Closed geodesics with prescribed intersection numbers

Let $(\Sigma, g)$ be a closed, oriented, negatively curved surface, and fix pairwise disjoint simple closed geodesics $\gamma_{\star,1}, \dots \gamma_{\star, r}$. We give an asymptotic growth as $L \to +\infty$ of the number of primitive closed geodesic of length less than $L$ intersecting $\gamma_{\star,j}$ exactly $n_j$ times, where $n_1, \dots, n_r$ are fixed nonnegative integers. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting $\Sigma$ along $\gamma_{\star,1}, \dots, \gamma_{\star, r}$ and by using the theory of Pollicott-Ruelle resonances for open systems.


Introduction
Let (Σ, g) be a closed oriented negatively curved Riemannian surface and denote by P the set of its oriented primitive closed geodesics. For L > 0 define N (L) = #{γ ∈ P, (γ) L}, where (γ) is the length of a geodesic γ. Then a classical result obtained by Margulis [Mar69] states that N (L) ∼ e hL hL as L → +∞, where h > 0 is the topological entropy of the geodesic flow of (Σ, g).
Denote by SΣ the unit tangent bundle of (Σ, g) and by (ϕ t ) t∈R the associated geodesic flow, acting on SΣ. Let π : SΣ → Σ be the natural projection. We denote by h j > 0 (j = 1, . . . , q) the entropy of the open system (Σ j , g| Σ j ), that is, the topological entropy of the flow ϕ restricted to the trapped set where the closure is taken in SΣ.
The number h ω is the maximum of the entropies of the surfaces encountered by any γ ∈ P satisfying ω(γ) ∼ ω, while d ω is the number of times any such γ will encounter a surface whose entropy is equal to h ω (for example, in Figure 1, if the entropy h 2 of Σ 2 is the greatest, we have h(ω) = h 2 and d(ω) = 3, as γ travels three times through Σ 2 ).

CLOSED GEODESICS WITH PRESCRIBED INTERSECTION NUMBERS 3
Note that Theorem 1 can be deduced from Theorem 2 by summing over admissible paths ω with n(ω) = n where n ∈ N r is fixed. We refer to §10 for a slightly more precise statement.
For the sake of simplicity, and to make the exposition clearer, we will deal in the major part of this article with the case r = 1. The case r > 1 will be then obtained by identical techniques, as described in §10. Thus from now on and unless stated otherwise, we will assume that we are given only a simple closed geodesic γ and we set N (n, L) = # {γ ∈ P, (γ) L, i(γ, γ ) = n} .
In this context, our result reads as follows.
(a) Suppose that γ is not separating, that is Σ \ γ is connected. Then there exists c > 0 such that for each n ∈ N, where h ∈]0, h[ is the entropy of the geodesic flow of the open system (Σ \ γ ).
(b) Suppose that γ separates Σ in two surfaces Σ 1 and Σ 2 . Let h j ∈]0, h[ denote the entropy of the open system (Σ j , g| Σ j ) and set h = max(h 1 , h 2 ). Then there is c > 0 such that for each n ∈ N we have as L → +∞, As before, the entropy h is defined as the topological entropy of the geodesic flow restricted to the trapped set where the closure is taken in SΣ.
(i) The case n = 0 is well known and follows from the growth rate of periodic orbits of Axiom A flows obtained by Parry-Pollicott [PP83] (see §2.4). However, to the best of our knowledge, the result is new for n > 0. (ii) Using a classical large deviation result by Kifer [Kif94] and Bonahon's intersection form [Bon86], we are in fact able to show that a typical closed geodesic γ satisfies i(γ, γ ) ≈ I (γ) for some I > 0 not depending on γ (see Proposition 9.1 below for a precise statement). In particular Theorem 3 is a statement about very uncommon closed geodesics.
Theorem 4. Let n 1. For any f ∈ C ∞ (∂) the limit exists, where for any γ ∈ P, I (γ) = {(x, v) ∈ Sγ, x ∈ γ } is the set of incidence vectors of γ along γ . This formula defines a probability measure µ n on∂, whose support is contained in Γ n .
We will give a full description of c and µ n in terms of Pollicott-Ruelle resonant states of the geodesic flow of (Σ , g) for the resonance h in §7. Here Σ is the compact surface with boundary obtained by cutting Σ along γ (see §2.4).
As a first step (which is of independent interest, see Corollary 6), we prove that the family s → S(s) extends to a meromorphic family of operators S(s) : C ∞ (∂) → D (∂) on the whole complex plane (here D (∂) denotes the space of distributions on∂), whose poles are contained in the set of Pollicott-Ruelle resonances of the geodesic flow of the surface with boundary (Σ , g) (see §2.5 for the definition of those resonances). In this context, the existence of such resonances follows from the work of Dyatlov-Guillarmou [DG16]. By using the microlocal structure of the resolvent of the geodesic flow provided by [DG16], we are moreover able to prove that for any χ ∈ C ∞ c (∂ \Sγ ), the composition (χS(s)) n is well defined for any n 1, as well as its super flat trace (meaning that we also look at the action of S(s) on differential forms, see §3.4) which reads tr s [(χS(s)) n ] = n i(γ,γ )=n where the products runs over all closed geodesics (not necessarily primitive) γ with i(γ, γ ) = n and # (γ) is the primitive length of γ ; this formula is a consequence of the Atiyah-Bott trace formula [AB67]. Furthermore, using a priori bounds on the growth of N (n, L) (obtained in §4), we prove that s → tr s [(χS(s)) n ] has a pole of order n at s = h, provided that χ has enough support. Then letting the support of 1 − χ being very close to Sγ , and estimating the growth of geodesics intersecting n times γ with at least one small angle, we are able to derive Theorem 3 from a classical Tauberian theorem of Delange [Del54].
Application to geodesic billards. We finally state a result on the growth number of periodic trajectories of the billard problem associated to a negatively curved surface with totally geodesic boundary, which follows from the methods used to prove Theorem 3.
Corollary 5. Let (Σ , g ) be a negatively curved surface with totally geodesic boundary. For any n ∈ N and L > 0 we denote by N (n, L) the number of closed billiard trajectories on (Σ , g ) (that is, geodesic trajectories that bounce on ∂Σ according to Descartes' law) with exactly n rebounds, and with length not greater than L. Then there is c > 0 such that where h is the entropy of the open system (Σ , g ).
Related works. As mentioned before, the case n = 0 follows from the work Parry-Pollicott [PP83] which is based on important contributions of Bowen [Bow72,Bow73], as the geodesic flow on (Σ , g) can be seen as an Axiom A flow (see Lemma 2.4 below and [DG16, §6.1]). For counting results on non compact Riemann surfaces, see also Sarnak [Sar80], Guillopé [Gui86], or Lalley [Lal89]. We refer to the work of Paulin-Pollicott-Schapira [PPS12] for counting results in more general settings.
We also mention a result by Pollicott [Pol85] which says that, if (Σ, g) is of constant curvature −1 and if γ is not separating, for some I > 0, which means that, the average intersection number between γ and geodesics of length not greater than L is about I L. We show that this also holds in our context (see §9.2).
Lalley [Lal88], Pollicott [Pol91] and Anantharaman [Ana00] investigated the asymptotic growth of the number of closed geodesics satisfying some homological constraints (see also Philips-Sarnak [PS87] and Katsuda-Sunada [KS88] for the constant curvature case). They show that for any homology class ξ ∈ H 1 (Σ, Z), we have #{γ ∈ P, (γ) L, [γ] = ξ} ∼ Ce hL /L g+1 for some C > 0 independent of ξ, where g is the genus of Σ and h > 0 is the entropy of the geodesic flow of (Σ, g). Such asymptotics are obtained by studying L-functions associated to some characters of H 1 (Σ, Z). However our problem is very different in nature; indeed, fixing a constraint in homology boils down to fixing algebraic intersection numbers whereas here we are interested in geometric intersection numbers. This makes L-funtions not well suited for this situation.
In the context of hyperbolic surfaces, Mirzhakani [Mir08,Mir16] computed the asymptotic growth of closed geodesics with prescribed self intersection numbers. Namely, for any k ∈ N, we have where i(γ, γ) denote the self-intersection number of γ (see also [ES16]).
Organization of the paper. The paper is organized as follows. In §2 we introduce some geometrical and dynamical tools. In §3 we introduce the dynamical scattering operator which is a central object in this paper and we compute its flat trace. In §4 we prove a priori bounds on N (n, L). In §5 we use a Tauberian argument to estimate certain quantities. In §6 we prove Theorem 3. In §7 we prove Theorem 4. In §8 we explain how the methods described above apply to the billard problem. In §9 we show that a typical closed geodesic γ satisfies i(γ, γ ) ≈ I (γ) for some I > 0. Finally in §10 we extend the results to the case where we are given more than one closed geodesic.
Acknowledgements. I am grateful to Colin Guillarmou for a lot of insightful discussions and for his careful reading of many versions of the present article. I also thank Frédéric Paulin for his help concerning the geometrical lemma 4.7. This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 725967).

Geometrical preliminaries
We recall here some classical geometrical and dynamical notions, and introduce the Pollicott-Ruelle resonances that arise in our setting.

Structural equations.
Here we recall some classical facts from [ST76, §7.2] about geometry of surfaces. Denote by M = SΣ = {(x, v) ∈ T Σ, v g = 1} the unit tangent bundle of Σ, by X the geodesic vector field on M , that is the generator of the geodesic flow ϕ = (ϕ t ) t∈R of (Σ, g), acting on M . We have the Liouville one-form α on M defined by Then α is a contact form (that is, α ∧ dα is a volume form on M ) and it turns out that X is the Reeb vector field associated to α, meaning that where ι denote the interior product.
We also set β = R * π/2 α where for θ ∈ R, R θ : M → M is the rotation of angle θ in the fibers; finally we denote by ψ the connection one-form, that is the unique one-form on M satisfying where V is the vertical vector field, that is, the vector field generating (R θ ) θ∈R . Then (α, β, ψ) is a global frame of T * M , and we denote H the vector field on M such that (X, H, V ) is the dual frame of (α, β, ψ). We then have the commutation relations where κ is the Gauss curvature of (Σ, g).
2.2. The Anosov property. It is well known [Ano67] that the flow (ϕ t ) has the Anosov property, that is, for any z ∈ M , there is a splitting which depends continuously on z, and with the following property. For any norm · on T M , there exists C, ν > 0 such that In fact E s (z) ⊕ E u (z) = ker α(z) and there exists two continuous functions r ± : M → R such that ±r ± > 0 and Moreover r ± satisfy the Ricatti equation ±Xr ± + r 2 ± + κ • π = 0, where κ is the curvature of Σ.

YANN CHAUBET
We will denote by T * M = E * 0 ⊕ E * s ⊕ E * u the splitting defined by (here the bundle RX is denoted by E 0 ) Note that this decomposition does not coincide with the usual dual decomposition, but it is motivated by the fact that covectors in E * s (resp. E * u ) are exponentially contracted in the future (resp. in the past). Also, we will often consider the symplectic lift of ϕ t , where − denotes the inverse transpose. We have the following lemma (see [DR20, §3.2]).
2.4. Cutting the surface along γ . As mentioned in the introduction, we may see Σ \ γ as the interior of a compact surface Σ with boundary consisting of two copies of γ . By gluing two copies of the annulus U obtained in the preceding subsection on each component of the boundary of Σ , we construct a slightly larger surface Σ δ ⊃ Σ whose boundary is identified with the boundary of U (see Figure 2).
Lemma 2.4. The surface Σ δ has strictly convex boundary, in the sense that the second fundamental form of the boundary ∂Σ δ with respect to its outward normal pointing vector is strictly negative.
The geodesic flow ϕ on M induces a flow on M δ = SΣ δ which we still denote by ϕ. We define Figure 2. The surfaces Σ (on the left) and Σ δ (on the right), in the case where γ is not separating. In Σ, the darker region corresponds to the neighborhood π(U ) of γ .
the first exit times in the future and in the past. We also set and we define the operators R ±,δ (s) by which are well defined whenever Re(s) 1 (note that our convention of R ±,δ (s) differs from [Gui17]). Then (L X ± s)R ±,δ (s) = Id Ω • c (M δ ) , and for any (u Because the boundary of Σ δ is strictly convex, it follows from [DG16, Proposition 6.1] that the family of operators R ± (s) extends to a meromorphic family of operators Here, we denoted where WF is the classical Hörmander wavefront set [Hör90]. Near any s 0 ∈ C, we have the development where Y ±,δ (s) is holomorphic near s = s 0 , and Π ±,δ (s 0 ) is a finite rank projector satisfying where we identified Π ±,δ (h) and its Schwartz kernel.

The scattering operator
In this section we introduce the dynamical scattering operator S ± (s) associated to our problem. By relating the scattering operator to the resolvent described above, we are able to compute its wavefront set. In consequence we obtain that the composition (χS ± (s)) n is well defined for χ ∈ C ∞ c (∂ \ ∂ 0 ), and we give a formula for its flat trace. For each x ∈ ∂Σ , let ν(x) be the normal outward pointing vector to the boundary of Σ , and set 3.1. First definitions. For any z ∈ M = SΣ we define the exit time of z in the future and past by ± (z) = inf{t > 0, ϕ ±t (z) ∈ ∂} and we set Γ ± = {z ∈ M, ∓ (z) = +∞}.
Then Γ + (resp. Γ − ) is the set of points of M which are trapped in the past (resp. in the future). The scattering map S ± : and satisfies S ± • S ∓ = Id ∂ ± \Γ ± . For s ∈ C, the scattering operator is the space of continuous forms on ∂), by declaring that S ± (s)ω(z) = S * ∓ ω(z)e −s ± (z) if z ∈ ∂ ± \ Γ ± and S ± (s)ω(z) = 0 otherwise. Indeed, this follows from the fact that there is C > 0 such that where ω ∞ is the uniform norm on C 0 (M, • T * M ).
3.2. The scattering operator via the resolvent. In this paragraph we will see that S ± (s) can be computed in terms of the resolvent. More precisely, we have the following result.
Proposition 3.2. For any Re(s) large enough we have An immediate consequence is the Corollary 6. The scattering operator → S ± (s) : Ω • (∂) → D • (∂) extends as a meromorphic family of s ∈ C with poles of finite rank, with poles contained in the set of Pollicott-Ruelle resonances of L X , that is, the set of poles of s → R ±,δ (s).
Before proving Proposition 3.2, we start by an intermediate result.
Remark 3.4. Note that Proposition 3.2 is not a direct consequence of Lemma 3.3. Indeed, the operator Q ε,± = e ±εs ι * ι X ϕ * ∓ε R +,δ (s)ι * could hide some singularities near Γ ± ; Proposition 3.2 tells us that is it not the case, at least for Re(s) large enough.
The existence of such an ε follows from the negativeness of the curvature. Let Then φ is smooth (since supp ψ ∩ Γ − = ∅) and (L X + s)φ = 0, and we have since ι X ψ = 0 and ψ has no support near ∂ +,δ . Now we let Φ : . We can replace X by −X to obtain the desired formula for S + (s), which concludes.
Thenũ is continuous and we claim that f n →ũ in D 0 (∂) when n → +∞. Indeed, notice that .
Looking at the geodesic equation for the metric (2.4), we see that ±X 2 ρ > 0 if ±ρ > 0; thus we may separate each interval [a k (z), b k (z)] into two subintervals on which |ρ | > 0 and change variables to get By (3.1), we have a k (z) kL for any k. Therefore we obtain, since each f n is continuous, by negativeness of the curvature. Thus f n (z) → 0 as n → +∞ for any z ∈ ∂ − , and by dominated convergence we have Thus, up to replacing u by χ − u, we have by Lemma 3.3 By what precedes there is C > 0 such that for any n 1 Summarizing the above facts, we obtain that for n 1 big enough, one has 4Cη.
Thus f n →ũ in D 0 (∂), which concludes the proof.
3.3. Composing the scattering maps. Recall that ∂ has two connected components ∂ (1) and ∂ (2) that we can identify in a natural way. We denote by ψ : ∂ → ∂ the map exchanging those components via this identification (in particular ψ(∂ ± ) = ∂ ∓ ), and we setS . Then for any n 1, the composition χS ± (s) Proof. We first prove the lemma for n = 2. According to [Hör90, Theorem 8.2.14], it suffices to show that where ∆ ε , Υ ε are defined in the proof of Lemma 2.5. As χ is supported far from ∂ 0 , we have (ϕ ε (z ), z ) / ∈ ∂ × ∂ for any z ∈ supp χ, and for any η ∈ T * z M δ such that X(z ), η = 0, we have , which is not possible. Now we have sin(θ) = 0 for z / ∈ ∂ 0 . As a consequence (3.2) is true, since supp χ ∩ ∂ 0 = ∅. This concludes the case n = 2.
By [Hör90, Theorem 8.2.14] we also have the bound where 0 denote the zero section in T * ∂. This formula gives that the set B 2 defined by and (z , η, z , 0) ∈ WF(χS ± (s)) ∪ B 1 .
Finally, we get, by definition of Υ ε , . This implies that B 2 can intersect E * ∓,∂ only in a trivial way. Indeed, for any t ε and (z, for z ∈ M representing both ϕ t (z) and ψ(ϕ t (z)). Thus A ∩ B 2 = ∅, which shows that χS ± (s) 3 is well defined. We may iterate this process to obtain that (χS ± (s)) n is well defined for every n 1.
3.4. The flat trace of the scattering operator. Let A : Then the flat trace of A is defined as where π j : ∂ × ∂ → ∂ is the projection on the j-th factor (j = 1, 2). In fact we have where tr is the transversal trace of Attiyah-Bott [AB67] and A k is the operator The purpose of this section is to compute the flat trace of S ± (s). In what follows, for any closed geodesic γ : R/Z → Σ, we will denote the set of incidence vectors of γ along γ , and . For any n 1, the operator (χS ± (s)) n has a well defined flat trace and for Re(s) big enough we have where the sum runs over all closed geodesics γ of (Σ, g) (not necessarily primitive) such that i(γ, γ ) = n. Here (γ) is the length of γ and # (γ) its primitive length.
Proof. The proof that the intersection WF (χS ± (s)) n ∩ ∆ is empty is very similar to the arguments we already gave, for example in Lemma 3.5. Since it might be repetitive, we shall omit it.
As L → +∞, the right hand side of (3.10) converges to , since for any closed geodesic γ : It remains to see that i * ∆ K χ,±,n (s), 1 −χ L → 0 as L → +∞. Note that Lemma 3.7 gives (3.13) . Also for any s ∈ C with Re(s) > 0 we have (3.14) Let N ∈ N such that ι * ∆ K χ,±,n (s 0 ) extends as a continuous linear form on C N (∂). Then Lemma 3.7 and (3.13) imply that if Re(s) is large enough, the product e −s ±,n (·) ι * ∆ K χ,±,n (s 0 ) is well defined and This equality is a consequence of (3.14) and Lemma A.1, since we can take s arbitrarily large to make exp(−s ±,n (·)) ∈ C N (∂) for any N > 0.
As a consequence we have the Corollary 7. The function s → η ±,χ,n (s) defined for Re(s) 1 by the right hand side of (3.5) extends to a meromorphic function on the whole complex plane.
To prove Theorem 3 we now want to use a standard Tauberian argument near the first pole of η ±,χ,n to obtain the growth of N (n, L). Indeed, it is known (see §5) that s → R ±,δ (s) has a pole at s = h . However since η ±,χ,n is given by the trace of the restriction to ∂ of R ±,δ , it is not clear a priori that η ±,χ,n will have the right behavior at s = h . However in the next section we obtain some priori bounds on N (n, L); this will imply that η ±,χ,n has indeed a pole at s = h of order n.

A priori bounds on the growth of geodesics with fixed intersection number with γ
The purpose of this section is to get a priori bounds on N (1, L) (and N (2, L) in the case where γ is separating), using Parry-Pollicott's bound for Axiom A flows [PP83].
Choose some point x ∈ γ . Let g the genus of Σ and (a 1 , b 1 , . . . , a g , b g ) the natural basis of generators of Σ, so that the fundamental group of Σ is the finitely presented group given by π 1 (Σ) = a 1 , b 1 , . . . , a g , b g , [a 1 where we set π 1 (Σ) = π 1 (Σ, x ). 4.1. The case γ is not separating. Note that the bound given in Theorem 3 is actually N (1, L) ∼ c e h L . We could obtain a bound of this order with the methods presented in §4.2 ; however the bounds given by Proposition 4.1 are sufficient for our purpose.
Up to applying a diffeomorphism to Σ, we may assume that γ is represented by a g ∈ π 1 (Σ). In particular, Σ is a surface of genus g − 1 with 2 punctures and the fundamental group π 1 (Σ ) = π 1 (Σ , x ) (here x is some choice of point on ∂Σ ) is the free group given by a 1 , b 1 , . . . , a g . Let Σ denote the universal cover of Σ and let x ∈ Σ such that π(x ) = x where π : Σ → Σ is the natural projection. Then π 1 (Σ) acts on Σ by deck transformations and we set where the distance comes from the metric π * g on Σ . Note that if γ [w] denotes the unique geodesic in the free homotopy class of w (which is represented by the conjugacy class [w]), we have (γ [w] ) (w). We also denote wl(w) = inf n 0, α 1 . . . α n = w, α j ∈ {a k , a −1 k , b k , b −1 k , k = 1, . . . , g − 1} ∪ {a g , a −1 g } the word length of an element w ∈ π 1 (Σ ). It follows from the Milnor-Švarc lemma [BH13,Proposition 8.19] that for some constant D > 0 we have Also recall that we have the classical orbital counting (see e.g. [Rob03]) for some A > 0, where h is the topological entropy of the geodesic flow (Σ , g) restricted to the trapped set (see the introduction).
Lemma 4.2. Take w, w ∈ π 1 (Σ ). Then [b g w] = [b g w ] (as conjugacy classes of π 1 (Σ)) if and only if w = b −1 g a n g b g w a −n g in π 1 (Σ) for some n ∈ Z.
Proof. If w = b −1 g a n g b g w a −n g , then clearly b g w and b g w are conjugated in π 1 (Σ, x ). Reciprocally, assume that [b g w] = [b g w ], and take smooth paths γ and γ representing b g w and b g w . Then there is a smooth homotopy H : [0, 1] × R/Z → Σ such that H(0, ·) = γ and H(1, ·) = γ . We may assume that H is transversal to γ so that H −1 (γ ) is a smooth submanifold of [0, 1] × R/Z. It is clear that we may deform a little bit the paths γ and γ (in π 1 (Σ, x )) so that γ and γ intersect transversaly γ exactly once, so that H It is immediate to check thatH realizes an homotopy between γ and γ withH(s, 0) ∈ γ for any s ∈ [0, 1]. Thus, we obtain b g w = a −n g b g w a n g for some n ∈ Z.
Proof of Proposition 4.1. In what follows, C is a constant that may change at each line. For any w ∈ π 1 (Σ ) and n ∈ Z, we have by (4.1) (a n g b g w a −n g ) In particular, for any L and w such that (w ) L, we have n ∈ Z, (a n g b g w a −n g ) L CL + C. Now for w ∈ π 1 (Σ ) set C w = {a n g b g wa −n g , n ∈ Z} ⊂ π 1 (Σ ) and denote by C the set of such classes. For C ∈ C we set (C) = inf w∈C (w). Now by Lemma 4.2 we have an injective map where P 1 denotes the set of primitive geodesics γ such that i(γ, γ ) = 1. In particular we get with (4.3) and (4.2) which concludes the proof.
4.1.2. Upper bound. Each γ ∈ P 1 with (γ) L lies in the free homotopy class of b ±1 g w for some w ∈ π 1 (Σ , x ) and (w) L + C. In particular (4.2) gives the bound for large L. Now let γ ∈ P 2 with (γ) L. Then γ is in the conjugacy class of some concatenation b ±1 g w b ±1 g w , where w , w ∈ π 1 (Σ ) satisfy (w ) + (w ) L + C. Thus

YANN CHAUBET
we get Iterating this process we finally get, for large L, N (n, L) CL n−1 exp(h L).

4.2.
The case γ is separating. Assume now that γ is separating and write Σ \ γ = Σ 1 Σ 2 where the surfaces Σ j are connected. Up to applying a diffeomorphism to Σ, we may assume that γ represents the class Here g 1 is the genus of the surface Σ 1 , and the genus g 2 of Σ 2 satisfies g 1 + g 2 = g.
For any w ∈ π 1 (Σ), we will denote by [w] its conjugacy class, and γ w the unique geodesic of Σ such that γ w is isotopic to any curve in w (in fact we will often identify [w] and γ w ). Let ( Σ,g) be the universal cover of (Σ, g), and choosex ∈ Σ some lift of x . Then π 1 (Σ) acts as deck transformations on Σ and we will denote (w) = dist Σ (x , wx ), w ∈ π 1 (Σ).
As in the preceding section, we have the orbital counting (see e.g. [Rob03]) #{w j ∈ π 1 (Σ j ), (w j ) L} ∼ A j e h j L , L → ∞, j = 1, 2, for some A 1 , A 2 > 0. 4.2.1. Lower bound. Unlike the case γ not separating, we will need a sharp lower bound. Namely, we prove here the following result.
Proposition 4.3. Assume that γ is separating, and that h 1 = h 2 = h . Then there is C > 0 such that for L large enough, Let us briefly describe the strategy used to prove Proposition 4.3. We denote by P(Σ j ) the set of primitive closed geodesics of Σ j . Then we know that where N j (L) = #{γ ∈ P(Σ j ), (γ) L}. In particular we have for any L large enough for some constant C > 0. Therefore we have for L large enough Now note that (4.5) implies that N (k + 1) − N (k) C e h k h k for any k large enough.
Therefore we get for L large enough (with some different constant C) As a consequence, if given geodesics γ j ∈ P(Σ j ), we are able to construct about (γ 1 ) (γ 2 ) new geodesics of Σ, intersecting γ exactly twice and of length bounded by (γ 1 ) + (γ 2 ), then Proposition 4.3 will follow. An idea would be to choose w j ∈ π 1 (Σ j ) such that [w j ] represents γ j , and to consider the geodesics given by the conjugacy classes [w 2w1 ] wherew j is a cyclic permutation of the word w j (there are about (γ j ) of those). However this process may not be injective (see Lemma 4.6), and so more work is needed. We start by the following lemma, which shows that the described procedure will give us indeed geodesics intersecting γ exactly twice, provided the geodesics γ j are not multiples of γ .
For j = 1, 2 let c j (s), s ∈ [0, 1], be paths contained in γ depending continuously on s and linking T j (s) to x , such that c j (0) = c j . Then the construction ofH shows that and reversing the role of τ 1 and τ 2 in the constructions made above,
Before starting the proof of Proposition 4.3, we state a technical result that will be useful to show that there are not too many elements w j , w j ∈ π 1 (Σ j ) such that [w 2 w 1 ] = [w 2 w 1 ]. For any element w j ∈ π 1 (Σ j ), we denote by ([w j ]) its translation length, that is ([w]) = inf x∈Σ dΣ(x, ww).
Of course this length coincide with the length of γ w .
Lemma 4.7. There exists C > 0 such that the following holds. For any w ∈ π 1 (Σ j ), there exists n w ∈ Z such that Proof. The method presented here was inspired by Frédéric Paulin. We fix j ∈ {1, 2} and denote w = w ,j . First note that if w = w k for some k ∈ Z then the conclusion is clear with n w = −k and C = 0. Next assume that w = w k for any k. In particular w is not the trivial element and is thus hyperbolic. Let (Σ,g) denote the universal cover of (Σ, g) ; then π 1 (Σ) acts as deck transformations on (Σ,g). For any w ∈ π 1 (Σ) \ {1}, we denote by (here z denotes any point inΣ) the two distinct fixed points of w in the boundary at infinity ∂ ∞Σ ofΣ. We also denote by A w the translation axis of w, that is, the unique complete geodesic of (Σ,g) converging towards w + (resp. w − ) in the future (resp. in the past). As i(w, w ) = 0 (since w is not a power of w which represents the boundary of Σ j ), we have A w ∩ A w = ∅. Moreover, w ± / ∈ {w ,− , w ,+ } (indeed if it were the case, then w would be equal to some power of w , as π 1 (Σ) acts properly and discontinuously onΣ).
In a first step, we will assume that the family (w ,− , w − , w + , w ,+ ) is cyclically ordered in ∂ ∞Σ S 1 , and we denote by w ↑ w this property. Consider z ∈ A w and z ∈ A w such that dist(A w , A w ) = dist(z, z ). For any x = y ∈Σ, we denote by [x, y] the unique geodesic segment joining x to y, by (x, y) the unique complete oriented geodesic ray passing through x and y and by (x, y) ± the future and past endpoints of (x, y). Then we claim that the following holds (see Figure 3): (1) For any n ≥ 1 we have dist(z, w n wz) n ([w ]) + ([w]); (2) There is c > 0, independent of w satisfying w ↑ w , such that for any n 1, the angle (taken in [0, π]) between the segments [w −1 w −n z, z] and [z, w n wz] is greater than c (denoted α on Figure) To see that (1) holds, first note that the segment [z, w n wz] intersects [w n z, w n z ], because w ↑ w , and denote by z the intersection point. Then, as [w n z, w n z ] is orthogonal to A w , we have ([z, z ]) n ([w ]). Moreover, as both [w n z, w n z ] and [w n wz , w n wz] are orthogonal to w n A w , we have dist(z , w n wz) ([w]).
We prove (2) as follows. We have a decomposition in connected sets where w ,± ∈ F ± . Then since w ↑ w , we have w n wz ∈ F + for any n 1 (since wz ∈ F 0 ∪ F + ) and thus the angle α w between [z , z] and [z, w n wz] is greater than the angle α z between (z, z ) and the ray joining z to (w z, w z ) + . Now α z only depends on z (and not on w), and we set c = inf y∈Aw α y > 0 (indeed y → α y is continuous and α y = α w y for any y ∈ A w ). As w −1 w −n z ∈ F − , we get (2). Now it is a classical fact from the theory of CAT(−κ) spaces (κ > 0) that the following holds. For c > 0 as above, there is C > 0 such that if η ∈ π 1 (Σ) \ {1} and z ∈Σ satisfy that the angle (taken in [0, π]) between [η −1 z, z] and [z, ηz] is greater or equal than c, then (η) dist(z, ηz) − C (see for example [PPS12, Lemma 2.8]). Applying this to η = w n w, we get with (1) and (2) (4.9) Here C does not depend on w such that w ↑ w . Now note that for n > 0 one has 2 w ↑ w =⇒ w n w ↑ w . (4.10) Moreover, for n > 0 large enough (depending on w), we have 3 Therefore, if n w = inf{n ∈ Z, w n w ↑ w } we have, for any n 0, w n w nw w ↑ w and w −n w nw−1 w ↑ w −1 .
Applying (4.9) with w replaced by w nw w we get Now applying (4.9) with w replaced by w −1 and w replaced by w nw−1 w, we get The Lemma easily follows from the last two estimates, up to changing C and C and replacing n w by n w − 1.
Proof of Proposition 4.3. Let j ∈ {1, 2}. For any primitive geodesic γ j ∈ P(Σ j ), we choose some w γ j ∈ π 1 (Σ j ) such that γ j corresponds to the conjugacy class [w γ j ]. We may assume that wl( As π 1 (Σ j ) is free, the element w γ j is unique up to cyclic permutations. We denote n j (γ j ) = wl([w γ j ]) ; then the Milnor-Švarc lemma implies, for any γ j ∈ P(Σ j ) and which gives (γ j ) Dn j (γ j ) + D, γ j ∈ P(Σ j ). (4.11) Our goal is now the following. Starting from geodesics γ j ∈ P(Σ j ), j = 1, 2, we want to construct about (γ 1 ) (γ 2 ) distinct geodesics in P, by considering the conjugacy classes [w γ 2w γ 1 ] wherew γ j runs over all cyclic permutation of w γ j . However, as explained before, this process may conduct to produce several times the same geodesic in P (recall Lemma 2 Indeed, we have (w n w) + = lim k→+∞ (w n w) k · w + ⊂ [w + , w ,+ ] (the interval joining w + to w ,+ in 4.6) so we are led to show estimates on the growth number of families of geodesics, as follows. For any γ j ∈ P(Σ j ), we define the family of conjugacy classes Here a class [w] is said to be primitive if the closed geodesic corresponding to [w] is primitive. We denote by C j the set of such families, and for each C j ∈ C j we set The minimum exists by Lemma 4.7. We have the following Lemma 4.8. There is C > 0 such that for L big enough, Proof. By Lemma 4.7 we have for any γ j ∈ P(Σ j ) LetÑ j (L) = #{C j ∈ C j , (C j ) L}. Then an Abel transformation gives and thus L k=1Ñ j (k) C exp(h L)/L for large L. On the other hand we have for Therefore, if M is big enough, we have for any L large enough For any C j ∈ C j , we choose some class [w C j ] ∈ C j such that (C j ) = ([w C j ]). Also w C j ∈ π 1 (Σ j ) may be chosen cyclically reduced, meaning that wl(w C j ) = wl([w C j ]). Let W C j ⊂ π 1 (Σ j ) denote the set of cyclic permutations of w C j Then |W C j | = wl([w C j ]) since w C j is primitive (see [LS62]).

YANN CHAUBET
Lemma 4.9. For any C j ∈ C j , there exists a subset W C j ⊂ W C j with and the following property. For any p, q ∈ Z and w ∈ W C j , Proof. We prove the lemma for j = 1. Let C 1 ∈ C 1 ; we set g = g 1 , w = w ,1 and W = W C 1 to simplify notations. For w ∈ W , we will say that w is of type A if (w ) p w(w ) q ∈ W C 1 for some p, q ∈ Z \ (0, 0). If w is of type A, then exactly 2g(|p| + |q|) 2 simplifications occur in the word w = (w ) p w(w ) q , since wl(w ) = 4g. As w = a 1 b 1 a −1 1 b −1 1 · · · a −1 g b −1 g and at least 2 simplifications occur in w', we see that w has necessarily one of the following forms : Denote n = wl(w) and w = u 1 · · · u n with u j ∈ {a k , b k , a −1 k , b −1 k , k = 1, . . . , g}. Set w k = u σ k (1) · · · u σ k (n) for k ∈ N where σ is the permutation sending (1, . . . , n) to (n, 1, . . . , n − 1), so that W = {w k , k = 1, . . . , n}.
Assume that w k is of type A. If w k is of the form (5) or (6), it is clear that w k+1 cannot be of type A. If w k is of the form (3) or (4), and if w k+1 is of type A, w k+1 is necessarily the form (5) or (6), so that w k+2 cannot be of type A. Finally assume that w k is of the form (1) or (2). Then we see that w k+1 cannot be of the form (1) or (2) except if g = 1. Therefore if w k+1 is still of type A and g > 1, it has one of the forms (3), (4), (5) or (6) and w k+2 or w k+3 is not of type A by what precedes. We showed that if g > 1 and w k is of type A, one of the words w k+1 , w k+2 , w k+3 is not of type A. Therefore by denoting W the set of words which are not of type A, the conclusion of the lemma holds. Now suppose g = 1 so that w = a 1 b 1 a −1 1 b −1 1 . If w k is of type A and not of the form (1) or (2), then w k+1 or w k+2 is not of type A by what precedes. Thus we assume that w k is of the form (1) or (2), but not of the form (3), (4), (5) or (6) (such words will be called of type B). In particular, we have w k = · · · u σ k (n) with u σ k (n) = b −1 1 , a −1 1 (as g = 1). Thus, in the word (w ) p w k (w ) q , simplifications can only occur between (w ) p and w, and it is not hard to see that #(O k ) 2 where satisfies the conclusion of the lemma.
Using Lemmas 4.5 and 4.6 we thus obtain that the map is injective. Moreover for any (w 1 , w 2 ) we have ([w 2 w 1 ]) ([w 1 ])+ ([w 2 ])+4 diam Σ+2. Indeed, let γ j ∈ P(Σ j ) denote the unique geodesic corresponding to the class [w j ] for j = 1, 2. Then we may find a smooth curveγ j based at x such thatγ j = w j as elements of π 1 (Σ j ) and (γ j ) ([w j ]) + 2 diam Σ + 1 (for example by removing some appropriate small piece of γ j and link the endpoints of the cutted curve to x ). Thus ([w 2 w 1 ]) (γ 2γ1 ) (γ 1 ) + (γ 2 ) + 4 diam Σ + 2 = ([w 1 ]) + ([w 2 ]) + 4 diam Σ + 2. We thus obtain, with R = 4 diam Σ + 2 and C being a constant changing at each line, , 2 (this follows by Lemma 4.9 and (4.11)) and the fact that (w j ) = (C j ) for any w j ∈ W j , and where By Lemma 4.8 we have and from this it is not hard to see that A(L) Le h L as L → +∞. Moreover, using (4.12) and similar techniques we used to obtain (4.7) (for example by noting that there is C such thatÑ j (L) −Ñ j (L − C) Ce h L /L for any L large enough, whereÑ j (L) = #{C j , (C j ) L}, as it follows from (4.12)) we get for L large enough which concludes the proof of Proposition 4.3.

Upper bound.
Each γ ∈ P 2 with (γ) L is in the conjugacy class w 1 w 2 for some w j ∈ π 1 (Σ j ) with (w 1 ) + (w 2 ) L + C. Therefore (4.2) implies Iterating this process we obtain (with C depending on n)
Here the class w j w εK is identified with the unique geodesic contained in the free homotopy class of w j w εK . Note that Ψ K is well defined: for different choices of c j , we would obtain w k j w j w −k j instead of w j for some k j ∈ Z; however the class w k j w j w −k j w εK coincides with w j w εK . Moreover, the image of Ψ K is indeed contained in P n . Indeed, by similar techniques used to prove Lemma 4.5, one can show that the geodesic [w j w εK ] intersects γ exactly n times, as γ does (adding turns aroung γ does not change the intersection number with γ ).
In particular, if γ : R/ (γ)Z → Σ is a closed geodesic intersecting γ exactly n times, and with at least one intersection angle smaller than η, then we can write γ ∼ w ±K w (4.14) for some w ∈ π 1 (Σ), satisfying (w ) (γ) − K (γ ) + C (for some C > 0 independent of γ). Here a ∼ b means that a is freely homotopic to b. As before, the unique geodesic contained in the free homotopy class of w intersects γ exactly n times (removing some turns around γ does not change the intersection number).
Proof. Because χΠ ±,∂ is of rank one, it follows that tr s ((χΠ ±,∂ ) n ) = c ± (χ) n for any n 1 (since the flat trace of finite rank operator coincide with its usual trace) and thus We set η n,χ (s) = tr s ((χS ± (s)) n ), and g n, Now if G n,χ (s) = 0 +∞ g n,χ (t)e −ts dt, a simple computation leads to where the last equality comes from Proposition 3.6. Because one has the expansion η n, Then applying the Tauberian theorem of Delange [Del54, Théorème III], we have Now note that, if P n is the set of primitive closed geodesics γ with i(γ, γ ) = n one has As a consequence we have lim inf t→+∞ N ± (n, χ, t) n!h t (c ± (χ)t) n e h t 1. ( which gives with (5.3) lim sup t→+∞ N ± (n, χ, t) n! (c ± (χ)t) n h t e h t σ.

Proof of theorem 3
In this section we prove Theorem 3. We will apply the asymptotic growth we obtained in the last section to some appropriate sequence of functions in C ∞ c (∂ \ ∂ 0 ). Let Then χ η ∈ C ∞ c (∂ \∂ 0 ) for any η > 0 small. The function χ η forgets about the trajectories passing at distance not greater than η from the "glancing" Sγ .
6.1. The case γ is not separating. Recall from §4 that we have the a priori bounds for L large enough. This estimate implies the following fact 4 : 4 Indeed, if it does not hold, then there is ε > 0 such that for any L 0 > 0 there is L 1 such that for any n 0, it holds ε < N (1, L 1 + nL 0 ) N (1, L 1 + (n + 1)L 0 ) , which gives N (1, L 1 + (n + 1)L 0 )ε n < N (1, L 1 ) for each n. As L 0 can be chosen arbitrarily, we see that (6.1) cannot hold.
6.2.1. The case h 1 = h 2 . In that case recall from §4 that we have the bound for L large enough. In particular, using Lemma 4.10 and §5.2.1 we may proceed exactly as in §6.1 to obtain where c = lim η→0 c ± (χ η ).
6.2.2. The case h 1 = h 2 = h. In that case recall from §4 that we have the bound for L large enough. In particular, using Lemma 4.10 and §5.2.2 we may proceed exactly as in §6.1 to obtain , assuming for simplicity that γ is not separating. By §2.5 we may write, since Π ±,δ (h ) is of rank one by §5.1, , with supp(u ± , s ± ) ⊂ Γ ±,δ and u ± , s ∓ ∈ ker(ι X ). Using the Guillemin trace formula [Gui77] and the Ruelle zeta function ζ Σ , we see that the Bowen-Margulis measure µ 0 (see [Bow72]) of the open system (M δ , ϕ t ), which is given by Bowen's formula On the other hand we have by definition of Π ±,∂ ,
We will now prove Theorem 4 which says that for any f ∈ C ∞ (S γ Σ) the limit exists and defines a probability measure µ n on S γ Σ supported in Γ n . We will also prove that, in the separating case, where c > 0 is the constant appearing in Theorem 3. Note that here we identify f as its lift p * f which is a function on ∂, so that the above formula makes sense (recall that p : SΣ → SΣ is the natural projection which identifies both components of ∂SΣ = ∂).
We have of course such a formula in the non separating case but we omit it here.
Proof of Theorem 4. Let f ∈ C ∞ (S γ Σ). Then reproducing the arguments in the proof of Proposition 3.6, we get for Re(s) big enough, where χ η is defined in §6 and I (γ, χ η ) = I ,± (γ, χ η ) (see §5; this does not depend on ± as the function F used to construct χ η is even). Now we may proceed exactly as in §5, replacing g n,χ (t) by to obtain that the limit (7.1) exists, and is equal to lim η→0 c −n Res s=h tr s (f (χ ηS± (s)) n ) provided γ is separating. Finally, if f ∈ C ∞ c (S γ Σ \ Γ n ) then there is L > 0 such that n (z) L, z ∈ supp(f ).
In particular for any γ ∈ P such that i(γ, γ ) = n and (γ) > L, we have f (z) = 0 for any z ∈ I (γ). This shows that µ n (f ) = 0 and the support condition for µ n follows.

Application to geodesic billards
We prove here Corollary 5. Take (Σ , g ) a compact oriented negatively curved surface with totally geodesic boundary ∂Σ . We can double the surface to obtain a closed surface Σ, g), and the doubled metric g which is smooth outside ∂Σ (it is of class C 3−ε near ∂Σ for every ε > 0). However the geodesic flow on (Σ, g) remains C 1 and Anosov, and one can see that the construction of the scattering operator is still valid in this context 5 , as well as the considerations on its wavefront set. Now ∂Σ is a disjoint union of closed geodesics γ ,1 , . . . , γ ,r , and the two open surfaces Σ , Σ which are the connected components of Σ \ ∂Σ are smooth and have same entropy. Now, instead ofS ± (s) = ψ * • S ± (s), consider where R : ∂ → ∂ is the reflexion according to the Fresnel-Descartes' law. Note that although the geodesic flow is only C 1 , the operatorŜ ± (s) is a weighted version of the transfer operator of the map z → R(S ∓ (z)), which is smooth where it is defined. Thus as in §3 6 , for any χ ∈ C ∞ c (∂ \ ∂ 0 ), we have the trace formula tr s (χŜ ± (s)) n = 2n but here the sum runs over all closed oriented billard trajectories of Σ with n rebounds (here we have a factor 2 since we count each trajectory twice as the manifold is doubled), and B(γ) is the set of inward pointing vectors in ∂ given by the rebounds of γ. Moreover it is clear that, to each oriented periodic billiard trajectory of Σ with two rebounds, correspond exactly two closed geodesics of Σ intersecting exactly twice ∂Σ . The methods given in §4 that led to an priori bound on the number of closed geodesics intersecting exactly two times γ extends in the context of the multicurve (γ ,1 , . . . , γ ,r ) given by ∂Σ , for example by choosing a point x ∈ γ ,1 and composing elements of π 1 (Σ , x ) with elements of π 1 (Σ \ Σ , x ) as in §4. Thus we get an a priori lower bound for the number of closed billiard trajectories with two rebounds and as in §5 the order of the pole of tr s (χ ηŜ± (s)) 2 at s = h (the entropy of the open system (Σ , g)) is exactly 5 Indeed we may embed Σ into a slightly larger smooth surface Σ δ with strictly convex boundary to prove (exactly as before) that the scattering operator S ± (s) (which does not depend on the extension !) extends meromorphically to the whole complex plane. 6 We can check the needed wavefront properties by using the fact that the geodesic flow of the doubled surface is still Anosov, as in §3.

YANN CHAUBET
two for small η, which implies that the pole of tr s (χ ηŜ± (s)) n is exactly n for every n (as the residue ofŜ ± (s) at s = h is of rank one). Thus reproducing the arguments of §6 we get Corollary 5.

A large deviation result
The goal of this last section, which is independent of the rest of this paper, is to prove the following result, which is a consequence of a classical large deviation result by Kifer [Kif94].
Proposition 9.1. There exists I > 0 such that the following holds. For any ε > 0, there is C, δ > 0 such that for large L In fact, I = 4i(m, δ γ ) where i is the Bonahon's intersection form [Bon86], δ γ is the Dirac measure on γ in andm is the renormalized Bowen-Margulis measure on M (here we see the intersection form as a function on the space of ϕ-invariant measures on SΣ, as described below). Lalley [Lal96] showed a similar result for self-intersection numbers; see also [PS06] for self intersection numbers with prescribed angles. 9.1. Bonahon's intersection form. Let M ϕ (SΣ) be the set of finite positive measures on SΣ invariant by the geodesic flow, endowed with the vague topology. For any closed geodesic γ, we denote by δ γ ∈ M ϕ (SΣ) the Lebesgue measure of γ parametrized by arc length (thus of total mass (γ)). Let µ ∈ M ϕ (SΣ) be the Liouville measure, that is, the measure associated to the volume form 1 2 α ∧ dα.
(ii) Note that the formulae for i(µ, µ) and i(µ, δ γ ) differ from [Bon88] ; it is due to our convention since here the Liouville measure µ corresponds to twice the Liouville current considered in [Bon88].
9.2. Large deviations. For any ν ∈ M ϕ (SΣ) we denote by h(ν) the measure-theoretical entropy of ϕ with respect to ν. Then we have the following result.
Proposition 9.4 (Kifer [Kif94]). Let F ⊂ M 1 ϕ (SΣ) be a closed set, where M 1 ϕ (SΣ) is the set of ϕ-invariant probability measures on SΣ. Then where h is the entropy of the geodesic flow.
10.1. Notations. For any j = 1, . . . , q, we denote by h j > 0 the topological entropy of the open system (Σ j , g| Σ j ), and by B j the set of indexes i such that γ ,i is a boundary component of Σ j . We decompose B j as where S j is the set of indexes i such that γ ,i lies in B j for some j = j, and O j = B j \S j . In fact S j (resp. O j ) is the set of shared (resp. unshared) boundary components of Σ j .
This quantity represents the number of times a curve has to travel through Σ j if it intersects n i times γ ,i .
For any admissible path ω = (u, v) we denote n(ω) = (n 1 , . . . , n r ) where we set n i = #{ , u = i}. An admissible path ω will be called primitive if every non trivial cyclic permutation of ω is distinct from ω. An element n ∈ N r will be called admissible if n = n(ω) for some admissible path ω. For any admissible n ∈ N r we set h n = max{h j , n, Σ j > 0} and d n = h j =hn n, Σ j .
The number h n is the maximum of the entropies encountered by a closed geodesic γ satisfying i(γ, γ ) = n i for i = 1, . . . , r, while d n is the number of times γ will travel through a surface Σ j with h j = h n .
Moreover, as in §5.1, the following holds. For any such that h(v ) = h n(ω) we have for some operatorΠ ±,∂v satisfying that F u +1 χΠ ±,∂v F u is of rank one.