_{i} ∈ T (p_{i}, n_{i}). The class τ̃_{i} is determined up to the action of Mod(p_{i}, n_{i}) but once we fix it and specify an oriented edge γ̃_{i} of τ̃_{i} corresponding to γ′_{i}, the isotopy class ω̃_{i} ∈ N(τ̃_{i}) of ω_{i} is uniquely determined by ω̃, τ̃, and γ̃. Thus we obtain a map CUT [τ̃, γ̃; τ̃_{1}, γ̃_{1}, …, τ̃_{k}, γ̃_{k}] : N(τ̃) → N(τ̃_{1}) × … × N(τ̃_{k}), which is obviously affine.

The gluing operation is inverse to cutting. Let X_{1}, …, X_{k} be translation surfaces and suppose each X_{i} has a distinguished oriented saddle connection γ_{i}. The gluing operation is defined if the holonomy vectors of γ_{1}, …, γ_{k} coincide. For any X_{i} choose a triangulation τ_{i} by disjoint saddle connections including γ_{i}. Then decompose X_{i} into plane triangles according to the partition τ_{i}. We assume that all triangles obtained by decomposing surfaces X_{1}, …, X_{k} are disjoint. Let T_{i}^{+} denote the triangle that corresponds to the cell of τ_{i} bounded by γ_{i} and lying to the right of γ_{i}. By T_{i}^{−} denote the triangle corresponding to the other cell of τ_{i} bounded by γ_{i}. Now reglue the triangles in the following way. Let the side of T_{i}^{+} parallel to γ_{i} be glued to the side of T_{i−1}^{−} parallel to γ_{i−1} (if i = 1, we assume T_{0}^{−} = T_{k}^{−} and γ_{0} = γ_{k}); this is possible as γ_{i} and γ_{i−1} have the same holonomy vector. All other sides of triangles are glued together as before. After the gluing we obtain a single translation surface X equipped with a triangulation τ by a maximal set of disjoint saddle connections. Let γ′_{i} (1 ≤ i ≤ k) be the edge of τ that corresponds to glued sides of T_{i}^{+} and T_{i−1}^{−}. Then γ′_{1}, …, γ′_{k} are homologous saddle connections of X. The surface X contains no more saddle connections homologous to γ′_{1}, …, γ′_{k} provided for any X_{i} there is no saddle connection homologous to γ_{i}. The area of X is the sum of areas of X_{1}, …, X_{k}.

Now suppose the translation structure ω_{i} of each X_{i}, 1 ≤ i ≤ k, belongs to some Ω(p_{i}, n_{i}). Let τ̃_{i} ∈ T (p_{i}, n_{i}) be the affine equivalence class of τ_{i} and ω̃_{i} ∈ N(τ̃_{i}) be the isotopy class of ω_{i}. Let γ̃_{i} be the oriented edge of τ̃_{i} corresponding to γ_{i}. Then the translation structure of X is isomorphic to some ω ∈ Ω(p, n), where p and n are determined by τ̃_{1}, …, τ̃_{k} and γ̃_{1}, …, γ̃_{k}. The triangulation τ corresponds to a triangulation in a class τ̃ ∈ T (p, n) that is determined up to the action of Mod(p, n). Once we fix τ̃ and specify its oriented edge γ̃ corresponding to γ′_{1}, the isotopy class ω̃ ∈ N(τ̃) of ω is uniquely determined by ω̃_{i}, τ̃_{i}, γ̃_{i}, i = 1, …, k. Thus we obtain a map GLU[τ̃_{1}, γ̃_{1}, …, τ̃_{k}, γ̃_{k}; τ̃, γ̃] : U → N(τ̃) that is defined on the set U of (ω̃_{1}, …, ω̃_{k}) ∈ N(τ̃_{1})×…×N(τ̃_{k}) such that holω̃_{1}(γ̃_{1}) = … = holω̃_{k}(γ̃_{k}). Note that U is an affine submanifold of Q(p_{1}, n_{1})×…×Q(p_{k}, n_{k}) and GLU[τ̃_{1}, γ̃_{1}, …, τ̃_{k}, γ̃_{k}; τ̃, γ̃] is an affine map.

We proceed to the collapsing operation. Let X be a translation surface with a distinguished oriented saddle connection γ. It is assumed that X has no saddle connection homologous to γ. Also, we require that X be not a torus with one singular point. Let τ be a triangulation of X by disjoint saddle connections including γ. By T^{+} and T^{−} denote two triangles of τ bounded by γ. Assume that T^{+} lies to the right of γ while T^{−} lies to the left. The saddle connection γ is collapsed as follows. We decompose X into plane triangles according to the partition τ and discard two triangles corresponding to T^{+} and T^{−}. Then the remaining triangles are modified in a canonical way so that they can be glued together into another translation surface. In general, the collapsing operation is well-defined if γ is much shorter than the other edges of τ and angles of any triangle of τ not bounded by γ are not too small (precise conditions will be given later). Let T_{0} = T^{+}, T_{1}, …, T_{k} = T^{−} be a sequence of triangles of τ such that any two neighboring triangles have common edge different from γ and the length of the sequence is the least possible. By γ_{i} (1 ≤ i ≤ k) denote a common edge of triangles T_{i−1} and T_{i} different from γ. Let γ_{+} be the edge of T^{+} different from γ and γ_{1}, and γ_{−} be the edge of T^{−} different from γ and γ_{k}. First consider the case when k = 1, that is, T^{+} and T^{−} have two common edges γ and γ_{1}. Since X is not a torus with one singular point, γ_{+} ≠ γ_{−}. Clearly, γ_{+} and γ_{−} are homologous saddle connections bounding a regular periodic cylinder. The collapsing operation in this case consists of removing the cylinder. Namely, after decomposing X and discarding two triangles only two sides of the remaining triangles cannot be glued as before. These sides correspond to homologous saddle connections γ_{+} and γ_{−}, therefore they can be glued by translation. The other sides are glued as before and we obtain a translation surface X_{0}.

Now consider the case when k > 1, i.e., γ is the only common edge of T^{+} and T^{−}. Then the edges γ_{1}, γ_{+}, γ_{k}, and γ_{−} are all distinct. Let T_{+} be the triangle of τ bounded by γ_{+} and distinct from T^{+}. Let T_{−} be the triangle bounded by γ_{+} and distinct from T^{−}. Note that T_{+} ≠ T_{1} as otherwise the edge of T_{+} different from γ_{+} and γ_{1} would be homologous to γ. Similarly, T_{−} ≠ T_{k−1}. Since the sequence T_{0}, T_{1}, …, T_{k} is as short as possible, it does not contain triangles T_{+} and T_{−}. Moreover, triangles T_{0}, T_{1}, …, T_{k} are all distinct and so are saddle connections γ_{1}, …, γ_{k}. Decompose the translation surface X into disjoint plane triangles according to the triangulation τ. For any i, 1 ≤ i ≤ k − 1, let T′_{i} be the plane triangle corresponding to T_{i}. By γ′_{i} and γ′′_{i+1} denote the sides of T′_{i} corresponding to edges γ_{i} and γ_{i+1}, respectively. By A_{i} denote the common endpoint of γ′_{i} and γ′′_{i+1}. Further, let T′_{+} and T′_{−} be the plane triangles corresponding to T_{+} and T_{−}. Let γ′_{+} denote the side of T′_{+} corresponding to γ_{+} and γ′_{−} denote the side of T′_{−} corresponding to γ_{−}. After discarding plane triangles corresponding to T^{+} and T^{−}, the sides γ′_{1}, γ′_{+}, γ′′_{k}, and γ′_{−} lose sides they were glued to. We are going to glue γ′_{1} to γ′_{+} and γ′′_{k} to γ′_{−} while the other sides of the remaining triangles will be glued as before. To make this possible, we shall modify triangles T′_{1}, …, T′_{k−1}. Let v denote the holonomy vector of γ. The saddle connections γ_{1} and γ_{+} can be oriented so that their holonomy vectors differ by v or −v. It follows that we can move the vertex A_{1} of the triangle T′_{1} by v or −v so that sides γ′_{1} and γ′_{+} can be glued by translation. However this changes the side γ′′_{1} and it cannot be glued to γ′_{2} anymore. To fix this problem, we move the vertex A_{2} of T′_{2} by v or −v, and so on. Finally each A_{i} is moved by v or −v. Note that each time the choice of a vector (v or −v) is uniquely determined by the configuration of triangles T_{0}, T_{1}, …, T_{k} and saddle connections γ, γ_{+}, γ_{1}, …, γ_{k}. It is assumed that none of the triangles T′_{1}, …, T′_{k−1} degenerates while being modified, i.e., each A_{i} does not cross the straight line containing the opposite side of T′_{i}; otherwise the collapsing operation is not defined. Besides, we assume that all plane triangles are still disjoint after modifications. Then we can glue sides in each pair by translation and obtain a translation surface X_{0}. Indeed, this follows from the above for all pairs except γ′′_{k} and γ′_{−}. The sides γ′′_{k} and γ′_{−} can be glued by translation since the other pairs can.

Now suppose the translation structure ω of X belongs to some Ω(p, n). Let τ̃ ∈ T (p, n) be the affine equivalence class of τ and ω̃ ∈ N(τ̃) be the isotopy class of ω. Let γ̃ be the oriented edge of τ̃ corresponding to γ. Then the translation structure of X_{0} is isomorphic to some ω_{0} ∈ Ω(p_{0}, n_{0}), where p_{0} and n_{0} are determined by τ̃ and γ̃. By construction ω_{0} is equipped with a triangulation τ_{0} by disjoint saddle connections. We distinguish two edges of τ_{0}. In the case k > 1, let e_{r} be the edge obtained by gluing the side γ′_{1} to γ′_{+} and e_{l} be the edge obtained by gluing γ′′_{k} to γ′_{−}. In the case k = 1, let e_{r} = e_{l} be the edge obtained by gluing together the sides corresponding to γ_{+} and γ_{−}. We orient e_{r} and e_{l} so that pairs of vectors (hol_{ω_{0}}(e_{r}), hol_{ω}(γ)) and (hol_{ω_{0}}(e_{l}), hol_{ω}(γ)) induce the standard orientation in ℝ^{2}. Let τ̃_{0} ∈ T (p_{0}, n_{0}) be the affine equivalence class of τ_{0}. τ̃_{0} is determined by τ̃ and γ̃ up to the Mod(p_{0}, n_{0}) action on T (p_{0}, n_{0}). Let T̃_{0}, …, T̃_{k} be triangles of τ̃ corresponding to T_{0}, …, T_{k} and γ̃_{1}, …, γ̃_{k} be edges of τ̃ corresponding to γ_{1}, …, γ_{k}. Note that T̃_{0}, …, T̃_{k} and γ̃, γ̃_{1}, …, γ̃_{k} are defined up to the H_{0}(p, n) action on M_{p}. Once we fix τ̃_{0} and specify its oriented edges ẽ_{r}, ẽ_{l} corresponding to e_{r}, e_{l}, the isotopy class ω̃_{0} of ω_{0} is uniquely determined by ω̃, τ̃, γ̃, T̃_{0}, …, T̃_{k}, γ̃_{1}, …, γ̃_{k}. Thus we obtain a map COL[τ̃, γ̃; T̃_{0}, …, T̃_{k}; γ̃_{1}, …, γ̃_{k}; τ̃_{0}, ẽ_{r}, ẽ_{l}] : U → N(τ̃_{0}), where U ⊂ N(τ̃). This map is affine, namely, there exists a linear mapping f : H^{1}(M_{p}, Z_{n}; ℝ^{2}) → H^{1}(M_{p_{0}}, Z_{n_{0}}; ℝ^{2}) such that dev(COL[τ̃, γ̃; T̃_{0}, …, T̃_{k}; γ̃_{1}, …, γ̃_{k}; τ̃_{0}, ẽ_{r}, ẽ_{l}](ω̃)) = f(dev(ω̃)) for any ω̃ ∈ U. It follows from the construction that U is the set of ω̃ ∈ N(τ̃) such that f(dev(ω̃)) ∈ dev(N(τ̃_{0})) (cf. the proof of Lemma 6.1).

The inserting operation is inverse to collapsing. Let X_{0} be a translation surface and τ_{0} be a triangulation of X_{0} by disjoint saddle connections. Let e_{r}, e_{l} be oriented edges of τ_{0} and v_{r}, v_{l} be their holonomy vectors. Finally, let v be a nonzero vector in ℝ^{2} such that pairs of vectors (v_{r}, v) and (v_{l}, v) induce the standard orientation in ℝ^{2}. The inserting operation results in a translation surface X that has a saddle connection with holonomy vector v. In general, the operation is defined if v is short enough (precise conditions will be given later). First consider the case when e_{l} = e_{r}.

Let T_{+} and T_{−} be triangles of τ_{0} bounded by e_{r}. We assume that the orientation of e_{r} agrees with the counterclockwise orientation of the boundary of T_{+}. Decompose X_{0} into disjoint plane triangles according to the partition τ_{0}. Let T′_{+} and T′_{−} be the plane triangles corresponding to T_{+} and T_{−}. Let γ′_{+} and γ′_{−} denote the sides of T′_{+} and T′_{−}, respectively, corresponding to e_{r}. Add a parallelogram B_{1}B_{2}B_{3}B_{4} such that B_{1} − B_{2} = B_{4} − B_{3} = v, B_{3} − B_{2} = B_{4} − B_{1} = v_{r}. Now glue all triangles and the parallelogram together as follows. The side B_{2}B_{1} of the parallelogram is glued to its side B_{3}B_{4}. The sides B_{1}B_{4} and B_{2}B_{3} are glued to γ′_{+} and γ′_{−}, respectively. All the other sides are glued as before. We obtain a translation surface X.

Now consider the case e_{l} ≠ e_{r}. Choose a sequence T_{1}, …, T_{k} of triangles of τ_{0} such that e_{r} is an edge of T_{1}, e_{l} is an edge of T_{k}, any two neighboring triangles have common edge different from e_{r} and e_{l}, and the length of the sequence is the least possible. By γ_{i} (1 ≤ i ≤ k − 1) denote a common edge of triangles T_{i} and T_{i+1} different from e_{r} and e_{l}. Also, let γ_{0} = e_{r} and γ_{k} = e_{l}. Let T_{+} be the triangle of τ_{0} bounded by e_{r} and distinct from T_{1}. Let T_{−} be the triangle bounded by e_{l} and distinct from T_{k}. Since the sequence T_{1}, …, T_{k} is as short as possible, it does not contain T_{+} and T_{−}. Also, triangles T_{1}, …, T_{k} are all distinct and so are saddle connections γ_{1}, …, γ_{k−1}. Decompose X_{0} into disjoint plane triangles according to the partition τ_{0}. For any i, 1 ≤ i ≤ k, let T′_{i} be the plane triangle corresponding to T_{i}. By γ′_{i−1} and γ′′_{i} denote the sides of T′_{i} corresponding to edges γ_{i−1} and γ_{i}, respectively. By A_{i} denote the common endpoint of γ′_{i−1} and γ′′_{i}. Further, let T′_{+} and T′_{−} be the plane triangles corresponding to T_{+} and T_{−}. Let γ′_{+} denote the side of T′_{+} corresponding to e_{r} and γ′_{−} denote the side of T′_{−} corresponding to e_{l}. We add two more plane triangles T′_{r} and T′_{l} with the following properties. First of all, T′_{r} and T′_{l} have oriented sides γ′_{r} and γ′_{l} equal to v as vectors. Moreover, T′_{r} lies to the right of γ′_{r} while T′_{l} lies to the left of γ′_{l}. Further, T′_{r} has a side e′_{+} that can be glued to the side γ′_{+} of T′_{+} while T′_{l} has a side e′_{−} that can be glued to the side γ′_{−} of T′_{−}. By e′_{r} denote the side of T′_{r} different from γ′_{r} and e′_{+}. By e′_{l} denote the side of T′_{l} different from γ′_{l} and e′_{−}. Now we are going to glue γ′_{r} to γ′_{l}, e′_{+} to γ′_{+}, e′_{−} to γ′_{−}, e′_{r} to γ′_{0}, e′_{l} to γ′′_{k}, while the other sides of the triangles will be glued as before. At this point the sides e′_{r} and γ′_{0} cannot be glued by translation but we can move the vertex A_{1} of the triangle T′_{1} by v or −v so that the gluing is possible. However this changes the side γ′′_{1} and it cannot be glued to γ′_{1} anymore. To fix this problem, we move the vertex A_{2} of T′_{2} by v or −v, and so on. Finally each A_{i} is moved by v or −v, where the choice of a vector is uniquely determined by the configuration of triangles T_{1}, …, T_{k} and saddle connections γ_{0}, γ_{1}, …, γ_{k}. It is assumed that all plane triangles remain disjoint after modifications. The inserting operation is defined if none of the triangles T′_{1}, …, T′_{k} degenerates while being modified. Then we can glue sides in each pair by translation and obtain a translation surface X. Indeed, this follows from the above for all pairs except e′_{l} and γ′′_{k}. The sides e′_{l} and γ′′_{k} can be glued by translation since the other pairs can.

Now suppose the translation structure ω_{0} of X_{0} belongs to some Ω(p_{0}, n_{0}). Let τ̃_{0} ∈ T (p_{0}, n_{0}) be the affine equivalence class of τ_{0} and ω̃_{0} ∈ N(τ̃_{0}) be the isotopy class of ω_{0}. Let ẽ_{r} and ẽ_{l} be oriented edges of τ̃_{0} corresponding to e_{r} and e_{l}. Then the translation structure of X is isomorphic to some ω ∈ Ω(p, n), where p and n depend on τ̃_{0}, ẽ_{l}, and ẽ_{r}. In both cases, e_{l} = e_{r} and e_{l} ≠ e_{r}, the surface X is equipped with a partition τ by disjoint saddle connections. Moreover, τ has a distinguished oriented edge γ with holonomy vector v. If e_{l} ≠ e_{r} then τ is a triangulation in a class τ̃ ∈ T (p, n) that is determined by τ̃_{0}, ẽ_{l}, and ẽ_{r} up to the Mod(p, n) action. If e_{l} = e_{r} then one of cells of τ is a quadrilateral. We can make τ into a triangulation by adding a diagonal of the quadrilateral. Two triangulations obtained this way are not affine equivalent but they correspond to the same element of MT (p, n). Hence the affine equivalence class τ̃ of any of them is determined up to the action of Mod(p, n). Let us fix τ̃ and specify its oriented edge γ̃ corresponding to γ. In the case e_{l} = e_{r} the isotopy class ω̃ ∈ N(τ̃) of ω is then uniquely determined by v, ω̃_{0}, τ̃_{0}, and ẽ_{r} = ẽ_{l}. In the case e_{l} ≠ e_{r}, let T̃_{1}, …, T̃_{k} be triangles of τ̃_{0} corresponding to T_{1}, …, T_{k} and γ̃_{1}, …, γ̃_{k−1} be edges of τ̃_{0} corresponding to γ_{1}, …, γ_{k−1}. Then ω̃ is uniquely determined by v, ω̃_{0}, τ̃_{0}, ẽ_{r}, ẽ_{l}, T̃_{1}, …, T̃_{k}, γ̃_{1}, …, γ̃_{k−1}. In any case we get a map INS[τ̃_{0}, ẽ_{r}, ẽ_{l}; T̃_{1}, …, T̃_{k}; γ̃_{1}, …, γ̃_{k−1}; τ̃, γ̃] : U_{0} → N(τ̃) defined on a set U_{0} ⊂ ℝ^{2}×N(τ̃_{0}). The map INS[D] (here D stands for the set of parameters of the operation) is affine, namely, there exists a linear mapping f_{0} : ℝ^{2}×H^{1}(M_{p_{0}}, Z_{n_{0}}; ℝ^{2}) → H^{1}(M_{p}, Z_{n}; ℝ^{2}) such that dev(INS[D](v, ω̃_{0})) = f_{0}(v, dev(ω̃_{0})) for any (v, ω̃_{0}) ∈ U_{0}. It follows from the construction that U_{0} is the set of (v, ω̃_{0}) ∈ ℝ^{2} × N(τ̃_{0}) such that f_{0}(v, dev(ω̃_{0})) ∈ dev(N(τ̃)).

Now we define more complex operation on translation surfaces. Suppose X_{0} is a translation torus with n singular points such that the translation structure ω_{0} of X_{0} belongs to Ω(1, n) and the isotopy class ω̃_{0} of ω_{0} belongs to P^{(n)} ⊂ N(τ^{(n)}). Further, let X_{1}, …, X_{k} be translation surfaces. The triangulation in the class τ^{(n)} of X_{0} has distinguished oriented edge L_{1} in the homotopy class L^{(n)}. By v denote the holonomy vector of L_{1}. First the cutting operation is applied to the surface X_{0} and the edge L_{1}. We yield an ordered sequence of n translation tori X_{01}, …, X_{0n}. Further for any X_{j}, 1 ≤ j ≤ k, we insert a saddle connection with holonomy vector v by applying an inserting operation. Assuming the inserting operation is well defined, we get a translation surface X′_{j}. Suppose l_{1}, …, l_{k} are integers such that 0 ≤ l_{1} ≤ … ≤ l_{k} ≤ n. We merge two sequences X_{01}, …, X_{0n} and X′_{1}, …, X′_{k} together in such a way that X_{0i} appears before X′_{j} if and only if i ≤ l_{j}. Each surface in the new sequence has a distinguished saddle connection with holonomy vector v. Finally we apply the gluing operation and obtain a translation surface X.

Now suppose the translation structure ω_{j} of every X_{j}, 1 ≤ j ≤ k, belongs to some Ω(p_{j}, n_{j}). Let ω, ω_{01}, …, ω_{0n}, ω′_{1}, …, ω′_{k} denote the translation structures of X, X_{01}, …, X_{0n}, X′_{1}, …, X′_{k}, respectively. It is no loss to assume that each of these translation structures also belongs to some Ω(p′, n′). Let the tilde denote the isotopy class of a translation structure. Then (ω̃_{01}, …, ω̃_{0n}) = CUT [τ^{(n)}, L^{(n)}; D_{0}](ω̃_{0}) for a set D_{0} of parameters. Further, ω̃′_{j} = INS[τ_{j}, D_{j}; τ′_{j}, γ′_{j}](v, ω̃_{j}), where τ_{j} ∈ T (p_{j}, n_{j}) and D_{j} is a set of parameters. Finally,

ω̃ = GLU[D′; τ, γ](ω̃_{01}, …, ω̃_{0l_{1}}, ω̃′_{1}, …, ω̃′_{k}, ω̃_{0,l_{k}+1}, …, ω̃_{0n})

for another set D′ of parameters. Thus we yield a map

OP R[n; τ_{1}, D_{1}, …, τ_{k}, D_{k}; l_{1}, …, l_{k}; τ, γ] : U^{∗} → N(τ)

defined on an open set U^{∗} ⊂ P^{(n)} × N(τ_{1}) × … × N(τ_{k}). We shall use the notation OP R[n; D], that is, D stands for all the other parameters. Suppose ω̃_{0} ∈ P^{(n)} and ω̃_{j} ∈ N(τ_{j}), j = 1, …, k. Then (ω̃_{0}, ω̃_{1}, …, ω̃_{k}) ∈ U^{∗} if and only if INS[τ_{j}, D_{j}; τ′_{j}, γ′_{j}](holω̃_{0}(L^{(n)}), ω̃_{j}) is well defined for 1 ≤ j ≤ k.

Lemma 8.3 The map OP R[n; D] is affine, injective, and volume preserving.

Proof. Suppose ω = OP R[n; D](ω_{0}, ω_{1}, …, ω_{k}). By construction every triangle of triangulations τ^{(n)}, τ_{1}, …, τ_{k} is assigned a triangle of τ while every edge is assigned one or two edges of τ. Let γ be an oriented edge of τ^{(n)} or τ_{j}, 1 ≤ j ≤ k, and γ′ be a corresponding edge of τ. If γ is an edge of τ^{(n)} then hol_{ω}(γ′) = hol_{ω_{0}}(γ). If γ is an edge of τ_{j} then hol_{ω}(γ′) = hol_{ω_{j}}(γ) + c_{γ} hol_{ω_{0}}(L^{(n)}), where c_{γ} ∈ {−1, 0, 1} depends only on γ.

Let γ_{01} = L^{(n)}, γ_{02}, …, γ_{0s_{0}} be a maximal set of edges of τ^{(n)} that do not divide the surface. For any j, 1 ≤ j ≤ k, let γ_{j1}, …, γ_{js_{j}} be a maximal set of edges of τ_{j} that do not divide the surface. Let γ′_{ji} denote an edge of τ assigned to γ_{ji}. It is easy to observe that the edges γ′_{ji}, 1 ≤ i ≤ s_{j}, 0 ≤ j ≤ k, are all distinct and comprise a maximal set of edges of τ that do not divide the surface. Let K = s_{0} + s_{1} + ⋯ + s_{k}. Define maps f_{1} : N(τ^{(n)}) × N(τ_{1}) × … × N(τ_{k}) → (ℝ^{2})^{K} and f_{2} : N(τ) → (ℝ^{2})^{K} by

f_{1}(ω_{0}, ω_{1}, …, ω_{k}) = (hol_{ω_{0}}(γ_{01}), …, hol_{ω_{0}}(γ_{0s_{0}}), hol_{ω_{1}}(γ_{11}), …, hol_{ω_{k}}(γ_{ks_{k}})),
f_{2}(ω) = (hol_{ω}(γ′_{01}), …, hol_{ω}(γ′_{0s_{0}}), hol_{ω}(γ′_{11}), …, hol_{ω}(γ′_{ks_{k}})).

Both maps are affine. By Lemmas 6.1 and 6.2, f_{1} and f_{2} are injective and volume preserving. By the above there exists a linear map F : (ℝ^{2})^{K} → (ℝ^{2})^{K} such that f_{2}(OP R[n; D](ω^{∗})) = F (f_{1}(ω^{∗})) for any ω^{∗} ∈ U^{∗}. Given v_{1}, …, v_{K} ∈ ℝ^{2}, one has F (v_{1}, v_{2}, …, v_{K}) = (v_{1}, v_{2} + c_{2}v_{1}, …, v_{K} + c_{K}v_{1}), where c_{2}, …, c_{K} ∈ {−1, 0, 1} are constants. It follows that F is volume preserving. Then OP R[n; D] is a volume preserving affine map. It is injective since f_{1} and F are injective.

Suppose U_{0} ⊂ ℝ^{2} and U_{j} ⊂ N(τ_{j}) ∩ Q_{1}(p_{j}, n_{j}), j = 1, …, k. Consider the set P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) consisting of ω ∈ N(τ) such that a(ω) ≤ 1 and ω = OP R[n; D](ω_{0}, t_{1}ω_{1}, …, t_{k}ω_{k}), where ω_{0} ∈ P^{(n)}(U_{0}), ω_{j} ∈ U_{j} and t_{j} > 0 for 1 ≤ j ≤ k. Given i ∈ {1, …, n} and σ ∈ [0, 1), we let ω ∈ P_{σ,i}^{(n)}[D](U_{0}; U_{1}, …, U_{k}) if, in addition, a_{i}(ω_{0}) > σa(ω).

Lemma 8.4 Let U_{0} ⊂ ℝ^{2} \ {(0, 0)} and U_{j} ⊂ N(τ_{j}) ∩ Q_{1}(p_{j}, n_{j}), 1 ≤ j ≤ k, be nonempty open subsets. Assume U_{0} is bounded and the closure of each U_{j}, 1 ≤ j ≤ k, is a compact subset of N(τ_{j}). Further assume that for any v ∈ U_{0} and ω_{j} ∈ U_{j} (1 ≤ j ≤ k) the j^{th} inserting operation is defined on (tv, ω_{j}) when t > 0 is small enough. Then (a)

lim_{ϵ→0} ϵ^{−2}μ(P^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k})) = μ^{∗}(P^{(n)}(U_{0}; U_{1}, …, U_{k})) > 0,

where μ is the canonical measure on N(τ) and μ^{∗} denotes the product of the canonical measures on Q(1, n), Q(p_{1}, n_{1}), …, Q(p_{k}, n_{k});

(b) for any σ ∈ [0, 1) and i ∈ {1, …, n},

lim_{ϵ→0} \frac{μ(P_{σ,i}^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k}))}{μ(P^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k}))} = (1 − σ)^{K−2},

where K is half of the dimension of N(τ);

(c) if, in addition, U_{0}, U_{1}, …, U_{k} are connected sets and tU_{0} ⊂ U_{0} for 0 < t < 1, then the set P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) is connected.

Proof. Consider an inserting operation f_{0} = INS[τ_{0}, e_{r}, e_{l}; D_{0}; τ′, γ′]. Here D_{0} is empty if e_{r} = e_{l}; otherwise D_{0} stands for a sequence T_{1}, …, T_{s} of triangles and a sequence γ_{1}, …, γ_{s−1} of edges of τ_{0}. The operation is defined on a set U_{0}^{∗} ⊂ ℝ^{2} × N(τ_{0}). Suppose v ∈ ℝ^{2} and ω ∈ N(τ_{0}). Let Λ denote the euclidean area form on ℝ^{2}. If (v, ω) ∈ U_{0}^{∗}, then Λ(hol_{ω}(e_{r}), v) > 0 and Λ(hol_{ω}(e_{l}), v) > 0. In the case e_{r} = e_{l}, these conditions determine the set U_{0}^{∗}. In the case e_{r} ≠ e_{l}, there are also s nondegeneracy conditions. They can be expressed in the form Λ(hol_{ω}(L_{1i}) + v, hol_{ω}(L_{2i})) > 0, i = 1, …, s, where L_{1i}, L_{2i} are certain oriented edges of τ_{0} such that Λ(hol_{ω}(L_{1i}), hol_{ω}(L_{2i})) > 0 for any ω ∈ N(τ_{0}). Namely, L_{1i} and L_{2i} are edges of the triangle T_{i} such that L_{1i} is in the sequence e_{r}, γ_{1}, …, γ_{s−1}, e_{l} while L_{2i} is not. It follows that U_{0}^{∗} is an open set. Furthermore, if (v, ω) ∈ U_{0}^{∗} then (tv, tω) ∈ U_{0}^{∗} for any t > 0 and (tv, ω) ∈ U_{0}^{∗} for 0 < t < 1. If Λ(hol_{ω}(e_{r}), v) > 0 and Λ(hol_{ω}(e_{l}), v) > 0 then (tv, ω) ∈ U_{0}^{∗} provided t > 0 is small enough. Each triangle T of τ_{0} corresponds to a triangle T′ of τ′. Suppose (v, ω) ∈ U_{0}^{∗} and ω′ = f_{0}(v, ω). If T is not in the sequence T_{1}, …, T_{s} then the area of T with respect to ω equals the area of T′ with respect to ω′. Otherwise the two areas may differ but the difference is at most C_{ω}|v|/2, where C_{ω} is the maximum length of edges of τ_{0} with respect to ω. Besides, there are two triangles of τ′ bounded by γ′ that are not associated to triangles of τ_{0}. Their areas with respect to ω′ do not exceed C_{ω}|v|/2. It follows that |a(ω′) − a(ω)| ≤ m_{0}C_{ω}|v|, where m_{0} is half of the number of triangles of τ′.

Now consider the map f = OP R[n; D] defined on the set U^{∗} ⊂ P^{(n)} × N(τ_{1}) × … × N(τ_{k}). Suppose (ω_{0}, ω_{1}, …, ω_{k}) ∈ U^{∗}. By the above (tω_{0}, tω_{1}, …, tω_{k}) ∈ U^{∗} for any t > 0, (tω_{0}, ω_{1}, …, ω_{k}) ∈ U^{∗} for 0 < t < 1, and (ω_{0}, t_{1}ω_{1}, …, t_{k}ω_{k}) ∈ U^{∗} for t_{1}, …, t_{k} ≥ 1. The set U_{0} is bounded, i.e., it is contained in the disk B(C_{0}) for some C_{0} > 0. Since the closure of each U_{j}, 1 ≤ j ≤ k, is a compact subset of N(τ_{j}), U_{j} has finite volume in Q_{1}(p_{j}, n_{j}). Also, there exist C, c, δ > 0 such that for any triangulation in the class τ_{j} of a translation surface in an isotopy class ω_{j} ∈ U_{j} all edges are of length at most C and at least c while all angles of each triangle are not less than δ. Since for any v ∈ U_{0} and ω_{j} ∈ U_{j} the j^{th} inserting operation is defined on (tv, ω_{j}) when t > 0 is small enough, there exists a constant ϵ_{0} > 0 depending on C_{0}, c, δ such that the j^{th} inserting operation is defined on (ϵ_{0}v, ω_{j}). It follows that P^{(n)}(ϵ_{0}U_{0}) × U_{1} × … × U_{k} ⊂ U^{∗}.

For any (ω_{0}, ω_{1}, …, ω_{k}) ∈ P^{(n)} × N(τ_{1}) × … × N(τ_{k}) let a^{∗}(ω_{0}, ω_{1}, …, ω_{k}) = a(ω_{0}) + a(ω_{1}) + ⋯ + a(ω_{k}). Suppose ω = f(ω^{∗}) for some ω^{∗} = (ω_{0}, t_{1}ω_{1}, …, t_{k}ω_{k}), where ω_{0} ∈ P^{(n)}(ϵU_{0}), ϵ > 0, ω_{j} ∈ U_{j} and t_{j} > 0 for 1 ≤ j ≤ k. By the above |a^{∗}(ω^{∗}) − a(ω)| ≤ mCt_{max}| hol_{ω_{0}}(L^{(n)})|, where t_{max} = max(t_{1}, …, t_{k}) and m is half of the number of triangles of the triangulation τ. Moreover, | hol_{ω_{0}}(L^{(n)})| ≤ C_{0}ϵ. If a^{∗}(ω^{∗}) ≤ 1 then t_{max} ≤ 1. If a(ω) ≤ 1 then |a^{∗}(ω^{∗}) − a(ω)| > t_{max}^{2} − 1; it follows that t_{max} < 2 whenever ϵ ≤ 3(2mC_{0}C)^{−1}. Hence there exist C_{1}, ϵ_{1} > 0 such that |a^{∗}(ω^{∗}) − a(ω)| ≤ C_{1}ϵ if a^{∗}(ω^{∗}) ≤ 1 or a(ω) ≤ 1 and ϵ ≤ ϵ_{1}.

For any ϵ, α > 0 let Y (ϵ, α) be the set of (ω_{0}, …, ω_{k}) ∈ P^{(n)} ×ℝ^{+}U_{1} ×…×ℝ^{+}U_{k} such that hol_{ω_{0}}(L^{(n)}) ∈ ϵU_{0} and a^{∗}(ω_{0}, …, ω_{k}) ≤ α. Observe that tY (ϵ, α) = Y (tϵ, t^{2}α) for any t > 0, where by definition t(ω_{0}, …, ω_{k}) = (tω_{0}, …, tω_{k}). Clearly, Y (ϵ, 1) = P^{(n)}(ϵU_{0}; U_{1}, …, U_{k}). It was shown in the proof of Lemma 8.2 that μ^{∗}(P^{(n)}(ϵU_{0}; U_{1}, …, U_{k})) = 𝔪(ϵU_{0})c^{∗}, where c^{∗} > 0 depends on U_{1}, …, U_{k}, and n. Hence μ^{∗}(Y (ϵ, α)) = α^{K}μ^{∗}(Y (α^{−1/2}ϵ, 1)) = α^{K−1}ϵ^{2}𝔪(U_{0})c^{∗}.

For any ϵ > 0 let Y_{0}(ϵ) = f^{−1}(P^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k})) ∩ P^{(n)}(ϵU_{0}; U_{1}, …, U_{k}), Y_{1}(ϵ) = f^{−1}(P^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k})) \ Y_{0}(ϵ), Y_{2}(ϵ) = P^{(n)}(ϵU_{0}; U_{1}, …, U_{k}) \ Y_{0}(ϵ), Y_{3}(ϵ) = Y_{2}(ϵ)∩U^{∗}, Y_{4}(ϵ) = Y_{2}(ϵ)\U^{∗}. If ω^{∗} ∈ Y_{1}(ϵ) then a^{∗}(ω^{∗}) > 1 and a(f(ω^{∗})) ≤ 1. If ω^{∗} ∈ Y_{3}(ϵ) then a(f(ω^{∗})) > 1 and a^{∗}(ω^{∗}) ≤ 1. In both cases, |a^{∗}(ω^{∗}) − 1| ≤ C_{1}ϵ provided ϵ ≤ ϵ_{1}. Therefore Y_{1}(ϵ)∪Y_{3}(ϵ) is a subset of Y (ϵ, 1+C_{1}ϵ) while it is disjoint from Y (ϵ, α) for α < 1 − C_{1}ϵ. It follows that μ^{∗}(Y_{1}(ϵ) ∪ Y_{3}(ϵ)) ≤ ((1 + C_{1}ϵ)^{K−1} − (1 − C_{1}ϵ)^{K−1})ϵ^{2}𝔪(U_{0})c^{∗} if C_{1}ϵ < 1. Hence ϵ^{−2}μ^{∗}(Y_{1}(ϵ) ∪ Y_{3}(ϵ)) → 0 as ϵ → 0.

It is easy to see that P^{(n)}(ϵU_{0}; U_{1}, …, U_{k}) ⊂ P^{(n)}(ϵU_{0}) × Ũ_{1} × … × Ũ_{k}, where Ũ_{j} = {tω | ω ∈ U_{j}, 0 < t ≤ 1}. Let μ_{0}, μ_{1}, …, μ_{k} be the canonical measures on Q(1, n), Q(p_{1}, n_{1}), …, Q(p_{k}, n_{k}), respectively. Then μ_{0}(P^{(n)}(ϵU_{0})) = 𝔪(ϵU_{0})c_{n} = ϵ^{2}𝔪(U_{0})c_{n}, where c_{n} > 0 is a constant (see the proof of Lemma 8.2). Besides, μ_{j}(Ũ_{j}) < ∞. Suppose (ω_{0}, …, ω_{k}) ∈ Y_{4}(ϵ). Then (ω_{0}, …, ω_{k}) ∉ U^{∗}, hence ω_{j} ∈ ϵϵ_{0}^{−1}Ũ_{j} for some 1 ≤ j ≤ k. Since μ_{j}(ϵϵ_{0}^{−1}Ũ_{j}) = (ϵ/ϵ_{0})^{2(2p_{j}+n_{j}−1)}μ_{j}(Ũ_{j}), it follows that ϵ^{−2}μ^{∗}(Y_{4}(ϵ)) → 0 as ϵ → 0. Thus we have proved that ϵ^{−2}μ^{∗}(Y_{1}(ϵ) ∪ Y_{2}(ϵ)) → 0 as ϵ → 0. Then lim_{ϵ→0} ϵ^{−2}μ^{∗}(Y_{0}(ϵ)) = 𝔪(U_{0})c^{∗} > 0. Since f is injective and volume preserving map, statement (a) of the lemma follows.

Suppose ω^{∗} ∈ Y_{0}(ϵ)\Y (ϵ, C_{1}ϵ^{1/2}), where ϵ ≤ ϵ_{1}. Then |a(f(ω^{∗}))−a^{∗}(ω^{∗})| ≤ C_{1}ϵ ≤ ϵ^{1/2}a^{∗}(ω^{∗}). If f(ω^{∗}) ∈ P_{σ,i}^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k}) for some σ > 0 and i, then ω^{∗} ∈ P_{(1−ϵ^{1/2})σ,i}^{(n)}(ϵU_{0}; U_{1}, …, U_{k}) assuming ϵ < 1, else ω^{∗} ∉ P_{(1+ϵ^{1/2})σ,i}^{(n)}(ϵU_{0}; U_{1}, …, U_{k}) assuming (1 + ϵ^{1/2})σ < 1. Therefore

μ^{∗}(P_{(1+ϵ^{1/2})σ,i}^{(n)}(ϵU_{0}; U_{1}, …, U_{k})) − μ^{∗}(Y (ϵ, C_{1}ϵ^{1/2}) ∪ Y_{2}(ϵ)) ≤
μ(P_{σ,i}^{(n)}[D](ϵU_{0}; U_{1}, …, U_{k})) ≤
μ^{∗}(P_{(1−ϵ^{1/2})σ,i}^{(n)}(ϵU_{0}; U_{1}, …, U_{k})) + μ^{∗}(Y (ϵ, C_{1}ϵ^{1/2}) ∪ Y_{1}(ϵ)).

Since ϵ^{−2}μ^{∗}(Y (ϵ, C_{1}ϵ^{1/2}) ∪ Y_{1}(ϵ) ∪ Y_{2}(ϵ)) → 0 as ϵ → 0, statement (b) of the lemma follows from its statement (a) and Lemma 8.2.

Now assume U_{0}, U_{1}, …, U_{k} are connected sets. It is easy to observe that the set P^{(n)}({v}) is connected for any nonzero v ∈ ℝ^{2}. Let g_{v} be a unique element of SL(2, ℝ) such that g_{v}v = (0, 1) and g_{v}v_{0} = (1, 0) for some v_{0} orthogonal to v. Then g_{v}P^{(n)}({v}) = P^{(n)}({(0, 1)}). Since g_{v} depends continuously on v, it follows that P^{(n)}(U_{0}) is connected whenever U_{0} ⊂ ℝ^{2} \ {(0, 0)} is connected. Suppose ω′, ω′′ ∈ P^{(n)}[D](U_{0}; U_{1}, …, U_{k}). Then f^{−1}(ω′) = (ω′_{0}, t′_{1}ω′_{1}, …, t′_{k}ω′_{k}) and f^{−1}(ω′′) = (ω′′_{0}, t′′_{1}ω′′_{1}, …, t′′_{k}ω′′_{k}), where ω′_{0}, ω′′_{0} ∈ P^{(n)}(U_{0}), ω′_{j}, ω′′_{j} ∈ U_{j} and t′_{j}, t′′_{j} > 0 for 1 ≤ j ≤ k. There exist continuous paths ω_{0} : [0, 1] → P^{(n)}(U_{0}) and ω_{j} : [0, 1] → U_{j}, j = 1, …, k, such that ω_{j}(0) = ω′_{j} and ω_{j}(1) = ω′′_{j} for 0 ≤ j ≤ k. Pick t_{0} > 0 such that t′_{j}, t′′_{j} ≥ t_{0} and ϵ_{0}t_{0} ≤ 1. For any u ∈ [0, 1] let

ω^{∗}(u) = (ϵ_{0}t_{0}ω_{0}(u), ((1 − u)t′_{1} + ut′′_{1})ω_{1}(u), …, ((1 − u)t′_{k} + ut′′_{k})ω_{k}(u)).

Then ω^{∗}(u) ∈ U^{∗} and depends continuously on u. In particular, u ↦ a(f(ω^{∗}(u))) is a continuous function on [0, 1], hence it is bounded. Note that a(f(tω^{∗}(u))) = a(tf(ω^{∗}(u))) = t^{2}a(f(ω^{∗}(u))) for any t > 0. Therefore a(f(t_{1}ω^{∗}(u))) ≤ 1 for some 0 < t_{1} < 1 and all u. Assume tU_{0} ⊂ U_{0} for 0 < t < 1. Then f(t_{1}ω^{∗}(u)) ∈ P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) for all u ∈ [0, 1], hence f(t_{1}ω^{∗}(0)) is joined to f(t_{1}ω^{∗}(1)) by a continuous path in P^{(n)}[D](U_{0}; U_{1}, …, U_{k}). Furthermore, P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) contains f(tω^{∗}(0)) for t_{1} ≤ t ≤ 1 and f(ϵω′_{0}, t′_{1}ω′_{1}, …, t′_{k}ω′_{k}) for ϵ_{0}t_{0} ≤ ϵ ≤ 1. Therefore ω′ can be joined to f(t_{1}ω^{∗}(0)). Similarly, ω′′ can be joined to f(t_{1}ω^{∗}(1)). Finally, ω′ can be joined to ω′′ within P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) by a continuous path. Thus P^{(n)}[D](U_{0}; U_{1}, …, U_{k}) is connected.

For any ϵ > 0 let Ω^{ϵ}(p, n) be the set of translation structures in Ω(p, n) admitting a saddle connection of length at most ϵ. Further, for any κ ≥ ϵ let Ω^{ϵ,κ}(p, n) be the set of translation structures admitting a saddle connection of length at most ϵ and no nonhomologous saddle connections of length at most κ. Clearly, Ω^{ϵ,κ}(p, n) ⊂ Ω^{ϵ}(p, n) and both sets are invariant under the H(p, n) action. Let Q^{ϵ}(p, n) and Q^{ϵ,κ}(p, n) denote the subsets of Q(p, n) corresponding to Ω^{ϵ}(p, n) and Ω^{ϵ,κ}(p, n). Let MQ^{ϵ}(p, n) and MQ^{ϵ,κ}(p, n) denote the corresponding subsets of MQ(p, n). By Q_{≤1}(p, n) denote the set of ω ∈ Q(p, n) such that a(ω) ≤ 1. The set MQ_{≤1}(p, n) ⊂ MQ(p, n) is defined in a similar way. Now we let Q_{≤1}^{ϵ}(p, n) = Q^{ϵ}(p, n) ∩ Q_{≤1}(p, n), Q_{≤1}^{ϵ,κ}(p, n) = Q^{ϵ,κ}(p, n) ∩ Q_{≤1}(p, n), MQ_{≤1}^{ϵ}(p, n) = MQ^{ϵ}(p, n) ∩ MQ_{≤1}(p, n), MQ_{≤1}^{ϵ,κ}(p, n) = MQ^{ϵ,κ}(p, n)∩MQ_{≤1}(p, n). Similarly, we define sets Q_{1}^{ϵ}(p, n), Q_{1}^{ϵ,κ}(p, n), MQ_{1}^{ϵ}(p, n), and MQ_{1}^{ϵ,κ}(p, n).

Theorem 8.5 ([MS]) Let μ_{0} and μ denote the canonical measures on MQ_{1}(p, n) and MQ(p, n), respectively. There exists c_{p,n} > 0 such that μ_{0}(MQ_{1}^{ϵ}(p, n)) ≤ c_{p,n}ϵ^{2} and μ(MQ_{≤1}^{ϵ}(p, n)) ≤ c_{p,n}ϵ^{2} for any ϵ > 0, and μ_{0}(MQ_{1}^{ϵ}(p, n) \ MQ_{1}^{ϵ,κ}(p, n)) ≤ c_{p,n}ϵ^{2}κ^{2} and μ(MQ_{≤1}^{ϵ}(p, n) \ MQ_{≤1}^{ϵ,κ}(p, n)) ≤ c_{p,n}ϵ^{2}κ^{2} for any κ ≥ ϵ.

For any Delaunay triangulation piece M(τ, ≺) ⊂ Q(p, n) we define a canonical ordered basis Γ = (γ_{1}, …, γ_{2p+n−1}) for the group H_{1}(M_{p}, Z_{n}; ℤ). Namely, for any j, 1 ≤ j ≤ 2p + n − 1, γ_{j} is the least (with respect to ≺) homology class of an edge of τ that is not a linear combination of γ_{1}, …, γ_{j−1}. The homology classes γ_{1}, …, γ_{2p+n−1} are determined up to multiplying by ±1. By X_{1} denote the set of vectors v ∈ ℝ^{2} such that hol_{ω}(γ_{1}) = v for some ω ∈ M(τ, ≺) ∩ Q_{≤1}(p, n). Given vectors v_{1}, …, v_{k−1} ∈ ℝ^{2}, 1 < k ≤ 2p + n − 1, let X_{k}(v_{1}, …, v_{k−1}) denote the set of v ∈ ℝ^{2} such that hol_{ω}(γ_{1}) = v_{1}, …, hol_{ω}(γ_{k−1}) = v_{k−1}, and hol_{ω}(γ_{k}) = v for some ω ∈ M(τ, ≺) ∩ Q_{≤1}(p, n).

Theorem 8.6 ([MS]) X_{1} is contained in the disk B(\sqrt{8/π}). Any X_{k}(v_{1}, …, v_{k−1}) is contained in the union of B(\sqrt{8/π}) and finitely many rectangles of area 1, where the number of rectangles is bounded by a constant depending on M(τ, ≺).

It is easy to see that a shortest saddle connection L_{0} of a translation surface M is a Delaunay edge. If all shortest saddle connections of M are homologous to L_{0}, then any shortest saddle connection of those nonhomologous to L_{0} is also a Delaunay edge. In view of this remark, Lemma 6.5, and Fubini’s theorem, Theorem 8.5 is a corollary of Theorem 8.6. Further notice that MQ_{1}^{ϵ}(p, n) = MQ_{1}(p, n) for ϵ ≥ \sqrt{8/π}. Hence Theorem 6.6 is a corollary of Theorem 8.5. Theorems 6.6, 8.5, and 8.6 were proved in Section 10 of the paper [MS] (although only the first of them was explicitly formulated in [MS]).

Let τ ∈ T (p, n) and e be an edge of τ. Suppose ω ∈ N(τ). Pick a translation structure ω′ in the isotopy class ω and let τ′ be the triangulation by disjoint saddle connections of ω′ such that (ω′, τ′) ∈ τ. Let e′ be the edge of τ′ corresponding to e. By θ_{e,τ}(ω) denote the sum of two angles opposite e′ in two triangles of τ′ bounded by e′. It follows from the proof of Lemma 6.5 that θ_{e,τ}(ω) is well defined and depends continuously on ω. For any δ > 0 let M_{δ}(τ) denote the set of ω ∈ N(τ) such that θ_{e,τ}(ω) < π − δ for every edge e of τ. By Proposition 5.3, M_{δ}(τ) ⊂ M(τ) and M(τ) = ⋃_{δ>0} M_{δ}(τ).

Suppose τ ∈ T (p, n) and γ is an edge of τ. For any ϵ, δ > 0 and κ > ϵ let S(τ, γ; ϵ, κ, δ) denote the set of ω ∈ Q_{≤1}^{ϵ,κ}(p, n) ∩ M(τ) such that | hol_{ω}(γ)| ≤ ϵ and for any edge e of τ we have θ_{e,τ}(ω) < π − δ unless two triangles of τ bounded by e have two common edges while their third edges are homologous to γ. By S(τ; ϵ, κ, δ) denote the union of the sets S(τ, γ; ϵ, κ, δ) over all edges γ of τ.

Lemma 8.7 Given Δ > 0, there exist κ, δ > 0 such that

μ̃(Q_{≤1}^{ϵ}(p, n) ∩ M(τ)) − μ̃(S(τ; ϵ, κ, δ)) ≤ Δϵ^{2}

for all ϵ < κ, where μ̃ denotes the canonical measure on Q(p, n).

Proof. By Lemma 6.5, each Delaunay triangulation halfpiece M^{h}(τ, ≺) projects injectively to MQ(p, n). Up to a set of zero volume, M(τ) is the union of finitely many halfpieces. Hence Theorem 8.5 implies there exists c_{τ} > 0 such that μ̃(Q_{≤1}^{ϵ}(p, n) ∩ M(τ)) − μ̃(Q_{≤1}^{ϵ,κ}(p, n) ∩ M(τ)) ≤ c_{τ}ϵ^{2}κ^{2} for κ ≥ ϵ > 0. Choose κ > 0 such that c_{τ}κ^{2} ≤ Δ/2.

Suppose ω′ is a translation structure in a class ω ∈ M(τ) ∩ Q_{≤1}^{ϵ,κ}(p, n), ϵ < κ, and let γ be an edge of τ such that | hol_{ω}(γ)| ≤ ϵ. Given a Delaunay cell T of ω′, there exist d > 0 and a map ı : B(d) → M_{p} such that ı is a translation with respect to ω′ and T = ı(T′), where T′ is the interior of a plane triangle inscribed in B(d). Note that d is the distance from the point q = ı(0) to the set of singular points of ω′. By Lemma 5.4, if d ≥ \sqrt{2/π} then q belongs to a periodic cylinder C of length at most d^{−1}. It is easy to see that if the triangle T is disjoint from C then all edges of T are not longer than the length of C. As ω ∈ Q^{ϵ,κ}(p, n), the length of C is either at most ϵ or at least κ. In the former case, C is a regular cylinder bounded by saddle connections homologous to γ and T ⊂ C. In the latter case, d ≤ κ^{−1} and all edges of T are of length at most 2κ^{−1}.

Suppose x_{1}, x_{2}, x_{3} ∈ ℝ^{2} are vertices of a triangle. For any δ > 0 let Π(x_{1}, x_{2}, x_{3}; δ) be the set of x ∈ ℝ^{2} such that π − δ < ∠x_{1}xx_{2} + ∠x_{1}x_{3}x_{2} < π and x is separated from x_{3} by the straight line passing through x_{1} and x_{2}. Π(x_{1}, x_{2}, x_{3}; δ) is a domain bounded by two circles intersecting at x_{1} and x_{2}, one of them being circumscribed about the triangle x_{1}x_{2}x_{3}. If we fix the length l of the segment x_{1}x_{2}, the radius r of the circle passing through x_{1}, x_{2}, x_{3}, and δ, then the area of Π(x_{1}, x_{2}, x_{3}; δ) is bound to take at most two values. Let α(l, r, δ) denote either of them. It is easy to see that for any l_{0}, r_{0} > 0 we have sup_{l≥l_{0}} sup_{r≤r_{0}} α(l, r, δ) → 0 as δ → 0.

Let γ and e be edges of τ. If two triangles bounded by e have two common edges while their third edges are homologous to γ, then for any ω ∈ N(τ) the edge e crosses a cylinder of periodic geodesics of ω homologous to γ. Assume this is not the case. For 0 < ϵ < κ and δ > 0 let S′(τ, γ, e; ϵ, κ, δ) denote the set of ω ∈ M(τ)∩Q_{≤1}^{ϵ,κ}(p, n) such that | hol_{ω}(γ)| ≤ ϵ and θ_{e,τ}(ω) > π − δ. If e is homologous to γ then θ_{e,τ}(ω) ≤ 2π/3 for all ω ∈ S′(τ, γ, e; ϵ, κ, δ) since the angle opposite the shortest side of a triangle does not exceed π/3. Hence S′(τ, γ, e; ϵ, κ, δ) is empty for δ ≤ π/3. Now suppose e and γ are not homologous. Let T_{1} and T_{2} be triangles of τ bounded by e. Let e_{1} be a side of T_{1} different from e and e_{2} be a side of T_{2} different from e and not homologous to e_{1}. If an edge of T_{1} or T_{2} is homologous to γ, it is no loss to assume e_{1} is homologous to γ. Suppose ω ∈ S′(τ, γ, e; ϵ, κ, δ). Once hol_{ω}(e) and hol_{ω}(e_{1}) are fixed, the holonomy vector hol_{ω}(e_{2}) belongs to a set Π(x_{1}, x_{2}, x_{3}; δ) of area α(l, r, δ), where l ≥ κ and r ≤ r_{1} = max(\sqrt{2/π}, κ^{−1}). If any edge of T_{1} is homologous to an edge of T_{2}, we can say more (cf. the proof of Lemma 6.5). Namely, once hol_{ω}(e) is fixed, the holonomy vector hol_{ω}(e_{2}) belongs to a set Π(x_{1}, x_{2}, x_{3}; δ/2) of area α(l, l/2, δ/2), where l = | hol_{ω}(e)|. By Lemma 6.2, there is a sequence γ_{1}, …, γ_{2p+n−1} of edges of τ whose homology classes comprise a basis for H_{1}(M_{p}, Z_{n}; ℤ). It can be assumed without loss of generality that this sequence contains an edge homologous to γ, the edges e, e_{2}, and e_{1} unless the latter one is homologous to an edge of T_{2}. By the above | hol_{ω}(γ_{i})| ≤ 2r_{1} unless γ_{i} crosses a cylinder of periodic geodesics homologous to γ. In the latter case we have hol_{ω}(γ_{i}) = t hol_{ω}(γ)+v, where −1 < t < 1, v is orthogonal to hol_{ω}(γ), and |v|·| hol_{ω}(γ)| ≤ 1. It follows that hol_{ω}(γ_{i}) belongs to a rectangle of area 4 depending on hol_{ω}(γ). Since e_{2} is not homologous to γ, Fubini’s theorem implies μ̃(S′(τ, γ, e; ϵ, κ, δ)) ≤ πϵ^{2}(4πr_{1}^{2})^{m−2} sup_{l≥κ} sup_{r≤r_{1}} α(l, r, δ), where m = 3(2p−2+n) is the number of edges of τ. For any δ′ > δ the set M(τ) ∩ Q_{≤1}^{ϵ,κ}(p, n) \ S(τ; ϵ, κ, δ) is contained in the union of at most m^{2} sets of the form S′(τ, γ, e; ϵ, κ, δ′). Therefore we can choose δ so that μ̃(M(τ)∩Q_{≤1}^{ϵ,κ}(p, n))−μ̃(S(τ; ϵ, κ, δ)) ≤ Δϵ^{2}/2 for all ϵ < κ.

Suppose ω is a translation structure. For any σ ∈ [0, 1) and R > 0 let N_{0}(ω, σ, R) denote the number of periodic cylinders of ω of length at most R and of area greater than σa(ω). By s(ω) denote the length of the shortest saddle connection of ω. Further, let N_{0}^{∗}(ω, σ, R) = N_{0}(ω, σ, s(ω)) if R ≥ s(ω) and N_{0}^{∗}(ω, σ, R) = 0 if 0 < R < s(ω). The numbers N_{0}(ω, σ, R), N_{0}^{∗}(ω, σ, R), and s(ω) do not change when ω is replaced by an isomorphic structure, hence they are well defined for ω ∈ Q(p, n) and ω ∈ MQ(p, n).

Proposition 8.8 Let M be a connected component of the moduli space MQ(p, n). By M_{0} denote the set of ω ∈ M such that a(ω) ≤ 1. Then there exists c(M) > 0 such that for any σ ∈ [0, 1),

lim_{ϵ→0} \frac{1}{ϵ^{2}} ∫_{M_{0}} N_{0}^{∗}(ω, σ, ϵ) dμ(ω) = (1 − σ)^{K−2}c(M),

where K is half of the dimension of M and μ denotes the canonical measure on M.

Proof. For any translation structure ω, N_{0}^{∗}(ω, σ, ϵ) is at most the number of saddle connections of length s(ω) of ω. As such saddle connections are Delaunay edges, they are disjoint by Proposition 5.2. Then Proposition 5.1 implies there exists C > 0 such that N_{0}^{∗}(ω, σ, ϵ) ≤ C for all ω ∈ MQ(p, n).

By M′_{0} denote the subset of M_{0} corresponding to translation structures that have a regular periodic cylinder bounded by the shortest saddle connection. Clearly, N_{0}^{∗}(ω, σ, ϵ) = 0 for ω ∈ M_{0} \ M′_{0}. Let π_{0} : Q(p, n) → MQ(p, n) be the natural projection and μ̃ be the canonical measure on Q(p, n). First consider the case p = 1. In this case M_{0} = MQ_{≤1}(1, n). It is easy to see that the set π_{0}(P^{(n)}(B(ϵ) ∩ ℝ_{+}^{2})) contains almost all elements of M′_{0} ∩ MQ^{ϵ}(1, n) for any ϵ > 0. There exists a mapping class ϕ ∈ Mod(p, n) of order n such that ϕτ^{(n)} = τ^{(n)} and ϕ sends L^{(n)} to a homologous edge. Any set of the form P^{(n)}(U) is invariant under ϕ. Suppose ω, ω′ ∈ P^{(n)}(B(ϵ)∩ℝ_{+}^{2})∩Q^{ϵ,κ}(p, n), κ > ϵ > 0. Then π_{0}(ω) = π_{0}(ω′) only if ω′ = ϕ^{m}ω for some m. Also, in this case N_{0}^{∗}(ω, σ, ϵ) equals the number of indices i such that ω ∈ P_{σ,i}^{(n)}(B(ϵ) ∩ ℝ_{+}^{2}). As shown in the proof of Lemma 8.2, μ̃(P_{σ,i}^{(n)}(B(ϵ) ∩ ℝ_{+}^{2})) = (1 − σ)^{n−1}𝔪(B(ϵ) ∩ ℝ_{+}^{2})c = (1 − σ)^{n−1}πcϵ^{2}/2, where c > 0. Since ϵ^{−2}μ̃(P^{(n)}(B(ϵ)) \ Q^{ϵ,ϵ}(1, n)) → 0 as ϵ → 0, it follows that

lim_{ϵ→0} ϵ^{−2} ∫_{M_{0}} N_{0}^{∗}(ω, σ, ϵ) dμ(ω) = (1 − σ)^{n−1}πc/2.

We proceed to the case p > 1. Consider an operation f_{0} = OP R[n_{0}; D] = OP R[n_{0}; D′; τ, γ], where τ ∈ T (p, n) and γ is an edge of τ, and a family of sets Y_{0}(ϵ) = P^{(n_{0})}[D](ϵU_{0}; U_{1}, …, U_{k}), ϵ > 0, where U_{0}, U_{1}, …, U_{k} satisfy the assumptions of Lemma 8.4. Further assume that U_{0} ⊂ B(1) and μ̃(Y_{0}(ϵ)∩M(τ))/μ̃(Y_{0}(ϵ)) → 1 as ϵ → 0. Clearly, Y_{0}(ϵ) ⊂ Q_{≤1}^{ϵ}(p, n). Any translation structure in an isotopy class ω ∈ Y_{0}(ϵ) has n_{0} regular cylinders of periodic geodesics homologous to γ. Suppose ω ∈ Q_{≤1}^{ϵ,κ}(p, n) for some κ > ϵ. Then the cylinders homologous to γ are the only periodic cylinders of length s(ω). It follows that N_{0}^{∗}(ω, 0, ϵ) = n_{0} while N_{0}^{∗}(ω, σ, ϵ) is equal to the number of indices i such that ω ∈ P_{σ,i}^{(n_{0})}[D](ϵU_{0}; U_{1}, …, U_{k}). Therefore

|∫_{Y_{0}(ϵ)} N_{0}^{∗}(ω, σ, ϵ) dμ̃(ω) − ∑_{i=1}^{n_{0}} μ̃(P_{σ,i}^{(n_{0})}[D](ϵU_{0}; U_{1}, …, U_{k}))| ≤

max(C, n_{0})μ̃(Y_{0}(ϵ) \ Q_{≤1}^{ϵ,ϵ}(p, n)).

Note that μ̃(M(τ) ∩ Q_{≤1}^{ϵ}(p, n) \ Q_{≤1}^{ϵ,ϵ}(p, n)) ≤ c_{τ}ϵ^{4}, where c_{τ} depends on τ (see the proof of Lemma 8.7). By assumption, μ̃(Y_{0}(ϵ)\M(τ))/μ̃(Y_{0}(ϵ)) → 0 as ϵ → 0. Hence Lemma 8.4 implies there exists c_{0} > 0 such that

lim_{ϵ→0} ϵ^{−2} ∫_{Y_{0}(ϵ)} N_{0}^{∗}(ω, σ, ϵ) dμ̃(ω) = (1 − σ)^{K−2}c_{0}.

For any Δ > 0 we wish to find finitely many maps f_{i} = OP R[n_{i}; D_{i}] and families of sets Y_{i}(ϵ) = P^{(n_{i})}[D_{i}](ϵU_{i0}; U_{i1}, …, U_{ik_{i}}), i = 1, …, m, that satisfy the conditions imposed above on f_{0} and Y_{0}(ϵ). Besides, we require that π_{0}(Y_{i}(ϵ)) ⊂ M_{0}, each Y_{i}(ϵ) be invariant under a mapping class ϕ_{i} ∈ Mod(p, n) of finite order o_{i}, ϵ^{−2}|μ(π_{0}(∪_{i}Y_{i}(ϵ))) − ∑_{i} o_{i}^{−1}μ̃(Y_{i}(ϵ))| → 0 as ϵ → 0, and μ(M′_{0} ∩ MQ^{ϵ}(p, n) \ π_{0}(∪_{i}Y_{i}(ϵ))) ≤ Δϵ^{2} for sufficiently small ϵ. First let us show how this helps to prove the proposition. By the above

lim_{ϵ→0} ϵ^{−2} ∫_{Y_{i}(ϵ)} N_{0}^{∗}(ω, σ, ϵ) dμ̃(ω) = (1 − σ)^{K−2}c_{i}

for some c_{i} > 0. Given ω ∈ M_{0}, let ω ∈ Y′(ϵ) if ω ∈ π_{0}(Y_{i}(ϵ)) ∩ π_{0}(Y_{j}(ϵ)) for some i ≠ j, let ω ∈ Y′′(ϵ) if more than o_{i} elements of some Y_{i}(ϵ) are mapped to ω by π_{0}, and let ω ∈ Y′′′(ϵ) if ω ∈ π_{0}(Y_{i}(ϵ)) for some i but π_{0}^{−1}(ω) ∩ Y_{i}(ϵ) has less than o_{i} elements. Any translation structure in an isomorphy class ω ∈ Y′′′(ϵ) admits nontrivial automorphism, therefore μ(Y′′′(ϵ)) = 0. Observe that μ(π_{0}(∪_{i}Y_{i}(ϵ))) + μ(Y′(ϵ)) ≤ ∑_{i} μ(π_{0}(Y_{i}(ϵ))) and ∑_{i} μ(π_{0}(Y_{i}(ϵ))) + (max_{i} o_{i})^{−1}μ(Y′′(ϵ)) ≤ ∑_{i} o_{i}^{−1}μ̃(Y_{i}(ϵ)). The above assumptions imply ϵ^{−2}μ(Y′(ϵ) ∪ Y′′(ϵ)) → 0 as ϵ → 0. Since N_{0}^{∗}(ω, σ, ϵ) is uniformly bounded and there are only finitely many families Y_{i}(ϵ), it follows that

lim_{ϵ→0} ϵ^{−2} ∫_{π_{0}(∪_{i}Y_{i}(ϵ))} N_{0}^{∗}(ω, σ, ϵ) dμ(ω) = (1 − σ)^{K−2} ∑_{i} o_{i}^{−1}c_{i}.

Moreover, π_{0}(∪_{i}Y_{i}(ϵ)) ⊂ M_{0} and

∫_{M_{0}\π_{0}(∪_{i}Y_{i}(ϵ))} N_{0}^{∗}(ω, σ, ϵ) dμ(ω) ≤ CΔϵ^{2}

for small ϵ. As Δ can be chosen arbitrarily small, the proposition follows.

It remains to fetch the needed maps f_{i} and sets Y_{i}(ϵ). The maps will be of the form f_{i} = OP R[n_{i}; D_{i}] = OP R[n_{i}; D̃_{i1}, …, D̃_{ik_{i}}; l̃_{i}; τ_{i}, γ_{i}], D̃_{ij} = (τ_{ij}, e_{r,ij}, e_{l,ij}; D_{ij}), l̃_{i} = (l_{i1}, …, l_{ik_{i}}), where τ_{i} ∈ T (p, n). Recall that γ_{i} is an oriented edge of τ_{i}, e_{r,ij} and e_{l,ij} are oriented edges of τ_{ij}, and l̃_{i} is a sequence of integers. We require that if ϕτ_{i} = τ_{i′} for some ϕ ∈ Mod(p, n) then τ_{i} = τ_{i′}. Moreover, if ϕτ_{i} = τ_{i′} and ϕ(γ_{i}) is homologous to γ_{i′} then τ_{i} = τ_{i′} and γ_{i} = γ_{i′}. If τ_{ij}, τ_{i′j′} ∈ T (p′, n′) and there exists ϕ ∈ Mod(p′, n′) such that ϕτ_{ij} = τ_{i′j′}, ϕ(e_{r,ij}) = e_{r,i′j′}, ϕ(e_{l,ij}) = e_{l,i′j′}, then D̃_{ij} = D̃_{i′j′}. We assume f_{1}, f_{2}, … is a maximal list of maps satisfying the above conditions. It is easy to see that the list is finite.

Suppose τ ∈ T (p′, n′) and e_{r}, e_{l} are oriented edges of τ. Choose an inserting operation f = INS[τ, e_{r}, e_{l}; D; τ′, γ′]. For any v ∈ S^{1} and δ > 0 we define U_{f}(τ, e_{r}, e_{l}; v, δ) as the set of ω ∈ N(τ) such that f(tv, ω) ∈ M_{δ}(τ′) for all sufficiently small t > 0. Assume f(tv, ω) is defined for small t > 0, that is, pairs of vectors (hol_{ω}(e_{r}), v) and (hol_{ω}(e_{l}), v) induce the standard orientation in ℝ^{2}. Let β_{r} (resp. β_{l}) denote the angle between hol_{ω}(e_{r}) (resp. hol_{ω}(e_{l})) and v. Let T_{r}^{+} be the triangle of τ such that e_{r} bounds T_{r}^{+} and the orientation of e_{r} agrees with the counterclockwise orientation of the boundary of T_{r}^{+}. Let T_{r}^{−} be the other triangle of τ bounded by e_{r}. Let θ_{r}^{+} and θ_{r}^{−} denote the measures relative to translation structures in the class ω of the angles of T_{r}^{+} and T_{r}^{−} opposite e_{r}. Similarly, we introduce triangles T_{l}^{+}, T_{l}^{−} and angles θ_{l}^{+}, θ_{l}^{−}. In the case e_{r} ≠ e_{l}, the set U_{f}(τ, e_{r}, e_{l}; v, δ) does not depend on the choice of f. Namely, ω ∈ U_{f}(τ, e_{r}, e_{l}; v, δ) if and only if ω ∈ M_{δ}(τ) and β_{r} + θ_{r}^{+}, π − β_{r} + θ_{r}^{−}, π − β_{l} + θ_{l}^{+}, β_{l} + θ_{l}^{−} are less than π − δ. In the case e_{r} = e_{l}, there are two principal choices. The triangle T_{r}^{+} corresponds to a triangle T′ of τ′. Let e′ be the edge of T′ corresponding to e_{r} and T′_{0} be the triangle separated from T′ by e′. Then γ′ is an edge of T′_{0}. If T′_{0} lies to the right of γ′ then ω ∈ U_{f}(τ, e_{r}, e_{l}; v, δ) if and only if ω ∈ M_{δ}(τ) and β_{r} + θ_{r}^{+}, β_{r} + θ_{r}^{−}, 2(π − β_{r}) are less than π − δ. Otherwise ω ∈ U_{f}(τ, e_{r}, e_{l}; v, δ) if and only if ω ∈ M_{δ}(τ) and π − β_{r} + θ_{r}^{+}, π − β_{r} + θ_{r}^{−}, 2β_{r} are less than π − δ. In particular, if the angle between v, v′ ∈ S^{1} does not exceed δ′ < δ then U_{f}(τ, e_{r}, e_{l}; v, δ) ⊂ U_{f}(τ, e_{r}, e_{l}; v′, δ − δ′).

Choose δ > 0 such that π/δ is an integer and divide the semicircle S^{1} ∩ ℝ_{+}^{2} into arcs of equal length δ. Suppose u is the interior of one of the arcs. Let U_{0u} = {tv | v ∈ u, 0 < t < 1}. By v_{0} denote the midpoint of u. For 1 ≤ j ≤ k_{i} let f_{ij} denote the j^{th} inserting operation of those used when building f_{i}. Given κ > 0, let U_{ijuκ} = U_{f_{ij}}(τ_{ij}, e_{r,ij}, e_{l,ij}; v_{0}, δ) ∩ Q_{1}(p_{ij}, n_{ij}) \ Q_{1}^{κ}(p_{ij}, n_{ij}), where τ_{ij} ∈ T (p_{ij}, n_{ij}). Finally, let Y_{iuκ}(ϵ) = P^{(n_{i})}[D_{i}](ϵU_{0u}; U_{i1uκ}, …, U_{ik_{i}uκ}), ϵ > 0. Now we shall check whether the families of sets Y_{iuκ}(ϵ) satisfy the above conditions.

Suppose ω ∈ U_{ijuκ}. Any Delaunay cell of a translation structure in the class ω is isometric to the interior of a triangle inscribed in a circle of radius d. Since ω ∈ Q_{1}(p_{ij}, n_{ij}) \ Q_{1}^{κ}(p_{ij}, n_{ij}), it follows from Lemma 5.4 that d ≤ d_{0}, where d_{0} = max(\sqrt{2/π}, κ^{−1}). In particular, all Delaunay edges are of length at least κ and at most 2d_{0}. Then each angle θ of a Delaunay triangle satisfies sin θ ≥ κ/(2d_{0}). Since U_{ijuκ} ⊂ M(τ_{ij}), it follows that the closure of U_{ijuκ} is a compact subset of N(τ_{ij}). Note that f_{ij} = INS[D̃_{ij}; τ′_{ij}, γ′_{ij}] for some τ′_{ij} ∈ T (p′_{ij}, n′_{ij}). There exists ϵ_{0} = ϵ_{0}(κ, δ) > 0 such that f_{ij}(v, ω) ∈ M_{δ/4}(τ′_{ij}) ∩ Q^{ϵ_{0},κ/2}(p′_{ij}, n′_{ij}) for all v ∈ ϵ_{0}U_{0u} and ω ∈ U_{ijuκ}.

Suppose ω = GLU[τ̃_{1}, γ̃_{1}, …, τ̃_{k}, γ̃_{k}; τ̃, γ̃](ω_{1}, …, ω_{k}) and γ̃_{j} (1 ≤ j ≤ k) is a unique shortest edge of τ̃_{j} relative to ω_{j}. Any edge e of τ̃_{j} different from γ̃_{j} is assigned a unique edge ẽ of τ̃ so that θ_{e,τ̃_{j}}(ω_{j}) = θ_{ẽ,τ̃}(ω). Since the angle opposite the shortest side of a triangle does not exceed π/3, we have θ_{γ̃,τ̃}(ω) ≤ 2π/3 and θ_{γ̃_{j},τ̃_{j}}(ω_{j}) ≤ 2π/3. By Proposition 5.3, ω ∈ M(τ̃) if and only if ω_{j} ∈ M(τ̃_{j}) for 1 ≤ j ≤ k. Further suppose ω′ = GLU[τ̃′_{1}, γ̃′_{1}, …, τ̃′_{k′}, γ̃′_{k′}; τ̃′, γ̃′](ω′_{1}, …, ω′_{k}), where γ̃′_{j} is a unique shortest edge of τ̃′_{j} relative to ω′_{j}. Assume that ω ∈ M(τ̃), ω′ ∈ M(τ̃′), and hol_{ω}(γ̃), hol_{ω′}(γ̃′) ∈ ℝ_{+}^{2}. Then ω is isomorphic to ω′ if and only if k = k′ and for some 1 ≤ k_{0} ≤ k the classes ω_{1}, …, ω_{k} are isomorphic to ω′_{k_{0}}, …, ω′_{k}, ω′_{1}, …, ω′_{k_{0}−1}, respectively.

All mapping classes ϕ_{i} ∈ T (p, n) such that ϕ_{i}τ_{i} = τ_{i} and ϕ_{i} sends γ_{i} to a homologous edge form a cyclic subgroup. Assume ϕ_{i} generates this subgroup. The order o_{i} of ϕ_{i} divides k_{i} and n_{i}. For any ω = f_{i}(ω_{0}, ω_{1}, …, ω_{k_{i}}) one has ωϕ_{i}^{−1} = f_{i}(ω_{0}ϕ^{−1}, ω_{k}, … ω_{k_{i}}, ω_{1}, …, ω_{k−1}), where 1 ≤ k ≤ k_{i} and ϕ ∈ Mod(1, n_{i}) is such that ϕτ^{(n_{i})} = τ^{(n_{i})} and ϕ(L^{(n_{i})}) is homologous to L^{(n_{i})}. It follows that each Y_{iuκ}(ϵ) is invariant under ϕ_{i}.

Let ω = f_{ij}(v, ω_{0}), ω′ = f_{i′j′}(v′, ω′_{0}), where ω_{0} ∈ M(τ_{ij}), ω′_{0} ∈ M(τ_{i′j′}), and v, v′ ∈ ℝ_{+}^{2}. Suppose |v|, |v′| ≤ ϵ and ω ∈ M(τ′_{ij}) ∩ Q^{ϵ,κ′}(p′_{ij}, n′_{ij}), ω′ ∈ M(τ′_{i′j′}) ∩ Q^{ϵ,κ′}(p′_{i′j′}, n′_{i′j′}) for some κ′ > ϵ. Then ω is isomorphic to ω′ if and only if i = i′, j = j′, v = v′, and ω_{0} = ω′_{0}. Now let ω ∈ Y_{iuκ}(ϵ), ω′ ∈ Y_{i′u′κ}(ϵ). Suppose ω ∈ M(τ_{i})∩Q^{ϵ,κ′}(p, n) and ω′ ∈ M(τ_{i′})∩Q^{ϵ,κ′}(p, n) for some κ′ > ϵ. Then it follows from the above that π_{0}(ω) = π_{0}(ω′) only if i = i′, u = u′, and ω is mapped to ω′ by an iterate of ϕ_{i}.

Using techniques of the proof of Lemma 8.4, we derive from the above that as ϵ → 0, μ̃(Y_{iuκ}(ϵ) ∩ M(τ_{i}))/μ̃(Y_{iuκ}(ϵ)) → 1, ϵ^{−2}|μ̃(Y_{iuκ}(ϵ)) − o_{i}μ(π_{0}(Y_{iuκ}(ϵ)))| → 0, and ϵ^{−2}μ(π_{0}(Y_{iuκ}(ϵ)) ∩ π_{0}(Y_{i′u′κ}(ϵ))) → 0 unless i = i′ and u = u′.

Suppose ω ∈ S(τ, γ; ϵ, 2κ, 2δ) for some τ ∈ T (p, n). As S(τ, γ; ϵ, 2κ, 2δ) does not depend on the orientation of γ, we orient γ so that hol_{ω}(γ) belongs to the closure of ℝ_{+}^{2}. Then hol_{ω}(γ) is in the closure of ϵU_{0u} for an arc u. Assume hol_{ω}(γ) ∈ ϵU_{0u}. Note that this assumption holds for almost all elements of S(τ, γ; ϵ, 2κ, 2δ). First we apply a cutting operation to ω and γ. Let (ω′_{1}, …, ω′_{k}) = CUT [τ, γ; τ′_{1}, γ′_{1}, …, τ′_{k}, γ′_{k}](ω), where τ′_{j} ∈ T (p′_{j}, n′_{j}), 1 ≤ j ≤ k. Clearly, each τ′_{j} has no edge homologous to γ′_{j}. In the case p′_{j} = n′_{j} = 1, it is no loss to assume that τ′_{j} = τ^{(1)}, γ′_{j} = L^{(1)}; then ω′_{j} ∈ P^{(1)}. If none of ω′_{j} belongs to Q(1, 1) then π_{0}(ω) ∉ M′_{0}. Assume this is not the case. Choose j such that (p′_{j}, n′_{j}) ≠ (1, 1) (this is possible as p > 1). Then ω′_{j} ∈ S(τ′_{j}, γ′_{j}; ϵ, 2κ, 2δ) ⊂ M_{2δ}(τ′_{j}) ∩ Q_{≤1}^{ϵ,2κ}(p′_{j}, n′_{j}) assuming 2δ < π/3. There exists ϵ_{1} = ϵ_{1}(κ, δ) > 0 such that any collapsing operation of the form COL[τ′_{j}, γ′_{j}; D′; τ̃, ẽ_{r}, ẽ_{l}] is defined on S(τ′_{j}, γ′_{j}; ϵ, 2κ, 2δ) for ϵ ≤ ϵ_{1}. If ω̃ = COL[τ′_{j}, γ′_{j}; D′; τ̃, ẽ_{r}, ẽ_{l}](ω′_{j}) then ω′_{j} = g(hol_{ω}(γ), ω̃), where g = INS[τ̃, ẽ_{r}, ẽ_{l}; D; τ′_{j}, γ′_{j}] for a set D of parameters. Consequently, ω can be obtained by an operation OP R[D̃; τ, γ]. Assuming ϵ is small enough, one has ω̃ ∈ U_{g}(τ̃, ẽ_{r}, ẽ_{l}; v_{0}, δ), where v_{0} is the midpoint of u. Also, ω̃ = tω̃_{0}, where t > 0, a(ω̃_{0}) = 1, and s(ω̃_{0}) > κ. Note that D depends on D′, moreover, any D corresponds to some D′. It follows that we can choose the cutting and the collapsing operations so that g = f_{i′j′} for some i′, j′. Then ω̃_{0} ∈ U_{i′j′uκ}. The conditions imposed above on f_{1}, f_{2}, … imply that ωϕ^{−1} ∈ Y_{iuκ}(ϵ) for some i and ϕ ∈ Mod(p, n) such that ϕτ = τ_{i} and ϕ(γ) is homologous to γ_{i}. Thus for small ϵ the set π_{0}(S(τ, γ; ϵ, 2κ, 2δ)) ∩ M′_{0} is contained in π_{0}(∪_{i,u}Y_{iuκ}(ϵ)) up to a subset of zero volume. Since the Mod(p, n) action on T (p, n) has only finitely many orbits, it follows from Lemma 8.7 that we can choose κ and δ so that μ(M′_{0} ∩ MQ^{ϵ}(p, n) \ π_{0}(∪_{i,u}Y_{iuκ}(ϵ))) ≤ Δϵ^{2} for small ϵ.

The only condition not verified yet is π_{0}(Y_{iuκ}(ϵ)) ⊂ M_{0}. It surely holds when the moduli space MQ(p, n) is connected. If this is not the case, we have to modify the construction slightly. Suppose U′_{j}, 1 ≤ j ≤ k_{i}, is a connected component of the open set U_{ijuκ}. Then sets Y (ϵ) = P^{(n_{i})}[D_{i}](ϵU_{0u}; U′_{1}, …, U′_{k_{i}}) are connected by Lemma 8.4. Hence sets π_{0}(Y (ϵ)) are either contained in M_{0} or disjoint from M_{0}. Clearly, Y_{iuκ}(ϵ) is the disjoint union of at most countably many sets of the form Y (ϵ). The mapping class ϕ_{i} permutes these sets. Now we replace Y_{iuκ}(ϵ) by a finite number of subsets of the form Y (ϵ). For any Δ > 0 this finite number can be chosen so that the union of the other subsets has volume at most Δϵ^{2}. Then we discard sets whose projections to MQ(p, n) are disjoint from M_{0}. Also, among sets Y (ϵ), Y (ϵ)ϕ_{i}^{−1}, Y (ϵ)ϕ_{i}^{−2}, … we discard all but one. The construction is completed.

Proof of Proposition 7.4. Let C be a connected component of MQ_{1}(p, n) and μ_{0} be the canonical measure on C. The formula (11) can be rewritten as follows:

b(σ, ϵ) = \frac{1}{μ_{0}(C)} ∫_{C} N_{0}(ω, σ, ϵ) dμ_{0}(ω).

Besides, let

b^{∗}(σ, ϵ) = \frac{1}{μ_{0}(C)} ∫_{C} N_{0}^{∗}(ω, σ, ϵ) dμ_{0}(ω).

First we estimate the difference b(σ, ϵ) − b^{∗}(σ, ϵ). Clearly, N_{0}(ω, 0, ϵ) ≥ N_{0}(ω, σ, ϵ) ≥ N_{0}^{∗}(ω, σ, ϵ) ≥ 0 for all ω ∈ C. If ω ∉ MQ_{1}^{ϵ}(p, n) then N_{0}(ω, σ, ϵ) = 0. If ω ∈ MQ_{1}^{ϵ,κ}(p, n) for some κ > ϵ, then N_{0}(ω, σ, ϵ) = N_{0}^{∗}(ω, σ, ϵ). It follows that for any κ > ϵ,

0 ≤ b(σ, ϵ) − b^{∗}(σ, ϵ) ≤ \frac{1}{μ_{0}(C)} ∫_{C∩MQ_{1}^{ϵ}(p,n)\MQ_{1}^{ϵ,κ}(p,n)} N_{0}(ω, 0, ϵ) dμ_{0}(ω).

By Theorem 7.3, there exist c_{0}, κ_{0} > 0 such that N_{0}(ω, 0, ϵ) ≤ c_{0}(ϵ/s(ω))^{3/2} for all ω ∈ C and ϵ < κ_{0}. Suppose ω ∈ C ∩ MQ_{1}^{ϵ}(p, n) \ MQ_{1}^{ϵ,κ}(p, n), where ϵ < κ ≤ κ_{0}. Obviously, s(ω) ≤ ϵ, hence 4^{−k−1}ϵ < s(ω) ≤ 4^{−k}ϵ for an integer k ≥ 0. Then ω ∈ MQ_{1}^{4^{−k}ϵ}(p, n) \ MQ_{1}^{4^{−k}ϵ,κ}(p, n) and N_{0}(ω, 0, ϵ) ≤ (4^{k+1})^{3/2}c_{0}. It follows that

b(σ, ϵ) − b^{∗}(σ, ϵ) ≤ \frac{c_{0}}{μ_{0}(C)} ∑_{k=0}^{∞} 8^{k+1}μ_{0}(MQ_{1}^{4^{−k}ϵ}(p, n) \ MQ_{1}^{4^{−k}ϵ,κ}(p, n))

provided ϵ < κ ≤ κ_{0}. Furthermore, Theorem 8.5 implies that

b(σ, ϵ) − b^{∗}(σ, ϵ) ≤ \frac{c_{0}c_{p,n}}{μ_{0}(C)} ∑_{k=0}^{∞} 8^{k+1}4^{−2k}ϵ^{2}κ^{2} = \frac{16c_{0}c_{p,n}}{μ_{0}(C)} ϵ^{2}κ^{2},

where c_{p,n} > 0 is a constant. As κ can be chosen arbitrarily small, it follows that

lim_{ϵ→0} ϵ^{−2}(b(σ, ϵ) − b^{∗}(σ, ϵ)) = 0.

Now we estimate b^{∗}(σ, ϵ). By C_{∞} denote the connected component of MQ(p, n) containing C. Clearly, C_{∞} = {tω | ω ∈ C, t > 0}. For any t > 0 let C_{t} denote the set of ω ∈ C_{∞} such that a(ω) ≤ t. Let μ be the canonical measure on C_{∞}. For all ω ∈ C and t > 0 we have a(tω) = t^{2}, s(tω) = t · s(ω), and N_{0}^{∗}(ω, σ, ϵ) = N_{0}^{∗}(tω, σ, tϵ). It follows that

b^{∗}(σ, ϵ) = \frac{1}{μ(C_{1})} ∫_{C_{1}} N_{0}^{∗}(ω, σ, ϵ\sqrt{a(ω)}) dμ(ω).

For any σ ∈ [0, 1) and positive ϵ, t, α let

b_{0}(σ, ϵ; t, α) = ∫_{C_{t}} N_{0}^{∗}(ω, σ, ϵ\sqrt{α}) dμ(ω).

Denote by K half of the dimension of C_{∞}. By Proposition 8.8, there exists c > 0 such that ϵ^{−2}b_{0}(σ, ϵ; 1, 1) → (1 − σ)^{K−2}c as ϵ → 0. For arbitrary t and α we have

b_{0}(σ, ϵ; t, α) = t^{K} ∫_{C_{1}} N_{0}^{∗}(t^{1/2}ω, σ, ϵ\sqrt{α}) dμ(ω) = t^{K}b_{0}(σ, ϵ\sqrt{α/t}; 1, 1),

therefore

lim_{ϵ→0} ϵ^{−2}b_{0}(σ, ϵ; t, α) = t^{K−1}α(1 − σ)^{K−2}c.

Suppose 0 < t_{1} < t_{2}. Then

b_{0}(σ, ϵ; t_{2}, t_{1}) − b_{0}(σ, ϵ; t_{1}, t_{1}) ≤
∫_{C_{t_{2}}\C_{t_{1}}} N_{0}^{∗}(ω, σ, ϵ\sqrt{a(ω)}) dμ(ω) ≤ b_{0}(σ, ϵ; t_{2}, t_{2}) − b_{0}(σ, ϵ; t_{1}, t_{2}).

It follows that for any integer k > 0 we have b_{k}^{−}(σ, ϵ) ≤ μ(C_{1})b^{∗}(σ, ϵ) ≤ b_{k}^{+}(σ, ϵ), where

b_{k}^{−}(σ, ϵ) = ∑_{i=1}^{k}(b_{0}(σ, ϵ; i/k, (i − 1)/k) − b_{0}(σ, ϵ; (i − 1)/k, (i − 1)/k)),

b_{k}^{+}(σ, ϵ) = ∑_{i=1}^{k}(b_{0}(σ, ϵ; i/k, i/k) − b_{0}(σ, ϵ; (i − 1)/k, i/k)).

By the above,

lim_{ϵ→0} ϵ^{−2}b_{k}^{−}(σ, ϵ) = (1 − σ)^{K−2}c ∑_{i=1}^{k} \frac{i − 1}{k}((\frac{i}{k} )^{K−1} − (\frac{i − 1}{k})^{K−1}),

lim_{ϵ→0} ϵ^{−2}b_{k}^{+}(σ, ϵ) = (1 − σ)^{K−2}c ∑_{i=1}^{k} \frac{i}{k} ((\frac{i}{k} )^{K−1} − (\frac{i − 1}{k})^{K−1}).

In particular, lim_{ϵ→0} ϵ^{−2}(b_{k}^{+}(σ, ϵ) − b_{k}^{−}(σ, ϵ)) = k^{−1}(1 − σ)^{K−2}c. As k can be chosen arbitrarily large, this implies that

lim_{ϵ→0} ϵ^{−2}b^{∗}(σ, ϵ) = (1 − σ)^{K−2} \frac{c}{μ(C_{1})} ∫_{0}^{1} t dt^{K−1}.

Since ϵ^{−2}(b(σ, ϵ) − b^{∗}(σ, ϵ)) → 0 as ϵ → 0, it follows that

lim_{ϵ→0} \frac{b(σ, ϵ)}{b(0, ϵ)} = lim_{ϵ→0} \frac{b^{∗}(σ, ϵ)}{b^{∗}(0, ϵ)} = (1 − σ)^{K−2}.

It remains to notice that K = 2p + n − 1.

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Pidstryhach Institute for Applied Problems of Mechanics and Mathematics of Ukrainian NAS, vul. Naukova 3b, 79060 Lviv, Ukraine

   E-mail address: vorobets@complex.lviv.ua