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# Closed geodesics with prescribed intersection numbers

Yann Chaubet

► To cite this version:

Yann Chaubet. Closed geodesics with prescribed intersection numbers. Geometry & Topology Monographs, 2024, 28 (2), pp.701 - 758. 10.2140/gt.2024.28.701. hal-05125193

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https://nantes-universite.hal.science/hal-05125193v1

Submitted on 23 Jun 2025

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Geometry &
Topology

Volume 28 (2024)

Closed geodesics with prescribed intersection numbers

YANN CHAUBET
<figure><img src="image_4.png" /></figure>
# Closed geodesics with prescribed intersection numbers

YANN CHAUBET

Let $(\Sigma, g)$ be a closed oriented negatively curved surface, and fix a simple closed geodesic $\gamma_\star$. We give the asymptotic growth as $L \to +\infty$ of the number of primitive closed geodesics of length less than $L$ intersecting $\gamma_\star$ exactly $n$ times, where $n$ is fixed positive integer. This is done by introducing a dynamical scattering operator associated to the surface with boundary obtained by cutting $\Sigma$ along $\gamma_\star$ and by using the theory of Pollicott–Ruelle resonances for open systems.

37D40

## 1 Introduction

Let $(\Sigma, g)$ be a closed oriented connected negatively curved Riemannian surface, and denote by $\mathcal{P}$ the set of its oriented primitive closed geodesics. For $L > 0$ define

$$N(L) = \#\{\gamma \in \mathcal{P} : \ell(\gamma) \le L\},$$

where, for $\gamma \in \mathcal{P}$, we denote by $\ell(\gamma)$ its length. Then a classical result obtained by Margulis [31] states that

$$N(L) \sim \frac{e^{hL}}{hL} \quad \text{as } L \to \infty,$$

where $h > 0$ is the topological entropy of the geodesic flow of $(\Sigma, g)$.

Our purpose here is to provide a similar asymptotic result for closed geodesics satisfying certain intersection constraints. Namely, let $\gamma_\star$ be a simple closed geodesic of $(\Sigma, g)$. For any $\gamma \in \mathcal{P}$, we denote by $i(\gamma, \gamma_\star)$ the geometric intersection number between $\gamma$ and $\gamma_\star$ (see Section 2.1), and we set

$$N(n, L) = \#\{\gamma \in \mathcal{P} : \ell(\gamma) \le L \text{ and } i(\gamma, \gamma_\star) = n\}.$$

We first state a result assuming $\gamma_\star$ is not separating, in the sense that $\Sigma \setminus \gamma_\star$ is connected.

**Theorem 1** Assume that $\gamma_\star$ is not separating. Then there are $c_\star > 0$ and $h_\star \in ]0, h[$ such that, for any $n \ge 1$,

$$(1-1) \qquad N(n, L) \sim \frac{(c_\star L)^n}{n!} \frac{e^{h_\star L}}{h_\star L} \quad \text{as } L \to \infty.$$

The number $h_\star$ in the above statement is the topological entropy of the geodesic flow $(\varphi_t)$ of $(\Sigma, g)$ when restricted to the trapped set

$$K_\star = \overline{\{(x, v) \in S\Sigma : \pi(\varphi_t(x, v)) \in \Sigma \setminus \gamma_\star \text{ for } t \in \mathbb{R}\}},$$
where the closure is taken in $S\Sigma$ and $\pi: S\Sigma \to \Sigma$ is the natural projection. Also, we provide in Section 7 a description of the constant $c_\star$ in terms of the Pollicott–Ruelle resonant states of the geodesic flow of the compact surface with boundary $\Sigma_\star$ obtained by cutting $\Sigma$ along $\gamma_\star$.

By using a classical large deviation result by Kifer [25] and Bonahon’s intersection form [6], one is able to show that a typical closed geodesic $\gamma$ satisfies $i(\gamma, \gamma_\star) \approx I_\star \ell(\gamma)$ for some $I_\star > 0$ not depending on $\gamma$ (see Proposition 8.1 for a precise statement). In particular, Theorem 1 is a statement about very uncommon closed geodesics.

The asymptotics (1-1) for $n=0$ is well known and follows from the work of Dal’bo [12] and from the growth rate of periodic orbits of axiom A flows obtained by Parry and Pollicott [35] (see Section 2.5). However, to the best of our knowledge, the result is new for $n > 0$. Note that it would be tempting to sum the right-hand side of (1-1) over $n$ in order to recover the asymptotic growth of $N(L)$ — for example, one could hope that $h_\star + c_\star = h$ — but if $L$ is fixed, the left-hand side of (1-1) vanishes whenever $n$ is large enough, and it is very unlikely that such an equality holds.

If $\gamma_\star$ is separating then $i(\gamma, \gamma_\star)$ is even, and we have the following result:

**Theorem 2** Suppose that $\gamma_\star$ separates $\Sigma$ in two surfaces, $\Sigma_1$ and $\Sigma_2$. Let $h_j \in ]0, h[$ denote the entropy of the open system $(\Sigma_j, g|_{\Sigma_j})$ and set $h_\star = \max(h_1, h_2)$. Then there is $c_\star > 0$ such that, for each $n \ge 1$, as $L \to +\infty$,

$$N(2n, L) \sim \begin{cases} \frac{(c_\star L)^n e^{h_\star L}}{n! h_\star L} & \text{if } h_1 \neq h_2, \\ 2 \frac{(c_\star L^2)^n e^{h_\star L}}{(2n)! h_\star L} & \text{if } h_1 = h_2. \end{cases}$$

As before, the number $h_j$ is defined as the topological entropy of the geodesic flow restricted to the trapped set

$$K_j = \{(x, v) \in S\Sigma : \pi(\varphi_t(x, v)) \in \Sigma_j \setminus \gamma_\star \text{ for } t \in \mathbb{R}\},$$

where the closure is taken in $S\Sigma$.

We also have an equidistribution result, as follows. Set

$$\partial_\star = \{(x, v) \in S\Sigma : x \in \gamma_\star\} \quad \text{and} \quad \Gamma = S\gamma_\star \cup \{z \in \partial_\star : \varphi_t(z) \in S\Sigma \setminus \partial_\star \text{ for } t > 0\},$$

where $S\gamma_\star = \{(x, v) \in \partial_\star : v \in T_x\gamma_\star\}$. We define the scattering map $S: \partial_\star \setminus \Gamma \to \partial_\star$ by

$$S(z) = \varphi_{\ell(z)}(z), \quad \ell(z) = \inf\{t > 0 : \varphi_t(z) \in \partial_\star\} \quad \text{for } z \in \partial_\star \setminus \Gamma.$$

For any $n \in \mathbb{N}_{\ge 1}$ we set

$$\Gamma_n = \partial_\star \setminus \{z \in \partial_\star \setminus \Gamma : S^k(z) \in \partial_\star \setminus \Gamma \text{ for } k = 1, \dots, n-1\},$$

which is a closed set of Lebesgue measure zero, and

$$\ell_n(z) = \ell(z) + \dots + \ell(S^{n-1}(z)) \quad \text{for } z \in \partial_\star \setminus \Gamma_n.$$
**Theorem 3** Assume that $\gamma_*$ is not separating and let $n \ge 1$. For any $f \in C^\infty(\partial_*)$, the limit

$$ \lim_{L \to +\infty} \frac{1}{N(n, L)} \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma, \gamma_*) = n}} \frac{1}{\#I_*(\gamma)} \sum_{z \in I_*(\gamma)} f(z) $$

exists, where, for any $\gamma \in \mathcal{P}$, the set $I_*(\gamma) = \{(x, v) \in S\gamma : x \in \gamma_*\}$ consists of the incidence vectors of $\gamma$ along $\gamma_*$. This formula defines a probability measure $\mu_n$ on $\partial_*$, whose support is contained in $\Gamma_n$.

Of course, a similar statement holds even if $\gamma_*$ is separating, though we will not explicitly state it here. As for $c_*$, we will provide a full description of $\mu_n$ in terms of the Pollicott–Ruelle resonant states of the geodesic flow of $(\Sigma_*, g)$ for the resonance $h_*$ in Section 7. Here, as before, $\Sigma_*$ is the compact surface with boundary obtained by cutting $\Sigma$ along $\gamma_*$ (see Section 2.5).

## Strategy of proof

A key ingredient used in the proof of Theorems 1, 2 and 3 is the scattering operator $S(s): C^\infty(\partial_*) \to C^\infty(\partial_* \setminus \Gamma)$, which is defined by

$$ S(s)f(z) = f(S(z))e^{-s\ell(z)} \quad \text{for } z \in \partial_* \setminus \Gamma \text{ and } s \in \mathbb{C}. $$

As a first step (which is of independent interest; see the corollary on page 714), we prove that, for any $\chi \in C_c^\infty(\partial_* \setminus S\gamma_*)$, the family $s \mapsto \chi S(s)\chi$ extends to a meromorphic family of operators $S(s): C^\infty(\partial_*) \to D'(\partial_*)$ on the whole complex plane (here $D'(\partial_*)$ denotes the space of distributions on $\partial_*$), whose poles are contained in the set of Pollicott–Ruelle resonances of the geodesic flow of the surface with boundary $(\Sigma_*, g)$; see Section 2.6 for the definition of those resonances. In this context, the existence of such resonances follows from the work of Dyatlov and Guillarmou [15], and we relate $S(s)$ with the resolvent of the geodesic flow (see Proposition 3.2). By using the microlocal structure of the resolvent of the geodesic flow provided by [15], we are moreover able to prove that the composition $(\chi S(s)\chi)^n$ is well defined for any $n \ge 1$, as well as its superflat trace (meaning that we also look at the action of $S(s)$ on differential forms, see Section 3.4), which reads

$$ (1-2) \qquad \mathrm{tr}_s^b[(\chi S(s)\chi)^n] = n \sum_{i(\gamma, \gamma_*)=n} \frac{\ell^\#(\gamma)}{\ell(\gamma)} e^{-s\ell(\gamma)} \prod_{z \in I_*(\gamma)} \chi^2(z), $$

where the products runs over all closed geodesics (not necessarily primitive) $\gamma$ with $i(\gamma, \gamma_*) = n$, and $\ell^\#(\gamma)$ is the primitive length of $\gamma$. This formula will be obtained by using the Atiyah–Bott trace formula [3], though our scattering map $S$ has singularities that we have to deal with. Furthermore, using a priori bounds on the growth of $N(n, L)$ (obtained in Section 4 by purely geometric techniques coming from the theory of CAT(-1) spaces), we prove that $s \mapsto \mathrm{tr}_s^b[(\chi S(s)\chi)^n]$ has a pole of order $n$ at $s = h_*$ provided that $\chi$ has enough support. For this step, we crucially use the fact that the asymptotics for $N(0, L)$ is already known, although we could recover it by using the modern techniques introduced in [15] without going
through the scattering maps. Finally, letting the support of $1 - \chi$ be very close to $S\gamma_*$, and estimating the growth of geodesics having $n$ intersections with $\gamma_*$ with at least one small angle, we are able to derive Theorems 1 and 2 from a classical Tauberian theorem of Delange [14].

## Related works

As mentioned before, the case $n = 0$ follows from work of Parry and Pollicott [35] which is based on important contributions of Bowen [9; 10], as the geodesic flow on $(\Sigma_*, g)$ can be seen as an axiom A flow; see Lemma 2.5 below and [15, Section 6.1]. For counting results on noncompact Riemann surfaces, see also the works of Sarnak [43], Guillopé [21] or Lalley [27]. We refer to the work of Paulin, Pollicott and Schapira [37] for counting results in more general settings.

We also mention a result by Pollicott [39] which says that, if $(\Sigma, g)$ is of constant curvature $-1$ and if $\gamma_*$ is not separating,

$$ (1-3) \qquad \frac{1}{N(L)} \sum_{\substack{\gamma \in \mathcal{P} \\ \ell(\gamma) \le L}} i(\gamma, \gamma_*) \sim I_* L $$

for some $I_* > 0$. Roughly speaking, this means that the average intersection number between $\gamma_*$ and closed geodesics of length not greater than $L$ is about $I_*L$. We will show that this result also holds in our context (see Section 8.2).

Lalley [26], Pollicott [40] and Anantharaman [1] investigated the asymptotic growth of the number of closed geodesics satisfying some homological constraints (see also Phillips and Sarnak [38] and Katsuda and Sunada [24] for the constant curvature case). They showed that, for any homology class $\xi \in H_1(\Sigma, \mathbb{Z})$,

$$ \#\{\gamma \in \mathcal{P} : \ell(\gamma) \le L \text{ and } [\gamma] = \xi\} \sim Ce^{hL}/L^{g+1} $$

for some $C > 0$ independent of $\xi$, where $g$ is the genus of $\Sigma$ and $h > 0$ is the topological entropy of the geodesic flow of $(\Sigma, g)$. Such asymptotics are obtained by studying $L$-functions associated to some characters of $H_1(\Sigma, \mathbb{Z})$. 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. In particular, $L$-functions are not well suited for this situation.

In the context of hyperbolic surfaces (ie surfaces with constant negative curvature $-1$), Mirzakhani [32; 33] computed the asymptotic growth of closed geodesics with prescribed self-intersection numbers. Namely, for any $k \in \mathbb{N}$,

$$ \#\{\gamma \in \mathcal{P} : \ell(\gamma) \le L \text{ and } i(\gamma, \gamma) = k\} \sim c_k L^{6(g-1)}, $$

where $i(\gamma, \gamma)$ denotes the self-intersection number of $\gamma$; see also Erlandsson and Souto [17].

Note that our scattering map $S$ defined above shares some similarities with the Sinai billiard map [44]. Similarly to the map $S$, which is not defined on the singularity set $\Gamma$, the billiard map is not continuous near some singular set consisting in grazing trajectories. In particular, it is plausible that recent functional analytic techniques developed by Baladi, Demers and Liverani [5] (see also Baladi and Demers [4]), as
the Sinai billiard map could be used to define an intrinsic spectrum of resonances for the transfer operator associated to S (without going through the resolvent of the geodesic flow of SΣ⋆).

We finally mention that the techniques presented herein allow one to obtain the asymptotic growth of closed geodesics for which several intersection numbers (with a family pairwise disjoint simple closed curves) are prescribed. However, such an extension requires more work, and for simplicity we will focus here on the case where we are given only one simple geodesic. The aforementioned generalization will be the subject of subsequent work.

## Organization of the paper

The paper is organized as follows. In Section 2 we introduce some geometric and dynamical tools. In Section 3 we introduce the dynamical scattering operator, which is a central object in this paper, and we compute its flat trace. In Section 4 we prove a priori bounds on N(n, L). In Section 5 we use a Tauberian argument to estimate certain quantities. In Section 6 we prove Theorems 1 and 2. In Section 7 we prove Theorem 3. Finally, in Section 8 we show that a typical closed geodesic γ satisfies $i(γ, γ_⋆) \approx I_⋆ℓ(γ)$ for some $I_⋆ > 0$.

## 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 CAT(-1) spaces and Léo Bénard, Mihajlo Cekić, Malo Jézéquel, Gerhard Knieper, Thibault Lefeuvre, Julien Marché and Gabriel Rivière for helpful comments and discussions. Finally, I warmly thank the referee for numerous remarks and suggestions that led to a significant improvement of this manuscript. This project has received funding from the European Research Council under the European Union's Horizon 2020 research and innovation programme (grant agreement 725967).

# 2 Geometric preliminaries

We recall here some classical geometric and dynamical notions, and introduce the Pollicott-Ruelle resonances that will arise in our situation. Throughout the article, $(\Sigma, g)$ will denote a closed connected oriented Riemannian surface of negative curvature.

## 2.1 Geometric intersection numbers

For any two loops $\alpha, \beta: \mathbb{R}/\mathbb{Z} \to \Sigma$, the geometric intersection number between $\alpha$ and $\beta$ is defined by

$$i(\alpha, \beta) = \inf_{\alpha' \sim \alpha, \beta' \sim \beta} |\alpha \cap \beta|,$$

where the infimum runs over all loops $\alpha'$ and $\beta'$ freely homotopic to $\alpha$ and $\beta$, respectively, and

$$|\alpha \cap \beta| = \{(\tau, \tau') \in (\mathbb{R}/\mathbb{Z})^2 : \alpha(\tau) = \beta(\tau')\}.$$
It is well known that, in every nontrivial free homotopy class of loops $c$, there is a unique oriented closed geodesic $\gamma_c \in c$ which minimizes the length among curves in $c$. In fact, closed geodesics also minimize intersection numbers, as follows:

**Lemma 2.1** Let $\gamma_1$ and $\gamma_2$ be any two nontrivial oriented closed geodesics, and assume that $\gamma_1$ (resp. $\gamma_2$) is not freely homotopic to a power of $\gamma_2$ (resp. $\gamma_1$). Then

$$i(\gamma_1, \gamma_2) = |\gamma_1 \cap \gamma_2|.$$

The above result is rather classical, but for the reader's convenience we provide a proof in Appendix A.

## 2.2 Structural equations

Here we recall some classical facts from [45, Section 7.2] about geometry of surfaces. Denote by $M = S\Sigma = \{(x, v) \in T\Sigma : \|v\|_g = 1\}$ the unit tangent bundle of $\Sigma$, and by $X$ the geodesic vector field on $M$, that is, the generator of the geodesic flow $\varphi = (\varphi_t)_{t \in \mathbb{R}}$ of $(\Sigma, g)$, acting on $M$. The Liouville one-form $\alpha$ on $M$ is defined by

$$\langle \alpha(z), \eta \rangle = \langle d_{(x,v)}\pi(\eta), v \rangle \quad \text{for } z = (x, v) \in M \text{ and } \eta \in T_{(x,v)}M,$$

where $\pi: M \to \Sigma$ is the natural projection. Then $\alpha$ is a contact form (that is, $\alpha \wedge d\alpha$ is a volume form on $M$) and it turns out that $X$ is the Reeb vector field associated to $\alpha$, meaning that

$$\iota_X \alpha = 1 \quad \text{and} \quad \iota_X d\alpha = 0,$$

where $\iota$ denotes the interior product.

We also set $\beta = R_{\pi/2}^*\alpha$, where, for $\theta \in \mathbb{R}$, $R_\theta: M \to M$ is the rotation of angle $\theta$ in the fibers. Finally we denote by $\psi$ the connection one-form, defined as the unique one-form on $M$ satisfying

$$\iota_V \psi = 1, \quad d\alpha = \psi \wedge \beta \quad \text{and} \quad d\beta = -\psi \wedge \alpha,$$

where $V$ is the vertical vector field, that is, the vector field generating $(R_\theta)_{\theta \in \mathbb{R}}$. Then $(\alpha, \beta, \psi)$ is a global frame of $T^*M$, and we denote by $H$ the unique vector field on $M$ such that $(X, H, V)$ is the dual frame of $(\alpha, \beta, \psi)$. We then have the commutation relations

$$[V, X] = H, \quad [V, H] = -X \quad \text{and} \quad [X, H] = (\kappa \circ \pi)V,$$

where $\kappa$ is the Gauss curvature of $(\Sigma, g)$.

## 2.3 The Anosov property

It is known, by the work of Anosov [2], that the flow $(\varphi_t)$ is hyperbolic. That is, for any $z \in M$ there is a $d\varphi_t$-invariant splitting

$$T_z M = \mathbb{R}X(z) \oplus E_s(z) \oplus E_u(z)$$
which depends continuously on $z$, and has the property that, for any norm $\|\cdot\|$ on $TM$, there exist $C, v > 0$ such that

$$ \|d\varphi_t(z)v\| \le Ce^{-vt}\|v\| \quad \text{for } v \in E_s(z), t \ge 0 \text{ and } z \in M, $$

and

$$ \|d\varphi_{-t}(z)v\| \le Ce^{-vt}\|v\| \quad \text{for } v \in E_u(z), t \ge 0 \text{ and } z \in M. $$

In fact, $E_s(z) \oplus E_u(z) = \ker \alpha(z)$ and there exist two continuous functions $r_{\pm}: M \to \mathbb{R}$ such that $\pm r_{\pm} > 0$ and

$$ E_s(z) = \mathbb{R}(H(z) + r_{-}V(z)) \quad \text{and} \quad E_u(z) = \mathbb{R}(H(z) + r_{+}V(z)) \quad \text{for } z \in M. $$

Moreover, the functions $r_{\pm}$ are differentiable along the flow direction, and they satisfy the Riccati equation

$$ \pm Xr_{\pm} + r_{\pm}^2 + \kappa \circ \pi = 0, $$

where $\kappa$ is the curvature of $\Sigma$.

We will denote by $T^*M = E_0^* \oplus E_s^* \oplus E_u^*$ the splitting defined by

$$ E_0^*(E_u \oplus E_s) = 0, \quad E_s^*(E_s \oplus E_0) = 0, \quad E_u^*(E_u \oplus E_0) = 0. $$

(Here the bundle $\mathbb{R}X$ is denoted by $E_0$.) Then we have $E_0^* = \mathbb{R}\alpha$ and

$$ (2-1) \qquad E_s^* = \mathbb{R}(r_{-\beta} - \psi), \quad E_u^* = \mathbb{R}(r_{+\beta} - \psi). $$

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) by the symplectic lift $\Phi_t$ of $\varphi_t$, which is defined by

$$ (2-2) \qquad \Phi_t(z, \xi) = (\varphi_t(z), d\varphi_t(z)^{-\top} \cdot \xi) \quad \text{for } (z, \xi) \in T^*M \text{ and } t \in \mathbb{R}, $$

where $^{-\top}$ denotes the inverse transpose. We have the following lemma:

**Lemma 2.2** [13, Section 3.2] If $t \ne 0$, we have $\iota_V \Phi_t(\beta) \ne 0$ and $\iota_H \Phi_t(\psi) \ne 0$.

## 2.4 A nice system of coordinates

In what follows, we write

$$ \partial_{\star} = \{(x, v) \in M : x \in \gamma_{\star}\} = S\Sigma|_{\gamma_{\star}}. $$

**Lemma 2.3** There exists a tubular neighborhood $U$ of $\partial_{\star}$ in $M$, and coordinates $(\tau, \rho, \theta)$ on $U$ with

$$ U \simeq (\mathbb{R}/\ell_{\star}\mathbb{Z})_{\tau} \times (-\delta, \delta)_{\rho} \times (\mathbb{R}/2\pi\mathbb{Z})_{\theta}, $$

where $\ell_{\star}$ is the length of $\gamma_{\star}$, and such that

$$ |\rho(z)| = \mathrm{dist}_g(\pi(z), \gamma_{\star}) \quad \text{and} \quad S_z \Sigma = \{(\tau(z), \rho(z), \theta) : \theta \in \mathbb{R}/2\pi\mathbb{Z}\} \quad \text{for } z \in U. $$

Moreover, in these coordinates, on $\{\rho = 0\}$,

$$ X = \cos(\theta)\partial_{\tau} + \sin(\theta)\partial_{\rho}, \quad H = -\sin(\theta)\partial_{\tau} + \cos(\theta)\partial_{\rho}, \quad V = \partial_{\theta}, $$

and

$$ \alpha = \cos(\theta) \, d\tau + \sin(\theta) \, d\rho, \quad \beta = -\sin(\theta) \, d\tau + \cos(\theta) \, d\rho, \quad \psi = d\theta. $$
**Proof** For $\tau \in \mathbb{R}/\ell_{\star}Z$ we set $(x_{\tau}, v_{\tau}) = \varphi_{\tau}(\gamma_{\star}(0), \dot{\gamma}_{\star}(0))$. We now define, for $\delta > 0$ small enough,

$$ \Psi(\tau, \rho, \theta) = R_{\theta-\pi/2}\varphi_{\rho}(x_{\tau}, v(x_{\tau})) \quad \text{for } (\tau, \rho, \theta) \in \mathbb{R}/\ell_{\star}Z \times (-\delta, \delta) \times \mathbb{R}/2\pi Z, $$

where $R_{\eta}: S\Sigma \to S\Sigma$ is the rotation of angle $\eta$ and $v(x_{\tau}) = R_{\pi/2}v_{\tau}$. Then $d\Psi(\tau, 0, \theta)$ is injective for any $\tau$ and $\theta$. Indeed, $\partial_{\tau}(\pi \circ \Psi)(\tau, 0, \theta) = v_{\tau}$ and $\partial_{\rho}(\pi \circ \Psi)(\tau, 0, \theta) = v(x_{\tau})$. Thus $d\Psi(\tau, 0, \theta): \mathbb{R}\partial_{\tau} \oplus \mathbb{R}\partial_{\rho} \to T\Sigma$ is injective. Moreover, $\partial_{\theta}(\pi \circ \Psi)(\tau, 0, \theta) = 0$ and $\partial_{\theta}\Psi(\tau, 0, \theta) = V(\Psi(\tau, 0, \theta)) \neq 0$. Thus $d\Psi(\tau, 0, \theta)$ is injective for any $\tau$ and $\theta$, and furthermore, if $\delta > 0$ is small enough, $\Psi: U \to M$ is an immersion. In particular, since $(\tau, \theta) \mapsto \Psi(\tau, 0, \theta)$ is clearly injective, we obtain that $\Psi|_{U}$ is a diffeomorphism onto its image provided that $\delta$ is chosen small enough.

Because $V = \partial_{\theta}$ and $\iota_{V}\alpha = \iota_{V}\beta = 0$, we may write $\alpha(\tau, 0, \theta) = a(\tau, \theta) d\tau + b(\tau, \theta) d\rho$ and $\beta(\tau, 0, \theta) = a'(\tau, \theta) d\tau + b'(\tau, \theta) d\rho$ for some smooth functions $a, a', b$ and $b'$. Now, since $d\alpha = \psi \wedge \beta$, we obtain $\mathcal{L}_{V}\alpha = \iota_{V} d\alpha = \beta$, and similarly $\mathcal{L}_{V}\beta = -\alpha$. Thus, $a' = \partial_{\theta}a, b' = \partial_{\theta}b$ and

$$ \partial_{\theta}^{2}a + a = 0, \quad \partial_{\theta}^{2}b + b = 0. $$

In consequence, $a(\tau, \theta) = a_1(\tau) \cos \theta + a_2(\tau) \sin \theta$ and $b(\tau, \theta) = b_1(\tau) \cos \theta + b_2(\tau) \sin \theta$ for some smooth functions $a_1, a_2, b_1$ and $b_2$. Moreover, by definition of the coordinates $(\tau, \rho, \theta)$, one has

$$ (2-3) \qquad X(\tau, 0, 0) = \partial_{\tau} \quad \text{and} \quad X(\tau, 0, \frac{1}{2}\pi) = \partial_{\rho}. $$

Therefore $a_1 = b_2 = 1$ and $a_2 = b_1 = 0$. We thus get the desired formulae for $\alpha$ and $\beta$. Now, writing $\psi = a'' d\tau + b'' d\rho + d\theta$ and using $\mathcal{L}_{V}\psi = 0$, we obtain $\partial_{\theta}a'' = \partial_{\theta}b'' = 0$. As $\iota_{X}\psi = 0$ we obtain $a'' = b'' = 0$ by (2-3). The formulae for $X$, $H$ and $V$ follow. $\square$

**Remark 2.4** If $\tilde{\partial} = \{\rho = 0\}$, then, for any $z = (\tau, 0, \theta) \in \partial$,

$$ T_z \tilde{\partial} = \mathbb{R}V(z) \oplus \mathbb{R}(\cos(\theta)X(z) - \sin(\theta)H(z)) \quad \text{and} \quad N_z^* \tilde{\partial} = \mathbb{R}(\sin(\theta)\alpha(z) + \cos(\theta)\beta(z)). $$

## 2.5 Cutting the surface along $\gamma_{\star}$

As mentioned in the introduction, we may see $\Sigma \setminus \gamma_{\star}$ as the interior of a compact surface $\Sigma_{\star}$ with boundary consisting of two copies of $\gamma_{\star}$. By gluing two copies of the annulus $U$ obtained in the preceding subsection on each component of the boundary of $\Sigma_{\star}$, we construct a slightly larger surface $\Sigma_{\delta} \supset \Sigma_{\star}$ whose boundary is identified with the boundary of $U$ (see Figure 1).

**Lemma 2.5** The surface $\Sigma_{\delta}$ has strictly convex boundary, in the sense that the second fundamental form of the boundary $\partial\Sigma_{\delta}$ with respect to its outward normal pointing vector is strictly negative.

**Proof** In the coordinates $(\tau, \rho)$ given by Lemma 2.3, the metric $g$ has the form

$$ (2-4) \qquad d\rho^2 + f(\tau, \rho) d\tau^2 $$

for some $f > 0$ satisfying $\partial_{\rho}f(\tau, 0) = 0$. Indeed, if $\nabla$ is the Levi-Civita connection, one has

$$ \frac{d}{d\rho} \langle \partial_{\rho}, \partial_{\tau} \rangle = \langle \nabla_{\partial_{\rho}} \partial_{\rho}, \partial_{\tau} \rangle + \langle \partial_{\rho}, \nabla_{\partial_{\rho}} \partial_{\tau} \rangle = \langle \partial_{\rho}, \nabla_{\partial_{\tau}} \partial_{\rho} \rangle = \frac{1}{2} \frac{d}{d\tau} \langle \partial_{\rho}, \partial_{\rho} \rangle = 0, $$
<figure><img src="image_5.png" /><figcaption>Figure 1: The surfaces Σ (on the left) and Σ<sub>δ</sub> (on the right) in the case where γ<sub>⋆</sub> is not separating. In Σ, the darker region corresponds to the neighborhood π(U) of γ<sub>⋆</sub>.</figcaption></figure>

since ∇<sub>∂<sub>ρ</sub></sub> ∂<sub>ρ</sub> = 0 (indeed, ρ ↦→ (τ, ρ) is a geodesic curve). Thus ⟨∂<sub>τ</sub>, ∂<sub>ρ</sub>⟩ = ⟨∂<sub>τ</sub>, ∂<sub>ρ</sub>⟩|<sub>ρ=0</sub> = 0. In particular, g has the form (2-4) with f(τ, ρ) = ⟨∂<sub>τ</sub>, ∂<sub>τ</sub>⟩, and we have ∂<sub>ρ</sub>f(τ, 0) = ∂<sub>ρ</sub>⟨∂<sub>τ</sub>, ∂<sub>τ</sub>⟩ = 2∂<sub>τ</sub>⟨∂<sub>ρ</sub>, ∂<sub>τ</sub>⟩|<sub>ρ=0</sub> = 0 (indeed, since τ ↦→ (τ, 0) is a geodesic curve, ∇<sub>∂<sub>τ</sub></sub> ∂<sub>τ</sub> = 0 on {ρ = 0}). In those coordinates, the scalar curvature reads

$$ \kappa(\tau, \rho) = \frac{-\partial_{\rho}^{2} f(\tau, \rho)}{f(\tau, \rho)}. $$

As κ < 0, we get ∂<sub>ρ</sub><sup>2</sup> f > 0, which gives ±∂<sub>ρ</sub> f > 0 on {±ρ > 0}. The second fundamental form of ∂Σ<sub>δ</sub> with respect to ∂<sub>ρ</sub> is defined by

$$ \langle \nabla_{\partial_{\tau}} \partial_{\tau}, \partial_{\rho} \rangle = -\frac{1}{2} \partial_{\rho} f(\tau, \rho), $$

which concludes the proof, since ∂<sub>ρ</sub> is outward pointing (resp. inward pointing) on {ρ = δ} (resp. {ρ = −δ}). □

**Lemma 2.6** In the coordinates given by Lemma 2.3,

$$ \pm X^2 \rho > 0 \quad \text{on} \quad \{\pm \rho > 0\}. $$

**Proof** Since, in the coordinates (τ, ρ), the metric g has the form (2-4), the Christoffel symbols of g are given by

$$ \Gamma^\rho_{\rho\rho} = \Gamma^\rho_{\tau\rho} = 0 \quad \text{and} \quad \Gamma^\rho_{\tau\tau} = -\frac{1}{2} \partial_\rho f. $$

In particular, if t ↦→ (τ(t), ρ(t)) is a geodesic path,

$$ \ddot{\rho}(t) - \frac{1}{2} \partial_{\rho} f(\tau(t), \rho(t)) = 0. $$

Because ∂<sub>ρ</sub>f(τ, 0) = 0 and −∂<sub>ρ</sub><sup>2</sup>f/f = κ < 0, we obtain that ±∂<sub>ρ</sub>f > 0 whenever ±ρ > 0. □

## 2.6 The resolvent of the geodesic flow for open systems

In what follows, we denote by Ω<sup>•</sup>(M<sub>δ</sub>) the set of differential forms on M<sub>δ</sub> and by Ω<sub>c</sub><sup>•</sup>(M<sub>δ</sub>) the elements of Ω<sup>•</sup>(M<sub>δ</sub>) whose support is contained in the interior of M<sub>δ</sub>. Here M<sub>δ</sub> = SΣ<sub>δ</sub> is the unit tangent bundle
of $\Sigma_\delta$. The set of currents on $M_\delta$, denoted by $D^\bullet(M_\delta)$, is defined as the topological dual of $\Omega_c^\bullet(M_\delta)$.
Note that we have an inclusion $\Omega^\bullet(M_\delta) \hookrightarrow D^\bullet(M_\delta)$ via the pairing

$$ \langle u, v \rangle = \int_{M_\delta} u \wedge v \quad \text{for } u, v \in \Omega^\bullet(M_\delta). $$

The geodesic flow $\varphi$ on $M$ induces a flow on $M_\delta = S\Sigma_\delta$, which we still denote by $\varphi$. We set

$$ \partial_\pm M_\delta = \{(x, v) \in \partial M_\delta : \pm \langle v, v_\delta(x) \rangle > 0\} \quad \text{and} \quad \partial_0 M_\delta = \{(x, v) \in \partial M_\delta : \pm \langle v, v_\delta(x) \rangle = 0\}, $$

where $v_\delta(x)$ is the unit vector orthogonal to $\partial\Sigma_\delta$, based at $x$, and pointing outward. Next, define

$$ \ell_{\pm, \delta}(z) = \inf\{t > 0 : \varphi_{\pm t}(z) \in \partial M_\delta\} \quad \text{for } z \in \text{int}(M_\delta) \cup \partial_\mp M_\delta, $$

and $\ell_{\pm, \delta}(z) = 0$ for $z \in \partial_\pm M_\delta \cup \partial_0 M_\delta$, where $\text{int}(M_\delta)$ denotes the interior of $M_\delta$. The numbers $\ell_{\pm, \delta}(z)$ are the first exit times of $z$ in the future and in the past. We also set

$$ \Gamma_{\pm, \delta} = \{z \in M_\delta : \ell_{\mp}(z) = +\infty\} \quad \text{and} \quad K_\delta = \Gamma_\delta^+ \cap \Gamma_\delta^-, $$

and we define the operators $R_{\pm, \delta}(s)$ by

$$ (2-5) \qquad R_{\pm, \delta}(s)\omega(z) = \pm \int_0^{\ell_{\mp, \delta}(z)} \varphi_{\mp t}^* \omega(z) e^{-ts} \, dt \quad \text{for } z \in M_\delta \text{ and } \omega \in \Omega_c^\bullet(M_\delta), $$

which are well defined as operators from $\Omega_c^\bullet(M_\delta)$ to $C(M_\delta, \Lambda^\bullet T^*M_\delta)$ whenever $\text{Re}(s) \gg 1$, where $C(M_\delta, \Lambda^\bullet T^*M_\delta)$ denotes the space of continuous differential forms on $M_\delta$. Note that our convention of $R_{\pm, \delta}(s)$ differs from that of [18]. The operator $R_{+, \delta}(s)$ (resp. $R_{-, \delta}(s)$) is the resolvent of $\mathcal{L}_X$ in the future (resp. in the past) for the spectral parameter $s$. More precisely,

$$ (2-6) \qquad (\mathcal{L}_X \pm s) R_{\pm, \delta}(s) = \text{Id}_{\Omega_c^\bullet(M_\delta)}, $$

and for any $(u, v) \in \Omega_c^\bullet(M_\delta \setminus \Gamma_{-, \delta}) \times \Omega_c^\bullet(M_\delta \setminus \Gamma_{+, \delta})$,

$$ (2-7) \qquad \int_{M_\delta} (R_{+,\delta}(s)u) \wedge v = - \int_{M_\delta} u \wedge R_{-,\delta}(s)v. $$

Indeed, for such $u$ and $v$, there is $L > 0$ such that

$$ (2-8) \qquad \text{supp}(u) \subset \{\ell_{+, \delta} \le L\} \quad \text{and} \quad \text{supp}(v) \subset \{\ell_{-, \delta} \le L\}. $$

In particular, the forms $R_{+, \delta}(s)u$ and $R_{-, \delta}(s)v$ are smooth up to the boundary of $M_\delta$. Indeed, (2-8) implies that, for any $z \in M_\delta$ and $t \in [0, \ell_{-, \delta}(z)]$,

$$ \varphi_{-t}^* u(z) \ne 0 \Rightarrow t \le L. $$

Therefore, for any $z \in M_\delta$,

$$ R_{+,\delta}(s)u(z) = \int_0^{\ell_{-,\delta}(z)} \varphi_{-t}^* u(z) e^{-ts} \, dt = \int_0^{\min(\ell_{-,\delta}(z), L+1)} \varphi_{-t}^* u(z) e^{-ts} \, dt, $$

and thus $R_{+,\delta}u$ is smooth, since $\varphi_{-t}^*u(z) = 0$ if $L \le t \le \ell_{-,\delta}(z)$. Similarly, $R_{-,\delta}(s)v$ is smooth. Finally, note that $\text{supp}(R_{+,\delta}(s)u) \cap \partial M_\delta \subset \partial_+ M_\delta$ and $\text{supp}(R_{-,\delta}(s)v) \cap \partial M_\delta \subset \partial_- M_\delta$. In particular, Stokes' formula and (2-6) imply (2-7).
Because the boundary of $\Sigma_\delta$ is strictly convex, it follows from [15, Proposition 6.1] that the family of operators $R_\pm(s)$ extends to a meromorphic family of operators

$$R_{\pm,\delta}(s) : \Omega_c^\bullet(M_\delta) \to D'^\bullet(M_\delta)$$

satisfying

$$(2-9) \qquad \mathrm{WF}'(R_{\pm,\delta}(s)) \subset \Delta(T^*M_\delta) \cup \Upsilon_{\pm,\delta} \cup (E_{\pm,\delta}^* \times E_{\mp,\delta}^*),$$

where $\Delta(T^*M_\delta)$ is the diagonal in $T^*M_\delta \times T^*M_\delta$,

$$\Upsilon_{\pm,\delta} = \{(\Phi_t(z, \xi), (z, \xi)) \in T^*(M_\delta \times M_\delta) : 0 \le \pm t \le \ell_{\pm,\delta}(z) \text{ and } \langle X(z), \xi \rangle = 0\},$$

and where

$$E_{+,\delta}^* = E_u^*|_{\Gamma_+^+}, \quad E_{-,\delta}^* = E_s^*|_{\Gamma_+^-}.$$

Here, we write

$$\mathrm{WF}'(R_{\pm,\delta}(s)) = \{(z, \xi, z', \xi') \in T^*(M_\delta \times M_\delta) : (z, \xi, z', -\xi') \in \mathrm{WF}(R_{\pm,\delta}(s))\},$$

where WF is the classical Hörmander wavefront set [23, Section 8]. In fact, by (2-9) we mean that $s \mapsto R_\pm(s)$ is meromorphic as a map $\mathbb{C} \to D'_{\Gamma_\pm}(M_\delta \times M_\delta)$ — we identify $R_\pm(s)$ and its Schwartz kernel — where $\Gamma_\pm$ is given by the right-hand side of (2-9), $\Gamma'_\pm = \{(z, \xi, z', -\xi') : (z, \xi, z', -\xi') \in \Gamma_\pm\}$, and where

$$D'_{\Gamma_\pm}(M_\delta \times M_\delta) = \{R \in D'(M_\delta \times M_\delta) : \mathrm{WF}(R) \subset \Gamma'_\pm\}$$

is endowed with its natural topology; see [23, Definition 8.2.2].

Near any $s_0 \in \mathbb{C}$, we have the expansion

$$R_{\pm,\delta}(s) = Y_{\pm,\delta}(s) + \sum_{j=1}^{J(s_0)} \frac{(X \pm s_0)^{j-1} \Pi_{\pm,\delta}(s_0)}{(s-s_0)^j},$$

where $Y_{\pm,\delta}(s)$ is holomorphic near $s = s_0$ and $\Pi_{\pm,\delta}(s_0)$ is a finite-rank projector satisfying

$$\mathrm{WF}'(\Pi_{\pm,\delta}(s_0)) \subset E_{\pm,\delta}^* \times E_{\mp,\delta}^* \quad \text{and} \quad \mathrm{supp}(\Pi_{\pm,\delta}(s_0)) \subset \Gamma_\pm^\pm \times \Gamma_\mp^\mp,$$

where we identified $\Pi_{\pm,\delta}(s_0)$ and its Schwartz kernel.

## 2.7 Restriction of the resolvent on the geodesic boundary

For any $\varepsilon > 0$, define the open sets

$$A_{\pm,\varepsilon} = \{\ell_{\pm,\delta} > \varepsilon\} \cap \{\ell_{\mp,\delta} > 0\} \subset \mathrm{int}(M_\delta),$$

and notice that, if $\varepsilon$ is small, $M_\delta/2 \subset A_{\pm,\varepsilon}$. Then we have diffeomorphisms $\varphi_{\pm\varepsilon}: A_{\pm,\varepsilon} \to A_{\mp,\varepsilon}$, which induce maps

$$\varphi_{\pm\varepsilon}^* : D'^\bullet(A_{\mp,\varepsilon}) \to D'^\bullet(A_{\pm,\varepsilon}).$$
Using a slight abuse of notation, we will still denote by $\varphi_{\pm\varepsilon}^* : D'^*(M_\delta) \to D'^*(A_{\pm,\varepsilon})$ the composition of
$\varphi_{\pm\varepsilon}^*$ with the inclusion $D'^*(M_\delta) \hookrightarrow D'^*(A_{\mp,\varepsilon})$, which is given by the restriction. Let

$$
\partial = \partial(S\Sigma_*) = \{(x, v) \in M_\delta : x \in \gamma_* \sqcup \gamma_*\}
$$

and $\partial_0 = S\gamma_* \sqcup S\gamma_* \subset \partial$.

**Lemma 2.7** For any $\varepsilon > 0$ small enough, we have

$$
WF(\varphi_{\pm\varepsilon}^{*} R_{\pm,\delta}(s)) \cap N^{*}(\partial \times \partial) = \emptyset,
$$

where

$$
N^*(\partial \times \partial) = \{(z', \xi', z, \xi) \in T^*(M_\delta \times M_\delta) : \langle \xi', T_{z'}\partial \rangle = \langle \xi, T_z\partial \rangle = 0\}.
$$

**Proof** We prove the statement for $R_{+,\delta}(s)$. By (2-9) and multiplicativity of wavefront sets (see [23, Theorem 8.2.14]),

$$
(2-10) \qquad WF'(\varphi_{-\epsilon}^* R_{+,\delta}(s)) \subset \Delta_{\epsilon} \cup \Upsilon_{+,\delta}^{\epsilon} \cup (E_{+,\delta}^* \times E_{-,\delta}^*),
$$

where

$$
\Delta_{\varepsilon} = \{(\Phi_{\varepsilon}(z, \xi), (z, \xi)) : (z, \xi) \in T^{*}M_{\delta}\}
$$

and

$$
\Upsilon_{+,\delta}^{\epsilon} = \{(\Phi_t(z, \xi), (z, \xi)) : \epsilon \le t \le \ell_{+,\delta}(z), \langle X(z), \xi \rangle = 0\}.
$$

Now assume that there is $\Xi = (z', \xi', z, \xi)$ lying in

$$
N^*(\partial \times \partial) \cap (\Delta_{\varepsilon} \cup \Upsilon_{+,\delta}^{\varepsilon} \cup (E_{+,\delta}^* \times E_{-,\delta}^*)).
$$

If $\Xi \in \Delta_\varepsilon$, then necessarily $z, z' \in \partial_0$, because $\varphi_\varepsilon(\partial \setminus \partial_0) \cap \partial = \emptyset$ whenever $\varepsilon > 0$ is smaller than the
injectivity radius of the manifold.<fnref n="1" /> We thus have $\xi \in N_z^* \partial = \mathbb{R}\beta(z)$ by Remark 2.4; now $\Phi_\varepsilon(\beta(z))$ does
not lie in $\mathbb{R}\beta(\varphi_\varepsilon(z))$ by Lemma 2.2, and therefore $\xi = 0$.

If $\Xi \in \Upsilon_{+,\delta}^{\varepsilon}$, then there is $T \ge \varepsilon$ such that $\Phi_T(z, \xi) = (z', \xi')$ with $\langle \xi, X(z) \rangle = 0$. However, by Remark 2.4, if $(z, \xi) \in N_z^* \partial$ and $\langle \xi, X(z) \rangle = 0$, then $z \in \partial_0$. Thus by what precedes, $\xi = 0$.

Finally, (2-1) and Remark 2.4 imply that $N^*\partial \cap E_{\pm,\delta}^* \subset \{0\}$. Thus we have shown that

$$
WF'(\varphi_{-\epsilon}^{*} R_{+,\delta}(s)) \cap N^{*}(\partial \times \partial) = \emptyset,
$$

which is equivalent to the conclusion of the lemma.<fnref n="2" />

<figure><img src="image_6.png" /></figure>

**Remark 2.8** This estimate together with [23, Theorem 8.2.4] imply that the operator $\iota^* \iota_X \varphi_{\mp\varepsilon}^* R_{\pm,\delta}(s) \iota_*$ is well defined and satisfies

$$
WF(\iota^* \iota_X \varphi_{\mp\varepsilon}^* R_{\pm,\delta}(s) \iota_*) \subset d(\iota \times \iota)^\top WF(\varphi_{\mp\varepsilon}^* R_{\pm,\delta}(s)),
$$

<footnote n="1">¹Let $x \in \partial\Sigma$. If $(x, v) \in \partial \setminus \partial_0$ satisfies that $(y, w) = \varphi_\varepsilon(x, v) \in \partial$, then the exponential map at $x$ is not injective on the closed ball $B(0, \varepsilon) \subset T_x\Sigma$ of radius $\varepsilon$, since $\pi(\varphi_{\varepsilon'}(x, v')) = y$ for some $v' \in S_x\Sigma$ tangent to $\partial\Sigma$ and some $\varepsilon' \in [0, \varepsilon]$. This follows from the fact that $\partial\Sigma$ is totally geodesic.</footnote>

<footnote n="2">²Since the set $\{(z, \xi, z', \xi') : (z, \xi, z', -\xi') \in N^*(\partial \times \partial)\}$ coincides with $N^*(\partial \times \partial)$, we may use WF or WF' interchangeably.</footnote>
where $ι: ∂ ↦→ M_δ$ and $ι × ι: ∂ × ∂ ↦→ M_δ × M_δ$ are the inclusions. Indeed, the Schwartz kernel of $ι^*ι_X ϕ_{∓ε}^* R_{+,δ}(s)ι_*$ coincides with the pullback by $ι × ι$ of the kernel of $ι_X ϕ_{∓ε}^* R_{+,δ}(s)$. It also follows from [23, Theorem 8.2.14] that the operator $ι^*ι_X ϕ_{∓ε}^* R_{+,δ}(s)$ maps

$$ \mathcal{D}_{N^*\partial}^{\bullet}(M_\delta) \to \mathcal{D}^{\bullet}(\partial) $$

continuously.

Here the pushforward $ι_*: Ω^*(\partial) → \mathcal{D}^{\bullet+1}(M_δ)$ is defined as follows. If $u ∈ Ω^k(\partial)$, we define the current $ι_*u ∈ \mathcal{D}^{k+1}(M_δ)$ by

$$ \langle ι_*u, v \rangle = \int_{\partial} u \wedge ι^*v, \quad v ∈ Ω^{n-k-1}(M_δ). $$

# 3 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, the composition $(\chi S_±(s))^n$ is well defined for $χ ∈ C_c^∞(\partial \setminus \partial_0)$, and we give a formula for its flat trace.

For each $x ∈ ∂Σ_*$, let $v(x)$ be the normal outward pointing vector to the boundary of $Σ_*$, and set

$$ \partial_{\pm} = \{(x, v) \in \partial : \pm \langle v(x), v \rangle_g > 0\}. $$

## 3.1 First definitions

We define the exit times in the future and in the past by

$$ \ell_{\pm}(z) = \inf\{t > 0 : \varphi_{\pm t}(z) \in \partial\} \quad \text{for } z \in M \setminus (\partial_{\pm} \cup \partial_0), $$

and we declare that $\ell_{\pm}(z) = \infty$ whenever $z \in \partial_{\pm} \cup \partial_0$. Then we set

$$ \Gamma_{\pm} = \{z \in M : \ell_{\mp}(z) = +\infty\}. $$

The set $\Gamma_+$ (resp. $\Gamma_-$) is the set of points of $M$ which are trapped in the past (resp. in the future). The scattering map $S_±: \partial_{\mp} \setminus \Gamma_{\mp} → \partial_{\pm} \setminus \Gamma_{\pm}$ is defined by

$$ S_{\pm}(z) = \varphi_{\pm \ell_{\pm}(z)}(z) \quad \text{for } z \in \partial_{\mp} \setminus \Gamma_{\mp}, $$

and satisfies $S_± \circ S_\mp = \text{Id}_{\partial_{\pm} \setminus \Gamma_{\pm}}$. For $s \in \mathbb{C}$, the scattering operator

$$ S_{\pm}(s): \Omega_c^{\bullet}(\partial_{\mp} \setminus \Gamma_{\mp}) \to \Omega_c^{\bullet}(\partial_{\pm} \setminus \Gamma_{\pm}) $$

is given by

$$ S_{\pm}(s)\omega = (S_{\mp}^{*}\omega)e^{-s\ell_{\mp}(\cdot)} \quad \text{for } \omega \in \Omega_c^{\bullet}(\partial_{\mp} \setminus \Gamma_{\mp}). $$

**Remark 3.1** If $\text{Re}(s)$ is large enough, $S_±(s)$ extends as a map

$$ C^0(\partial, \wedge^\bullet T^*\partial) \to C^0(\partial, \wedge^\bullet T^*\partial), $$
where $C^0(\partial, \Lambda^\bullet T^*\partial)$ is the space of continuous forms on $\partial$, by declaring that

$$S_{\pm}(s)\omega(z) = S_{\mp}^{*}\omega(z)e^{-s\ell_{\mp}(z)} \quad \text{if } z \in \partial_{\pm} \setminus \Gamma_{\pm}$$

and $S_{\pm}(s)\omega(z) = 0$ otherwise. Indeed, by Lemma 3.8 and (3-16), there is $C > 0$ such that

$$\|S_{\pm}^{*}\omega(z)\| \le C e^{C\ell_{\mp}(z)} \|\omega\|_{\infty} \quad \text{for } z \in \partial_{\pm} \setminus \Gamma_{\pm} \text{ and } \omega \in \Omega^{\bullet}(M),$$

where $\|\omega\|_{\infty}$ is the uniform norm on $C^0(M, \Lambda^\bullet T^*M)$.

## 3.2 The scattering operator via the resolvent

In this section we will see that $S_{\pm}(s)$ can be computed in terms of the resolvent. More precisely, we have the following result:

**Proposition 3.2** For any $\text{Re}(s)$ large enough,

$$S_{\pm}(s) = (-1)^N e^{\pm \varepsilon s} \ell^*_{\ell_X} \varphi_{\mp \varepsilon}^* R_{\pm, \delta}(s) \ell_*$$

as maps $\Omega_c^*(\partial \setminus \partial_0) \to D'^*(\partial)$, where $N: \Omega^*(\partial) \to \mathbb{N}$ is the degree operator. That is, $N(w) = k$ if $w$ is a $k$-form.

As a consequence of this proposition, Remark 2.8 and the continuity of the pullback [23, Theorem 8.2.4],

$$(\ell \times \ell)^*: D'_{\Gamma_{\pm, \varepsilon}}(M_{\delta} \times M_{\delta}) \to D'(\partial \times \partial),$$

where $\Gamma_{\pm, \varepsilon}$ is the right-hand side of (2-10), we get:

**Corollary** The scattering operator $s \mapsto S_{\pm}(s): \Omega^*(\partial \setminus \partial_0) \to D'(\partial)$ extends as a meromorphic family of $s \in \mathbb{C}$ 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 $s \mapsto R_{\pm, \delta}(s)$.

Before proving Proposition 3.2, we start with an intermediate result:

**Lemma 3.3** We have $S_{\pm}(s) = (-1)^N e^{\pm \varepsilon s} \ell^*_{\ell_X} \varphi_{\mp \varepsilon}^* R_{\pm, \delta}(s) \ell_*$ as maps

$$\Omega_c^*(\partial_{\mp} \setminus \Gamma_{\mp}) \to D'^*(\partial_{\pm} \setminus \Gamma_{\pm}).$$

**Remark 3.4** (i) Proposition 3.2 is not a direct consequence of Lemma 3.3. Indeed, the operator $Q_{\varepsilon, \pm}(s) = (-1)^N e^{\pm \varepsilon s} \ell^*_{\ell_X} \varphi_{\mp \varepsilon}^* R_{\pm, \delta}(s) \ell_*$ could hide some singularities near $\Gamma_{\pm}$; Proposition 3.2 tells us that this is not the case, at least far from $\partial_0$.

(ii) A consequence of Proposition 3.2 is that $Q_{\varepsilon, \pm}(s)$ is identically zero on $\partial_{\pm}$ (in the sense that $Q_{\varepsilon, \pm}(s)u = 0$ whenever $\text{supp}(u) \subset \partial_{\pm}$), as is the case for $S_{\pm}(s)$. This can be seen directly from using the fact that

$$\text{supp}(\varphi_{\mp \varepsilon}^* R_{\pm, \delta}(s) \ell_* u) \subset \{\varphi_t(z) : z \in \text{supp}(u) \text{ and } \varepsilon \le \pm t \le \ell_{\pm, \delta}(z)\}.$$
**Proof** Let $u \in \Omega_c^*(\partial_- \setminus \Gamma_-)$, and $U' \subset \partial_-$ be a neighborhood of supp $u$ such that $\bar{U}'$ does not intersect $\partial_0$. Let $\varepsilon > 0$ be small enough that

$$z \in \partial_- \implies \ell_+(z) > \varepsilon.$$

The existence of such an $\varepsilon$ follows from the fact that, for each $x \in \partial\Sigma$, the exponential map $\exp_x: T_x\Sigma \to \Sigma$ is injective on $B(0, \varepsilon) \subset T_x\Sigma$ whenever $\varepsilon > 0$ is small enough (independent of $x$). Note also that, for every $z \in \partial_-$,

$$\pi(\varphi_t(z)) \in \Sigma_\delta \setminus \Sigma_\star \quad \text{for } -\ell_{-,\delta}(z) < t < 0,$$

by Lemma 2.6. Next, let us set

$$U = \{(t, z) \in \mathbb{R} \times U' : -\ell_{-, \delta}(z) < t < \varepsilon\}.$$

Then $U$ is diffeomorphic to a tubular neighborhood of $U'$ in $M_\delta$ via $(t, z) \mapsto \varphi_t(z)$.<fnref n="3" /> Let $\chi \in C^\infty(\mathbb{R})$ be such that $\chi \equiv 1$ near $]-\infty, 0]$ and $\chi \equiv 0$ on $]\frac{1}{2}\varepsilon, +\infty[$. Set, in the above coordinates,

$$\psi(t, z) = \chi(t)e^{-ts}u(z) \in \wedge^\bullet T^*_{(t,z)}M_\delta,$$

where we see $u(z)$ as a form in $T^*_{(t,z)}M$ by declaring $\iota_{\partial_t}u(z) = 0$. We extend $\psi$ by 0 on $M$, and we set

$$\phi = \psi - R_{+,\delta}(s)(\mathcal{L}_X + s)\psi.$$

Then $\phi$ is smooth by (2-5), since $\text{supp } \psi \cap \Gamma_- = \emptyset$. Moreover $(\mathcal{L}_X + s)\phi = 0$, and we have

$$\phi|_{\partial_-} = u \quad \text{and} \quad \phi|_{\partial_+} = S_+(s)u,$$

where $S_+(s) = S_+(s)|_{\Omega_c^*(\partial_- \setminus \Gamma_-)}$. Let $h \in \Omega_c^*(M_\delta \setminus \Gamma_{+,\delta})$, so that $R_{-,\delta}(s)h$ is smooth (see the discussion following (2-7)). We have, by (2-6) and (2-7),

$$\begin{align*}
\int_{M_\delta} \phi \wedge h &= \int_{M_\delta} \psi \wedge h - \int_{M_\delta} R_{+,\delta}(s)(\mathcal{L}_X + s)\psi \wedge h = \int_{M_\delta} \psi \wedge h + \int_{M_\delta} (\mathcal{L}_X + s)\psi \wedge R_{-,\delta}(s)h \\
&= \int_{M_\delta} \psi \wedge h - \int_{M_\delta} \psi \wedge (\mathcal{L}_X - s)R_{-,\delta}(s)h + \int_{\partial M_\delta} \iota_X(\psi \wedge R_{-,\delta}(s)h) \\
&= \int_{\partial M_\delta} \iota_X(\psi \wedge R_{-,\delta}(s)h) = (-1)^{\deg \psi} \int_{\partial_{-,\delta}} \psi \wedge \iota_X R_{-,\delta}(s)h,
\end{align*}$$

since $\iota_X \psi = 0$ and $\psi$ has no support near $\partial_{+,\delta}$. Now we let $\Phi: \partial_- \to \partial_{-,\delta}$ be defined by $\Phi(z) = \varphi_{-\ell_{-,\delta}(z)}(z)$. Assume that the support of $h$ does not intersect $U$. Then a change of variable gives

$$\Phi^*(\iota_X R_{-,\delta}(s)h)|_{\partial_{-,\delta}} = \iota_X R_{-,\delta}(s)h e^{-s\ell_{-,\delta}(\cdot)}.$$

As we have $\Phi^*(\psi|_{\partial_{-,\delta}}) = (\psi|_{\partial_-})e^{+s\ell_{-,\delta}(\cdot)} = ue^{+s\ell_{-,\delta}(\cdot)}$ by definition of $\psi$, we obtain

$$(3-1) \qquad \int_{M_\delta} \phi \wedge h = (-1)^{\deg u} \int_{\partial_-} u \wedge \iota^*(\iota_X R_{-,\delta}(s)h).$$

Now because $(\mathcal{L}_X - s)R_{-,\delta}(s)h = h$, we get $(\mathcal{L}_X - s)R_{-,\delta}(s)h = 0$ near $U$, and thus $\varphi_\varepsilon^* R_{-,\delta}(s)h = e^{\varepsilon s} R_{-,\delta}(s)h$ near $U$. Let $v \in \Omega_c^*(\partial_+ \setminus \Gamma_+)$. Then $\bar{U} \cap \text{supp}(v) = \emptyset$ (because $\text{supp}(v) \subset \partial_+ \setminus \Gamma_+$). As

<footnote n="3">³The map $G: (t, z) \mapsto \varphi_t(z)$ is clearly smooth on $U$. By Lemma 2.6, $t \mapsto \rho(\varphi_t(z))$ is strictly increasing for $z \in \partial_-$. Therefore, by uniqueness of the integral curves of $X$, we see that $G$ is injective. The inverse of $G$ is given by $G^{-1}(z') = (t(z'), z(z'))$, where $t(z') = \inf\{t \ge 0 : \varphi_t(z') \in \partial\}$ and $z(z') = \varphi_{-t(z')}(z')$, which is smooth on $G(U)$ by the implicit function theorem.</footnote>
WF($l_*v$) $\subset N^*\partial$, we may find $h_n \in \Omega_c^*(M_\delta \setminus \Gamma_+, \delta)$, for $n \in \mathbb{N}$, such that $h_n \to l_*v$ in $D_{N^*\partial}^*(M_\delta)$, and with the property that $\text{supp}(h_n) \cap \bar{U} = \emptyset$.<fnref n="4" /> Then applying (3-1) to $h = h_n$ and letting $n \to \infty$ yields<fnref n="5" />

$$\int_{\partial_+} (S_+(s)u) \wedge v = (-1)^{\deg u} e^{-\epsilon s} \int_{\partial_-} u \wedge l_*^*l_X \varphi_\epsilon^* R_{-,\delta}(s)l_*v,$$

because $\phi|_{\partial_+} = S_+(s)u$. Since $\int_{\partial_+} S_+(s)u \wedge v = \int_{\partial_-} u \wedge S_-(s)v$, we obtain

$$S_-(s) = (-1)^{\deg u} e^{-\epsilon s} l_*^*l_X \varphi_\epsilon^* R_{-,\delta}(s)l_*$$

as maps $\Omega_c^*(\partial_+ \setminus \Gamma_+) \to \Omega_c^*(\partial_- \setminus \Gamma_-)$. We can replace $X$ by $-X$ to obtain the desired formula for $S_+(s)$. □

**Proof of Proposition 3.2** Let $u \in \Omega^*(\partial \setminus \partial_0)$ and write $u = u(\tau, \theta) \in T^*_{(\tau,\theta)}\partial$. Let $\chi \in C_c^\infty(\mathbb{R}, [0, 1])$ be such that $\int_{\mathbb{R}} \chi = 1$, $\chi(0) \neq 0$, $\chi \equiv 0$ on $\mathbb{R} \setminus [-\frac{1}{2}\delta, \frac{1}{2}\delta]$ and $\chi > 0$ on $[-\frac{1}{2}\delta, \frac{1}{2}\delta]$. For $n \in \mathbb{N}_{\ge 1}$ we set $\chi_n = n\chi(n\cdot)$, so that $\chi_n$ converges to the Dirac measure on $\mathbb{R}$ as $n \to +\infty$. We define $u_n \in \Omega_c^*(M_\delta)$ in the $(\tau, \rho, \theta)$ coordinates by

$$u_n = \chi_n(\rho)u(\tau, \theta) \wedge d\rho.$$

Then $u_n \to (-1)^N l_*u$ in $D_{N^*\partial}^*(M_\delta)$, since $\partial = \{\rho = 0\}$. In particular, setting

$$f_n = l_*^* \varphi_{-\epsilon}^* l_X R_{+,\delta}(s) u_n \quad \text{for } n \ge 1,$$

Remark 2.8 gives that $f_n \to (-1)^N l_*^* \varphi_{-\epsilon}^* l_X R_{+,\delta}(s)l_*u$ in $D^*(\partial)$. Moreover, if $\text{Re}(s)$ is large enough, then for any $n \in \mathbb{N}$, we have $(-1)^N l_*^* \varphi_{-\epsilon}^* l_X R_{+,\delta}(s)u_n \in C^0(M_\delta, \wedge^\bullet T^*M_\delta)$ and thus $f_n \in C^0(\partial, \wedge^\bullet T^*\partial)$. Then we claim that $f_n \to S_+(s)u$ is in $D^*(\partial \setminus \partial_0)$ when $n \to +\infty$, where we recall that

$$S_+(s)u(z) = \begin{cases} S_-^*u(z)e^{-s\ell_-(z)} & \text{if } z \in \partial_+ \setminus \Gamma_+, \\ 0 & \text{if not.} \end{cases}$$

Let $F = \{|\rho| \le \frac{1}{2}\delta\}$. Since the neighborhood $\{|\rho| < \frac{1}{2}\delta\}$ is strictly convex, there exists $L > 0$ such that, for any $z \in F$ and $T > 0$ with $\varphi_{-T}(z) \in F$, we have

$$(3-2) \qquad \varphi_{-T}(z) \notin F \quad \text{for all } t \in ]0, T[ \quad \implies \quad T \ge L.$$

Next, take $z \in \partial_+ \setminus \Gamma_+$. Then the set $\{t \in [\epsilon, \ell_{-\delta}(z)] : \varphi_{-t}(z) \in F\}$ is a finite union of closed intervals, say

$$\{t \ge \epsilon : \varphi_{-t}(z) \in F\} = \bigcup_{k=0}^{K(z)} [a_k(z), b_k(z)],$$

with $a_k(z) \le b_k(z) \le +\infty$ and $b_k(z) < a_{k+1}(z)$ for every $k$. We set $\rho(t) = \rho(\varphi_{-t}(z))$ for any $t \ge 0$, and we take any smooth norm $\|\cdot\|$ on $\wedge^\bullet T^*M_\delta$. Note that $u_n = \chi_n(\rho)u_1$. Moreover, if $z \in M_\delta$ and $t < \ell_{-\delta}(z)$, we have

$$(3-3) \qquad \|\varphi_{-t}^* u_1(z)\| \le C \|u_1(\varphi_{-t}z)\| \exp(C|t|)$$

<footnote n="4">⁴For example, we may take $h_n(\rho, \tau, \theta) = \chi_n(\rho)v(\tau, \theta) \wedge d\rho$, where $\chi_n \in C_c^\infty(]-\delta, \delta])$ converges to the Dirac measure.</footnote>

<footnote n="5">⁵Here we use that $l_*^*l_X \varphi_\epsilon^* R_{-,\delta}(s)h_n \to l_*^*l_X \varphi_\epsilon^* R_{-,\delta}(s)l_*v$ in $D^*(\partial)$ as $n \to \infty$ by Remark 2.8, since $h_n \to l_*v$ in $D_{N^*\partial}^*(M_\delta)$.</footnote>
for some $C > 0$. Let $\theta_0 > 0$ small and $h \in C^\infty(M_\delta, [0, 1])$ such that $h = 1$ on supp $u_1$ and

$$ (3-4) \qquad h(\tau, \rho, \theta) = 0 \quad \text{when } \mathrm{dist}(\theta, \pi\mathbb{Z}) < \theta_0. $$

(Such an $h$ exists if $\theta_0$ is small enough, since $u \in \Omega^\bullet(\partial \setminus \partial_0)$.) Then there is $c = c(\theta_0) > 0$ such that $|X\rho| \ge c$ on supp $h$, by Lemma 2.3. In particular, if $\mathrm{Re}(s) > C$, then, by (3-3) and (3-4),

$$ \begin{align*} \|f_n(z)\| &\le \int_{\epsilon}^{\ell_{-\delta}(z)} (\chi_n \circ \rho)(\varphi_{-t}(z)) \|\varphi_{-t}^*(\iota_X u_1)(z)\| e^{-ts} \, dt \\ &\le C \|u\|_{\infty} \sum_{k=0}^{K(z)} e^{(C-s)a_k(z)} \int_{a_k(z)}^{b_k(z)} \chi_n(\rho(t)) h(\varphi_{-t}(z)) \, dt \\ &\le C c^{-1} \|u\|_{\infty} \sum_{k=0}^{K(z)} e^{(C-s)a_k(z)} \int_{a_k(z)}^{b_k(z)} \chi_n(\rho(t)) |X\rho(\varphi_{-t}(z))| \, dt. \end{align*} $$

Of course, for $t < \ell_{-\delta}(z)$, we have $X\rho(\varphi_{-t}(z)) = \rho'(t)$. Moreover, by Lemma 2.6, $\pm X^2\rho > 0$ if $\pm\rho > 0$. Thus we may separate each interval $[a_k(z), b_k(z)]$ into two subintervals on which $|\rho'| > 0$, and change variables to get

$$ \int_{a_k(z)}^{b_k(z)} \chi_n(\rho(t)) |\rho'(t)| \, dt \le 2 \int_{\mathbb{R}} \chi_n(\rho) \, d\rho \le 2. $$

By (3-2), $a_k(z) \ge kL$ for any $k$. Therefore we obtain

$$ (3-5) \qquad \|f_n(z)\| \le \frac{2\|u\|_{\infty}}{1 - e^{(C-\mathrm{Re}(s))L}} \quad \text{for } z \in \partial_+ \setminus \Gamma_+ \text{ and } n \ge 1. $$

Moreover, if $z \in \partial_-$, we have that $t \mapsto \rho(\varphi_{-t}(z))$ is strictly increasing for any $z \in \partial_-$ by Lemma 2.6. Thus we may reproduce the argument made above to obtain that (3-5) also holds for $z \in \partial_-$. Finally, it is shown in [18, Section 2.4] that $\mathrm{Leb}(\Gamma_+ \cap \partial_+) = 0.6$ In particular, since each $f_n$ is a continuous, (3-5) holds for any $z \in (\partial_+ \cup \partial_-) \setminus \Gamma_+ = \partial$.

Next, let $v \in \Omega^\bullet(\partial)$. By Lemma 2.6, the set $\{\varphi_{-t}(z) : t \ge \varepsilon\}$ is included in $\{\rho \ge \rho(\varphi_{-\varepsilon}(z))\}$ for any $z \in \partial_-$. In particular, as $\mathrm{supp}(u_n) \to \partial$ when $n \to \infty$, we have $f_n(z) \to 0$ for $z \in \partial_-$. By dominated convergence we get, as $n \to \infty$,

$$ \int_{\partial_{-}} f_{n} \wedge v \to 0. $$

Next, let $\eta > 0$, and $\chi_\pm \in C_c^\infty(\partial_\pm \setminus \Gamma_\pm)$ such that

$$ (3-6) \qquad \chi_{-} = 1 \quad \text{on } \mathrm{supp}(\chi_{+} \circ S_{+}) \quad \text{and} \quad \mathrm{vol}(\mathrm{supp}(1 - \chi_{+})) < \eta. $$

Such functions exist, as $\mathrm{Leb}(\Gamma_+ \cap \partial) = 0$. We have

$$ \int_{\partial_+} f_n \wedge v = \int_{\partial_+} \chi_+ f_n \wedge v + \int_{\partial_+} (1 - \chi_+) f_n \wedge v. $$

<footnote n="6">⁶Actually, Section 2.4 of [18] says that $\mathrm{Leb}(\Gamma_{+,\delta} \cap \partial_{+,\delta}) = 0$. However, $J_\delta : z \mapsto \varphi_{\ell_{+,\delta}(z)}(z)$ realizes a local diffeomorphism $\partial_+ \to J_\delta(\partial_{+,\delta})$, and we have $J_\delta(\Gamma_+) \subset \Gamma_{+,\delta}$.</footnote>
Note that $f_n = \tilde{f}_n$ on supp $\chi_+$, where $\tilde{f}_n$ is defined exactly as $f_n$, replacing $u$ by $\tilde{u} = \chi_-u \in \Omega^\bullet(\partial_- \setminus \Gamma_-)$.
By Lemma 3.3, $Q_{\varepsilon,+}(s)\tilde{u} = S_+(s)\tilde{u}$, and since $\tilde{f}_n \to Q_{\varepsilon,+}(s)\tilde{u}$, we have

$$ \int_{\partial_+} \chi_+ f_n \wedge v = \int_{\partial_+} \chi_+ \tilde{f}_n \wedge v \to \int_{\partial_+} \chi_+ S_+(s) \tilde{u} \wedge v = \int_{\partial_+} \chi_+ S_+(s) u \wedge v, $$

where we used that $S_+(s)u = S_+(s)\tilde{u}$ on supp $\chi_+$. On the other hand, as the forms $f_n$ are uniformly bounded by (3-5) and the discussion below, there is $C > 0$ such that, for any $n \ge 1$,

$$ \left| \int_{\partial_+} (1 - \chi_+) S_+(s) u \wedge v \right| < C \eta \quad \text{and} \quad \left| \int_{\partial_+} (1 - \chi_+) f_n \wedge v \right| < C \eta, $$

where we used the second part of (3-6). Summarizing the above facts, we obtain that, for $n \ge 1$ big enough,

$$ \left| \int_{\partial} f_n \wedge v - \int_{\partial} S_+(s) u \wedge v \right| \le 4C \eta. $$

Thus, $f_n \to S_+(s)u$ in $D^\bullet(\partial)$. $\square$

### 3.3 Composing the scattering maps

Recall that $\partial$ has two connected components $\partial^{(1)}$ and $\partial^{(2)}$ that we can identify in a natural way. We denote by $\psi: \partial \to \partial$ the map exchanging those components via this identification (in particular, $\psi(\partial_\pm) = \partial_\mp$), and we set

$$ \tilde{S}_{\pm}(s) = \psi^* \circ S_{\pm}(s). $$

Also we denote by $\Psi = T^*\partial \to T^*\partial$ the symplectic lift of $\psi$ to $T^*\partial$; that is,

$$ \Psi(z, \xi) = (\psi(z), d\psi_z^{-\top}\xi) \quad \text{for } (z, \xi) \in T^*\partial. $$

**Lemma 3.5** Let $\chi \in C_c^\infty(\partial \setminus \partial_0)$. Then for any $n \ge 1$, the composition $(\chi \tilde{S}_\pm(s)\chi)^n$, which is well defined from $C^0(\partial, \wedge^\bullet T^*\partial)$ to $C^0(\partial, \wedge^\bullet T^*\partial)$ for Re($s$) large and holomorphic with respect to $s$ by Remark 3.1, admits a meromorphic continuation as a family of operators $\Omega^\bullet(\partial) \to D^\prime(\partial)$.

**Proof** We prove the lemma for $S_+(s)$. First, assume that $n = 2$. According to [23, Theorem 8.2.14], it suffices to show that $A_1 \cap B_1 = \emptyset$, where for $n \ge 1$ we set

$$ \begin{align} A_n &= \{(z, \xi) : (z', 0, z, \xi) \in \mathrm{WF}'((\chi \tilde{S}_{\pm}(s))^n) \text{ for some } z' \in \partial\}, \tag{3-7} \\ B_n &= \{(z, \xi) : (z, \xi, z', 0) \in \mathrm{WF}((\chi \tilde{S}_{\pm}(s))^n) \text{ for some } z' \in \partial\}. \notag \end{align} $$

By Proposition 3.2 and Remark 2.8,

$$ (3-8) \qquad \mathrm{WF}'(\chi S_+(s)\chi)|_{\mathrm{supp}(\chi \times \chi)} \subset d(\iota \times \iota)^\top (\Delta_\varepsilon \cup \Upsilon_{\varepsilon, \delta}^\varepsilon \cup (E_{\varepsilon, \delta}^* \times E_{-\varepsilon, \delta}^*)) $$

where $\Delta_\varepsilon$ and $\Upsilon_{\varepsilon, \delta}^\varepsilon$ are defined as in the proof of Lemma 2.7. Note that in the coordinates of Lemma 2.3, $\iota(z) = (\tau, 0, \theta) \in \partial$ for any $z = (\tau, \theta) \in \partial$, and thus

$$ d\iota^\top(z, \eta) = \eta_\tau d\tau + \eta_\theta d\theta \quad \text{for } \eta = \eta_\tau d\tau + \eta_\rho d\rho + \eta_\theta d\theta \in T_z^*M. $$
As $\chi$ is supported far from $\partial_0$, we have $(\varphi_\epsilon(z'), z') \notin \partial \times \partial$ for any $z' \in \text{supp } \chi$ (see for example Lemma 2.6), and, for any $\eta \in T_{z'}^*M_\delta$ such that $\langle X(z'), \eta \rangle = 0$, we have

$$ (3-9) \qquad \mathrm{d}\iota^\top(z', \eta) = 0 \implies \eta = 0 $$

by Lemma 2.3, since $\partial_0 = \{(\tau, 0, \theta) : \theta \in \pi\mathbb{Z}\}$. This implies that $A_1$ is contained in $E_{-,\partial}^*$, while $B_1$ is contained in $\Psi(E_{+,\partial}^*)$ where $E_{+,\partial}^* = (\mathrm{d}\iota)^\top(E_{+,\delta}^*)$. Now we claim that $\Psi(E_{+,\partial}^*) \cap E_{-,\partial}^* \subset \{0\}$ far from $\partial_0$. By Lemma 2.3 and Section 2.3, for any $z = (\tau, 0, \theta) \in \partial^{(j)} \cap \Gamma_\pm$,

$$ E_{+,\partial}^*(z) = \mathbb{R}(\mathrm{d}\iota)_z^\top (r_+(z)\beta(z) - \psi(z)) = \mathbb{R}(-\sin(\theta)r_+(z) \mathrm{d}\tau - \mathrm{d}\theta), $$

since $\iota(\tau, \theta) = (\tau, 0, \theta)$. Then $r_+(\psi(z)) \neq r_-(z)$ for all $z$. Indeed, the contrary would mean that $E_s(z') \cap E_u(z') \neq \{0\}$ for some $z' \in M$ (represented by both $z$ and $\psi(z)$ in $M_\delta$), which is not possible. Now we have $\sin(\theta) \neq 0$ for $z \notin \partial_0$. As a consequence, (3-7) is true, since $\text{supp } \chi \cap \partial_0 = \emptyset$. This concludes the case $n = 2$, and by [23, Theorem 8.2.14] we also have the bound

$$ \mathrm{WF}'((\chi \tilde{S}_+(s)\chi)^2) \subset (\mathrm{WF}'(\chi \tilde{S}_+(s)\chi) \circ \mathrm{WF}'(\chi \tilde{S}_+(s)\chi)) \cup (B_1 \times \emptyset) \cup (\emptyset \times A_1), $$

where $\emptyset$ denotes the zero section in $T^*\partial$, with $A_1 \subset E_{-,\partial}^*$ and $B_1 \subset \Psi(E_{+,\partial}^*)$, and where, for any conical subsets $\Upsilon_1, \Upsilon_2 \subset T^*(M \times M)$, we write

$$ \Upsilon_1 \circ \Upsilon_2 = \{(x_1, \xi_1, x_2, \xi_2) : (x_1, \xi_1, y, \eta) \in \Upsilon_1 \text{ and } (y, \eta, x_2, \xi_2) \in \Upsilon_2 \text{ for some } (y, \eta)\}. $$

Note that, if we set

$$ E_{s, \partial_\pm}^* = \mathrm{d}\iota^\top(E_s^*|_{\partial_\pm}) \quad \text{and} \quad E_{u, \partial_\pm}^* = \mathrm{d}\iota^\top(E_u^*|_{\partial_\pm}), $$

$$ \text{we have } A_1 \subset E_{s, \partial_-}^* \text{ and } B_1 \subset \Psi(E_{u, \partial_+}^*) = E_{u, \partial_-}^*. $$

We proceed by induction, assuming that, for some $n \ge 2$, the composition $(\chi \tilde{S}_\pm(s))^n$ is well defined with the bound

$$ (3-10) \qquad \mathrm{WF}'((\chi \tilde{S}_+(s))^n) \subset (\mathrm{WF}'(\chi \tilde{S}_+(s)\chi)^{n-1} \circ \mathrm{WF}'(\chi \tilde{S}_+(s)\chi)) \cup (B_{n-1} \times \emptyset) \cup (\emptyset \times A_1), $$

and that $A_{n-1} \subset E_{s, \partial_-}^*$ and $B_{n-1} \subset E_{u, \partial_-}^*$. This formula implies that the set $A_n$ is included in

$$ \{(z, \xi) \in T^*\partial : (z', 0, z'', \eta) \in \mathrm{WF}'((\chi \tilde{S}_+(s)\chi)^{n-1}) \text{ and } (z'', \eta, z, \xi) \in \mathrm{WF}'(\chi \tilde{S}_+(s)\chi) \text{ for some } z', z'' \in \partial\} \cup A_1. $$

We have $A_{n-1} \subset E_{s, \partial_-}^*$, and note that $\Psi(E_{+,\partial}^*) \subset E_{u, \partial_-}^*$ and $E_{u, \partial_-}^* \cap E_{s, \partial_-}^* = \{0\}$. Moreover, as mentioned above, $\varphi_\epsilon(z') \notin \partial$ whenever $z' \in \text{supp}(\chi)$. Thus we obtain, by (3-8),

$$ A_n \subset \{(z, \xi) : (z'', \eta, z, \xi) \in \mathrm{d}(\iota \times \iota)^\top (\Upsilon_{+,\delta}^\epsilon) \text{ for some } \eta \in \Psi(E_{s,\partial_-}^*)\} \cup A_1. $$

Now suppose $(z'', \eta, z, \xi) \in \mathrm{d}(\iota \times \iota)^\top (\Upsilon_{+,\delta}^\epsilon)$ with $z'', z \in \text{supp } \chi$. Note that $\Psi(E_{s,\partial_-}^*) = E_{s,\partial_+}^*$ and thus, if $\eta \in \Psi(E_{s,\partial_-}^*) \cap \mathrm{d}(z'')^\top \ker X(z'')$, then $\eta = \mathrm{d}(z'')^\top \tilde{\eta}$ for some $\tilde{\eta} \in E_s^*(z'')$ by (3-9). Since $E_s^*$ is preserved by $\Phi_{-t}$, we obtain $(z, \xi) \in \mathrm{d}\iota^\top(E_s^*)$. In particular, this yields $A_n \subset E_{s, \partial_-}^*$. Reversing the roles
of $(\chi \tilde{S}_+(s))^{n-1}$ and $\chi \tilde{S}_+(s)$ in (3-10), we get that $B_n$ is included in
$\{(z, \xi) \in T^*\partial : (z, \xi, z', -\eta) \in \text{WF}(\chi \tilde{S}_+(s)\chi) \text{ and } (z', \eta, z'', 0) \in \text{WF}((\chi \tilde{S}_+(s)\chi)^{n-1}) \text{ for some } z', z'' \in \partial\} \cup B_1$.

Proceeding as above, one gets $B_n \subset E_{u,\partial_-}^*$ . Finally, $B_n \cap A_1 = \emptyset$, since $E_{u,\partial_-}^* \cap E_{s,\partial_-}^*$ on supp $\chi$ by (3-9). As a consequence, the composition $(\chi \tilde{S}_+(s)\chi)^{n+1} = (\chi \tilde{S}_+(s)\chi)^n \circ (\chi \tilde{S}_+(s)\chi)$ is well defined by [23, Theorem 8.2.14], and (3-10) holds with $n$ replaced by $n+1$. $\square$

**Remark 3.6** Using (3-10) inductively, one can actually show that WF'((χ̃S+(s)χ)^n) is contained in d(ι̂ × ι̂)⊤Γ_{ε,+}, where

$\tilde{\Gamma}_{\varepsilon,+} = \{(\hat{\Phi}_t(z, \xi), (z, \xi)) : z, \hat{\phi}_t(z) \in S\Sigma|_{\gamma_*} \cap \hat{l}(\text{supp } \chi), \langle X(z), \xi \rangle = 0, t \ge \varepsilon\} \cup (E_u^* \times E_s^*)|_{\text{supp}(\chi \times \chi)}$.
Here (and only here), in order to avoid confusion, we denote by $\hat{\phi}$ (resp. $\hat{\Phi}_t$) the complete geodesic flow on $M = S\Sigma$ (resp. the symplectic lift of the geodesic flow on $T^*M$), and by $\hat{l}: \partial \to S\Sigma|_{\gamma_*} \hookrightarrow M$ the identification of both components of $\partial$.

**3.4 The flat trace of the scattering operator**

Let $A: \Omega^\bullet(\partial) \to \mathcal{D}^\bullet(\partial)$ be an operator such that $WF'(A) \cap \Delta(T^*\partial) = \emptyset$, where $\Delta(T^*\partial)$ is the diagonal in $T^*(\partial \times \partial)$. Then by [23, Theorem 8.2.4], the pullback $\iota_\Delta^* K_A$ is well defined, where $\iota_\Delta: z \mapsto (z, z)$ is the diagonal inclusion and $K_A \in \mathcal{D}^3(\partial \times \partial)$ is the Schwartz kernel of $A$, defined by

$$
\int_{\partial} Au \wedge v = \int_{\partial \times \partial} K_A \wedge \pi_1^* u \wedge \pi_2^* v \quad \text{for } u, v \in \Omega^\bullet(\partial),
$$

where $\pi_j : \partial \times \partial \to \partial$ is the projection on the $j^{th}$ factor (for $j = 1, 2$). We then define the (super)flat trace of $A$ by

$$
-\mathrm{tr}_s^b A = \langle \iota_{\Delta}^* K_A, 1 \rangle.
$$

In fact, one can show that

$$
(3-11) \qquad -\mathrm{tr}_s^b(A) = \sum_{k=0}^{2} (-1)^k \mathrm{tr}^b(A_k),
$$

where tr$^b$ is the transversal trace of Atiyah and Bott [3] and $A_k$ is the operator

$$
A_k : C^\infty(\partial, \wedge^k T^*\partial) \to \mathcal{D}'(\partial, \wedge^k T^*\partial)
$$

induced by $A$ on the space of $k$-forms (see also [16, Section 2.4] for an introduction to the flat trace).

The purpose of this section is to compute the flat trace of $S_{\pm}(s)$. In what follows, for any closed geodesic $\gamma: \mathbb{R}/\ell\mathbb{Z} \to \Sigma$, we will write

$$
I_{\star}(\gamma) = \{z \in S\Sigma|_{\gamma_{\star}} : z = (\gamma(\tau), \dot{\gamma}(\tau)) \text{ for some } \tau \in \mathbb{R}/\ell\mathbb{Z}\}
$$

for the set of incidence vectors of $\gamma$ along $\gamma_*$, and

$$
I_{\star, \pm}(\gamma) = p_{\star}^{-1}(I_{\star}(\gamma)) \cap \partial_{\mp},
$$

where $p_*: S\Sigma_* \to S\Sigma$ is the natural projection.
**Proposition 3.7** Let $\chi \in C_c^\infty(\partial \setminus \partial_0)$. For any $n \ge 1$, the operator $(\chi \tilde{S}_\pm(s))^n$ has a well-defined flat trace, and for $\mathrm{Re}(s)$ big enough,

$$ (3-12) \qquad \mathrm{tr}_s^b((\chi \tilde{S}_\pm(s)\chi)^n) = n \sum_{i(\gamma, \gamma_\star)=n} \frac{\ell^\#(\gamma)}{\ell(\gamma)} e^{-s\ell(\gamma)} \left( \prod_{z \in I_{\star, \pm}(\gamma)} \chi^2(z) \right)^{\ell(\gamma)/\ell^\#(\gamma)}, $$

where the sum runs over all (not necessarily primitive) closed geodesics $\gamma$ of $(\Sigma, g)$ such that $i(\gamma, \gamma_\star) = n$. Here $\ell(\gamma)$ is the length of $\gamma$ and $\ell^\#(\gamma)$ its primitive length.

This formula should be compared with the formula

$$ \mathrm{tr}_s^b((\chi f^*\chi)^n) = \sum_{\gamma \in \mathrm{Per}_n(f)} m^\#(\gamma) \operatorname{sgn}(\det(1-P_\gamma)) \left(\prod_{z \in \gamma} \chi^2(z)\right)^{n/m^\#(\gamma)}, $$

which is valid for any smooth Anosov diffeomorphism $f: Z \to Z$ of a closed manifold $Z$ and $\chi \in C^\infty(Z)$. Here $f^*: C^\infty(Z) \to C^\infty(Z)$ is the pullback operator, $\mathrm{Per}_n(f)$ is the set of $n$-periodic orbits of $f$, $m^\#(\gamma)$ is the minimal period of $\gamma$ and $P_\gamma$ is the linearized Poincaré map of $\gamma$ (that is, $P_\gamma = \mathrm{d}f(z)$ for $z \in \gamma$). Note that the above sum is finite, unlike the sum in (3-12). This is due to the fact that $S_\pm$ is singular at $\Gamma_\pm$, which allows $S_\pm$ to have an infinite number of $n$-periodic points.

**Proof** The proof that the intersection

$$ (3-13) \qquad \mathrm{WF}'((\chi \tilde{S}_\pm(s)\chi)^n) \cap \Delta(T^*\partial) $$

is empty follows from the estimate in Remark 3.6, since $E_u^* \cap E_s^* = \{0\}$ and $\mathrm{d}\hat{l}(z)^\top: \ker X(\hat{l}(z)) \to T_z^*\partial$ is injective for any $z \in \mathrm{supp}(\chi)$.

For any $n \ge 1$, we define the set $\tilde{\Gamma}_\pm^n \subset \partial$ by

$$ \mathcal{C}\tilde{\Gamma}_\pm^n = \{z \in \partial : (\tilde{S}_\pm)^k(z) \text{ is well defined for } k = 1, \dots, n\}, $$

where $\tilde{S} = \psi \circ S$. Equivalently,

$$ \tilde{\Gamma}_\pm^1 = \Gamma_\pm \quad \text{and} \quad \tilde{\Gamma}_\pm^{n+1} = \tilde{\Gamma}_\pm^n \cap (\tilde{S}_\mp)^n (\Gamma_\pm \setminus \tilde{\Gamma}_\mp^n) $$

for $n \ge 1$. Also, we set

$$ (3-14) \qquad \tilde{\ell}_{\pm,n}(z) = \ell_{\pm}(z) + \ell_{\pm}(\tilde{S}_{\pm}(z)) + \dots + \ell_{\pm}(\tilde{S}_{\pm}^{n-1}(z)) \quad \text{for } z \in \mathcal{C}\tilde{\Gamma}_{\pm}^{n}, $$

where $\ell_{\pm}(z) = \inf\{t > 0 : \varphi_{\pm t}(z) \in \partial\}$, with the convention that $\tilde{\ell}_{\pm,n}(z) = +\infty$ if $z \in \tilde{\Gamma}_\pm^n$. We will need the following:

**Lemma 3.8** Let $n \ge 1$. For any $k \ge 1$, there exists $C_{k,n} > 0$ such that

$$ \|d^k \ell_{\pm,n}(z)\| \le C_{k,n} \exp(C_{k,n} \ell_{\pm,n}(z)) \quad \text{for } z \in \mathcal{C}\tilde{\Gamma}_{\pm}^n. $$

**Proof** By induction on $n$, using (3-14) and the fact that $S_\pm(\mathcal{C}\tilde{\Gamma}_\pm^n) = \mathcal{C}\tilde{\Gamma}_\pm^{n-1}$, we see that the lemma reduces to proving the estimate

$$ (3-15) \qquad \|d^k \ell_{\pm}(z)\| \le C_k \exp(C_k \ell_{\pm}(z)) \quad \text{for } z \in \mathcal{C}\tilde{\Gamma}_{\pm}^1. $$
In what follows, $C_k$ is a constant depending only on $k$, which may change at each line. First, notice that $\|d^k\varphi_t(z)\| \le C_k e^{C_k|t|}$ for any $t \in \mathbb{R}$ and $z \in M_\delta$ such that $\varphi_t(z) \in M_\delta$, for some constant $C_k$; see for example [8, Proposition A.4.1]. Moreover,

$$dS_{\pm}(z) = d[\varphi_{\ell_{\pm}}(z)](z) + X(S_{\pm}(z)) d\ell_{\pm}(z) \quad \text{for } z \in \mathbb{C}\tilde{\Gamma}_{\pm}^{1}.$$

By induction we obtain that, for any $k$,

$$(3-16) \quad \|d^k S_{\pm}(z)\| \le C_k \exp(C_k \ell_{\pm}(z)) + C_k \sum_{j=1}^k \|d^j \ell_{\pm}(z)\|^{m_j} \quad \text{with } m_j \in \mathbb{N} \text{ for } j = 1, \dots, k$$

for any $z \in \mathbb{C}\tilde{\Gamma}_{\pm}^{1}$. Let $(\tau, \rho, \theta)$ be the coordinates defined near $\partial$ given by Lemma 2.3. Then $\rho(S_{\pm}(z)) = 0$ for $z \in \tilde{\Gamma}_{\pm}^{1}$, and thus

$$(3-17) \quad (X\rho)(S_{\pm}(z)) d\ell_{\pm}(z) = -d\rho(S_{\pm}(z)) \circ d[\varphi_{\ell_{\pm}}(z)](z) \quad \text{for } z \in \mathbb{C}\tilde{\Gamma}_{\pm}^{1}.$$

Let $z \notin \tilde{\Gamma}_{1}^{\pm}$; Lemma 2.3 gives

$$(3-18) \quad (X\rho)(S_{\pm}(z)) = \sin(\theta(S_{\pm}(z))).$$

Set $z' = S_{\pm}(z)$, and write $(\tau(t), \rho(t)) = \pi(\varphi_{\mp t}(z'))$, so that $\rho(0) = 0$. By the proof of Lemma 2.6, $t \mapsto |\rho(t)|$ is strictly increasing (indeed $z \notin \tilde{\Gamma}_{1}^{\pm}$ and thus $\dot{\rho}(0) = \pm X\rho(z') \ne 0$), and whenever $|\rho(t)| \le \frac{1}{2}\delta$,

$$(3-19) \quad \ddot{\rho}(t) = G(\tau(t), \rho(t))$$

for some smooth function $G \in C^{\infty}((\mathbb{R}/\ell_{\star}\mathbb{Z})_{\tau} \times [-\frac{1}{2}\delta, \frac{1}{2}\delta]_{\rho})$ satisfying $G(\tau, 0) = 0$ and $\partial_{\rho}G(\tau, \rho) > 0$. If $D = \sup|\partial_{\rho}G|$, we have $|G(\tau, \rho)| \le D|\rho|$ and thus $|\ddot{\rho}(t)| \le D|\rho(t)|$, with $\rho(0) = \ddot{\rho}(0) = 0$ and $\dot{\rho}(0) = \pm X\rho(S_{\pm}(z))$. By comparing the solution of (3-19) with the solutions of $\ddot{y}(t) = Dy(t)$, we obtain

$$|\rho(t)| \le |X\rho(z')| \operatorname{sh}(Dt).$$

In particular, $|\rho(t)| < \frac{1}{2}\delta$ whenever $|X\rho(S_{\pm}(z))|\operatorname{sh}(Dt) < \frac{1}{2}\delta$, and thus $\operatorname{sh}(D\ell_{\mp}(z')) \ge \frac{1}{2}\delta|X\rho(z')|$. By (3-18), we conclude that there is $C > 0$ such that

$$(3-20) \quad |\sin(\theta(S_{\pm}(z)))| \ge C \exp(-C\ell_{\pm}(z)) \quad \text{for } z \in \mathbb{C}\tilde{\Gamma}_{\pm}^{1}.$$

We therefore obtain, for any $z \in \tilde{\Gamma}_{1}^{\pm}$,

$$\|d\ell_{\pm}(z)\| \le C^{-1} \exp(C\ell_{\pm}(z)) \|d\rho(S_{\pm}(z))\| \cdot \|d[\varphi_{\ell_{\pm}}(z)](z)\| \le Ce^{C\ell_{\pm}(z)}.$$

Now, repeatedly using (3-16), (3-17) and (3-20), we obtain (3-15) by induction on $k$. $\square$

Consider $\tilde{\chi} \in C^{\infty}(\mathbb{R}, [0, 1])$ such that $\tilde{\chi} \equiv 1$ on $]-\infty, 1]$ and $\tilde{\chi} \equiv 0$ on $[2, +\infty[$, and set $\tilde{\chi}_L(z) = \tilde{\chi}(\ell_{\pm,n}(z) - L)$ for $z \in \partial$. Then $\tilde{\chi}_L \in C_c^{\infty}(\partial \setminus \tilde{\Gamma}_{\pm}^n)$, and by (3-11) we see that the Atiyah-Bott trace formula [3, Corollary 5.4] reads in our case

$$(3-21) \quad \langle i_{\Delta}^{*} K_{\chi, \pm, n}(s), \tilde{\chi}_{L} \rangle = \sum_{(\tilde{S}_{\mp})^{n}(z)=z} e^{-s\ell_{\pm,n}(z)} \tilde{\chi}_{L}(z) \prod_{k=0}^{n-1} \chi^2((\tilde{S}_{\mp})^{k}(z)),$$
where $K_{\chi, \pm, n}(s)$ is the Schwartz kernel of $(\chi S_{\pm}(s))^n$. Indeed, a simple computation (for example in the spirit of [16, Appendix B]<fnref n="7" />) shows that, for any diffeomorphism $f: \partial \to \partial$ with isolated nondegenerate fixed points,

$$ (3-22) \qquad \mathrm{tr}^b(F_k) = \sum_{f(z)=z} \frac{\mathrm{tr} \wedge^k \mathrm{d}f(z)}{|\det(1 - \mathrm{d}f(z))|}, $$

where $F_k: \Omega^k(\partial) \to \Omega^k(\partial)$ is defined by $F_k\omega = f^*\omega$ and $\wedge^k \mathrm{d}f(z)$ is the map induced by $\mathrm{d}f(z)$ on $\wedge^k T_z^*\partial$. Since $\sum_k (-1)^k \mathrm{tr}(\wedge^k \mathrm{d}f(z)) = \det(1 - \mathrm{d}f(z))$, it holds that

$$ (3-23) \qquad \mathrm{tr}_s^b(F) = \sum_k (-1)^{k+1} \mathrm{tr}^b(F_k) = - \sum_{f(z)=z} \mathrm{sgn} \det(1 - \mathrm{d}f(z)). $$

Now note that $\tilde{\chi}_L(\chi \tilde{S}_{\pm}(s)\chi)^n$ is by definition the operator given by

$$ (3-24) \qquad \omega \mapsto \tilde{\chi}_L(\cdot) \left( \prod_{k=1}^{n} (\chi \circ (\tilde{S}_{\mp})^k)(\chi \circ (\tilde{S}_{\mp})^{k-1}) \right) e^{-s \ell_{\pm, n}(\cdot)} (\tilde{S}_{\mp})^{n*} w. $$

Moreover, $\mathrm{sgn} \det(1 - \mathrm{d}(\tilde{S}_{\mp})^n(z)) = -1$ for any $z$ such that $(\tilde{S}_{\mp})^n(z) = z$. Indeed, for such a $z$, $\mathrm{d}(\tilde{S}_{\mp})^n(z)$ is conjugated to the linearized Poincaré map

$$ P_z = \mathrm{d}(\varphi_{\ell_{\pm, n}(z)})(z) |_{E^u(z) \oplus E^s(z)}, $$

which satisfies $\det(1 - P_z) < 0$ as the matrix of $P_z$ in the decomposition $E^u(z) \oplus E^s(z)$ reads $\begin{pmatrix} \lambda & 0 \\ 0 & \lambda^{-1} \end{pmatrix}$ for some $\lambda > 1$ (since $\varphi_t$ preserves the volume form $\alpha \wedge d\alpha$). Finally, by (3-13), the pairing in the left-hand side of (3-21) is well defined; moreover, the proof of (3-22) can be revisited for the operator (3-24) thanks to the introduction of our cutoff functions $\tilde{\chi}_L$ and $\chi$, yielding (3-21).

As $L \to +\infty$, the right-hand side of (3-21) converges to

$$ n \sum_{i(\gamma, \gamma_*)=n} \frac{\ell^\#(\gamma)}{\ell(\gamma)} e^{-s\ell(\gamma)} \left( \prod_{z \in I_{\star, \pm}(\gamma)} \chi^2(z) \right)^{\ell(\gamma)/\ell^\#(\gamma)}, $$

since for any closed geodesic $\gamma: \mathbb{R}/\mathbb{Z} \to \Sigma$ such that $i(\gamma, \gamma_*) = n$,

$$ \#\{z \in \partial : z = (\gamma(\tau), \gamma'(\tau)) \text{ for some } \tau\} = n \frac{\ell^\#(\gamma)}{\ell(\gamma)}. $$

Note that the sum converges whenever $\mathrm{Re}(s)$ is large enough by Margulis' asymptotic formula, given in the introduction. It remains to see that $\langle i_{\Delta}^* K_{\chi, \pm, n}(s), 1 - \tilde{\chi}_L \rangle \to 0$ as $L \to +\infty$. Note that Lemma 3.8 gives

$$ (3-25) \qquad \|d^k \tilde{\chi}_L\| \le C_k e^{C_k L}. $$

By Remark 3.1, if $s_0 > 0$ is large enough, one has $S_{\pm}(s_0): \Omega^\bullet(\partial) \to C^0(\partial, \wedge^\bullet T^*\partial)$. Also, for any $s \in \mathbb{C}$ with $\mathrm{Re}(s) > 0$,

$$ (3-26) \qquad S_{\pm}(s_0 + s)w = (S_{\pm}(s_0)w) e^{-s\ell_{\pm}(\cdot)} \quad \text{for } w \in \Omega^{\bullet}(\partial). $$

<footnote n="7">⁷Actually, in the aforementioned reference, the authors deal with flows, but the diffeomorphism case is even simpler.</footnote>
Let $N \in \mathbb{N}$ such that $i_{\Delta}^{*} K_{\chi, \pm, n}(s_0)$ extends as a continuous linear form on $C^{N}(\partial)$. Then applying Lemma 3.8, we see that if $\mathrm{Re}(s)$ is large enough, the function $\exp(-s\ell_{\pm, n}(\cdot))$ lies in $C^{N}(\partial)$. Thus, the product $e^{-s\ell_{\pm, n}(\cdot)} i_{\Delta}^{*} K_{\chi, \pm, n}(s_0)$ is well defined and by (3-25) we have

$$ \begin{aligned} |\langle e^{-s\ell_{\pm,n}(\cdot)} i_{\Delta}^{*} K_{\chi, \pm,n}(s_0), (1-\tilde{\chi}_L) \rangle| &= |\langle i_{\Delta}^{*} K_{\chi, \pm,n}(s_0), (1-\tilde{\chi}_L) e^{-s\ell_{\pm,n}(\cdot)} \rangle| \\ &\le C \|(1-\tilde{\chi}_L) e^{-s\ell_{\pm,n}(\cdot)}\|_{C^N(\partial)} \le C_N e^{(C_N - \mathrm{Re}(s))L}, \end{aligned} $$

since $\ell_{\pm,n} \ge L$ on $\mathrm{supp}(1-\tilde{\chi}_L)$. Therefore, to obtain that $\langle i_{\Delta}^{*} K_{\chi, \pm,n}(s_0+s), 1-\tilde{\chi}_L \rangle \to 0$ as $L \to +\infty$, it suffices to show that

$$ e^{-s\ell_{\pm,n}(\cdot)} i_{\Delta}^{*} K_{\chi, \pm,n}(s_0) = i_{\Delta}^{*} K_{\chi, \pm,n}(s_0+s). $$

This equality is a consequence of (3-26) and Lemma B.1, since we can take $s$ arbitrarily large. $\square$

Recall from Remark 3.6 that $s \mapsto (\chi \tilde{S}_{\pm}(s)\chi)^n$ admits a meromorphic continuation in $\mathcal{D}_{\Gamma_{\varepsilon, \pm}}^{3'}$ $(\partial \times \partial)$, where $\Gamma_{\varepsilon, \pm}$ does not intersect the conormal to the diagonal in $\partial \times \partial$. In particular:

**Corollary** The function $s \mapsto \eta_{\pm, \chi, n}(s)$ defined for $\mathrm{Re}(s) \gg 1$ by the right-hand side of (3-12) extends to a meromorphic function on the whole complex plane.

To prove Theorem 1, we wish to use a standard Tauberian argument near the first pole of $\eta_{\pm, \chi, n}$ to obtain the growth of $N(n, L)$. Indeed, it is known (see Section 5) that $s \mapsto R_{\pm, \delta}(s)$ has a simple pole at $s = h_{\star}$. However, since $\eta_{\pm, \chi, n}$ is given by the trace of the $n^{\mathrm{th}}$ self-composition of the restriction of $R_{\pm, \delta}$ to $\partial$, it is not clear a priori that $\eta_{\pm, \chi, n}$ will have a singularity at $s = h_{\star}$. In the next section we obtain some a priori bounds on $N(n, L)$; this will imply that $\eta_{\pm, \chi, n}$ indeed has a pole at $s = h_{\star}$, of order $n$.

# 4 A priori bounds on the growth of geodesics with fixed intersection number with $\gamma_{\star}$

The purpose of this section is to get a priori bounds on $N(1, L)$ — and $N(2, L)$ in the case where $\gamma_{\star}$ is separating — using Parry and Pollicott’s bound for axiom A flows [35].

Choose some point $x_{\star} \in \gamma_{\star}$. Let $g$ be the genus of $\Sigma$ and $(a_1, b_1, \dots, a_g, b_g)$ be a basis of generators of $\Sigma$, so that the fundamental group of $\Sigma$ is the finitely presented group given by

$$ (4-1) \qquad \pi_1(\Sigma) = \langle a_1, b_1, \dots, a_g, b_g, [a_1, b_1] \cdots [a_g, b_g] = 1 \rangle, $$

where we set $\pi_1(\Sigma) = \pi_1(\Sigma, x_{\star})$ for some choice of $x_{\star} \in \gamma_{\star}$ (see Figure 2 for the case where $\gamma_{\star}$ is not separating, and Figure 4 otherwise).
<figure><img src="image_7.png" /><figcaption>Figure 2: The generators $a_1, b_1, \dots, a_g, b_g$ of $\pi_1(\Sigma)$ (on the left) and the generators $a_1, b_1, \dots, a_g$ of $\pi_1(\Sigma_\star)$ (on the right) when $g=2$. Here $\gamma_\star$ is assumed to be not separating and is represented by $a_2$ in $\pi_1(\Sigma)$.</figcaption></figure>

## 4.1 The case $\gamma_\star$ is not separating

Up to applying a diffeomorphism to $\Sigma$, we may assume that $\gamma_\star$ is represented by $a_g \in \pi_1(\Sigma)$. The cut surface $\Sigma_\star$ is a topological surface of genus $g-1$ with 2 punctures, and the fundamental group$^8$ $\pi_1(\Sigma_\star) = \pi_1(\Sigma_\star, x_\star)$ is the free group given by $\langle a_1, b_1, \dots, a_g \rangle$, which follows from the fact that $\Sigma_\star$ is homotopically equivalent to a connected sum of $2g-1$ circles. We refer to Figure 2 for a picture of the generators and the choice of $x_\star$. By the presentation of $\pi_1(\Sigma)$ given above, we have

$$ (4-2) \qquad b_g a_g b_g^{-1} = a'_g \quad \text{where} \quad a'_g = [a_1, b_1] \cdots [a_{g-1}, b_{g-1}] a_g, $$

and note that $a'_g$ also defines an element of $\pi_1(\Sigma_\star)$.

**Lemma 4.1** The map $q_\star: \Sigma_\star \to \Sigma$ given by the identification of the boundary components of $\Sigma_\star$ induces a map $q_{\star,*}: \pi_1(\Sigma_\star) \to \pi_1(\Sigma)$, which is injective.

**Proof** Let $\langle a_g \rangle$ (resp. $\langle a'_g \rangle$) be the infinite cyclic subgroup of $\pi_1(\Sigma_\star)$ generated by $a_g$ (resp. $a'_g$). Then by (4-1) and (4-2), the group $\pi_1(\Sigma)$ is the HNN$^9$ extension $\pi_1(\Sigma_\star)*_\phi$ of $\pi_1(\Sigma_\star)$ with respect to the isomorphism $\phi: \langle a'_g \rangle \to \langle a_g \rangle$ given by $\phi(a'_g) = a_g$, that is, $\pi_1(\Sigma_\star)*_\phi$ is the finitely presented group defined by

$$ \pi_1(\Sigma_\star)*_\phi = \langle a_1, b_1, \dots, a_g, t : t^{-1}a'_g t = a_g \rangle; $$

see [30, Section IV.2]. Now the map $q_{\star,*}: \pi_1(\Sigma_\star) \to \pi_1(\Sigma)$ coincides with the natural map $\pi_1(\Sigma_\star) \to \pi_1(\Sigma_\star)*_\phi$, and this map is injective by [30, Theorem IV.2.1]. $\square$

We may see the cut surface $\Sigma_\star$ as the convex core of a complete, noncompact, negatively curved surface, with funnels. Indeed, by Lemma 4.1, the group $\pi_1(\Sigma_\star)$ can be thought of as a subgroup of $\pi_1(\Sigma)$, and the convex core of the infinite surface $\Sigma_\star^e = \pi_1(\Sigma_\star)\backslash\tilde{\Sigma}$ is canonically isometric to $\Sigma_\star$ (here $\tilde{\Sigma}$ is a universal cover of $\Sigma$). Another way to obtain this is by gluing two arbitrary funnels as follows. Recall that near each connected component of the boundary $\partial\Sigma_\star \subset \Sigma_\delta$ we have coordinates

<footnote>$^8$Here, in order not to burden the notation, we still denote by $x_\star \in \Sigma_\star$ a lift of $x_\star \in \Sigma$ by the natural map $q_\star: \Sigma_\star \to \Sigma$; see Figure 2.</footnote>

<footnote>$^9$HNN refers to the authors Graham Higman, Bernhard Neumann and Hanna Neumann [22].</footnote>
($\tau, \rho) \in \mathbb{R}/\ell_{\star}\mathbb{Z}_{\tau} \times [-\delta, \delta]_{\rho}$ given by Lemma 2.3, for which $\partial\Sigma_{\star} = \{\rho = 0\}$ and $\partial\Sigma_{\delta} = \{\rho = \delta\}$. In those coordinates, the metric has the form $d\rho^2 + f(\tau, \rho)d\tau^2$ for some smooth function $f$ satisfying $\partial_{\rho}f(\tau, 0) = 0$ and $\kappa(\tau, \rho) = -\partial_{\rho}^2 f(\tau, \rho)/f(\tau, \rho)$. Then we arbitrarily extend $f$ to a smooth function on $(\mathbb{R}/\ell_{\star}\mathbb{Z})_{\tau} \times [-\delta, +\infty[$ so that, for some constants $c, C > 0$,

$$c \le \frac{\partial_{\rho}^2 f}{f} \le C.$$

By gluing the funnels $(\mathbb{R}/\ell_{\star}\mathbb{Z}) \times [0, \infty[$ and $\Sigma_{\star}$ along the corresponding connected components, we obtain a complete negatively curved surface $\Sigma_{\star}^e$, whose metric in the funnels is given by $d\rho^2 + f(\tau, \rho)d\tau^2$. We will again denote by $(\varphi_t)$ the geodesic flow on the unit tangent bundle $S\Sigma_{\star}^e$ of $\Sigma_{\star}^e$.

Let $\tilde{\Sigma}_{\star}$ denote the universal cover of $\Sigma_{\star}^e$ and let $\tilde{x}_{\star} \in \tilde{\Sigma}_{\star}$ be such that $\pi(\tilde{x}_{\star}) = x_{\star}$, where $\pi: \tilde{\Sigma}_{\star} \to \Sigma_{\star}^e$ is the natural projection. Then $\pi_1(\Sigma_{\star}^e, x_{\star}) = \pi_1(\Sigma_{\star})$ acts on $\tilde{\Sigma}_{\star}$ by deck transformations so that $\Sigma_{\star}^e \simeq \pi_1(\Sigma_{\star})\backslash\tilde{\Sigma}_{\star}$. Moreover, Lemma 2.6 implies that the recurrent set of the geodesic flow on $S\Sigma_{\star}^e$ is compact and included in $S\Sigma_{\star}$; thus $\pi_1(\Sigma_{\star})$ is convex-cocompact in the sense of [12]. The aforementioned lemma also implies that every closed geodesic in $\Sigma_{\star}^e$ which is not contained in $\partial\Sigma_{\star}$ is actually contained in the interior of $\Sigma_{\star}$.

It is well known that there is a one-to-one correspondence between oriented closed geodesics on $\Sigma_{\star}^e$ (all of them belonging to $\Sigma_{\star}$) and the set of free homotopy classes of loops in $\Sigma_{\star}^e$. The latter set is itself in one-to-one correspondence with the set of conjugacy classes of $\pi_1(\Sigma_{\star})$. We set

$$\ell_{\star}(w) = \mathrm{dist}(\tilde{x}_{\star}, w\tilde{x}_{\star}) \quad \text{for } w \in \pi_1(\Sigma_{\star}),$$

where the distance comes from the metric $\pi^*g$ on $\tilde{\Sigma}_{\star}$. For any $w \in \pi_1(\Sigma_{\star})$, we denote by $[w]$ the associated conjugacy class of $\pi_1(\Sigma_{\star})$. Note that if $\gamma_{[w]}$ denotes the unique geodesic in the free homotopy class of $w$ (which is represented by the conjugacy class $[w]$), we have $\ell(\gamma_{[w]}) \le \ell_{\star}(w)$. We also denote by

$$(4-3) \qquad \mathrm{wl}(w) = \min\{n \ge 0 : w = \alpha_1 \cdots \alpha_n \text{ with } \alpha_j \in \mathcal{L}_g \setminus \{b_g, b_g^{-1}\}\}$$

the word length of an element $w \in \pi_1(\Sigma_{\star})$, where $\mathcal{L}_g = \bigcup_{k=1}^g \{a_k, a_k^{-1}, b_k, b_k^{-1}\}$. We will say that a word $\alpha_1 \cdots \alpha_k$ with $\alpha_j \in \mathcal{L}_g$ is reduced if $\alpha_j \ne (\alpha_{j+1})^{-1}$ for any $j = 1, \dots, k-1$. As $\pi_1(\Sigma_{\star})$ is free, for each $w \in \pi_1(\Sigma_{\star})$, there is exactly one reduced word $\alpha_1 \cdots \alpha_n$ such that $n = \mathrm{wl}(w)$; see [30, page 4]. It follows from the Milnor–Švarc lemma [11, Proposition I.8.19] that, for some constant $D > 0$,

$$(4-4) \qquad \frac{1}{D} \mathrm{wl}(w) - D \le \ell_{\star}(w) \le D \mathrm{wl}(w) + D \quad \text{for } w \in \pi_1(\Sigma_{\star}).$$

Also, as $\pi_1(\Sigma_{\star})$ is convex cocompact, we have the classical orbital counting (see [42, paragraphe 1.F and corollaire 2])

$$(4-5) \qquad \#\{w \in \pi_1(\Sigma_{\star}) : \ell_{\star}(w) \le L\} \sim Ae^{h_{\star}L} \quad \text{as } L \to \infty$$

for some $A > 0$, where $h_{\star} > 0$ is the topological entropy of the geodesic flow of $(\Sigma_{\star}^e, g)$ restricted to the trapped set

$$K_{\star}^{e} = \{(x, v) \in S\Sigma_{\star}^{e} : \varphi_{t}(x, v) \in S\Sigma_{\star} \text{ for } t \in \mathbb{R}\}.$$
In fact, $h_{\star} > 0$ also coincides with the entropy of the geodesic flow of $(\Sigma, g)$ restricted to the trapped set $K_{\star}$ mentioned in the introduction,

$$K_{\star} = \{(x, v) \in S\Sigma : \pi(\varphi_t(x, v)) \in \Sigma \setminus \gamma_{\star} \text{ for } t \in \mathbb{R}\},$$

where the closure is taken in $S\Sigma$ and $K_{\star}^{e} = p_{\star}^{-1}(K_{\star})$, where $p_{\star}: S\Sigma_{\star} \to S\Sigma$ is the natural map given by the identification of both components of $\partial S\Sigma_{\star}$.

### 4.1.1 Lower bound
In this section we will prove:

**Proposition 4.2** If $\gamma_{\star}$ is not separating, then there is $C > 0$ such that, for any $L$ large enough,

$$N(1, L) \geq C \frac{e^{h_{\star}L}}{L}.$$

Note that Theorem 1 actually gives $N(1, L) \sim c_{\star}e^{h_{\star}L}$, so Proposition 4.2 is not sharp. We could obtain a better bound with the methods presented in Section 4.2, which deals with the separating case; however, Proposition 4.2 will be sufficient for our purposes (see Remarks 5.2, 5.3 and 5.4).

**Lemma 4.3** Take $w, w' \in \pi_1(\Sigma_{\star})$. Then $[wb_g] = [w'b_g]$ as conjugacy classes of $\pi_1(\Sigma)$ if and only if $w = a_g^n w' a_g'^{-n}$ in $\pi_1(\Sigma_{\star})$ for some $n \in \mathbb{Z}$.

**Proof** If $w = a_g^n w' b_g a_g'^{-n} b_g^{-1}$, then clearly $wb_g$ and $w'b_g$ are conjugate in $\pi_1(\Sigma, x_{\star})$. Reciprocally, assume that $[wb_g] = [w'b_g]$. We may find smooth paths $\gamma$ and $\gamma'$ representing respectively the elements $wb_g$ and $w'b_g$, with $i(\gamma, \gamma_{\star}) = i(\gamma', \gamma_{\star}) = 1$ and such that the intersections $\gamma \cap \gamma_{\star}$ and $\gamma' \cap \gamma_{\star}$ are transverse. As $[wb_g] = [w'b_g]$, the loops $\gamma$ and $\gamma'$ lie in the same free homotopy class. Thus there is a smooth homotopy $H: [0, 1] \times \mathbb{R}/\mathbb{Z} \to \Sigma$ such that $H(0, \cdot) = \gamma$ and $H(1, \cdot) = \gamma'$. We may assume that $H$ is transverse to $\gamma_{\star}$ (see for example [20, Corollary, page 73]) in the sense that

$$dH(s, \tau)(T_{(s,\tau)}([0,1]\times \mathbb{R}/\mathbb{Z})) + T_{H(s,\tau)}\gamma_{\star} = T_{H(s,\tau)}\Sigma \quad \text{for } H(s, \tau) \in \gamma_{\star}.$$

In particular, $H^{-1}(\gamma_{\star})$ is a smooth submanifold of $[0, 1] \times \mathbb{R}/\mathbb{Z}$. As $\gamma$ and $\gamma'$ intersect $\gamma_{\star}$ transversally exactly once, $H^{-1}(\gamma_{\star}) \cap (\{j\} \times \mathbb{R}/\mathbb{Z}) = \{j\} \times \{[0]\}$ for $j = 0, 1$ (here $[0]$ is sent to $x_{\star}$ by both $\gamma$ and $\gamma'$). Thus, necessarily, there exists an embedding $F: [0, 1] \to [0, 1] \times \mathbb{R}/\mathbb{Z}$ such that $\mathrm{Im}(F) \subset H^{-1}(\gamma_{\star})$ and $F(j) = (j, [0])$ for $j = 0, 1$ (see Figure 3). Write $F = (S, T)$, and define

$$\tilde{H}(s, t) = H(S(s), [T(s) + t]) \quad \text{for } (s, t) \in [0, 1] \times [0, 1].$$

It is immediate to check that $\tilde{H}$ realizes a homotopy between $\gamma$ and $\gamma'$, and we have $\tilde{H}(s, 0) = H(F(s)) \in \gamma_{\star}$ for any $s \in [0, 1]$. For any $s$, let us denote by $c_s$ the path $[0, 1] \ni u \mapsto \tilde{H}(su, 0)$ which links $x_{\star}$ to $H(S(s), [T(s)])$ within $\gamma_{\star}$. The continuous family of paths $s \mapsto \gamma_s$, where $\gamma_s$ is given by the concatenation $c_s^{-1} \tilde{H}(s, \cdot) c_s$, realizes a continuous interpolation between $\gamma_0 = \gamma$ and $\gamma_1 = c_1^{-1} \gamma' c_1$. As $S(1) = 1$ and $T(1) = [0]$ we have $c_1(0) = c_1(1) = x_{\star}$, and since $c_1(u) \in \gamma_{\star}$ for each $u \in [0, 1]$ we get $c_1 = a_g^{-n}$ for some $n \in \mathbb{Z}$. This yields $wb_g = a_g^n w' b_g a_g'^{-n}$ in $\pi_1(\Sigma)$, and thus $w = a_g^n w' a_g'^{-n}$, where the equality stands in $\pi_1(\Sigma)$. By Lemma 4.1, this equality actually holds in $\pi_1(\Sigma_{\star})$. $\square$
<figure><img src="image_8.png" /><figcaption>Figure 3: Proof of Lemma 4.3. The path linking $(0, [0]) \in \{0\} \times \mathbb{R}/\mathbb{Z}$ to $(1, [0])$ is the image of $F$.</figcaption></figure>

**Proof of Proposition 4.2** In what follows, $C$ is a constant that may change at each line. For any $w \in \pi_1(\Sigma_\star)$ and $n \in \mathbb{Z}$, by (4-4),

$$ (4-6) \qquad \ell_*(a_g^n w a_g'^{-n}) \ge \frac{1}{D} \mathrm{wl}(a_g^n w a_g'^{-n}) - D. $$

Let $w'$ be the unique reduced word such that $w' = w a_g'^{-n}$. Then write $w' = a_g^{-k} w''$ for some $w''$, where $|k|$ is maximal, and note that necessarily $|k| \le \mathrm{wl}(w) + 1$, since $a_g' = [a_1, b_1] \cdots [a_{g-1}, b_{g-1}] a_g$. Then

$$ \mathrm{wl}(a_g^n w a_g'^{-n}) = |n| - |k| + \mathrm{wl}(w'') = |n| - 2|k| + \mathrm{wl}(w') \ge |n| - 2(\mathrm{wl}(w) + 1) + \mathrm{wl}(w'). $$

Now the triangle inequality for $\mathrm{wl}$ gives $(4(g-1)+1)|n| = \mathrm{wl}(a_g'^{-n}) \le \mathrm{wl}(w') + \mathrm{wl}(w^{-1})$, and thus we obtain $\mathrm{wl}(a_g^n w a_g'^{-n}) \ge C|n| - C \mathrm{wl}(w) - C$ for each $n$. Injecting this in (4-6) yields (for some different $C$)

$$ \ell_*(a_g^n w a_g'^{-n}) \ge C|n| - C \mathrm{wl}(w) - C \quad \text{for } n \in \mathbb{Z}. $$

In particular, for any $L$ and $w$ such that $\ell_*(w) \le L$, by (4-4),

$$ (4-7) \qquad |\{n \in \mathbb{Z} : \ell_*(a_g^n w a_g'^{-n}) \le L\}| \le CL + C. $$

Now, for $w \in \pi_1(\Sigma_\star)$ set $C_w = \{a_g^n w a_g'^{-n} : n \in \mathbb{Z}\} \subset \pi_1(\Sigma_\star)$, and denote by $\mathcal{C}$ the set $\{C_w : w \in \pi_1(\Sigma_\star)\}$. For $C \in \mathcal{C}$, we set $\ell_*(C) = \inf_{w \in C} \ell_*(w)$. Then by Lemma 4.3, we have a well-defined and injective map

$$ \{C \in \mathcal{C} : \ell_*(C) \le L\} \to \{\gamma \in \mathcal{P}_1 : \ell(\gamma) \le L + C\}, \quad C_w \mapsto [wb_g], $$

where $\mathcal{P}_1$ denotes the set of primitive geodesics $\gamma$ such that $i(\gamma, \gamma_\star) = 1$.<fnref n="10" /> In particular we get, with (4-7) and (4-5),

$$ \begin{align} (4-8) \qquad N(1, L) &\ge |\{C \in \mathcal{C} : \ell_*(C) \le L - C\}| \ge \frac{1}{CL+C} \sum_{\substack{C \in \mathcal{C} \\ \ell_*(C) \le L-C}} |\{w \in C : \ell_*(w) \le L - C\}| \nonumber \\ &= \frac{1}{CL+C} |\{w \in \pi_1(\Sigma_\star) : \ell_*(w) \le L - C\}| \ge \frac{1}{CL+C} \exp(h_*(L-C)), \end{align} $$

where the equality comes from the fact that $\pi_1(\Sigma_\star)$ is the disjoint union of the subsets $C$ with $C \in \mathcal{C}$. $\square$

<footnote n="10">¹⁰Each class $[wb_g]$ defines a geodesic in $\mathcal{P}_1$. Indeed, it follows from Lemma 2.1 that $i([wb_g], \gamma_\star) \le 1$. On the other hand, the absolute value of the algebraic intersection number between $wb_g$ and $a_g$ is 1, and this implies that there is at least one intersection point between $[wb_g]$ and $\gamma_\star$, since the algebraic intersection number is preserved by free homotopies.</footnote>
<figure><img src="image_9.png" /><figcaption>Figure 4: The generators $a_1, b_1, \dots, a_g, b_g$ of $\pi_1(\Sigma)$. Here $\gamma_*$ is assumed to be separating and $g_1 = g_2 = 1$.</figcaption></figure>

**4.1.2 Upper bound** Each $\gamma \in \mathcal{P}_1$ with $\ell(\gamma) \le L$ lies in the free homotopy class of $w'b_g^{\pm 1}$ for some $w' \in \pi_1(\Sigma_*, x_*)$ and $\ell_*(w) \le L + C$. In particular, (4-5) gives the bound

$$N(1, L) \le C \exp(h_* L)$$

for large $L$. Now let $\gamma \in \mathcal{P}_2$ with $\ell(\gamma) \le L$. Then we may find a deformation of the loop $\gamma$ into a loop $\gamma'$ which is represented by the conjugacy class of $wb_g^{\pm 1}w'b_g^{\pm 1}$ in $\pi_1(\Sigma)$ for some $w, w' \in \pi_1(\Sigma_*)$. This deformation can be made so that $\ell_*(w) + \ell_*(w') \le L + C$. Thus,

$$N(2, L) \le C \sum_{\substack{w, w' \in \pi_1(\Sigma_*) \\ \ell_*(w) + \ell_*(w') \le L + C}} 1 \le \sum_{k=0}^{L+C} C \exp(h_*k)C \exp(h_*(L+C-k)) \le C'L \exp(h_*L).$$

Iterating this process, we finally get, for large $L$,

$$N(n, L) \le CL^{n-1} \exp(h_*L).$$

## 4.2 The case $\gamma_*$ is separating

In this section we assume $\gamma_*$ is separating, and we write $\Sigma \setminus \gamma_* = \Sigma_1 \cup \Sigma_2$, where the surfaces $\Sigma_j$ are connected. Up to applying a diffeomorphism to $\Sigma$, we may assume that $\gamma_*$ represents the class

$$(4-9) \qquad [a_1, b_1] \cdots [a_{g_1}, b_{g_1}] = [a_g, b_g]^{-1} \cdots [a_{g_1+1}, b_{g_1+1}]^{-1} \in \pi_1(\Sigma)$$

(see Figure 4). Here $g_1$ is the genus of the surface $\Sigma_1$, and the genus $g_2$ of $\Sigma_2$ satisfies $g_1 + g_2 = g$.

We set $\pi_1(\Sigma) = \pi_1(\Sigma, x_*)$ and $\pi_1(\Sigma_j) = \pi_1(\Sigma_j, x_*)$ for $j = 1, 2$ (we see $\Sigma_j$ as a compact surface with boundary $\gamma_*$ so that $x_*$ lives on both surfaces). Then $\pi_1(\Sigma_1)$ and $\pi_1(\Sigma_2)$ are the free groups generated by $a_1, b_1, \dots, a_{g_1}, b_{g_1}$ and $a_{g_1+1}, b_{g_1+1}, \dots, a_g, b_g$, respectively, and we denote by $w_{\star,1}$ and $w_{\star,2}$ the two natural words given by (4-9) representing $\gamma_*$ in $\pi_1(\Sigma_1)$ and $\pi_1(\Sigma_2)$, respectively. Note that we have a well-defined map

$$\pi_1(\Sigma_1) \times \pi_1(\Sigma_2) \to \pi_1(\Sigma), \quad (w_1, w_2) \mapsto w_2w_1,$$

given by the composition of two curves.
**Lemma 4.4** For $j = 1, 2$, the map $q_{j,*}: \pi_1(\Sigma_j) \to \pi_1(\Sigma)$ induced by the inclusion $\Sigma_j \hookrightarrow \Sigma$ is injective.

**Proof** For $j = 1, 2$ let $\langle w_{\star, j} \rangle$ be the infinite cyclic group of $\pi_1(\Sigma_j)$ generated by $w_{\star, j}$, and let $\phi: \langle w_{\star, 1} \rangle \to \langle w_{\star, 2} \rangle$ be the isomorphism given by $\phi(w_{\star, 1}) = w_{\star, 2}$. By (4-1), the group $\pi_1(\Sigma)$ is the free product with amalgamation $\pi_1(\Sigma_1) *_{\phi} \pi_1(\Sigma_2)$, that is, the finitely presented group given by

$$ \pi_1(\Sigma_1) *_{\phi} \pi_1(\Sigma_2) = \{a_1, b_1, \dots, a_g, b_g : w_{\star, 1} = \phi(w_{\star, 1})\}; $$

see [30, Section IV.2]. With this representation, the map $q_{j,*}$ coincides with the natural map $\pi_1(\Sigma_j) \to \pi_1(\Sigma_1) *_{\phi} \pi_1(\Sigma_2)$, which is injective by [30, Theorem IV.2.6]. $\square$

For any $w \in \pi_1(\Sigma)$, we will denote by $[w]$ its conjugacy class and by $\gamma_w$ the unique geodesic of $\Sigma$ such that $\gamma_w$ is isotopic to any curve in $w$ (in fact we will often identify $[w]$ and $\gamma_w$). Let $(\tilde{\Sigma}, \tilde{g})$ be the universal cover of $(\Sigma, g)$, and choose $\tilde{x}_{\star} \in \tilde{\Sigma}$ some lift of $x_{\star}$. Then $\pi_1(\Sigma)$ acts as deck transformations on $\tilde{\Sigma}$ and we will write

$$ \ell_{\star}(w) = \mathrm{dist}_{\tilde{\Sigma}}(\tilde{x}_{\star}, w\tilde{x}_{\star}) \quad \text{for } w \in \pi_1(\Sigma). $$

As in the preceding subsection, we have the orbital counting

$$ (4-10) \qquad \#\{w_j \in \pi_1(\Sigma_j) : \ell_{\star}(w_j) \le L\} \sim A_j e^{h_j L} \quad \text{as } L \to \infty \text{ for } j = 1, 2 $$

for some $A_1, A_2 > 0$, where $h_j > 0$ is the topological entropy of the geodesic flow restricted to the trapped set

$$ K_j = \overline{\{ (x, v) \in S \Sigma_j^\circ : \varphi_t(x, v) \in S \Sigma_j^\circ \text{ for } t \in \mathbb{R} \}}, $$

where $\Sigma_j^\circ = \Sigma_j \setminus \partial\Sigma_j$ for $j = 1, 2$.

**4.2.1 Lower bound** Unlike the case where $\gamma_{\star}$ is not separating, we will need a better lower bound. Namely, we prove here the following result:

**Proposition 4.5** Assume that $\gamma_{\star}$ is separating and that $h_1 = h_2 = h_{\star}$. Then there is $C > 0$ such that, for $L$ large enough,

$$ (4-11) \qquad N(2, L) \ge \frac{C L e^{h_{\star} L}}{\log(L)^4}. $$

If $h_1 \ne h_2$ we have, for $L$ large enough and $h_{\star} = \max(h_1, h_2)$,

$$ (4-12) \qquad N(2, L) \ge \frac{C e^{h_{\star} L}}{\log(L)^2}. $$

Note that Theorem 2 gives $N(2, L) \sim C L e^{h_{\star} L}$ if $h_1 = h_2$ and $N(2, L) \sim C e^{h_{\star} L}$ if $h_1 \ne h_2$. In particular, Proposition 4.5 gives a bound which is sharp up to a logarithmic loss, whereas in Proposition 4.2, we had a linear loss. Indeed, obtaining a sharper bound is important here, because a linear defect would not be sufficient to obtain Theorem 2 in the case $h_1 = h_2$ — at least with our methods. If $h_1 \ne h_2$, a linear loss would nevertheless be sufficient, but our proof of (4-11) actually gives (4-12) without too much effort. We refer to Remarks 5.2, 5.3 and 5.4 for a more detailed discussion about the importance of (4-11).
The strategy to prove Proposition 4.5 is the following. We wish to construct enough closed geodesics intersecting $\gamma_\star$ exactly twice by considering conjugacy classes of the form $[w_2w_1]$ where $w_j \in \pi_1(\Sigma_j)$ for $j = 1, 2$. Lemma 4.6 will tell us that, if $w_j$ is not a power of $w_\star$, for $j = 1, 2$, then the closed geodesic representing $[w_2w_1]$ indeed intersects $\gamma_\star$ exactly twice. Next, in Lemma 4.7, we describe the injectivity defect of the map $(w_1, w_2) \mapsto [w_2w_1]$. Finally, in Proposition 4.8, we show that this injectivity defect is not too harmful in the sense that there are not too many $w_j$, $w'_j \in \pi_1(\Sigma_j)$ such that $[w_2w_1] = [w'_2w'_1]$. This will allow us to obtain the desired bound with a logarithmic loss.

**Lemma 4.6** For two elements $w_j \in \pi_1(\Sigma_j)$ for $j = 1, 2$, we have $i(\gamma_{w_2w_1}, \gamma_\star) = 2$ except if $w_j = w_{\star,j}^k$ in $\pi_1(\Sigma_j)$ for some $k \in \mathbb{Z}$ and $j \in \{1, 2\}$, in which case $i(\gamma_{w_2w_1}, \gamma_\star) = 0$.

**Proof** Let $\gamma: \mathbb{R}/\mathbb{Z} \to \Sigma$ be a smooth curve in the free homotopy class of $w_2w_1$ such that

$$ \{\tau \in \mathbb{R}/\mathbb{Z} : \gamma(\tau) \in \gamma_{\star}\} = \{\tau_1, \tau_2\} \quad \text{for some } \tau_1 \neq \tau_2 \in \mathbb{R}/\mathbb{Z}. $$

We may also choose $\gamma$ so that $\gamma|_{[\tau_1, \tau_2]}$ (resp. $\gamma|_{[\tau_2, \tau_1]}$) is homotopic to some representative $\gamma_1: [0, 1] \to \Sigma_1$ of $w_1$ (resp. some representative $\gamma_2: [0, 1] \to \Sigma_2$ of $w_2$) relative to $\gamma_\star$, meaning that there is a homotopy between $\gamma|_{[\tau_1, \tau_2]}$ and $\gamma_1$ with endpoints (not necessarily fixed) in $\gamma_\star$. Here $[\tau_1, \tau_2] \subset \mathbb{R}/\mathbb{Z}$ is the interval linking $\tau_1$ and $\tau_2$ in the counterclockwise direction.

As $\gamma_{w_2w_1}$ minimizes the quantity $i(\gamma, \gamma_\star)$ for $\gamma \in [\gamma_{w_2w_1}]$ (see Lemma 2.1) we have either $i(\gamma_{w_2w_1}, \gamma_\star) = 0$ or $i(\gamma_{w_2w_1}, \gamma_\star) = 2$. If $i(\gamma_{w_2w_1}, \gamma_\star) = 0$, then there exists a homotopy $H: [0, 1] \times \mathbb{R}/\mathbb{Z} \to \Sigma$ such that $H(0, \cdot) = \gamma$ and $H(1, \cdot) = \gamma$, so that $H(1, \tau) \notin \gamma_\star$ for any $\tau$. As in the proof of Lemma 4.3, we may assume that $H$ is transverse to $\gamma_\star$, in the sense that

$$ dH(s, \tau)(T_{(s,\tau)}([0,1]\times\mathbb{R}/\mathbb{Z})) + T_{H(s,\tau)}\gamma_{\star} = T_{H(s,\tau)}\Sigma \quad \text{for } H(s, \tau) \in \gamma_{\star}, $$

so that the preimage

$$ H^{-1}(\gamma_{\star}) \subset [0, 1] \times \mathbb{R}/\mathbb{Z} $$

is an embedded submanifold of $[0, 1] \times \mathbb{R}/\mathbb{Z}$ (see Figure 5). As $H^{-1}(\gamma_\star) \cap \{s=0\} = \{\tau_1, \tau_2\}$ and $H^{-1}(\gamma_\star) \cap \{s=1\} = \emptyset$, it follows that there is an embedding $F: [0, 1] \to [0, 1] \times \mathbb{R}/\mathbb{Z}$ such that $F(0) = (0, \tau_1)$, $F(1) = (0, \tau_2)$ and

$$ F(t) \in H^{-1}(\gamma_{\star}) \quad \text{for } t \in [0, 1]. $$

As $F$ is an embedding, $F$ is homotopic (by a homotopy which preserves the endpoints) either to $J_{[\tau_1, \tau_2]}$ or to $J_{[\tau_2, \tau_1]}$, where $J_{[\tau, \tau')}: [0, 1] \to [0, 1] \times \mathbb{R}/\mathbb{Z}$ is the natural map that sends $[0, 1]$ to $\{0\} \times [\tau, \tau']$. We may assume without loss of generality that $F \sim J_{[\tau_1, \tau_2]}$. In particular, writing $F = (S, T)$, the map $T$ is homotopic to $I_{[\tau_1, \tau_2]} = p_2 \circ J_{[\tau_1, \tau_2]}$, where $p_2: [0, 1] \times \mathbb{R}/\mathbb{Z} \to \mathbb{R}/\mathbb{Z}$ is the projection over the second factor. This means that there is $G: [0, 1] \times [0, 1] \to \mathbb{R}/\mathbb{Z}$ such that, for any $s, t \in [0, 1]$,

$$ G(s, 0) = \tau_1, \quad G(s, 1) = \tau_2, \quad G(0, t) = \tau_1 + t(\tau_2 - \tau_1) \quad \text{and} \quad G(1, t) = T(t). $$
<figure><img src="image_10.png" /><figcaption>Figure 5: Proof of Lemma 4.6. The path linking $(0, \tau_1)$ to $(0, \tau_2)$ is the image of $F$.</figcaption></figure>

Now we set $\tilde{H}(s, t) = H(sS(t), G(s, t))$ for $s, t \in [0, 1]$. Then

$$ \tilde{H}(0, t) = \gamma(\tau_1 + t(\tau_2 - \tau_1)) \quad \text{and} \quad \tilde{H}(1, t) = (H \circ F)(t) \quad \text{for } t \in [0, 1], $$

$$ \tilde{H}(s, 0) = H(0, \tau_1) = x_1 \quad \text{and} \quad \tilde{H}(s, 1) = H(0, \tau_2) = x_2 \quad \text{for } s \in [0, 1]. $$

We conclude that $t \mapsto \gamma|_{[\tau_1, \tau_2]}(\tau_1 + t(\tau_2 - \tau_1))$, and thus $\gamma_1$, is homotopic (relative to $\gamma_\star$) to some curve contained in $\gamma_\star$. Thus $w_1 = w_\star^k$, for some $k \in \mathbb{Z}$, in $\pi_1(\Sigma)$. As the inclusion $\pi_1(\Sigma_j) \to \pi_1(\Sigma)$ is injective by Lemma 4.4, the lemma follows. $\square$

Now, we need to understand when the geodesics given by $[w_2w_1]$ and $[w'_2w'_1]$ are the same. This is the purpose of the following:

**Lemma 4.7** Take $w_j, w'_j \in \pi_1(\Sigma_j)$ for $j = 1, 2$ such that $i(\gamma_{[w_2w_1]}, \gamma_\star) = 2$. Then $[w_2w_1] = [w'_2w'_1]$ as conjugacy classes of $\pi_1(\Sigma)$ if and only if there are $p, q \in \mathbb{Z}$ such that

$$ (4-13) \qquad w_2 = w_{\star,2}^p w'_2 w_{\star,2}^q \quad \text{and} \quad w_1 = w_{\star,1}^{-q} w'_1 w_{\star,1}^{-p}. $$

**Proof** Again, let $\gamma: \mathbb{R}/\mathbb{Z} \to \Sigma$ be a smooth curve intersecting $\gamma_\star$ transversely such that

$$ \{\tau \in \mathbb{R}/\mathbb{Z} : \gamma(\tau) \in \gamma_\star\} = \{\tau_1, \tau_2\} \quad \text{for some } \tau_1 \neq \tau_2 \in \mathbb{R}/\mathbb{Z}, $$

with $\gamma([\tau_1, \tau_2]) \subset \Sigma_1$ and $\gamma([\tau_2, \tau_1]) \subset \Sigma_2$. Let $x_j = \gamma(\tau_j)$ for $j = 1, 2$, and chose arbitrary paths $c_j$ contained in $\gamma_\star$ linking $x_j$ to $x_\star$. Note that all the preceding choices can be made so that the curve $\gamma_1 = c_2\gamma|_{[\tau_1, \tau_2]}c_1^{-1}$ (resp. $\gamma_2 = c_1\gamma|_{[\tau_2, \tau_1]}c_2^{-1}$) represents $w_\star^p w_1 w_\star^q$ (resp. $w_\star^{-q} w_2 w_\star^{-p}$) for some $p, q \in \mathbb{Z}$. We may proceed in the same way to obtain $\gamma'$, $\tau'_1, \tau'_2, c'_1, c'_2, p'$ and $q'$ so that the same properties hold with $w_1$ and $w_2$ replaced by $w'_1$ and $w'_2$. By hypothesis, $\gamma$ is freely homotopic to $\gamma'$. Thus we may find a smooth map $H: [0, 1] \times \mathbb{R}/\mathbb{Z} \to \Sigma$ such that $H(0, \cdot) = \gamma$ and $H(1, \cdot) = \gamma'$. As in Lemma 4.6, $H$ may be chosen to be transverse to $\gamma_\star$, so that

$$ H^{-1}(\gamma_\star) \subset [0, 1] \times \mathbb{R}/\mathbb{Z} $$
is a finite union of smooth embedded submanifolds of $[0, 1] \times \mathbb{R}/\mathbb{Z}$. Let $(x, \rho): \Sigma \to \mathbb{R}/\mathbb{Z} \times (-\varepsilon, \varepsilon)$ be coordinates near $\gamma_*$ such that $\{\rho=0\} = \gamma_*$ and $|\rho| = \text{dist}(\gamma_*, \cdot)$, and such that $\{(-1)^{j-1}\rho \ge 0\} \subset \Sigma_j$. As $H^{-1}(\gamma_*) \cap \{s=0\} = \{\tau_1, \tau_2\}$ and $H^{-1}(\gamma_*) \cap \{s=1\} = \{\tau'_1, \tau'_2\}$, we have two smooth embeddings $F_1, F_2: [0, 1] \to [0, 1] \times \mathbb{R}/\mathbb{Z}$ such that $F_j([0, 1]) \subset H^{-1}(\gamma_*)$ and $F_j(0) = (0, \tau_j)$ for $j=1, 2$, with $F_j(1) = \tau'_1$ or $\tau'_2$ (indeed we have $i(\gamma, \gamma_*) = 2$ and thus there is a path in $H^{-1}(\gamma_*)$ linking $\{s=0\}$ to $\{s=1\}$, since otherwise we could proceed as in the proof of Lemma 4.6 to obtain that $i(\gamma, \gamma_*) = 0$). In fact, $F_1(1) = (1, \tau'_1)$ and $F_2(1) = (1, \tau'_2)$, which we shall prove later. Set $F_j = (S_j, T_j)$ and

$$ \tilde{H}(s, t) = H((1-t)S_1(s) + tS_2(s), T_1(s) + t(T_2(s) - T_1(s))) \quad \text{for } s, t \in [0, 1]. $$

Then

$$ \tilde{H}(0, t) = \gamma(\tau_1 + t(\tau_2 - \tau_1)) \quad \text{and} \quad \tilde{H}(1, t) = \gamma'(\tau'_1 + t(\tau'_2 - \tau'_1)) \quad \text{for } t \in [0, 1], $$

$$ \tilde{H}(s, 0) = H(S_1(s), T_1(s)) \quad \text{and} \quad \tilde{H}(s, 1) = H(S_2(s), T_2(s)) \quad \text{for } s \in [0, 1]. $$

For $j=1, 2$, let $c_j(s)$, $s \in [0, 1]$ be paths, contained in $\gamma_*$ depending continuously on $s$ and linking $T_j(s)$ to $x_*$, such that $c_j(0) = c_j$. Then the construction of $\tilde{H}$ shows that

$$ c_2(0)\gamma|_{[\tau_1, \tau_2]}c_1(0)^{-1} \sim c_2(1)\gamma'|_{[\tau'_1, \tau'_2]}c_1(1)^{-1}, $$

and reversing the role of $\tau_1$ and $\tau_2$ in the constructions made above,

$$ c_1(0)\gamma|_{[\tau_2, \tau_1]}c_2(0)^{-1} \sim c_1(1)\gamma'|_{[\tau'_2, \tau'_1]}c_2(1)^{-1}. $$

Thus we obtain

$$ w_{*}^{p} w_{1} w_{*}^{q} = c_{2}(1) c_{2}^{\prime -1} w_{*}^{p'} w_{1}^{\prime} w_{*}^{q'} c_{1}^{\prime} c_{1}(1)^{-1} \quad \text{and} \quad w_{*}^{-q} w_{2} w_{*}^{-p} = c_{1}(1) c_{1}^{\prime -1} w_{*}^{-q'} w_{2} w_{*}^{-p'} c_{2}^{\prime} c_{2}(1)^{-1}, $$

which is the conclusion of Lemma 4.7 as the paths $c_1(1)c_1^{\prime -1}$ and $c_2(1)c_2^{\prime -1}$ are contained in $\gamma_*$ (and, again, the inclusions $\pi_1(\Sigma_j) \to \pi_1(\Sigma)$ for $j=1, 2$ are injective).

Thus it remains to show that $F_j(1) = (1, \tau'_j)$ for $j=1, 2$. We extend $\rho$ into a smooth function $\rho: \Sigma \to \mathbb{R}$ such that $(-1)^{j-1}\rho > 0$ on $\Sigma_j \setminus \gamma_*$. There exists a continuous path $G: [0, 1] \to ([0, 1] \times \mathbb{R}/\mathbb{Z}) \setminus H^{-1}(\gamma_*)$ such that

$$ G(0) \in \{0\} \times ]\tau_1, \tau_2[ \quad \text{and} \quad G(1) \in \{1\} \times (\mathbb{R}/\mathbb{Z} \setminus \{\tau'_1, \tau'_2\}). $$

(Indeed, otherwise it would mean that there is a continuous path in $[0, 1] \times \mathbb{R}/\mathbb{Z}$ linking $(0, \tau_1)$ to $(0, \tau_2)$, which would imply, as in Lemma 4.6, that $i(\gamma, \gamma_*) = 0$.) In particular, $\rho \circ H \circ G > 0$ since $\rho(H(0, \tau)) > 0$ for $\tau \in ]\tau_1, \tau_2[$. Thus necessarily $G(1) \in \{1\} \times ]\tau'_1, \tau'_2[$, since $\rho(H(1, \tau)) < 0$ for $\tau \in ]\tau'_2, \tau'_1[$. Now, as $\text{Im}(F_1) \cap \text{Im}(F_2) = \emptyset$ (again, if the intersection was not empty we could find a path linking $(0, \tau_1)$ to $(0, \tau_2)$), we have that $G(1)$ lies in $]T_1(1), T_2(1)[$. Since $(\rho \circ H \circ G)(1) > 0$, it follows that $T_1(1) = \tau'_1$ and $T_2(1) = \tau'_2$. $\square$

The above lemma motivates the next result:
**Proposition 4.8** There is a constant $C > 0$ such that the following holds. For any $w \in \pi_1(\Sigma_j)$ such that $w$ is not a power of $w_{\star,j}$, there are $p_w, q_w \in \mathbb{Z}$ such that if $w' = w_{\star,j}^{p_w} w w_{\star,j}^{q_w}$,

$$ (4-14) \qquad \ell_{\star}(w_{\star,j}^{p_w} w'_{\star,j} w_{\star,j}^{q_w}) \geq (|p| + |q|)\ell(\gamma_{\star}) + \ell_{\star}(w') - C \quad \text{for } p, q \in \mathbb{Z}. $$

In what follows, for any $x, y \in \tilde{\Sigma}$ we will denote by $[x, y]$ the unique geodesic segment joining $x$ and $y$. Before starting the proof of Proposition 4.8, we state a classical result valid in negatively curved spaces:

**Lemma 4.9** For each $\delta > 0$ there exists a constant $C > 0$ such that the following holds. For any sequence of geodesic segments $[x_0, x_1], [x_1, x_2], [x_2, x_3]$ in $\tilde{\Sigma}$ such that $\text{dist}(x_1, x_2) \geq \delta$ and such that the angle between $[x_{j-1}, x_j]$ and $[x_j, x_{j+1}]$ is equal to $\pm \frac{1}{2}\pi$ for $j = 1, 2$,

$$ (4-15) \qquad \text{dist}(x_0, x_3) \geq \text{dist}(x_0, x_1) + \text{dist}(x_1, x_2) + \text{dist}(x_2, x_3) - C. $$

We will need the following intermediate result:

**Fact 4.10** For any $\varepsilon > 0$ there is $C > 0$ such that, for any pairwise distinct points $x, y, z \in \tilde{\Sigma}$ such that the absolute value of the angle (taken in $]-\pi, \pi]$) between $[x, y]$ and $[y, z]$ is not smaller than $\varepsilon$, we have

$$ \text{dist}(x, z) \geq \text{dist}(x, y) + \text{dist}(y, z) - C. $$

**Proof** We prove the result by comparing $\tilde{\Sigma}$ with a model space of constant curvature, as follows. Let $a = \text{dist}(x, y)$, $b = \text{dist}(y, z)$, $c = \text{dist}(x, z)$ and $\gamma = \angle([x, y], [y, z])$. Let $\tilde{\Sigma}_k$ be a simply connected complete Riemannian surface with constant curvature $-k^2 < 0$ such that $\kappa \leq -k^2$ everywhere for some $k > 0$ (recall that $\kappa$ is the curvature of $\Sigma$). Consider any points $\bar{x}, \bar{y}, \bar{z} \in \tilde{\Sigma}_k$ such that

$$ \text{dist}_k(\bar{x}, \bar{y}) = a, \quad \text{dist}_k(\bar{y}, \bar{z}) = b \quad \text{and} \quad \angle([\bar{x}, \bar{y}], [\bar{y}, \bar{z}]) = \gamma, $$

where $\text{dist}_k$ is the distance in $\tilde{\Sigma}_k$, and set $\bar{c} = \text{dist}_k(x, z)$. Then by a classical trigonometric formula for spaces of constant negative curvature (see [11, I.2.7]),

$$ \text{ch}(k\bar{c}) = \text{ch}(ka)\text{ch}(kb) - \text{sh}(ka)\text{sh}(kb)\cos(\gamma). $$

As $\gamma \in ]-\pi, \pi] \setminus ]-\varepsilon, \varepsilon[$, we have $\cos(\gamma) \leq 1 - \eta$ for some $\eta \in ]0, 1[$ depending on $\varepsilon$. Thus

$$ \text{ch}(k\bar{c}) \geq \eta \text{ch}(ka) \text{ch}(kb). $$

Using $\frac{1}{2} \exp(t) \leq \text{ch}(t) \leq \exp(t)$ for $t \geq 0$, one gets

$$ \bar{c} \geq a + b + \frac{\log(\frac{1}{4}\eta)}{k}. $$

As the scalar curvature of $\tilde{\Sigma}$ is everywhere not greater than $-k^2$, the space $\tilde{\Sigma}$ is a CAT($-k^2$) space; see [11, Theorem II.4.1]. In particular, by comparison, one obtains $c \geq \bar{c}$ (see [11, Proposition II.1.7]), which concludes the proof. $\square$
<figure><img src="image_11.png" /><figcaption>Figure 6: Proof of Lemma 4.9.</figcaption></figure>

**Proof of Lemma 4.9** Let $x_0, x_1, x_2$ and $x_3$ be as in the statement. For $j = 0, 1, 2$ we set $d_j = \text{dist}(x_j, x_{j+1})$. We first assume one of the numbers $d_0$ or $d_2$ is not greater than $\delta$, say $d_0 \le \delta$. Then Fact 4.10 (applied with $x = x_1, y = x_2$ and $z = x_3$) yields $\text{dist}(x_1, x_3) \ge d_1 + d_2 - C$, and thus

$$ \text{dist}(x_0, x_3) \ge \text{dist}(x_1, x_3) - \text{dist}(x_0, x_1) \ge d_1 + d_2 + C - d_0 \ge d_0 + d_1 + d_2 + C - 2\delta. $$

Therefore we may assume that $d_0, d_2 \ge \delta$. Applying Fact 4.10 for the points $x_0, x_1$ and $x_2$ yields

$$ (4-16) \qquad \text{dist}(x_0, x_2) \ge d_0 + d_1 - C. $$

For any pairwise distinct $x, y, z \in \tilde{\Sigma}$, we denote by $\Delta(x, y, z)$ the triangle generated by $x, y$ and $z$. Then as $d_0, d_1 \ge \delta$, the triangle $\Delta(x_0, x_1, x_2)$ contains some triangle $\Delta(x, y, z)$ with a right angle at $y$ and $\text{dist}(x, y) = \text{dist}(y, z) = \delta$ (namely, $y = x_1, x \in [x_1, x_0]$ and $z \in [x_1, x_2]$). Clearly the area $|\Delta(x, y, z)|$ of $\Delta(x, y, z)$ is bounded from below by some constant $D > 0$ depending only on $\delta > 0$ (indeed, it suffices to verify this property for $x, y$ and $z$ lying in a compact set given by a finite union of fundamental domains of $\Sigma$). Therefore, $|\Delta(x_0, x_1, x_2)| \ge D$. Let $\alpha$ and $\beta$ be the angles of $\Delta(x_0, x_1, x_2)$ at $x_0$ and $x_2$, respectively (see Figure 6). Let $\tilde{\mu}_g$ bet the Riemannian measure of $\tilde{\Sigma}$, and $\tilde{\kappa}$ its scalar curvature. Then, by the Gauss-Bonnet formula [29, Theorem 9.3],

$$ \int_{\Delta(x_0, x_1, x_2)} \tilde{\kappa} \, d\tilde{\mu}_g + \frac{1}{2}\pi + (\pi - \alpha) + (\pi - \beta) = 2\pi. $$

This gives

$$ \beta \le \frac{1}{2}\pi - \alpha - k^2 |\Delta(x_0, x_1, x_2)| \le \frac{1}{2}\pi - k^2 D. $$

Therefore the angle between $[x_0, x_2]$ and $[x_2, x_3]$ is not smaller than $k^2 D$. In particular, we may apply Fact 4.10 to get $\text{dist}(x_0, x_3) \ge \text{dist}(x_0, x_2) + d_2 - C$ for some $C$ depending only on $k^2 D$. Combining this with (4-16), we conclude the proof. $\square$

**Proof of Proposition 4.8** We fix $j \in \{1, 2\}$ and write $w_* = w_{*,j}$ for simplicity. Let $w \in \pi_1(\Sigma_j)$ be such that $w \ne w_*^k$ for any $k$. Then $w$ is not the trivial element, and thus it is hyperbolic. Recall that $(\tilde{\Sigma}, \tilde{g})$ is
<figure><img src="image_12.png" /><figcaption>Figure 7: Proof of Proposition 4.8.</figcaption></figure>

the universal cover of $(\Sigma, g)$ and that $\pi_1(\Sigma)$ acts by deck transformations on $\tilde{\Sigma}$. For any $u \in \pi_1(\Sigma) \setminus \{1\}$, we denote by

$$u_{\pm} = \lim_{k \to +\infty} u^{\pm k}(z)$$

the two distinct fixed points of $u$ in the boundary at infinity $\partial_{\infty}\tilde{\Sigma}$ of $\tilde{\Sigma}$ (here $z$ denotes any point in $\tilde{\Sigma}$). We also denote by $A_u$ the translation axis of $u$, that is, the unique complete geodesic of $(\tilde{\Sigma}, \tilde{g})$ converging towards $u_+$ (resp. $u_-$) in the future (resp. in the past). Note that $A_{ww^*w^{-1}} = wA_{w^*}$. As the conjugacy classes $[ww^*w^{-1}]$ and $[w^*]$ both represent the geodesic $\gamma_*$, we have either $A_{w^*} = wA_{w^*}$ or $A_{w^*} \cap wA_{w^*} = \emptyset$. Since $w$ is not a power of $w^*$, we necessarily have $A_{w^*} \cap wA_{w^*} = \emptyset$. Write $\gamma_* = \{\varphi_s(z_*): s \in [0, \ell(\gamma_*)]\}$ for some $z_* = (x_*, v_*) \in M$. By hyperbolicity of the geodesic flow, there is $\delta > 0$ such that the following holds. For any $z \in M$ such that $\inf_{s \in \mathbb{R}} \text{dist}_M(z, \varphi_s(z_*)) \le \delta$,

$$(4-17) \qquad \varphi_{\ell(\gamma_*)}(z) = z \implies z = \varphi_s(z_*) \quad \text{for some } s \in \mathbb{R}.$$

As $\ell([ww^*w^{-1}]) = \ell([w^*]) = \ell(\gamma_*)$, we obtain

$$(4-18) \qquad \text{dist}(A_{w^*}, wA_{w^*}) \ge \delta.$$

Let $\tilde{x} \in A_{w^*}$ and $\tilde{y} \in wA_{w^*}$ be the unique points such that $\text{dist}(\tilde{x}, \tilde{y}) = \text{dist}(A_{w^*}, wA_{w^*})$, and take $p, q \in \mathbb{Z}$. Then $\text{dist}(\tilde{x}, \tilde{y}) \ge \delta$ by (4-18), and thus we may apply Lemma 4.9 with the sequence of geodesic segments $[w_*^{-p}\tilde{x}_*, \tilde{x}]$, $[\tilde{x}, \tilde{y}]$, $[\tilde{y}, ww_*^q\tilde{x}_*]$ to obtain

$$\text{dist}(ww_*^q\tilde{x}_*, w_*^{-p}\tilde{x}_*) \ge \text{dist}(ww_*^q\tilde{x}_*, \tilde{y}) + \text{dist}(\tilde{y}, \tilde{x}_*) + \text{dist}(\tilde{x}, w_*^{-p}\tilde{x}_*) - C$$

for some $C > 0$ independent of $w, p$ and $q$ (see Figure 7). Next, let $p_w, q_w \in \mathbb{Z}$ such that

$$\text{dist}(\tilde{x}, w_*^{-p_w}\tilde{x}_*) < \ell(\gamma_*) \quad \text{and} \quad \text{dist}(\tilde{y}, ww_*^q\tilde{x}_*) < \ell(\gamma_*).$$

Then, for any $p, q \in \mathbb{Z}$,

$$\text{dist}(\tilde{x}, w_*^{-p}\tilde{x}_*) \ge |p - p_w|\ell(\gamma_*) - \ell(\gamma_*) \quad \text{and} \quad \text{dist}(\tilde{y}, ww_*^q\tilde{x}_*) \ge |q - q_w|\ell(\gamma_*) - \ell(\gamma_*),$$
which yields

$$
\mathrm{dist}(w_{\star}^{p} w w_{\star}^{q} \tilde{x}_{\star}, \tilde{x}_{\star}) \geq (|p - p_{w}| + |q - q_{w}|)\ell(\gamma_{\star}) + \mathrm{dist}(\tilde{x}, \tilde{y}) - C - 2\ell(\gamma_{\star}).
$$

Finally, we note that

$$
\mathrm{dist}(\tilde{x}, \tilde{y}) \geq \mathrm{dist}(w w_{\star}^{q_w} \tilde{x}_{\star}, w_{\star}^{-p_w} \tilde{x}_{\star}) - 2\ell(\gamma_{\star}) = \ell_{\star}(w_{\star}^{p_w} w w_{\star}^{q_w}) - 2\ell(\gamma_{\star}). \quad \square
$$

Building on Lemmata 4.6 and 4.7 and Proposition 4.8, we prove Proposition 4.5:

**Proof of Proposition 4.5** In what follows, $C$ is a positive constant independent of $L$ that may change at each line. First, assume that $h_1 = h_2 = h_\star$. For $j = 1, 2$ we denote by $\langle w_{\star,j} \rangle = \{w_{\star,j}^n : n \in \mathbb{Z}\}$ the infinite cyclic subgroup of $\pi_1(\Sigma_j)$ generated by $w_{\star,j}$, and we set $\pi_1(\Sigma_j)_\star = \pi_1(\Sigma_j) \setminus \langle w_{\star,j} \rangle$. Since $\ell_\star(w_{\star,j}^n) = |n|\ell(\gamma_\star)$, there is $C$ such that, for any large $L$,

$$
(4-19) \qquad C^{-1} e^{h_{\star} L} \le N_{\star,j}(L) \le C e^{h_{\star} L}
$$

by (4-10), where $N_{\star,j}(L) = \#\{w \in \pi_1(\Sigma_j)_\star : \ell_\star(w) \le L\}$. For $w \in \pi_1(\Sigma_j)_\star$, we set

$$
C_w = \{w_{\star}^{p} w w_{\star}^{q} : p, q \in \mathbb{Z}\} \subset \pi_1(\Sigma_j)_{\star}.
$$

and we define $\mathscr{C}_j = \{C_w : w \in \pi_1(\Sigma_j)_\star\}$. Note that the elements $C \in \mathscr{C}_j$ are pairwise disjoint, and thus we have a partition $\coprod_{C \in \mathscr{C}_j} C$ of $\pi_1(\Sigma_j)_\star$. We also write

$$
\ell_{\star}(C) = \inf\{\ell_{\star}(w) : w \in C\} \quad \text{for } C \in \mathscr{C}_j \text{ with } j = 1, 2.
$$

Then Proposition 4.8 yields

$$
\#\{w \in C : \ell_{\star}(w) \le L\} \le C(L - \ell_{\star}(C) + C)^2
$$

for any $C \in \mathscr{C}_j$ such that $\ell_{\star}(C) \le L$. Thus

$$
N_{\star,j}(L) = \sum_{\substack{C \in \mathscr{C}_j \\ \ell_{\star}(C) \le L}} \#\{w \in C : \ell_{\star}(w) \le L\} \le C \sum_{\substack{C \in \mathscr{C}_j \\ \ell_{\star}(C) \le L}} (L - \ell_{\star}(C) + C)^2.
$$

Let $\beta > 0$ be a large number. Then

$$
(4-20) \quad \sum_{\substack{C \in \mathscr{C}_j \\ \ell_{\star}(C) \le L - \beta \log L}} (L - \ell_{\star}(C) + C)^2 \le (L + C)^2 \#\{C \in \mathscr{C}_j : \ell_{\star}(C) \le L - \beta \log L\}.
$$

However, using (4-19), we obtain

$$
\#\{C \in \mathscr{C}_j : \ell_{\star}(C) \le L - \beta \log L\} \le N_{\star,j}(L - \beta \log L) \le C L^{-h_{\star}\beta} e^{h_{\star}L}.
$$

In particular, if $h_\star\beta > 2$, and if $A_\beta(L)$ denotes the left-hand side of (4-20), we have the bound $A_\beta(L) \ll N_{\star,j}(L)$ as $L \to \infty$. Thus, for large $L$,

$$
C^{-1} N_{\star,j}(L) \le \sum_{\substack{C \in \mathscr{C}_j \\ \ell_{\star}(C) \in [L - \beta \log L, L]}} (L - \ell_{\star}(C) + C)^2 \le (\beta \log L + C)^2 \#\{C \in \mathscr{C}_j : \varepsilon L \le \ell(C) \le L\},
$$
where $\epsilon > 0$ is any small number. This finally yields, for any large $L$,

$$ (4-21) \qquad \#\{C \in \mathscr{C}_j : \epsilon L \le \ell(C) \le L\} \ge \frac{C^{-1}e^{h_*L}}{(\beta \log L + C)^2}. $$

For any $C \in \mathscr{C}_j$, we choose some $w_C \in C$ such that $\ell_*(w_C) = \ell_*(C)$. Then Lemmata 4.6 and 4.7 imply that we have a well-defined and injective map

$$ \mathscr{C}_1 \times \mathscr{C}_2 \to \{\gamma \in \mathcal{P} : i(\gamma, \gamma_*) = 2\}, \quad (C_1, C_2) \mapsto [w_{C_2} w_{C_1}] \equiv \gamma_{w_{C_2} w_{C_1}}. $$

Obviously, $\ell(\gamma w_2 w_1) \le \ell_*(w_1) + \ell_*(w_2)$ for any $w_1$ and $w_2$, and thus we get, for large $L$,

$$ N(2, L) \ge \#\{(C_1, C_2) \in \mathscr{C}_1 \times \mathscr{C}_2 : \ell_*(C_1) + \ell_*(C_2) \le L \text{ and } \ell_*(C_1), \ell_*(C_2) \ge \epsilon L\} $$

$$ \ge \sum_{\substack{C_1 \in \mathscr{C}_1 \\ \epsilon L \le \ell_*(C_1) \le L}} \#\{C_2 \in \mathscr{C}_2 : \epsilon L \le \ell_*(C_2) \le L - \ell_*(C_1)\} \ge \sum_{\substack{C_1 \in \mathscr{C}_1 \\ \epsilon L \le \ell_*(C_1) \le L}} \frac{C^{-1} e^{h_*(L - \ell_*(C_1))}}{(\beta \log(L - \ell_*(C_1)) + C)^2}. $$

For simplicity, in what follows we will use the notation $f(\ell) = C^{-1}e^{h_*\ell}/(\beta \log(\ell)+C)^2$ and $N(\mathscr{C}_1, L) = \#\{C \in \mathscr{C}_j : \epsilon L \le \ell(C) \le L\}$. Fix some large number $\mu > 0$. Note that, if $\mu$ is large enough, there is $C > 0$ (depending on $\mu$) such that, for any large $\ell$,

$$ (4-22) \qquad f(\ell + \mu) - f(\ell) \ge C^{-1} f(\ell). $$

There holds

$$ (4-23) \quad \begin{aligned} N(2, L) &\ge C^{-1} \sum_{k \in [\epsilon L / \mu, L / \mu]} (N(\mathscr{C}_1, k\mu) - N(\mathscr{C}_1, (k-1)\mu)) f(L - (k-1)\mu) \\ &\ge C^{-1} \sum_{k \in [\epsilon L / \mu + 1, L / \mu - 1]} N(\mathscr{C}_1, k\mu) (f(L - (k-1)\mu) - f(L - k\mu)) \\ &\phantom{\ge C^{-1} \sum_{k \in [\epsilon L / \mu + 1, L / \mu - 1]} N(\mathscr{C}_1, k\mu) (f(L - (k-1)\mu) - f(L - k\mu))} - N(\mathscr{C}_1, \epsilon L + \mu) f(L - \epsilon L), \end{aligned} $$

where we used an Abel transformation in the last inequality. Next, note that by (4-19), one has $N(\mathscr{C}_1, L) \le N_{\star,1}(L) \le C e^{h_*L}$. This yields

$$ (4-24) \qquad N(\mathscr{C}_1, \epsilon L + \mu) f(L - \epsilon L) = \mathcal{O}(e^{h_*L}) $$

as $L \to \infty$. On the other hand, (4-22) gives, for any large $L$,

$$ \begin{aligned} & \sum_{k \in [\epsilon L / \mu + 1, L / \mu - 1]} N(\mathscr{C}_1, k\mu) (f(L - (k-1)\mu) - f(L - k\mu)) \\ &\ge \sum_{k \in [\epsilon L / \mu + 1, L / \mu - 1]} N(\mathscr{C}_1, k\mu) f(L - k\mu) \\ &\ge C^{-1} \sum_{k \in [\epsilon L / \mu + 1, L / \mu - 1]} \frac{e^{h_*k\mu}}{(\beta \log(k\mu) + C)^2} \frac{e^{h_*(L-k\mu)}}{(\beta \log(L-k\mu) + C)^2} \ge \frac{C^{-1} L e^{h_*L} (1-\epsilon)}{2\mu(\log(L) + C)^4}. \end{aligned} $$

We conclude the proof of Proposition 4.5 for the case $h_1 = h_2$ by combining this last estimate with (4-23) and (4-24).
If $h_1 \neq h_2$, say $h_1 > h_2$ (the case $h_1 < h_2$ is identical), one is able to obtain the desired bound by considering, for example, the injective map $\mathscr{C}_1 \to \{\gamma \in \mathcal{P} : i(\gamma, \gamma_\star) = 2\}$ given by $C \mapsto [a_g w_c]$ and by using (4-21). $\square$

**4.2.2 Upper bound** Clearly, each $\gamma \in \mathcal{P}_2$ with $\ell(\gamma) \le L$ may be represented by the conjugacy class of $w_1w_2$ for some $w_j \in \pi_1(\Sigma_j)$ with $\ell_\star(w_1) + \ell_\star(w_2) \le L + C$. Therefore, (4-5) implies

$$ \begin{aligned} N(2, L) &\le \#\{(w_1, w_2) \in \pi_1(\Sigma_1) \times \pi_1(\Sigma_2) : \ell_\star(w_1) + \ell_\star(w_2) \le L + C\} \\ &\le \sum_{k=0}^{L+C} C \exp(h_1 k) \exp(h_2(L-k+C)), \end{aligned} $$

which gives, for large $L$, if $h_\star = \max(h_1, h_2)$,

$$ N(2, L) \le \begin{cases} CL \exp(h_\star L) & \text{if } h_1 = h_2, \\ C \exp(h_\star L) & \text{if } h_1 \neq h_2. \end{cases} $$

Iterating this process we obtain (with $C$ depending on $n$)

$$ N(2n, L) \le \begin{cases} CL^{2n-1} \exp(h_\star L) & \text{if } h_1 = h_2, \\ CL^{n-1} \exp(h_\star L) & \text{if } h_1 \neq h_2. \end{cases} $$

**4.3 Relative growth of closed geodesics with a small intersection angle**

For $x = \gamma_\star(\tau) \in \text{Im}(\gamma_\star)$, we let $v_\star(x) = \dot{\gamma}_\star(\tau)$. For any $\eta > 0$ small, we consider the number $N(n, \eta, L) = \#\mathcal{P}_{\eta,n}(L)$, where $\mathcal{P}_{\eta,n}(L)$ is the set of closed geodesics $\gamma: \mathbb{R}/\ell(\gamma)\mathbb{Z} \to \Sigma$ of length not greater than $L$, intersecting $\gamma_\star$ exactly $n$ times, and such that there is $t \in \mathbb{R}/\ell(\gamma)\mathbb{Z}$ with $\gamma(t) \in \text{Im}(\gamma_\star)$ and

$$ \text{angle}(\dot{\gamma}(t), v_\star(\gamma(t))) < \eta \quad \text{or} \quad \text{angle}(\dot{\gamma}(t), -v_\star(\gamma(t))) < \eta. $$

The purpose of this section is to prove the following estimate:

**Lemma 4.11** Let $n \ge 1$. For any $\varepsilon, L_0 > 0$, there exists $\eta_0 > 0$ such that, for any $\eta \in ]0, \eta_0[$ and any large $L$,

$$ (4-25) \qquad N(1, \eta, L) \le 4N(1, L - L_0) \quad \text{and} \quad N(n, \eta, L) \le \varepsilon L^{n-1} \exp(h_\star L) $$

if $\gamma_\star$ is not separating, and

$$ (4-26) \qquad N(2, \eta, L) \le 4N(2, L - L_0) \quad \text{and} \quad N(2n, \eta, L) \le \begin{cases} \varepsilon L^{2n-1} \exp(h_\star L) & \text{if } h_1 = h_2, \\ \varepsilon L^{n-1} \exp(h_\star L) & \text{if } h_1 \neq h_2, \end{cases} $$

if $\gamma_\star$ is separating.

**Proof** We first prove the lemma when $\gamma_\star$ is assumed not separating. Let $\gamma: [0, \ell(\gamma)] \to \Sigma$ be an element of $\mathcal{P}_{\eta,n}(L)$ parametrized by arc length. Let $0 \le t_1 < t_2 < \dots < t_n < \ell(\gamma)$ be such that $\gamma(t_j) \in \text{Im}(\gamma_\star)$. For every $j = 1, \dots, n$, we choose a path $c_j$ contained in $\text{Im}(\gamma_\star)$ of length not greater than $\ell(\gamma_\star)$ that links
$x_j = \gamma(t_j)$ to $x_\star$. Recall that we have a map $q_\star: \Sigma_\star \to \Sigma$ given by the identification of the boundary components of $\Sigma_\star$. Write $q_\star^{-1}(x_\star) = \{x_\star, \bar{x}_\star\}$, where we chose some $x_\star \in \Sigma_\star$ with $q_\star(x_\star) = x_\star$, as in Section 4.1. Then $\gamma$ is freely homotopic to the composition

$$w_1 w_2 \cdots w_n, \quad \text{where } w_j = c_{j+1} \gamma|_{[t_j, t_{j+1}]} c_j^{-1} \in \pi_1(\Sigma) \text{ for } j = 1, \dots, n,$$

with the convention that $t_{n+1} = \ell(\gamma)$ and $c_{n+1} = c_1$. Note also that

$$\ell_{\star}(w_j) \le |t_{j+1} - t_j| + 2\ell(\gamma_{\star}).$$

In fact, the elements $w_j$ actually define elements of the space $\pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$, that is, the space of equivalence classes of paths $c: [0, 1] \to \Sigma_\star$ with $c(0), c(1) \in \{x_\star, \bar{x}_\star\}$, where two paths are equivalent if they are homotopic via a homotopy preserving the endpoints. The space $\pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$ is not a group (we may not be able to concatenate two paths); however, we have a natural map $\pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\}) \to \pi_1(\Sigma)$. In particular, for any $u_1, \dots, u_n \in \pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$, the composition $u_n \cdots u_1$ is well defined in $\pi_1(\Sigma)$. For any $u \in \pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$, we will denote by $\ell_\star(u)$ the infimum of the lengths of curves in the equivalence class $u$.

Up to reparametrizing of $\gamma$, we may assume that $t_1 = 0$, and either $\angle(v, v_\star) < \eta$ or $\angle(v, -v_\star) < \eta$, where we set $x = \gamma(0)$, $v_\star = v_\star(x)$ and $v = \dot{\gamma}(0)$. We will first assume that $\angle(v, v_\star) < \eta$. Let $L_0 > 0$ be a large number and $\varepsilon > 0$ be small. By continuity of the geodesic flow $(\varphi_t)$, there is $\eta_0 > 0$ such that, if $\eta < \eta_0$,

$$\mathrm{dist}_M(\varphi_t(v), \varphi_t(v_\star)) \le \varepsilon \quad \text{for } t \in [0, L_0].$$

Let $K$ be a positive integer such that $K \in [L_0/\ell(\gamma_\star) - 1, L_0/\ell(\gamma_\star)]$, so that

$$\mathrm{dist}_{\Sigma}(\pi(\varphi_{K\ell(\gamma_\star)}(v)), x) < \varepsilon.$$

Let $c_K$ be a path in $\Sigma$ of length not greater than $\varepsilon$ linking $\pi(\varphi_{K\ell(\gamma_\star)}(v))$ and $x$. Then, if $\varepsilon > 0$ is small enough,<fnref n="11" />

$$c_1 c_K \gamma|_{[0, K\ell(\gamma_\star)]} c_1^{-1} = a_g^K \quad \text{in } \pi_1(\Sigma).$$

In particular, $w_1 = w'_1 a_g^K$ in $\pi_1(\Sigma)$, where $w'_1 = c_2 \gamma|_{[K\ell(\gamma_\star), t_2]} c_K^{-1} c_1^{-1}$. Note also that

$$\ell_{\star}(w'_1) \le |t_2 - K\ell(\gamma_{\star})| + 2\ell(\gamma_{\star}) + \varepsilon,$$

where $w'_1$ is seen as an element of $\pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$. Note that if we had assumed $\angle(v, -v_\star) < \eta$, we would have obtained the same factorization with $a_g^K$ instead of $a_g^K$. Next, let

$$A_{K,n}(L) = \left\{ (w_1, \dots, w_n) \in \pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})^n : \sum_{j=1}^n \ell_\star(w_j) \le L + (2n - K)\ell(\gamma_\star) + \varepsilon \right\},$$

<footnote n="11">¹¹If $\varepsilon > 0$ is small enough, we have the following property. For any $x \in \Sigma$ and $L > 0$, if we are given two paths $c, c': [0, L] \to \Sigma$ such that $c(0) = c'(0) = c(L) = c'(L) = x$ and $\mathrm{dist}_{\Sigma}(c(t), c'(t)) < \varepsilon$, then $c$ and $c'$ define the same element in $\pi_1(\Sigma, x)$.</footnote>
and consider the map $\Psi_{K,n,\pm}: A_{K,n}(L) \to \mathcal{P}$ given by $(w_1, \dots, w_n) \mapsto [w_1 \cdots w_na_g^{\pm K}]$. Then the discussion above shows that

$$ \mathcal{P}_{\eta,n}(L) \subset \mathrm{Im}(\Psi_{K,n,+}) \cup \mathrm{Im}(\Psi_{K,n,-}). $$

In particular, $N(n, \eta, L) \le 2\#A_{K,n}(L)$. Next, we obtain a bound on $A_{K,n}(L)$ as follows. Let $c_\star$ be a path connecting $\bar{x}_\star$ and $x_\star$ in $\Sigma_\star$, so that the image of $c_\star^{-1}$ in $\pi_1(\Sigma)$ is $b_g$ (see Figure 2). Then it is not hard to see that, for any $w \in \pi_1(\Sigma_\star, \{x_\star, \bar{x}_\star\})$, there is $u \in \pi_1(\Sigma_\star, x_\star)$ such that $w$ can be written as

$$ u, \ c_{\star}u, \ uc_{\star}^{-1} \quad \text{or} \quad c_{\star}uc_{\star}^{-1} $$

(depending on the endpoints of $w$), with $\ell_\star(u) \le \ell_\star(w) + 2\ell(c_\star)$. This immediately gives

$$ \#A_{K,1}(L) \le 4\#\{u \in \pi_1(\Sigma_\star) : \ell_\star(u) \le L\} \le C \exp(h_\star L). $$

As in Section 4.1.2, we obtain, for some $C_n > 0$ depending only $n$,

$$ \#A_{K,n}(L) \le C_n L^{n-1} \exp(h_\star(L - L_0)), $$

where we used that $K\ell(\gamma_\star) \ge L_0 - \ell(\gamma_\star)$. This proves the second part of (4-25). For the first part, we proceed as follows. With the notation of the proof of Proposition 4.5, one has well-defined maps

$$ \Psi_{K,1,\pm,r}, \Psi_{K,1,\pm,l} : \{C \in \mathcal{C} : \ell_\star(w) \le L - K\ell(\gamma_\star)\} \to \{\gamma \in \mathcal{P}_1 : \ell(\gamma) \le L + 2C\}, $$

given respectively by $C \mapsto [a_g^{\pm K}wb_g]$ and $C \mapsto [b_g^{-1}wa_g^{\pm K}]$, where $w$ is any element of $C$. Next, we remark that the above discussion implies that every $\gamma \in \mathcal{P}_{\eta,1}(L)$ can be written as

$$ [a_g^{\pm K}wb_g] \quad \text{or} \quad [b_g^{-1}wa_g^{\pm K}] $$

for some $w \in \pi_1(\Sigma_\star)$ with $\ell_\star(w) \le L - K\ell(\gamma_\star) + C$. Therefore the union of the images of the maps $\Psi_{K,1,\pm,r}$ and $\Psi_{K,1,\pm,l}$ contains $\mathcal{P}_\eta(L + 2C)$, and thus

$$ N(1, \eta, L) \le 4\#\{C \in \mathcal{C} : \ell_\star(w) \le L - K\ell(\gamma_\star) + 2C\} \le 4N(1, L - K\ell(\gamma_\star) + 3C), $$

where we used the first inequality of (4-8). This gives the first part of (4-25).

Next, assume that $\gamma_\star$ is separating. Then, as above, every $\gamma: [0, \ell(\gamma)] \to \Sigma$ such that $\gamma \in \mathcal{P}_{2n,\eta}(L)$ can be written as a composition $w_{1,1}w_{1,2}\cdots w_{1,n}w_{2,n}$ for some $w_{k,j} \in \pi_1(\Sigma_k)$ for $k=1,2$ and $j=1,2,\dots,n$, with

$$ \sum_{j=1}^{n} \ell_{\star}(w_{2,j}) + \ell_{\star}(w_{1,j}) \le \ell(\gamma) + 4n\ell(\gamma_{\star}). $$

Now, if $\eta$ is small, we may proceed as before to obtain (up to reparametrization of $\gamma$) that $w_{1,1} = w_{\star,1}^{\pm K}w'_{1,1}$ or $w_{1,1} = w'_{1,1}w_{\star,1}^{\pm K}$ for some $w'_{1,1} \in \pi_1(\Sigma_1)$ with

$$ \ell_\star(w'_{1,1}) \le \ell_\star(w_{1,1}) - K\ell(\gamma_\star) + C. $$
Here $K$ is a large number depending on $\eta$ (ie such that $K \to \infty$ as $\eta \to 0$) and $C > 0$ is a constant independent of $\gamma$ and $K$. Thus we get

$$N(2n, \eta, L) \le C \# \left\{ (w_{1,1}, w_{2,1}, \dots, w_{1,n}, w_{2,n}) : w_{k,j} \in \pi_1(\Sigma_k), \sum_{j=1}^n \ell_*(w_{1,j}) + \ell_*(w_{2,j}) \le L - K\ell(\gamma_*) + C_n \right\}.$$

Then we obtain the second part of (4-26) by proceeding as in Section 4.2.2. For the first part of (4-26), we proceed as follows. For $w_j \in \pi_1(\Sigma_j)_\star$, we define

$$C_{w_1, w_2} = \{(w'_1, w'_2) : [w'_1 w'_2] = [w_1 w_2]\}$$

and $\ell_*(C_{w_1, w_2}) = \inf\{\ell_*(w'_1) + \ell_*(w'_2) : (w'_1, w'_2) \in C_{(w_1, w_2)}\}$. We also introduce the notation $\mathscr{C}_{1,2} = \{C_{w_1, w_2} : w_j \in \pi_1(\Sigma_j)_\star\}$. By Lemmata 4.6 and 4.7, we have well-defined maps

$$\Psi_{K,1,\pm,r}, \Psi_{K,1,\pm,l} : \{C \in \mathscr{C}_{1,2} : \ell_*(C_{w_1,w_2}) \le L - K\ell(\gamma_*)\} \to \{\gamma \in \mathcal{P}_2 : \ell(\gamma) \le L\}$$

given respectively by $C \mapsto [w_1 w_{\star,1}^{\pm K} w_2]$ and $C \mapsto [w_{\star,1}^{\pm K} w_1 w_2]$. By the discussion above, the union of the images of those maps contains $\mathcal{P}_{2,\eta}(L)$. Therefore

$$N(2, \eta, L) \le 4 \# \{C \in \mathscr{C}_{1,2} : \ell_*(C_{w_1,w_2}) \le L - K\ell(\gamma_*)\} \le 4N(2, L - K\ell(\gamma_*)),$$

where we used Lemmata 4.6 and 4.7 again in the last inequality. The first part of (4-26) follows. $\square$

# 5 A Tauberian argument

The goal of this section is to give an asymptotic growth of the quantity

$$N_{\pm}(n, \chi, t) = \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma_*, \gamma) = n \\ \ell(\gamma) \le t}} I_{\star, \pm}(\gamma, \chi)$$

as $t \to +\infty$, where $\chi \in C_c^\infty(\partial \setminus \partial_0)$ and $I_{\star, \pm}(\gamma, \chi) = \prod_{z \in I_{\star, \pm}(\gamma)} \chi^2(z)$.

## 5.1 The case $\gamma_\star$ is not separating

By [15, Theorem 3 and Section 6.2], the zeta function

$$\zeta_{\Sigma_{\star}}(s) = \prod_{\gamma \in \mathcal{P}_{\star}} (1 - e^{-s\ell(\gamma)})$$

extends meromorphically to the whole complex plane, and moreover we may write

$$\frac{\zeta'_{\Sigma_{\star}}(s)}{\zeta_{\Sigma_{\star}}(s)} = \sum_{k=0}^{2} (-1)^k \text{tr}^{\flat}(e^{\pm \varepsilon s} \varphi_{\mp \varepsilon}^* R_{\pm, \delta}(s) |_{\Omega_c^k(M_\delta) \cap \ker \iota_X}),$$

where the flat trace is computed on $M_\delta$. Here $\mathcal{P}_\star$ denotes the set of primitive closed geodesics of $(\Sigma_\star, g)$. By [12], we may apply [35, Proposition 9] (see also [36, Theorem 9.1]) to obtain that $\zeta_{\Sigma_\star}$ is holomorphic
in {$\mathrm{Re}(s) \ge h_\star$}, except for a simple pole at $s = h_\star$, where $h_\star > 0$ is the topological entropy of the geodesic flow of $(\Sigma_\star, g)$ restricted to its trapped set. Write the Laurent expansion given in Section 2.6 of $R_{\pm,\delta}(s)$ near $s = h_\star$ as

$$R_{\pm,\delta}(s) = Y_{\pm,\delta}(s) + \frac{\Pi_{\pm,\delta}(h_\star)}{s-h_\star} + \sum_{j=2}^{J(h_\star)} \frac{(X \pm h_\star)^{j-1} \Pi_{\pm,\delta}(h_\star)}{(s-h_\star)^j} : \Omega_c^\bullet(M_\delta) \to \mathcal{D}^\bullet(M_\delta).$$

By [15, (5.8)], we have $\mathrm{tr}^b(e^{\pm\varepsilon h_\star} \varphi_{\pm\varepsilon}^* \Pi_{\pm,\delta}(h_\star)) = \mathrm{rank} \, \Pi_{\pm,\delta}(h_\star)$ and

$$\mathrm{tr}^b(\varphi_{\pm\varepsilon}^*(X \pm h_\star)^j \Pi_{\pm,\delta}(h_\star)) = 0 \quad \text{for } j = 1, \dots, J(h_\star) - 1.$$

We write $\Omega^k = \Omega_c^k(M_\delta)$ and $\Omega_0^k = \Omega^k \cap \ker \iota_X$. Then, by [18, Propositions 2.4 and 4.4], the map $s \mapsto R_{\pm,\delta}(s)|_{\Omega_0^0}$ has no pole in $\{\mathrm{Re}(s) > 0\}$. Since $\Omega_0^2 = \Omega_0^0 \wedge d\alpha$, and $R_{\pm,\delta}(s)|_{\Omega_0^2} = R_{\pm,\delta}(s)|_{\Omega_0^0} \wedge d\alpha$ (because $\varphi_t^*\alpha = \alpha$), it follows that $s \mapsto R_{\pm,\delta}(s)|_{\Omega_0^2}$ has no poles in $\{\mathrm{Re}(s) > 0\}$. In particular, the residue of $\zeta'_{\Sigma_\star}(s)/\zeta_{\Sigma_\star}(s)$ at $s = h_\star$ is given by $\mathrm{rank}(\Pi_{\pm,\delta}(h_\star)|_{\Omega_0^1})$, and since $\zeta_{\Sigma_\star}(s)$ has a simple pole at $s = h_\star$, this residue is equal to 1. Therefore,

$$\mathrm{rank}(\Pi_{\pm,\delta}(h_\star)|_{\Omega_0^1}) = 1.$$

In particular, $(X \pm h_\star)^j \Pi_{\pm,\delta} = 0$ for each $j = 1, \dots, J(h_\star) - 1$. As $R_{\pm,\delta}(s)$ commutes with $\iota_X$, it preserves the spaces $\Omega_0^k$. Writing $\Omega^k = \Omega_0^k \oplus \alpha \wedge \Omega_0^{k-1}$ we have, for any $w = u + \alpha \wedge v$ with $\iota_X u = 0$ and $\iota_X v = 0$,

$$\Pi_{\pm,\delta}(h_\star)|_{\Omega^2}(u + \alpha \wedge v) = \Pi_{\pm,\delta}(h_\star)|_{\Omega_0^2}(u) + \alpha \wedge \Pi_{\pm,\delta}(h_\star)|_{\Omega_0^1}(v).$$

Thus $\Pi_{\pm,\delta}(h_\star)|_{\Omega^2} = \alpha \wedge \iota_X \Pi_{\pm,\delta}(h_\star)|_{\Omega_0^1}$. By Proposition 3.2 and the fact that $\varphi_{\pm\varepsilon}^* \Pi_{\pm,\delta}(h_\star) = e^{\pm\varepsilon h_\star} \Pi_{\pm,\delta}(h_\star)$, we have, near $s = h_\star$,

$$(5-1) \qquad \chi \tilde{S}_{\pm}(s) \chi = \chi Y_{\pm}(s) \chi + \frac{\chi \psi^* \iota^* \iota_X \Pi_{\pm,\delta}(h_\star) \iota_* \chi}{s-h_\star},$$

where $s \mapsto Y_{\pm}(s)$ is holomorphic in a neighborhood of $h_\star$. We write

$$\Pi_{\pm,\partial} = \psi^* \iota^* \iota_X \Pi_{\pm,\delta}(h_\star) \iota_* : \Omega^\bullet(\partial) \to \mathcal{D}^\bullet(\partial).$$

Then, by what precedes, and since $\iota_X \Pi_{\pm,\delta}(h_\star)|_{\Omega^1} = 0$, we obtain that $\mathrm{rank}(\Pi_{\pm,\partial}) \le 1$. Finally, for any $\chi \in C_c^\infty(\partial \setminus \partial_0)$, we set

$$c_{\pm}(\chi) = \mathrm{tr}_s^b(\chi \Pi_{\pm,\partial} \chi).$$

**Lemma 5.1** Let $\chi \in C_c^\infty(\partial \setminus \partial_0)$ be such that $c_\pm(\chi) > 0$. Then

$$N_{\pm}(n, \chi, t) \sim \frac{(c_{\pm}(\chi)t)^n e^{h_{\star}t}}{n! h_{\star}t} \quad \text{as } t \to +\infty.$$

**Proof** Because $\chi \Pi_{\pm,\partial}$ is of rank one, it follows that $\mathrm{tr}_s^b((\chi \Pi_{\pm,\partial})^n) = c_\pm(\chi)^n$ for any $n \ge 1$ (since the flat trace of a finite-rank operator coincides with its usual trace), and thus

$$\mathrm{tr}_s^b((\chi \tilde{S}_{\pm}(s)\chi)^n) = \frac{c_{\pm}(\chi)^n}{(s-h_{\star})^n} + \mathcal{O}((s-h_{\star})^{-n+1}) \quad \text{as } s \to h_{\star}.$$
Note that here we implicitly used the fact that the flat trace of products of the form

$$ (5-2) \qquad (\chi Y_{\pm}(s)\chi)^{k_1} (\chi \Pi_{\pm,\partial\chi})^{\ell_1} (\chi Y_{\pm}(s)\chi)^{k_2} (\chi \Pi_{\pm,\partial\chi})^{\ell_2} \dots $$

makes sense. Indeed, note that both WF($\chi\Pi_{\pm,\partial\chi}$) and WF($\chi Y_{\pm}(s)\chi$) are contained in WF($\chi\tilde{S}_{\pm}(s)\chi$) by (5-1) and Cauchy's integral formula. Thus we may reproduce the proofs of Lemma 3.5, Remark 3.6 and Proposition 3.7 to obtain that the composition (5-2) is well defined and that its flat trace makes sense. Next, set $\eta_{n,\chi}(s) = \text{tr}_s^b((\chi\tilde{S}_{\pm}(s)\chi)^n)$ and

$$ g_{n,\chi}(t) = \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma, \gamma_*)=n}} \ell^\#(\gamma) \sum_{\substack{k \ge 1 \\ k \ell(\gamma) \le t}} I_{\star, \pm}(\gamma, \chi)^k \quad \text{for } t \ge 0. $$

Now, if $G_{n,\chi}(s) = \int_0^{+\infty} g_{n,\chi}(t)e^{-ts}dt$, a simple computation leads to

$$ G_{n,\chi}(s) = \frac{1}{s} \sum_{i(\gamma, \gamma_*)=n} \ell^\#(\gamma) e^{-s\ell(\gamma)} I_{\star, \pm}(\gamma, \chi)^{\ell(\gamma)/\ell^\#(\gamma)} = -\frac{\eta'_{n,\chi}(s)}{ns}, $$

where the last equality comes from Proposition 3.7. Using the expansion

$$ \eta'_{n,\chi}(s) = -nc_{\pm}(\chi)^n(s-h_*)^{-(n+1)} + \mathcal{O}((s-h_*)^{-n}) \quad \text{as } s \to h_*, $$

we obtain

$$ G_{n,\chi}(h_*s) = \frac{c_{\pm}(\chi)^n}{h_*^{n+2}(s-1)^{n+1}} + \mathcal{O}((s-h_*)^{-n}) \quad \text{as } s \to h_* $$

Then, applying the Tauberian theorem of Delange [14, théorème III],

$$ \frac{1}{h_*} g_{n,\chi}\left(\frac{t}{h_*}\right) \sim \frac{c_{\pm}(\chi)^n e^t}{h_*^{n+2} n!} t^n \quad \text{as } t \to +\infty, $$

and so

$$ (5-3) \qquad g_{n,\chi}(t) \sim \frac{(c_{\pm}(\chi)t)^n}{n!h_*} \exp(h_*t). $$

Now note that, if $\mathcal{P}_n$ is the set of primitive closed geodesics $\gamma$ with $i(\gamma, \gamma_*) = n$,

$$ g_{n,\chi}(t) \le \sum_{\substack{\gamma \in \mathcal{P}_n \\ \ell(\gamma) \le t}} \ell(\gamma) \left\lfloor \frac{t}{\ell(\gamma)} \right\rfloor I_{\star, \pm}(\gamma, \chi) \le tN(n, \chi, t). $$

As a consequence,

$$ (5-4) \qquad \liminf_{t \to +\infty} N_{\pm}(n, \chi, t) \frac{n!h_*t}{(c_{\pm}(\chi)t)^n e^{h_*t}} \ge 1. $$

For the other bound, we use the a priori bound, obtained in Section 4.1.2,

$$ (5-5) \qquad N_{\pm}(n, \chi, t) \le N(n, t) \le \frac{Ct^n e^{h_*t}}{n! h_*t} $$

to deduce that, for any $\sigma > 1$,

$$ (5-6) \qquad \limsup_{t \to +\infty} N_{\pm}\left(n, \chi, \frac{t}{\sigma}\right) \frac{n! h_*t}{t^n e^{h_*t}} = 0. $$
Now we may write

$$
\begin{align*}
(5-7) \quad N_{\pm}(n, \chi, t) &= N_{\pm}\left(n, \chi, \frac{t}{\sigma}\right) + \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma_{\star}, \gamma)=n \\ t/\sigma \le \ell(\gamma) \le t}} I_{\star, \pm}(\gamma, \chi) \\
&\le N_{\pm}\left(n, \chi, \frac{t}{\sigma}\right) + \frac{\sigma}{t} \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma_{\star}, \gamma)=n \\ t/\sigma \le \ell(\gamma) \le t}} I_{\star, \pm}(\gamma, \chi)\ell(\gamma) \le N_{\pm}\left(n, \chi, \frac{t}{\sigma}\right) + \frac{\sigma}{t} g_{n, \chi}(t),
\end{align*}
$$

which gives, with (5-3) and (5-6),

$$
\limsup_{t \to +\infty} N_{\pm}(n, \chi, t) \frac{n! h_{\star t}}{(c_{\pm}(\chi)t)^n e^{h_{\star t}}} \le \sigma.
$$

As $\sigma > 1$ is arbitrary, the lemma is proven.

□

**Remark 5.2** If we assume that $c_{\pm}(\chi) = 0$, then with the notation of the above proof, the map $s \mapsto \eta_{1,\chi}(s)$ has no pole on the line $\{\mathrm{Re}(s) = h_{\star}\}$. In particular, we may reproduce the arguments of the aforementioned proof, replacing $g_{n,\chi}(t)$ by $g_{n,\chi}(t) + \exp(h_{\star}t)$, to obtain that $s \mapsto \int_0^\infty (g_{n,\chi}(t) + \exp(h_{\star}t)) \exp(-ts) dt$ has a pole of order 1 at $s = h_{\star}$, which implies that $g_{n,\chi}(t) + \exp(h_{\star}t) \sim \exp(h_{\star}t)$ as $t \to \infty$. This gives $g_{n,\chi}(t) \ll_{t \to \infty} \exp(h_{\star}t)$, and hence

$$
N_{\pm}(1, \chi, t) \ll \frac{\exp(h_{\star}t)}{t} \quad \text{as } t \to \infty,
$$

where we used the last line of (5-7) and (5-5). Note that this bound is incompatible with the one provided
by Proposition 4.2; this will help us to prove that $c_{\pm}(\chi) > 0$, by showing that $N(1, t)$ can be controlled
by $N_{\pm}(1, \chi, t)$ whenever $\chi$ has enough support (see Section 6.1).

**5.2 The case $\gamma_*$ is separating**

In this case, $\Sigma_{\delta}$ consists of two surfaces, $\Sigma_{\delta}^{(1)}$ and $\Sigma_{\delta}^{(2)}$. We write $M_{\delta} = M_{\delta}^{(1)} \sqcup M_{\delta}^{(2)}$, where $M_{\delta}^{(j)} = S\Sigma_{\delta}^{(j)}$ for $j = 1, 2$, and $\partial = \partial^{(1)} \sqcup \partial^{(2)}$ with $\partial^{(j)} \subset M_{\delta}^{(j)}$. Note that, if $\tilde{S}_{\pm}^{(j)}(s)$ denotes the restriction of $\tilde{S}_{\pm}(s)$ to $\partial^{(j)}$, we have

$$
\tilde{S}_{\pm}^{(1)}(s) : \Omega^{\bullet}(\partial^{(1)}) \to \mathcal{D}^{\prime\bullet}(\partial^{(2)}) \quad \text{and} \quad \tilde{S}_{\pm}^{(2)}(s) : \Omega^{\bullet}(\partial^{(2)}) \to \mathcal{D}^{\prime\bullet}(\partial^{(1)}).
$$

As in Section 5.1,

$$
\chi \tilde{S}_{\pm}^{(j)}(s) \chi = \chi Y_{\pm}^{(j)}(s) \chi + \frac{\chi \Pi_{\pm, \partial}^{(j)} \chi}{s - h_j} \quad \text{as } s \to h_j,
$$

with rank($\Pi_{\pm, \partial}^{(j)}$) = 1. Here $Y_{\pm}^{(j)}(s)$ is holomorphic near $s = h_j$ and $h_j$ is the topological entropy of the geodesic flow of $\Sigma_{\delta}^{(j)}$. As before, fix $\chi \in C_c^{\infty}(\partial \setminus \partial_0)$.
**5.2.1 The case $h_1 \neq h_2$** We may assume $h_1 > h_2$, and we define

$$c_{\pm}(\chi) = \text{tr}_s^b(\chi \tilde{S}_{\pm}^{(2)}(h_1) \chi^2 \Pi_{\pm,\partial}^{(1)} \chi).$$

Because $\Pi_{\pm,\partial}^{(1)}$ is of rank one, $\text{tr}_s^b((\chi \tilde{S}_{\pm}^{(2)}(h_1) \chi^2 \Pi_{\pm,\partial}^{(1)} \chi)^n) = c_{\pm}(\chi)^n$ for any $n \ge 1$, and thus, by cyclicity of the flat trace (indeed the flat trace coincides with the real trace for operators of finite rank), as $s \to h_1$,

$$ \begin{aligned} \text{tr}_s^b((\chi \tilde{S}_{\pm}(s)\chi)^{2n}) &= \text{tr}_s^b((\chi \tilde{S}_{\pm}^{(1)}(s)\chi^2 \tilde{S}_{\pm}^{(2)}(s)\chi)^n + (\chi \tilde{S}_{\pm}^{(2)}(s)\chi^2 \tilde{S}_{\pm}^{(1)}(s)\chi)^n) \\ &= \frac{2c_{\pm}(\chi)^n}{(s-h_1)^n} + \mathcal{O}((s-h_1)^{-n+1}). \end{aligned} $$

Now we may proceed exactly as in Section 5.1 to obtain that, if $c_{\pm}(\chi) > 0$,

$$N_{\pm}(2n, \chi, t) \sim \frac{(c_{\pm}(\chi)t)^n e^{h_{\star}t}}{n! h_{\star}t} \quad \text{as } t \to +\infty.$$

**Remark 5.3** (continuation of Remark 5.2) If $h_1 \neq h_2$ and if we assume that $c_{\pm}(\chi) = 0$, then the map $s \mapsto \text{tr}_s^b((\chi \tilde{S}_{\pm}(s)\chi)^2)$ has no pole on the line $\{\text{Re}(s) = h_{\star}\}$. As in Remark 5.2, this yields

$$(5-8) \qquad N_{\pm}(2, \chi, t) \ll \frac{\exp(h_{\star}t)}{t} \quad \text{as } t \to \infty.$$

Again, the bound given in Proposition 4.5 is incompatible with (5-8)—in fact, even a weaker bound (say, a lower bound with a linear loss with respect to Theorem 2) would be incompatible with (5-8) for the case $h_1 \neq h_2$ —and this will imply that $c_{\pm}(\chi)$ is positive.

**5.2.2 The case $h_1 = h_2 = h_\star$** In that case, by writing $c_{\pm}(\chi) = \text{tr}_s^b(\chi \Pi_{\pm,\partial}^{(1)} \chi \Pi_{\pm,\partial}^{(2)})$, we have

$$\text{tr}_s^b((\chi \tilde{S}_{\pm}(s)\chi)^{2n}) = \frac{2c_{\pm}(\chi)^n}{(s-h_{\star})^{2n}} + \mathcal{O}((s-h_{\star})^{-2n+1}) \quad \text{as } s \to h_{\star}.$$

Again, provided that $c_{\pm}(\chi) \neq 0$, we may proceed exactly as in Section 5.1 to obtain

$$N_{\pm}(2n, \chi, t) \sim 2 \frac{(c_{\pm}(\chi)t^2)^n e^{h_{\star}t}}{(2n)! h_{\star}t}.$$

**Remark 5.4** (continuation of Remark 5.3) If $h_1 = h_2$ and $c_{\pm}(\chi) = 0$, then the function $s \mapsto \text{tr}_s^b((\chi \tilde{S}_{\pm}(s)\chi)^2)$ might have a pole at $s = h_{\star}$, of order at most 1. Therefore, reproducing the arguments of Section 5.1, we obtain

$$(5-9) \qquad N_{\pm}(2, \chi, t) = \mathcal{O}(\exp(h_{\star}t)) \quad \text{as } t \to \infty.$$

Note that here, assuming $c_{\pm}(\chi) = 0$ only wins us a factor of $t$ for the bound on $N_{\pm}(2, \chi, t)$ (with respect to the asymptotics of Theorem 2), whereas in Remarks 5.2 and 5.3 we could win a bit more. This is why we need a lower bound on $N(2, L)$ which is sharp up to a sublinear loss for the case where $h_1 = h_2$ (see Proposition 4.5 and the comments below). Indeed, we will see that $N(2, t)$ can be controlled by $N_{\pm}(2, \chi, t)$ whenever $\chi$ has enough support; hence, Proposition 4.5 will contradict (5-9), yielding again $c_{\pm}(\chi) > 0$ (see Section 6.2).
# 6 Proof of Theorems 1 and 2

In this section we prove Theorems 1 and 2. We will apply the asymptotic growth we obtained in the last section to some appropriate sequence of functions in $C_c^\infty(\partial \setminus \partial_0)$. Let $F \in C^\infty(\mathbb{R}, [0, 1])$ be an even function such that $F \equiv 0$ on $[-1, 1]$ and $F \equiv 1$ on $]-\infty, -2] \cup [2, +\infty[$. For any small $\eta > 0$, set

$$F_{\eta}(t) = \sum_{k \in \mathbb{Z}} F\left(\frac{t - k\pi}{\eta}\right).$$

Then $F_{\eta}$ is $2\pi$-periodic and it induces a function $F_{\eta}: \mathbb{R}/2\pi\mathbb{Z} \to \mathbb{R}_{\ge 0}$. In the coordinates from Lemma 2.3, we define

$$\chi_{\eta}(z) = F_{\eta}(\theta) \quad \text{for } z = (\tau, 0, \theta) \in \partial.$$

Then $\chi_{\eta} \in C_c^\infty(\partial \setminus \partial_0)$ for any $\eta > 0$ small; the function $\chi_{\eta}$ is introduced in order to forget about trajectories passing at distance not greater than $\eta$ from the “glancing set” $S\gamma_{\star}$.

## 6.1 The case $\gamma_{\star}$ is not separating

Recall from Section 4 that we have the a priori bounds

$$(6-1) \qquad C^{-1} \frac{e^{h_{\star} L}}{h_{\star} L} \le N(1, L) \le C e^{h_{\star} L}$$

for $L$ large enough. This estimate implies the following fact:<fnref n="12" />

$$\forall \varepsilon > 0 \exists L_0 > 0 \forall L_1 > 0 \exists L > L_1 \quad N(1, L - L_0) \le \varepsilon N(1, L).$$

In particular, we see with the first part of (4-25) in Lemma 4.11 that, for any $\eta > 0$ small enough,

$$(6-2) \qquad \liminf_{L \to +\infty} \frac{N(1, \eta, L)}{N(1, L)} \le \frac{1}{2},$$

where $N(1, \eta, L)$ is as defined in Section 4.3.

For $\eta > 0$ small and $L > 0$, neither $c_{\pm}(\chi_{\eta})$ nor $N_{\pm}(n, \chi_{\eta}, L)$ (see Section 5.1) depend on $\pm$, since $F$ is an even function. We denote them simply by $c(\eta)$ and $N(n, \chi_{\eta}, L)$, respectively. Then we claim that $c(\eta) > 0$ if $\eta > 0$ is small enough. Indeed, if $c(\eta) = 0$, then Remark 5.2 implies

$$(6-3) \qquad N(1, \chi_{\eta}, L) \ll \frac{\exp(h_{\star} L)}{h_{\star} L} \quad \text{as } L \to +\infty.$$

On the other hand, $N(1, L) = N(1, \chi_{\eta}, L) + R(\eta, L)$ with

$$R(\eta, L) = N(1, L) - N(1, \chi_{\eta}, L) \le N(1, 2\eta, L),$$

<footnote n="12">¹²If it does not hold, then there is an $\varepsilon > 0$ such that, for any $L_0 > 0$, there is an $L_1$ such that, for any $n \ge 0$, it holds that $\varepsilon < N(1, L_1 + nL_0)/N(1, L_1 + (n+1)L_0)$, which gives $N(1, L_1 + (n+1)L_0)\varepsilon^n < N(1, L_1)$ for each $n$. Now, if $L_0$ is large enough, we see that (6-1) cannot hold, by making $n \to \infty$.</footnote>
and thus, if $\eta$ is small enough, (6-2) gives

$$ \limsup_{L \to +\infty} \frac{N(1, \chi_{\eta}, L)}{N(1, L)} \ge \frac{1}{2}. $$

Since $C^{-1} \exp(h_{\star}L)/L \le N(1, L)$ for large $L$, (6-3) cannot hold, and thus $c(\eta) > 0$.

In particular, we can apply Lemma 5.1 to get $\lim_{L} N(n, \chi_{\eta}, L)(n!/(c(\eta)L)^n)(h_{\star}L/e^{h_{\star}L}) = 1$. As $N(n, L) \ge N(n, \chi_{\eta}, L)$, for $L$ large enough,

$$ C^{-1} \frac{L^n}{n!} \frac{e^{h_{\star}L}}{h_{\star}L} \le N(n, L) \le C \frac{L^n}{n!} \frac{e^{h_{\star}L}}{h_{\star}L} $$

(the upper bound comes from Section 4.1.2). Let $\varepsilon > 0$. Then the above estimate combined with the second part of (4-25) in Lemma 4.11 implies that, for $\eta > 0$ small enough,

$$ \limsup_{L} R(n, \eta, L) \frac{n! h_{\star} L}{L^n e^{h_{\star} L}} < \varepsilon, $$

where $R(n, \eta, L) = N(n, L) - N(n, \chi_{\eta}, L)$. Writing $N(n, \chi_{\eta}, L) \le N(n, L) \le N(n, \chi_{\eta}, L) + R(n, \eta, L)$, we obtain

$$ c(\eta)^n \le \liminf_{L} N(n, L) \frac{n! h_{\star} L}{L^n e^{h_{\star} L}} \le \limsup_{L} N(n, L) \frac{n! h_{\star} L}{L^n e^{h_{\star} L}} \le c(\eta)^n + \varepsilon $$

for any $\eta$ small enough (depending on $\varepsilon$!). As $\varepsilon > 0$ is arbitrary, we finally get

$$ N(n, L) \sim \frac{(c_{\star} L)^n e^{h_{\star} L}}{n! h_{\star} L} \quad \text{as } L \to +\infty, $$

where $c_{\star} = \lim_{\eta \to 0} c(\eta) < +\infty$ (the limit exists as $\eta \mapsto c(\eta)$ is nonincreasing and bounded by above by (6-1)).

## 6.2 The case $\gamma_{\star}$ is separating

### 6.2.1 The case $h_1 \neq h_2$ In this case, recall from Section 4 that we have the bound

$$ \frac{C^{-1} e^{h_{\star} L}}{\log(L)^2} \le N(2, L) \le C e^{h_{\star} L} $$

for $L$ large enough. In particular, using (4-26) in Lemma 4.11 and Remark 5.3, we may proceed exactly as in Section 6.1 to obtain

$$ N(2n, L) \sim \frac{(c_{\star} L)^n e^{h_{\star} L}}{n! h_{\star} L} \quad \text{as } L \to +\infty, $$

where $c_{\star} = \lim_{\eta \to 0} c_{\pm}(\chi_{\eta})$.

### 6.2.2 The case $h_1 = h_2 = h_{\star}$ In this case, recall from Section 4 that we have the bound

$$ \frac{C^{-1} L e^{h_{\star} L}}{\log(L)^4} \le N(2, L) \le C L e^{h_{\star} L} $$
for $L$ large enough. In particular, using Lemma 4.11 and Remark 5.4, we may proceed exactly as in Section 6.1 to obtain

$$N(2n, L) \sim 2 \frac{(c_{\star} L)^n e^{h_{\star} L}}{(2n)! h_{\star} L} \quad \text{as } L \to +\infty,$$

where $c_{\star} = \lim_{\eta \to 0} c_{\pm}(\chi_{\eta})$.

# 7 A Bowen–Margulis type measure

## 7.1 Description of the constant $c_{\star}$

In this subsection we describe the constant $c_{\star}$ in terms of Pollicott–Ruelle resonant states of the open system $(M_{\delta}, \varphi_t)$, assuming for simplicity that $\gamma_{\star}$ is not separating. By Section 2.6, since $\Pi_{\pm,\delta}(h_{\star})$ is of rank one (see Section 5.1), we may write

$$\Pi_{\pm,\delta}(h_{\star})|_{\Omega^1(M_{\delta})} = u_{\pm} \otimes (\alpha \wedge s_{\mp}) \quad \text{for } u_{\pm} \in \mathcal{D}_{E_{\pm,\delta}}^{1}(M_{\delta}) \text{ and } s_{\mp} \in \mathcal{D}_{E_{\mp,\delta}}^{1}(M_{\delta}),$$

with $\text{supp}(u_{\pm}, s_{\pm}) \subset \Gamma_{\pm,\delta}$ and $u_{\pm}, s_{\mp} \in \text{ker}(\iota_X)$. Using the Guillemin trace formula [19] and the Ruelle zeta function $\zeta_{\Sigma_{\star}}$, we see that the Bowen–Margulis measure $\mu_0$ (see [9]) of the open system $(M_{\delta}, \varphi_t)$, which is given by Bowen’s formula

$$\mu_0(f) = \lim_{L \to +\infty} \sum_{\substack{\gamma \in \mathcal{P}_{\delta} \\ \ell(\gamma) \le L}} \frac{1}{\ell(\gamma)} \int_0^{\ell(\gamma)} f(\gamma(\tau), \dot{\gamma}(\tau)) \, d\tau \quad \text{for } f \in C_c^{\infty}(M_{\delta}),$$

coincides with the distribution $f \mapsto \text{tr}_s^b(f \Pi_{\pm,\delta}(h)) = \int_{M_{\delta}} f u_{\pm} \wedge \alpha \wedge s_{\mp}$. Note that $\text{supp}(u_{\pm} \wedge \alpha \wedge s_{\mp}) \subset K_{\star}$, where $K_{\star} \subset S\Sigma_{\star}$ is the trapped set. On the other hand, by definition of $\Pi_{\pm,\delta}$,

$$c_{\star} = \lim_{\eta \to 0} \text{tr}_s^b(\chi_{\eta} \Pi_{\pm, \partial}) = - \lim_{\eta \to 0} \int_{\partial} \chi_{\eta} \psi^* \iota^* u_{\pm} \wedge \iota^* s_{\mp}.$$

## 7.2 A Bowen–Margulis type measure

In what follows we set $S_{\gamma_{\star}}\Sigma = \{(x, v) \in S\Sigma : x \in \gamma_{\star}\}$ and, for any primitive geodesic $\gamma: \mathbb{R}/\ell(\gamma)\mathbb{Z} \to \Sigma$,

$$I_{\star}(\gamma) = \{z \in S_{\gamma_{\star}}\Sigma : z = (\gamma(\tau), \dot{\gamma}(\tau)) \text{ for some } \tau\}.$$

For any $n \ge 1$, we define the set $\Gamma_n \subset S_{\gamma_{\star}}\Sigma$ by

$$\mathbb{C}\Gamma_n = \{z \in S_{\gamma_{\star}}\Sigma : (\tilde{S}_{\pm})^k(z) \text{ is well defined for } k = 1, \dots, n\}.$$

Also, we set $\ell_n(z) = \max(\ell_{+,n}(z), \ell_{-,n}(z))$, where

$$\ell_{\pm,n}(z) = \ell_{\pm}(z) + \ell_{\pm}(\tilde{S}_{\pm}(z)) + \dots + \ell_{\pm}(\tilde{S}_{\pm}^{n-1}(z)) \quad \text{for } z \in \mathbb{C}\Gamma_n,$$

and $\ell_{\pm}(z) = \inf\{t > 0 : \varphi_{\pm t}(z) \in S_{\gamma_{\star}}\Sigma\}$.

We will now prove Theorem 3, which says that, for any $f \in C^{\infty}(S_{\gamma_{\star}}\Sigma)$, the limit

$$(7-1) \qquad \mu_n(f) = \lim_{L \to +\infty} \frac{1}{N(n, L)} \sum_{\gamma \in \mathcal{P}_n} \frac{1}{n} \sum_{z \in I_{\star}(\gamma)} f(z)$$
exists and defines a probability measure $\mu_n$ on $S_{\gamma_*} \Sigma$ supported in $\Gamma_n$. We will also prove that, in the nonseparating case,

$$ (7-2) \qquad \mu_n(f) = c_*^{-n} \lim_{\eta \to 0} \mathrm{tr}_s^b(f(\chi_\eta \Pi_\pm, \partial\chi_\eta)^n), $$

where $c_* > 0$ is the constant appearing in Theorem 1. Note that here we identify $f$ with its lift $p_*^*f$ (which is a function on $\partial\Sigma$), so that the above formula makes sense (recall that $p_*: S\Sigma_* \to S\Sigma$ is the natural projection which identifies both components of $\partial S\Sigma_* = \partial\Sigma$). Of course, a similar formula holds in the nonseparating case, but we omit it here.

**Proof of Theorem 3** Let $f \in C^\infty(S_{\gamma_*} \Sigma)$ be a nonnegative function. Then, reproducing the arguments in the proof of Proposition 3.7, for Re(s) big enough,

$$ \mathrm{tr}_s^b(f(\chi_\eta \tilde{S}_\pm(s)\chi_\eta)^n) = \sum_{i(\gamma, \gamma_*)=n} \left( \sum_{z \in I_*(\gamma)} f(z) \right) e^{-s\ell(\gamma)} I_*(\gamma, \chi_\eta), $$

where $\chi_\eta$ is as defined in Section 6 and $I_*(\gamma, \chi_\eta) = I_{\star, \pm}(\gamma, \chi_\eta)$ (see Section 5; this does not depend on $\pm$, as the function $F$ used to construct $\chi_\eta$ is even). Now, as $f$ is nonnegative, we may proceed exactly as in Section 5, replacing $g_{n,\chi}(t)$ by

$$ g_{n, \chi_\eta, f}(t) = \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma, \gamma_*)=n}} \left( \sum_{z \in I_*(\gamma)} f(z) \right) \sum_{\substack{k \ge 1 \\ k \ell(\gamma) \le t}} I_*(\gamma, \chi_\eta) \quad \text{for } t \ge 0, $$

to obtain that

$$ (7-3) \quad \lim_{L \to \infty} \frac{n! h_* L}{L^n e^{h_* L}} \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma_*, \gamma)=n \\ \ell(\gamma) \le L}} \left( \sum_{z \in I_*(\gamma)} f(z) \right) I_*(\gamma, \chi_\eta) = \mathrm{Res}_{s=h_*} \mathrm{tr}_s^b (f(\chi_\eta \tilde{S}_\pm(s) \chi_\eta)^n). $$

We denote by $v_{n,\eta}(f)$ the left-hand side of (7-3). Then $\eta \mapsto v_{n,\eta}(f)$ is a nonnegative and nonincreasing function which is bounded by above by $nc_*^n \|f\|_\infty$ by Theorem 1. In particular, the formula

$$ \mu_n(f) = \lim_{\eta \to 0} \frac{1}{nc_*^n} v_{n,\eta}(f) \quad \text{for } f \in C^\infty(S_{\gamma_*} \Sigma, \mathbb{R}_{\ge 0}) $$

defines a measure $\mu_n$ on $S_{\gamma_*} \Sigma$ whose total mass is not greater than 1. In fact, its total mass is equal to 1, since, by definition of $c_*$,

$$ \mu_n(1) = \lim_{\eta \to 0} \frac{nc_*(\chi_\eta)^n}{nc_*^n} = 1. $$

Let $\varepsilon > 0$. Then, for each $f \in C^\infty(S_{\gamma_*} \Sigma, \mathbb{R}_{\ge 0})$, one has, by Lemma 4.11,

$$ \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma_*, \gamma)=n \\ \ell(\gamma) \le L}} \left( \sum_{z \in I_*(\gamma)} f(z) \right) (1 - I_*(\gamma, \chi_\eta)) \le nN(n, \eta, L) \|f\|_\infty \le \varepsilon nN(n, L) \|f\|_\infty $$
for large $L$ whenever $\eta$ is small enough. In particular, setting

$$ \mu_n^+(f) = \limsup_L \frac{A_f(n, L)}{nN(n, L)} \quad \text{and} \quad \mu_n^-(f) = \liminf_L \frac{A_f(n, L)}{nN(n, L)}, $$

where

$$ A_f(n, L) = \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma^*, \gamma)=n \\ \ell(\gamma) \le L}} \left( \sum_{z \in I_*(\gamma)} f(z) \right), $$

we see that, for each $\varepsilon > 0$ and $\eta$ small depending on $\varepsilon$,

$$ |\mu_n^{\pm}(f) - v_{n,\eta}(f)| \le \varepsilon \|f\|_{\infty}. $$

Indeed, setting

$$ A_f(n, \eta, L) = \sum_{\substack{\gamma \in \mathcal{P} \\ i(\gamma^*, \gamma)=n \\ \ell(\gamma) \le L}} \left( \sum_{z \in I_*(\gamma)} f(z) \right) I_*(\gamma, \chi_\eta), $$

we have

$$ \limsup_L \left| \left( \frac{1}{nN(n, L)} - \frac{n!L^n}{nc_*^n e^{h_*L}} \right) A_f(n, \eta, L) \right| = 0 $$

by Theorem 1, since $A_f(n, \eta, L) \le nN(n, L)$. Now we may let $\eta \to 0$ to get $|\mu_n^{\pm}(f) - \mu_n(f)| \le \varepsilon \|f\|_{\infty}$; since $\varepsilon$ is arbitrary, this yields $\mu_n^{\pm}(f) = \mu_n(f)$. This implies that the limit (7-1) exists, and moreover (7-2) holds by (7-3) (provided that $\gamma_*$ is not separating).

Next, take a general $f \in C^\infty(S_{\gamma_*} \Sigma)$, which we no longer assume to be nonnegative. Choose some smooth functions $f_{\delta, \pm}$, $\delta \in ]0, 1[$ with the property that $\|f - (f_{\delta,+} + f_{\delta,-})\|_{\infty} \le \delta$ and $\pm f_{\delta, \pm} \ge 0$, and write $f_{\delta} = f_{\delta,+} + f_{\delta,-}$. By nonnegativeness of $\pm f_{\delta, \pm}$, the arguments above imply that $A_{f_{\delta}}(n, L)/(nN(n, L)) \to \mu_n(f_{\delta})$ as $L \to \infty$. On the other hand, $|A_f(n, L) - A_{f_{\delta}}(n, L)| \le A_{|f-f_{\delta}|}(n, L) \le \delta n N(n, L)$. Letting $L \to \infty$, this yields

$$ \mu_n(f_{\delta}) - \delta \le \liminf_L \frac{A_f(n, L)}{nN(n, L)} \le \limsup_L \frac{A_f(n, L)}{nN(n, L)} \le \mu_n(f_{\delta}) + \delta. $$

Since $\mu_n(f_{\delta}) \to \mu_n(f)$ as $\delta \to 0$, (7-1) and (7-2) are valid for $f$.

Finally, if $f \in C_c^\infty(S_{\gamma_*} \Sigma \setminus \Gamma_n)$ then there is $L > 0$ such that

$$ \ell_n(z) \le L \quad \text{for } z \in \text{supp}(f). $$

In particular, for any $\gamma \in \mathcal{P}$ such that $i(\gamma, \gamma_*) = n$ and $\ell(\gamma) > L$, we have $f(z) = 0$ for any $z \in I_*(\gamma)$. This shows that $\mu_n(f) = 0$, and the support condition for $\mu_n$ follows. $\square$

# 8 A large deviation result

The goal of this section, which is independent of the rest of the paper, is to prove the following result, which is a consequence of a classical large deviation result by Kifer [25]:
**Proposition 8.1** There exists $I_* > 0$ such that the following holds. For any $\varepsilon > 0$, there are $C, \delta > 0$ such that, for large $L$,

$$ (8-1) \qquad \frac{1}{N(L)} \# \left\{ \gamma \in \mathcal{P} : \ell(\gamma) \le L \text{ and } \left| \frac{i(\gamma, \gamma_*)}{\ell(\gamma)} - I_* \right| \ge \varepsilon \right\} \le C \exp(-\delta L). $$

In fact, $I_* = 4i(\bar{m}, \delta_{\gamma_*})$, where $i$ is Bonahon's intersection form [6], $\delta_{\gamma_*}$ is the Dirac measure on $\gamma_*$ and $\bar{m}$ is the renormalized Bowen-Margulis measure on $M$ (here we see the intersection form as a function on the space of $\varphi$-invariant measures on $S\Sigma$, as described below). Lalley [28] showed a similar result for self-intersection numbers; see also [41] for self-intersection numbers with prescribed angles.

## 8.1 Bonahon's intersection form

Let $M_\varphi(S\Sigma)$ be the set of finite positive measures on $S\Sigma$ invariant by the geodesic flow, endowed with the vague topology. For any closed geodesic $\gamma$, we denote by $\delta_\gamma \in M_\varphi(S\Sigma)$ the Lebesgue measure of $\gamma$ parametrized by arc length (thus of total mass $\ell(\gamma)$). Let $\mu \in M_\varphi(S\Sigma)$ be the Liouville measure, that is, the measure associated to the volume form $\frac{1}{2}\alpha \wedge d\alpha$.

**Proposition 8.2** (Bonahon [7]; see also Otal [34]) There exists a continuous function

$$ i : M_{\varphi}(S\Sigma) \times M_{\varphi}(S\Sigma) \to \mathbb{R}_{+} $$

which is additive and positively homogeneous with respect to each variable and such that $i(\mu, \mu) = 2\pi \text{vol}(\Sigma)$ and

$$ i(\delta_{\gamma}, \delta_{\gamma'}) = i(\gamma, \gamma') \quad \text{and} \quad i(\mu, \delta_{\gamma}) = 2\ell(\gamma), $$

for any closed geodesics $\gamma$ and $\gamma'$.

**Remark 8.3** (i) Actually, Bonahon's intersection form is a pairing on the space of geodesic currents. This space is naturally identified with the space of $\varphi$-invariant measures on $S\Sigma$ which are also invariant by the flip $R: (x, v) \mapsto (x, -v)$. By $i(v, v')$ for general $v, v' \in M_\varphi(S\Sigma)$ we simply mean $i(\Phi(v), \Phi(v'))$ where $\Phi: v \mapsto v + R^*v$ (note that $\varphi_t R = R\varphi_{-t}$ for $t \in \mathbb{R}$).

(ii) The formulae for $i(\mu, \mu)$ and $i(\mu, \delta_\gamma)$ differ from [7]; this is due to our convention, since here the Liouville measure $\mu$ corresponds to twice the Liouville current considered in [7].

## 8.2 Large deviations

For any $v \in M_\varphi(S\Sigma)$ we denote by $h(v)$ the measure-theoretical entropy of $\varphi$ with respect to $v$. Then we have the following result:

**Proposition 8.4** (Kifer [25]) Let $F \subset M_\varphi^1(S\Sigma)$ be a closed set, where $M_\varphi^1(S\Sigma)$ is the set of $\varphi$-invariant probability measures on $S\Sigma$. Then

$$ \limsup_L \frac{1}{L} \log \frac{1}{N(L)} \# \left\{ \gamma \in \mathcal{P} : \ell(\gamma) \le L \text{ and } \frac{\delta_\gamma}{\ell(\gamma)} \in F \right\} \le \sup_{v \in F} h(v) - h, $$

where $h$ is the entropy of the geodesic flow.
**Proof of Proposition 8.1** We denote by $\bar{m} \in \mathcal{M}_{\varphi}^1(S\Sigma)$ the unique probability measure of maximal entropy, that is,

$$ \bar{m} = \lim_{L \to +\infty} \sum_{\substack{\gamma \in \mathcal{P} \\ \ell(\gamma) \le L}} \frac{\delta_{\gamma}}{\ell(\gamma)}, $$

where the convergence holds in the weak sense. Let $\varepsilon > 0$. Define

$$ F_{\varepsilon} = \{v \in \mathcal{M}_{\varphi}^{1}(S\Sigma) : |i(v, \delta_{\gamma_{\star}}) - i(\bar{m}, \delta_{\gamma_{\star}})| \ge \varepsilon\}. $$

Then $F_{\varepsilon}$ is closed in $\mathcal{M}_{\varphi}^1(S\Sigma)$, and thus compact by the Banach–Alaoglu theorem, and $\bar{m} \in \mathcal{C}F_{\varepsilon}$ so that $\delta = h - \sup_{v \in F_{\varepsilon}} h(v) > 0$. In particular, for large $L$,

$$ \frac{1}{N(L)} \# \left\{ \gamma \in \mathcal{P} : \frac{\delta_{\gamma}}{\ell(\gamma)} \in F_{\varepsilon} \right\} \le C \exp(-\delta' L) $$

for some $0 < \delta' < \delta$ and $C > 0$. Now, by Proposition 8.2, $\delta_{\gamma}/\ell(\gamma) \in F_{\varepsilon}$ gives $|i(\gamma, \gamma_{\star})/\ell(\gamma) - i(\bar{m}, \delta_{\gamma_{\star}})| \ge \varepsilon$. Let $I_{\star} = i(\bar{m}, \delta_{\gamma_{\star}})$. Then it is a well-known fact that $\bar{m}$ has full support in $S\Sigma$, which implies $I_{\star} > 0$ by definition of $i(\bar{m}, \delta_{\gamma_{\star}})$; see [34]. $\square$

**Remark 8.5** (i) It is not hard to see that Proposition 8.1 implies

$$ \frac{1}{N(L)} \sum_{\ell(\gamma) \le L} i(\gamma, \gamma_{\star}) \sim I_{\star} L $$

as $L \to +\infty$. Thus we recover [39, Theorem 4].

(ii) If $(\Sigma, g)$ is hyperbolic, then $\bar{m}$ is the renormalized Liouville measure and, with Proposition 8.2, we find

$$ I_{\star} = \frac{\ell(\gamma_{\star})}{2\pi^2(g-1)}. $$

(iii) If $\varepsilon < I_{\star}$ then every closed geodesic $\gamma$ which does not intersect $\gamma_{\star}$ satisfies $\delta_{\gamma}/\ell(\gamma) \in F_{\varepsilon}$. In particular, the right-hand side of (8-1) is bounded from below by $C \exp((h_{\star} - h)L)$, where we used that $N(0, L) \sim \exp(h_{\star}L)/h_{\star}L$ and $N(L) \sim \exp(hL)/hL$ as $L \to \infty$.

## Appendix A Closed geodesics minimize intersection numbers

In this section we prove Lemma 2.1. We proceed by contradiction and assume that $i(\gamma_1, \gamma_2) < |\gamma_1 \cap \gamma_2|$. As $\gamma_1$ and $\gamma_2$ are not powers of each other, the images of $\gamma_1$ and $\gamma_2$ intersect transversally (otherwise their images would coincide by uniqueness of the geodesic equation). Since $i(\gamma_1, \gamma_2) < |\gamma_1 \cap \gamma_2|$, we may find loops $\alpha_j : \mathbb{R}/\mathbb{Z} \to \Sigma$ for $j = 1, 2$ with $\alpha_j \sim \gamma_j$ and $|\alpha_1 \cap \alpha_2| < |\gamma_1 \cap \gamma_2|$, and we may moreover assume that $\alpha_1$ and $\alpha_2$ intersect transversally. Let $H_j : [0, 1] \times \mathbb{R}/\mathbb{Z} \to \Sigma$ for $j = 1, 2$ be smooth homotopies between $\gamma_j$ and $\alpha_j$, and define $H : [0, 1] \times \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z} \to \Sigma \times \Sigma$ by setting

$$ H(s, \tau_1, \tau_2) = (H_1(s, \tau_1), H_2(s, \tau_2)) \quad \text{for } (s, \tau_1, \tau_2) \in [0, 1] \times \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}. $$
Let $\Delta(\Sigma) = \{(x, x) : x \in \Sigma\}$ be the diagonal in $\Sigma$. Then $H(0, \cdot)$ and $H(1, \cdot)$ are transverse to $\Delta(\Sigma)$, in the sense that, for every $k = 0, 1$ and $(\tau_1, \tau_2) \in \mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}$ with $H(k, \tau_1, \tau_2) \in \Delta(\Sigma)$,

$$dH(k, \tau_1, \tau_2)T_{(k,\tau_1,\tau_2)}(\mathbb{R}/\mathbb{Z} \times \mathbb{R}/\mathbb{Z}) + T_{H(k,\tau_1,\tau_2)}\Delta(\Sigma) = T_{H(k,\tau_1,\tau_2)}(\Sigma \times \Sigma).$$

In particular, by [20, Corollary page 73] we may assume that $H$ is globally transverse to $\Delta(\Sigma)$, so that $H^{-1}(\Delta(\Sigma))$ is a smooth 1-dimensional submanifold of $[0, 1] \times (\mathbb{R}/\mathbb{Z})^2$. Now

$$|\gamma_1 \cap \gamma_2| = |H^{-1}(\Delta(\Sigma)) \cap \{0\} \times (\mathbb{R}/\mathbb{Z})^2| \quad \text{and} \quad |\alpha_1 \cap \alpha_2| = |H^{-1}(\Delta(\Sigma)) \cap \{1\} \times (\mathbb{R}/\mathbb{Z})^2|.$$

Since $|\gamma_1 \cap \gamma_2| > |\alpha_1 \cap \alpha_2|$ and because $H^{-1}(\Delta(\Sigma))$ is smooth, we may find a smooth path $c: [0, 1] \to [0, 1] \times (\mathbb{R}/\mathbb{Z})^2$ such that $c(0) \neq c(1)$ and

$$\mathrm{Im}(c) \subset H^{-1}(\Delta(\Sigma)) \quad \text{and} \quad c(0), c(1) \in \{0\} \times (\mathbb{R}/\mathbb{Z})^2.$$

Write $c = (S, T_1, T_2)$ for some smooth functions $S: [0, 1] \to [0, 1]$ and $T_j: [0, 1] \to \mathbb{R}/\mathbb{Z}$, and for $u \in [0, 1]$ define the path $c_u = (uS, T_1, T_2): [0, 1] \to [0, 1] \times (\mathbb{R}/\mathbb{Z})^2$. Let $x_k = H(c(k)) \in \Sigma$ for $k = 0, 1$. Then define the paths

$$\beta_{j,u} = \pi_j \circ H \circ c_u: [0, 1] \to \Sigma \quad \text{for } j = 1, 2 \text{ and } u \in [0, 1],$$

where $\pi_1, \pi_2: \Sigma \times \Sigma \to \Sigma$ are the projections over the first and second factor, respectively. As $c_1 = c$ and $\mathrm{Im}(c) \subset H^{-1}(\Delta(\Sigma))$, we have $\beta_{1,1} = \beta_{2,1}$. In particular, the paths $\beta_{1,0}$ and $\beta_{2,0}$ are homotopic within the space of curves linking $x_0$ and $x_1$, since for each $u$, one has $\beta_{j,u}(k) = x_k$ for $j = 1, 2$ and $k = 0, 1$. Moreover, the paths $\beta_{1,0}$ and $\beta_{2,0}$ are subpaths of $\gamma_1$ and $\gamma_2$, respectively, and are in particular geodesic paths. Let $\tilde{\Sigma}$ be a universal cover of $\Sigma$ and take $\tilde{x}_0 \in \tilde{\Sigma}$ a lift of $x_0$. For $j = 1, 2$, let $\tilde{\beta}_j: [0, 1] \to \tilde{\Sigma}$ be the unique lift of $\beta_{j,0}$ starting at $\tilde{x}_0$. Then $\tilde{\beta}_1(1) = \tilde{\beta}_2(1)$ since the paths $\beta_{j,0}$ for $j = 1, 2$ are homotopic in $\Sigma$ via a homotopy preserving endpoints. In particular, we have found two distinct geodesic segments of $\tilde{\Sigma}$ joining $\tilde{x}_0$ and $\tilde{\beta}_0(1)$ (the image of the paths $\tilde{\beta}_{j,0}$ for $j = 1, 2$ cannot coincide since $c(0) \neq c(1)$ and the intersection $\gamma_1 \cap \gamma_2$ is transverse). Thus the exponential map $\exp_{\tilde{x}_0}: T_{\tilde{x}_0} \tilde{\Sigma} \to \tilde{\Sigma}$ at $\tilde{x}_0$ is not a diffeomorphism, and $\tilde{\Sigma}$ cannot be negatively curved by virtue of the Cartan-Hadamard theorem (see for example [29, Theorem 11.5]). This completes the proof.

## Appendix B An elementary fact about pullbacks of distributions

**Lemma B.1** Let $K \in \mathcal{D}'(\mathbb{R}^d \times \mathbb{R}^d)$ be a compactly supported distribution. We assume that $WF(K) \subset \Gamma$, where $\Gamma \subset T^*(\mathbb{R}^d \times \mathbb{R}^d)$ is a closed conical subset such that

$$\Gamma \cap N^*\Delta = \emptyset, \quad \text{where } N^*\Delta = \{(x, \xi, x, -\xi) : (x, \xi) \in T^*\mathbb{R}^d\}.$$

In particular, the pullback $i^*K$, where $i: x \mapsto (x, x)$, is well defined. Then, for $N \in \mathbb{N}_{\ge 1}$ large enough, the following holds. Let $u \in C_c^N(\mathbb{R}^d)$ and assume that the pullback $i^*(\pi_1^*uK)$ is well defined, where $\pi_1: (x, x) \mapsto x$ is the projection on the first factor. Then

$$i^*(\pi_1^*u \cdot K) = u \cdot i^*K.$$
**Proof** Let $K_\epsilon \in C^\infty(\mathbb{R}^d \times \mathbb{R}^d)$, $\epsilon \in ]0, 1]$, be a sequence of distributions supported in a fixed compact set such that $K_\epsilon \to K$ in $D'_\Gamma(\mathbb{R}^d \times \mathbb{R}^d)$. Let $\Gamma' \subset T^*(\mathbb{R}^d \times \mathbb{R}^d)$ be an open conical subset containing $N^*\Delta$. As $K_\epsilon$ is compactly supported, we may assume that $|t-q| > \delta_0$ for any $(t, q) \in \Gamma \times \Gamma'$ such that $|t| = |q| = 1$ for some $\delta_0 > 0$. By definition of the convergence in $D'_\Gamma(\mathbb{R}^d \times \mathbb{R}^d)$ (see [23, Definition 8.2.2]), for every $N$ there is $C_N > 0$ such that, for any $\epsilon > 0$ small enough,

$$ (B-1) \qquad |\hat{K}_\epsilon(q)| \le C_N \langle q \rangle^{-N} \quad \text{for } q \in \Gamma'. $$

Let $\Gamma'' \subset \Gamma'$ be another open conical subset containing $N^*\Delta$, and let $\delta > 0$ be such that, for any $q \in \Gamma''$ and $t \in \mathbb{R}^{2d}$,

$$ (B-2) \qquad |t-q| < \delta|q| \implies t \in \Gamma'. $$

Then, for any $q \in \Gamma''$,

$$ \begin{aligned} (2\pi)^{2d} |\widehat{K_\epsilon \pi_1^* u}(q)| &\le \int_{\mathbb{R}_t^{2d}} |\hat{K}_\epsilon(t)| \cdot |\widehat{\pi_1^* u}(q-t)| dt \\ &\le \int_{|t-q| < \delta|q|} |\hat{K}_\epsilon(t)| \cdot |\widehat{\pi_1^* u}(q-t)| dt + \int_{|t-q| \ge \delta|q|} |\hat{K}_\epsilon(t)| \cdot |\widehat{\pi_1^* u}(q-t)| dt. \end{aligned} $$

Let $N_1, N_2 \in \mathbb{N}_{\ge 1}$ and $\langle t \rangle = \sqrt{1 + |t|^2}$. Then, using (B-1), (B-2) and Peetre's inequality, and assuming that $u \in C_c^{N_2}(\mathbb{R}^d)$ with $N_2 \ge 2d + 1$,

$$ \begin{aligned} \int_{|t-q| < \delta|t|} |\hat{K}_\epsilon(t)| \cdot |\widehat{\pi_1^* u}(q-t)| dt &\le C_{N_1, N_2} \int_{|t-q| < \delta|q|} \langle t \rangle^{-N_1} \langle q-t \rangle^{-N_2} dt \\ &\le C'_{N_1, N_2} \langle q \rangle^{-N_1+N_2} \int_{\mathbb{R}^d} \langle t \rangle^{-N_2} dt. \end{aligned} $$

On the other hand, if $k$ is the order of $K$ and $N_3 \in \mathbb{N}_{\ge 1}$ is such that $u \in C_c^{N_3}(\mathbb{R}^d)$, then

$$ \begin{aligned} \int_{|t-q| \ge \delta|q|} |\hat{K}_\epsilon(t)| \cdot |\widehat{\pi_1^* u}(q-t)| dt &\le C_{k, N_3} \int_{|t-q| \ge \delta|q|} \langle t \rangle^k \langle q-t \rangle^{-N_3} \\ &\le C'_{k, N_3} \langle q \rangle^{-N_3+(k+2d+1)} \int_{\mathbb{R}^{2d}} \langle t \rangle^{-2d-1} dt. \end{aligned} $$

Therefore, if $u \in C^N(\mathbb{R}^d)$ with $N = k + 2d + 1 + N'$,

$$ (B-3) \qquad (2\pi)^{2d} |\widehat{K_\epsilon \pi_1^* u}(q)| \le C_N \langle q \rangle^{-N'} \quad \text{for } q \in \Gamma''. $$

Note that, for $\varphi \in C_c^\infty(\mathbb{R}^d)$,

$$ \langle i^*(K_\epsilon \pi_1^* u), \varphi \rangle = \int_{\mathbb{R}_x^d} \varphi(x) \int_{\mathbb{R}_\xi^d \times \mathbb{R}_\eta^d} \widehat{K_\epsilon \pi_1^* u}(\xi, \eta) e^{ix(\xi+\eta)} \, d\xi \, d\eta \, dx. $$

Indeed, (B-3) shows that the integral in $(\xi, \eta)$ converges near $N^*\Delta$ if $N' \ge 2d + 1$, and far from $N^*\Delta$ we can use the stationary phase method to get enough convergence in $(\xi, \eta)$, so the above integral makes sense as an oscillatory integral and coincides with $\langle i^*(K_\epsilon \pi_1^* u), \varphi \rangle$, since this formula is obviously true if $u$ is smooth. Moreover, all the above estimates are uniform in $\epsilon$ and thus, letting $\epsilon \to 0$, we obtain the desired result, since obviously $i^*(K_\epsilon \pi_1^* u) = u(i^*K_\epsilon)$ for each $\epsilon \in ]0, 1]$. $\square$
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Institut de Mathématiques d'Orsay, Université Paris-Saclay
Orsay, France

Current address: Laboratoire de Mathématiques Jean Leray, Université de Nantes
Nantes, France

yann.chaubet@univ-nantes.fr

Proposed: Benson Farb

Seconded: Mladen Bestvina, Dmitri Burago

Received: 27 August 2021

Revised: 4 May 2022
# GEOMETRY & TOPOLOGY

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GEOMETRY & TOPOLOGY

Volume 28 Issue 2 (pages 497–1003) 2024

<table>
  <tbody>
    <tr>
      <td>On the top-weight rational cohomology of A<sub>g</sub></td>
      <td>497</td>
    </tr>
    <tr>
      <td>MADELINE BRANDT, JULIETTE BRUCE, MELODY CHAN, MARGARIDA MELO, GWYNETH MORELAND and COREY WOLFE</td>
      <td></td>
    </tr>
    <tr>
      <td>Algebraic uniqueness of Kähler-Ricci flow limits and optimal degenerations of Fano varieties</td>
      <td>539</td>
    </tr>
    <tr>
      <td>JIYUAN HAN and CHI LI</td>
      <td></td>
    </tr>
    <tr>
      <td>Valuations on the character variety: Newton polytopes and residual Poisson bracket</td>
      <td>593</td>
    </tr>
    <tr>
      <td>JULIEN MARCHÉ and CHRISTOPHER-LLOYD SIMON</td>
      <td></td>
    </tr>
    <tr>
      <td>The local (co)homology theorems for equivariant bordism</td>
      <td>627</td>
    </tr>
    <tr>
      <td>MARCO LA VECCHIA</td>
      <td></td>
    </tr>
    <tr>
      <td>Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories</td>
      <td>641</td>
    </tr>
    <tr>
      <td>HANNAH ALPERT and FEDOR MANIN</td>
      <td></td>
    </tr>
    <tr>
      <td>Closed geodesics with prescribed intersection numbers</td>
      <td>701</td>
    </tr>
    <tr>
      <td>YANN CHAUBET</td>
      <td></td>
    </tr>
    <tr>
      <td>On endomorphisms of the de Rham cohomology functor</td>
      <td>759</td>
    </tr>
    <tr>
      <td>SHIZHANG LI and SHUBHODIP MONDAL</td>
      <td></td>
    </tr>
    <tr>
      <td>The nonabelian Brill-Noether divisor on M&#773;<sub>13</sub> and the Kodaira dimension of R&#773;<sub>13</sub></td>
      <td>803</td>
    </tr>
    <tr>
      <td>GAVRIL FARKAS, DAVID JENSEN and SAM PAYNE</td>
      <td></td>
    </tr>
    <tr>
      <td>Orbit equivalences of R&#773;-covered Anosov flows and hyperbolic-like actions on the line</td>
      <td>867</td>
    </tr>
    <tr>
      <td>THOMAS BARTHELMÉ and KATHRYN MANN</td>
      <td></td>
    </tr>
    <tr>
      <td>Microlocal theory of Legendrian links and cluster algebras</td>
      <td>901</td>
    </tr>
    <tr>
      <td>ROGER CASALS and DAPING WENG</td>
      <td></td>
    </tr>
    <tr>
      <td>Correction to the article Bimodules in bordered Heegaard Floer homology</td>
      <td>1001</td>
    </tr>
    <tr>
      <td>ROBERT LIPSHITZ, PETER OZSVÁTH and DYLAN P THURSTON</td>
      <td></td>
    </tr>
  </tbody>
</table>