remind the reader that by definition, a saddle connection is represented by a single geodesic segment joining a pair of conical points, and not by a broken line of geodesic segments.)
Choose, for example, a flat torus R2/(Z⊕Z)as a surface S and mark a generic pair of points P0,P1 on this torus. Let v0 := P0P1. The set Vsc(S)⊂ R2 is the square lattice of the form {v0+ u}u∈Z⊕Z.
Similarly to the torus case, the set Vsc(S) is a discrete subset of R2. However, whenS is different from the torus the map from the set of saddle connections onS to Vsc(S) is not injective: different saddle connections may have the same holonomy. We define the multiplicity of an element v∈ Vsc(S) to be the number p of distinct saddle connections γ1, ..., γp such that hol(γi) = v. For example, for the surface S presented on the left of Figure 2 the vector (0,1) ∈ Vsc(S) has multiplicity four: the saddle connections β, γ, γ1, η chosen with appropriate orientation have holonomy(0,1).
In this example the saddle connections represented by β, γ, γ1, η are not pair-wise homologous. As indicated above, the fact that their holonomy coincides is not generic; deforming the surface slightly we see that the corresponding saddle connec-tions have different holonomy; see the right-hand-side picture at Figure 2. The situ-ation is different with the saddle connectionsγ and γ1, for they are homologous. The right-hand-side picture at Figure 2 confirms that even after any small deformation of the surface the corresponding saddle connections share the same holonomy vector hol(γ)=hol(γ1).
From the measure-theoretical point of view Proposition 3.1 allows us to assume from now on that if two saddle connections β, γ have the same holonomy, then γ and β are homologous. In particular, if β joins distinct saddles P1,P2, then γ joins the same pair of saddles. Since the surfaces as in the left-hand-side picture at Figure 2 form a set of measure zero we will ignore them in our considerations.
Somewhat surprisingly, higher multiplicity is very common even for a generic surface. For example, we will show that if the genus is at least 3, then counting sep-arately the vectors of multiplicity one in Vsc(S) of length at most L and the vectors of multiplicity two, we get quadratic growth in L for the number of elements of both multiplicities.
Multiplicity is not the only property distinguishing elements in Vsc(S): some of them correspond to closed saddle connections, other elements correspond to saddle connections joining distinct zeroes. We may also consider only thosev∈Vsc(S), which correspond to saddle connections joining some particular pair of saddles. We refine our consideration slightly more specifying the following data.
Suppose that we have precisely p homologous saddle connectionsγ1, ..., γp join-ing a zero z1 of order m1 to a zero z2 of order m2, see Figure 5 for a topological
picture. All the γi, 1≤ i≤ p have the same holonomy (where the orientation of each γi is from z1 to z2). By convention the cyclic order of the γi at z1 is clockwise in the orientation defined by the flat structure. Let the angle between γi and γi+1 at z1 be 2π(ai+1); let the angle between γi and γi+1 at z2 be 2π(ai +1). We call this data the configuration
C =(m1,m2,ai,ai)
of the p homologous saddle connections. If there is just a single saddle connection (p=1) then C =(m1,m2).
Convention 3. — We reserve the notion “configuration” for geometric types of pos-sible collections of saddle connections, and not for the collections themselves.
Now given a surface S and a configuration C, let VC(S) ⊂ Vsc(S) denote the vectorsv inR2such that there are precisely p saddle connectionsγ1, ..., γp forming the configuration of the type C and having holonomy hol(γi) = v. We want to compute the number of collections {γ1, ..., γp} of the type C having holonomy vector of length at most L. In other words, we want to compute the asymptotics as L → ∞ of the cardinality
|VC(S)∩B(L)|
of intersection of the discrete set VC(S) with the disc B(L)⊂R2 of radius Lcentered at the origin.
Now consider the second problem of this paper: counting closed saddle connec-tions.
We have a surfaceSand a saddle connectionγ1 joining a zeroz1 to itself. There may be other saddle connections γ2, ..., γm having the same holonomy vector as γ1. By the same argument as above we shall always assume that all such γi are homolo-gous.
Some of the γi may start and end at the same zero z1, the others may start and end at the other zeroes. Each γi returns to its zero at some angle θi which is an odd multiple ofπ. Some of the γi may bound cylinders filled with regular closed geodesics.
A configuration C describes all these geometric data; this notion will be formalized in Section 11.
We may have numerous collections of preciselyp homologous closed saddle con-nections corresponding to a configuration C, i.e. defining the prescribed number of cylinders, prescribed anglesθi, etc. As in the first problem for a surface Sand config-uration C let VC(S) denote the vectors vin R2 such that the saddle connections in C have holonomy hol(γ1) = · · · = hol(γp) = v equal to that vector. Again we shall
compute the asymptotics as L→ ∞ of
|VC(S)∩B(L)|, whereB(L) is the disc of radius L.
Theorem (A. Eskin, H. Masur; [EMa]). — For either of the problems, given a configu-ration C, there is a constant c=c(α,C) such that for almost all Sin any connected component of any stratum H1(α) one has
Llim→∞
|VC(S)∩B(L)|
πL2 =c(α,C).
(4)
The constant c(α,C) depends only on the connected component of the stratum and on the configura-tion C.
Versions of the above theorems where convergence a.e. is replaced by conver-gence in L1 are proved in [Ve4]. Thus the growth rate is quadratic for a generic surface. What is perhaps surprising is that this formula says that the growth rate is quadratic for a generic surface even for multiple homologous saddle connections. By contrast, a generic surface does not have any pairs of saddle connections that deter-mine vectors in the same direction but with different lengths, see Proposition 7.4 at the end of Section 7.
Remark 3.2. — The issue of higher multiplicity and specifying the angles is not explicitly addressed in [EMa]. However, the set VC(S) defined above, satisfies all of the axioms of that paper, so the asymptotic formula from [EMa] is applicable to the problems formulated above.
One of the main results of this paper is a description of all possible configura-tions C for any connected component of any stratum, and an evaluation of the cor-responding constants c(α,C).
We note that the lists of configurations and the values of the constants have been verified numerically. Beside direct considerations of flat surfaces we used the following implicit experiments. A formula based on the formula of Kontsevich (see [Ko]) ex-presses the sum of the Lyapunov exponents of the Teichmüller geodesic flow on each connected component of every stratum H(α) in terms of some rational function of α and of some specific linear combination of the constantsc(α,C)for the corresponding component of H(α). These Lyapunov exponents have been computed numerically and the constants achieved in this paper are consistent with these exponents and with Kontsevich’s formula.
3.3. Siegel–Veech formula. — Fix a topological type of a configurationC. To every