Proof. Since $C^\infty _0(M)$ is dense in $H^1(M)$ , we can assume without loss of generality that f is compactly supported with $\int _M f^2 = 1$ .

Observe that we can find $C^1$ functions $g,h:\mathbb {R}\rightarrow [0,1]$ so that

• ○ $g^2(x) + h^2(x)=1$ for any $x\in \mathbb {R}$ ,

• ○ $supp(g)\subseteq (-\infty ,1],\, supp(h)\subseteq (0,+\infty ]$ .

Given a positive constant $r>0$ , we define $u:=u_r(x),\alpha :=\alpha _r(x)\in C^1_0(M)$ satisfying the following properties, by using scalings of $g,h$ along equidistant sets to $\partial C(M)$ :

1. 1. $0\leq u,\alpha \leq 1$

2. 2. $supp(u) \subseteq B_{2r}(C(M))$

3. 3. $supp(\alpha ) \subseteq int(B^c_r(C(M)))$

4. 4. $u^2+\alpha ^2\equiv 1$

5. 5. $|\nabla u|,|\nabla \alpha |\leq \frac {C}{r}$ everywhere in M, for some constant C independent of r and M.

Similarly, for the thick-thin decomposition of M, we define functions $v:=v_r(x),\beta :=\beta _r(x)\in C^1_0(M)$ along equidistant sets to $\partial M^{>\epsilon }$ satisfying the following properties

1. 1. $0\leq v,\beta \leq 1$

2. 2. $supp(v) \subseteq B_{2r}(M^{>\epsilon })$

3. 3. $supp(\beta ) \subseteq int(B^c_r(M^{>\epsilon }))$

4. 4. $v^2+\beta ^2\equiv 1$

5. 5. $|\nabla v|,|\nabla \beta |\leq \frac {C}{r}$ everywhere in M, for some constant C independent of r and M.

Define then $f_1:=uvf,\,f_2:=u\beta f,\,f_3:=\alpha f$ which are in $C^1_0(M)$ . By the definitions of $u,\alpha ,v,\beta $ , we have

1. 1. $supp(f_1)\subseteq B_{2r}(M^{>\epsilon }\cap C(M))$

2. 2. $supp(f_2)\subseteq B^c_{r}(M^{>\epsilon })$

3. 3. $supp(f_3)\subseteq B^c_{r}(C(M))$

4. 4. $f^2_1+f^2_2+f^2_3=f^2$ .

We can expand $R(f_1)$ as

$$\begin{align*}R(f_1) = \bigg(\int_M f^2|\nabla (uv)|^2+2uvf\langle\nabla (uv), \nabla f \rangle + u^2v^2|\nabla f|^2\bigg)\bigg/\bigg(\int_M u^2v^2f^2\bigg). \end{align*}$$

Since $\nabla (uv) = u\nabla v + v\nabla u$ , then it follows that $|\nabla (uv)|\leq \frac {2C}{r}$ , and subsequently

$$\begin{align*}\int_M f^2|\nabla (uv)|^2 \leq \frac{4C^2}{r^2}. \end{align*}$$

By Cauchy-Schwarz, we also have that

$$\begin{align*}\bigg|\int_M 2uvf\langle\nabla (uv), \nabla f \rangle\bigg| \leq 2\int_M |\langle f\nabla(uv),\nabla f \rangle| \leq \frac{4C}{r}\sqrt{R(f)}. \end{align*}$$

Collecting these inequalities and defining $a:=\int _M f^2_1$ for convenience, we arrive to

(2) $$ \begin{align} R(f_1) \leq \bigg( \frac{4C^2}{r^2} + \frac{4C}{r}\sqrt{R(f)} + \int_M u^2v^2|\nabla f|^2\bigg)\bigg/a. \end{align} $$

Similarly, define $b:=\int _M f^2_2,\,c:=\int _M f^2_3$ . Then

(3) $$ \begin{align} R(f_2) \leq \bigg( \frac{4C^2}{r^2} + \frac{4C}{r}\sqrt{R(f)} + \int_M u^2\beta^2|\nabla f|^2\bigg)\bigg/b, \end{align} $$

(4) $$ \begin{align} R(f_3) \leq \bigg( \frac{4C^2}{r^2} + \frac{4C}{r}\sqrt{R(f)} + \int_M \alpha^2|\nabla f|^2\bigg)\bigg/c. \end{align} $$

By doing $a(2)+b(3)+c(4)$ , we obtain

(5) $$ \begin{align} aR(f_1)+bR(f_2)+cR(f_3) &\leq \bigg( \frac{12C^2}{r^2} + \frac{12C}{r}\sqrt{R(f)} + \int_M (u^2v^2 + u^2\beta^2 + \alpha^2)|\nabla f|^2\bigg)\nonumber\\ &= \bigg( \frac{12C^2}{r^2} + \frac{12C}{r}\sqrt{R(f)} + \int_M |\nabla f|^2\bigg)\nonumber\\ &\leq\bigg( \frac{12C^2}{r^2} + \frac{12C}{r}\sqrt{\mu} + \mu \bigg). \end{align} $$

Since $supp(f_2)\subseteq int(B^c_{r}(M^{>\epsilon })),\,supp(f_3)\subseteq int(B^c_{r}(C(M)))$ , we have by Lemmas 2.2 and 2.3 (or more precisely, by applying a combination of the lemmas on each component of M) that $R(f_2)\geq (\tanh r)^2(n-1)^2/4,\,R(f_3)\geq (\tanh r)^2(n-1)^2/4$ . Using these bounds together with the obvious bound $aR(f_1)\geq 0$ , we arrive to

$$\begin{align*}(b+c)\frac{(\tanh r)^2(n-1)^2}{4} \leq \frac{12C^2}{r^2} + \frac{12C}{r}\sqrt{\mu} + \mu, \end{align*}$$

$$\begin{align*}(b+c) \leq \frac{4}{(\tanh r)^2(n-1)^2}\bigg( \frac{12C^2}{r^2} + \frac{12C}{r}\sqrt{\mu} + \mu\bigg). \end{align*}$$

By the fact that $a+b+c=1$ , we obtain

$$\begin{align*}a \geq \frac{4}{(\tanh r)^2(n-1)^2}\bigg(\frac{(\tanh r)^2(n-1)^2}{4} - \frac{12C^2}{r^2} - \frac{12C}{r}\sqrt{\mu} - \mu\bigg). \end{align*}$$

The result follows from observing that for the left-hand side, we have $\int _{B_{2r}(M^{>\epsilon }\cap C(M))} f^2 \geq a$ , whereas the right-hand side depends only on $\mu ,r$ and converges to $1-\frac {4\mu }{(n-1)^2}>0$ as $r\rightarrow +\infty $ .

Proof. By Proposition 3.1, there exist $r>0$ and $\eta>0$ independent of k so that $\int _{B_{2r}(C(M_k)^{>\epsilon })} |f_k|^2\geq \eta $ . By elliptic regularity and strong convergence, we have that the Sobolev norms

$$\begin{align*}\Vert f_k \Vert_{W^{2,\ell}(B_{2r}(C(M_k)^{>\epsilon}))} \end{align*}$$

are uniformly bounded for any given $\ell $ . By the Rellich-Kondrachov compactness theorem, we can take a convergent subsequence with limit f in $B_{2r}(C(M)^{>\epsilon })$ in any $W^{2,\ell }$ norm. Taking $r\rightarrow +\infty $ and doing a Cantor diagonal argument, we have that $R(f) \leq \mu ,\, -\Delta _M f = R(f)f,\, \int _{M} |f|^2\geq \eta $ , which concludes the Lemma.

Proof. To prove the theorem, we will show the convergence of eigenspaces. Namely, let $V_k, V$ denote the linear spaces of functions generated by the eigenfunctions with eigenvalues in $Spec_\mu (M_k)$ and $ Spec_\mu (M)$ , which have a natural orthogonal decomposition by the eigenspaces of $Spec_\mu (M_k)$ and $Spec_\mu (M)$ . We show that $V_k\rightarrow V$ , in the following sense:

1. 1. Any function $f\in V$ can be obtained as the limit of a strongly convergent sequence $(f_k\in V_k)_{k\in \mathbb {N}}$ .

2. 2. Any sequence of families $( f_{l,k} \subset V_k)_{k\in \mathbb {N}}$ of orthonormal functions in $M_k$ converges strongly (after possibly taking a subsequence) to a linearly independent family of functions in M.

Item (1) implies that $\liminf _{k\rightarrow \infty } m_{\lambda , k}\geq m_\lambda $ , and Item (2) implies that $\limsup _{k\rightarrow \infty } m_{\lambda , k}\leq m_\lambda $ . Thus, it suffices to prove the convergence of eigenspaces. We first show Item (2). Suppose that $f_{1,k},\ldots f_{l,k}$ are orthonormal eigenfunctions of $M_k$ . By Lemma 3.2, we can assume they converge in compact sets to $f_1,\ldots ,f_l$ . If the functions $f_1,\ldots , f_l$ are not linearly independent in $L^2(M)$ , there exist real numbers $\alpha _1,\ldots ,\alpha _l$ not all vanishing so that $\alpha _1f_1+\ldots +\alpha _lf_l\equiv 0$ . Hence, $g_k = \alpha _1f_{1,k}+\ldots +\alpha _lf_{l, k}$ are functions in $H^1(M_k)$ with norm $\sqrt {\alpha _1^2+\ldots + \alpha _k^2}\neq 0$ . We can normalize $\Vert g_k\Vert _{L^2(M)}=1$ so that $R(g_k)\leq \mu $ , and since the limit of $g_k$ in compact sets is not identically zero from Proposition 3.1, we have a contradiction.

Now we prove Item (1). Assume that not all functions in V are obtained as limits of functions in $V_k$ . Let $V'$ be the proper maximal space in V, consisting of functions that can be obtained as limits. Assume that there exists an eigenfunction f of M with eigenvalue $\lambda $ , such that f is orthogonal to $V'$ . Approximate f in $H^1(M)$ by a compactly supported function $f_0$ , which is normalized so that $\int _M|f_0|^2=1$ and $R(f_0)$ is close to $\lambda $ . It follows that $\int _M f_0g =\int _M (f_0-f)g$ can be taken uniformly small for all $g\in V'$ with $\int _M |g|^2=1$ . Let $f^{k}_{0}$ be the pullback of $f_{0}$ in $M_k$ by the maps $\varphi _{k,i}$ from the definition of strong convergence. Then for sufficiently large k, we have that (after identifying the compact cores) $\int _{M_k} f_{0}^{k}g_k$ can be also taken uniformly small for any $g_k\in V_k$ with $\int _{M_k}|g_k|^2=1$ by Proposition 3.1. For large k, we also have that in $M_k$ the Rayleigh quotient $R(f^{k}_{0})$ is close to $\lambda $ . Denote then by $f_{0,k}$ the projection of $f_{0}^{k}$ perpendicular to $V_k$ . Then $R(f_{0, k})$ is also very close to $\lambda $ for sufficiently large k. Hence, this contributes to an eigenfunction in $M_k$ which does not belong to $V_k$ . However, by construction, $V_k$ is the linear space of functions generated by eigenfunctions with eigenvalues in $Spec_{\mu }(M_k)$ , which gives a contradiction. Therefore, any function $f\in V$ can be obtained as the limit of a strongly convergent sequence $(f_k\in V_k)$ .

Proof. By [Reference McMullenMcM99, Theorem 1.5] and Theorem 2.4, we have that $\lim _{k\rightarrow \infty } \lambda _0(M_k) = \lambda _0(M)$ . By Theorem 3.3, if $\lambda _{1}(M)\geq (n-1)^{2}/4$ , then $\liminf \lambda _{1}(M_k)\geq (n-1)^{2}/4$ for sufficiently large k, or if $\lambda _{1}(M)<(n-1)^{2}/4$ , we have that $\lim _{k\rightarrow \infty } \lambda _1(M_k) = \lambda _1(M)>\lambda _0(M)$ for sufficiently large k. In either case, the convergence of $s_1(M_k)$ to $s_1(M)$ follows.

Proof. This follows Dalbo-Otal-Peigne’s proof [Reference Dal’bo, Otal and PeignéDOP00] on the finiteness of $m_{\mathrm {BM}}$ . We first let $\epsilon>0$ be a constant which is smaller than the shortest geodesic in M. Take a fundamental domain F for the convex core of M in the universal cover $\mathbb {H}^{n}$ , and divide F as the thin part $F^{<\epsilon }$ (i.e., the intersection of F with the thin part of M) and the thick part $F^{>\epsilon }$ . Consider a component D of $F^{<\epsilon }$ , which must be a cuspidal component. Suppose that $\mathcal {H}$ is the corresponding horoball based at the parabolic fixed point $\xi $ , so that D is a fundamental domain for the parabolic subgroup $\mathcal {P}<\pi _1(M)$ that preserves $\mathcal {H}$ .

As detailed in [Reference Dal’bo, Otal and PeignéDOP00, page 118], we can bound $\tilde {m}_{\mathrm {BM}}$ in D by

$$\begin{align*}\tilde{m}_{\mathrm{BM}}(T^1D) \leq \sum_{p\in\mathcal{P}}\int_{\mathcal{D}\times p\mathcal{D}} c^\mu(d\eta^-d\eta^+) \int_{(\eta^-\eta^+)\cap\mathcal{H}}dt, \end{align*}$$

where $c^\mu (d\eta ^-d\eta ^+)=e^{-2\delta (\Gamma )(\eta ^-|\eta ^+)_{x}}d\mu _{x}(\eta ^{-})d\mu _{x}(\eta ^{+})$ for a given point $x\in \mathbb {H}^{n}$ and $\mathcal {D}\subseteq \partial _{\infty } \mathbb {H}^{n}\setminus \lbrace \xi \rbrace $ is a compact set, such that $\lbrace p\mathcal {D} \rbrace _{p\in \mathcal {P}}$ covers $\Lambda (M) \setminus \lbrace \xi \rbrace $ . The existence of the compact set $\mathcal {D}$ is ensured by the assumption that M is geometrically finite, hence, the parabolic fixed point $\xi $ is bounded [Reference BowditchBow93]. Now, let $\mathcal {P}_k$ be the elementary group in $M_k$ that converges to $\mathcal {P}$ , which is either parabolic or loxodromic. We discuss the proof that when the groups $\mathcal {P}_{k}$ are loxodromic. The argument for parabolic subgroups is similar.

Let $\mathcal {H}_k$ be a neighborhood of the geodesic $\xi ^-_k\xi ^+_k$ preserved by $\mathcal {P}_k$ so that $\mathcal {H}_k\rightarrow \mathcal {H}$ , $\xi ^\pm _k\rightarrow \xi $ . Since $M_k$ converges to M strongly, by [Reference McMullenMcM99], we can take $\mathcal {D}$ large enough so that $\lbrace p\mathcal {D} \rbrace _{p\in \mathcal {P}_k}$ covers $\Lambda (M_k)\setminus \lbrace \xi ^-_k, \xi ^+_k\rbrace $ . Hence, it follows that

$$\begin{align*}m^k_{\mathrm{BM}}(T^1(\mathcal{H}_k/\mathcal{P}_k)) \leq \sum_{p\in\mathcal{P}_k}\int_{\mathcal{D}\times p\mathcal{D}} c^\mu_k(d\eta^-d\eta^+) \int_{(\eta^-\eta^+)\cap\mathcal{H}_k}dt. \end{align*}$$

Assume without loss of generality that we can take a common point $x\in \mathcal {H}_k,\mathcal {H}$ . There exist compact set $K\subset \mathbb {H}^n$ and open neighborhood $V\subseteq \mathbb {H}^n$ of $\xi $ so that for k large, if the (oriented) geodesic $\eta ^-\eta ^+$ with $\eta ^-\in \mathcal {D}\cap \Lambda (\Gamma _k)$ and $\eta ^+\in p\mathcal {D}\cap \Lambda (\Gamma _k)$ intersects $\mathcal {H}_k$ , then the point of entry belongs to $K\cap \partial \mathcal {H}_k$ and $p^{-1}x$ belongs to V. In particular, such geodesic $\eta ^-\eta ^+$ verifies $0\leq (\eta ^-|\eta ^+)_x\leq diam(K)$ . Moreover, we have that $|\int _{(\eta ^-\eta ^+)\cap \mathcal {H}_{k}}dt - d(x,px)|<2diam(K)$ . Hence, there exists a constant $C>0$ depending only on $diam(K)$ so that

$$\begin{align*}m^k_{\mathrm{BM}}(T^1(\mathcal{H}_k/\mathcal{P}_k)) \leq C\,\left(\sum_{p\in\mathcal{P}^{\prime}_k} \mu^k_x(\mathcal{D})\mu^k_x(p\mathcal{D})(d(x,px)+C)\right), \end{align*}$$

where $\mu ^{k}_{x}$ denotes the Patterson-Sullivan measure on $M_k$ and $\mathcal {P}^{\prime }_k$ is the subset of $\lbrace p\in \mathcal {P}_k\,|\, p^{-1}x \in V \rbrace $ so that the summand $\int _{\mathcal {D}\times p\mathcal {D}} c^\mu _k(d\eta ^-d\eta ^+) \int _{(\eta ^-\eta ^+)\cap \mathcal {H}_k}dt$ is non-zero.

Recall that

$$\begin{align*}\mu^k_x(p\mathcal{D}) = \int_{\mathcal{D}} e^{-\delta(\Gamma_k)B_\eta(p^{-1}x,x)}\mu^k_x(d\eta), \end{align*}$$

so we would like to estimate $B_\eta (p^{-1}x,x)$ . Observe that as $\mathcal {H}_k$ is preserved by $\mathcal {P}_k$ , we have that if $\eta ^-\eta ^+$ is a geodesic with $\eta ^-\in \mathcal {D}\cap \Lambda (\Gamma _k)$ and $\eta ^+\in p\mathcal {D}\cap \Lambda (\Gamma _k)$ that intersects $\mathcal {H}_k$ , then the exit point of $\eta ^-\eta ^+$ from $\mathcal {H}_k$ belongs to $pK\cap \partial \mathcal {H}_k$ . By triangular inequality, we have that under such conditions $|\int _{(\eta ^-\eta ^+)\cap \mathcal {H}_{k}}dt - B_\eta (x,px)|<2diam(K)$ . Hence, for $p\in \mathcal {P}^{\prime }_k$ have $|B_\eta (p^{-1}x,x) - d(p^{-1}x,x)|\leq 4diam(K)$ . Combining this with our previous inequality (and making the domain of the sum bigger if necessary), we get

$$\begin{align*}m^k_{\mathrm{BM}}(T^1(\mathcal{H}_k/\mathcal{P}_k)) \leq C\, \left(\sum_{p\in\mathcal{P}_k, p^{-1}x\in V} (\mu^k_x(\mathcal{D}))^2e^{-\delta(\Gamma_k)d(p^{-1}x,x)}(d(x,px)+C)\right) \end{align*}$$

for some $C>0$ independent of $\epsilon $ and k.

We claim that the above discussion holds for smaller $\epsilon $ corresponding to a smaller neighborhood $V(\epsilon )$ for the same base point x. Consider a smaller thin part corresponding to $\epsilon '<\epsilon $ . The sets $\mathcal {H}_k, K$ vary with $\epsilon '$ , although it is clear that $\mathcal {H}_k(\epsilon ')\subset \mathcal {H}_k(\epsilon )$ and $diam(K(\epsilon ')) < diam(K(\epsilon ))$ . Hence, after taking a base point $y\in K(\epsilon ')$ , we have

$$\begin{align*}m^k_{\mathrm{BM}}(T^1(\mathcal{H}_k(\epsilon')/\mathcal{P}_k)) \leq C\,\left(\sum_{p\in\mathcal{P}_k, p^{-1}y\in V(\epsilon')} (\mu^k_y(\mathcal{D}))^2e^{-\delta(\Gamma_k)d(p^{-1}y,y)}(d(y,py)+C)\right) \end{align*}$$

for a constant $C>0$ independent of $\epsilon '$ and k.

The neighborhood $V(\epsilon ')$ is smaller and smaller as $\epsilon '\rightarrow 0$ , as if $\eta ^-\eta ^+$ intersects $\mathcal {H}_k(\epsilon ')$ , then it has to intersect $\mathcal {H}_k(\epsilon )$ . Then for the p summands considered for $\epsilon '$ , we have

$$\begin{align*}d(y,py)\leq d(x,px)+C',\quad \mu^k_y(\mathcal{D}) \leq C'e^{-\delta(\Gamma_k)d(x,y)}\mu^k_x(\mathcal{D}) \end{align*}$$

for $C'$ constant independent of $\epsilon '$ and k. We always have the bound $d(y,py)\geq d(x,px) - 2d(x,y)$ by triangular inequality and the fact that p is an isometry.

Putting altogether, we have that

(6) $$ \begin{align} m^k_{\mathrm{BM}}(T^1(\mathcal{H}_k(\epsilon')/\mathcal{P}_k)) \leq C"(\mu^k_x(\mathcal{D}))^2\,\left(\sum_{p\in\mathcal{P}_k, p^{-1}x\in V(\epsilon')} e^{-\delta(\Gamma_k)d(p^{-1}x,x)}(d(x,px)+C")\right) \end{align} $$

for a constant $C">0$ independent of $\epsilon '$ and k. Recall that when $\delta (\Gamma _k)$ is strictly bigger than $(n-1)/2$ , by [Reference McMullenMcM99, Theorem 6.1], the tails of the series $\sum _{p\in \mathcal {P}_k}e^{-\delta (\Gamma _k)d(p^{-1}x,x)}$ are uniformly small. Specifically, for any $\eta>0$ , there exists a neighborhood $U\subset \mathbb {H}^{n}$ of $\xi $ so that

$$\begin{align*}\sum_{p\in\mathcal{P}_k, \,px\subset U}e^{-\delta(\Gamma_k)d(p^{-1}x,x)} < \eta, \end{align*}$$

for k sufficiently large. We also have that the tails of the series $\sum _{p\in \mathcal {P}_k}e^{-\delta (\Gamma _k)d(p^{-1}x,x)}(d(x,px)+C")$ are uniformly small, as $d(x,px)$ is uniformly dominated by $e^{cd(p^{-1}x,x)}$ for any $c>0$ . Hence, by taking $\epsilon '$ sufficiently small, the right-hand side of (6) corresponds to a smaller tail of the series $\sum _{p\in \mathcal {P}_k}e^{-\delta (\Gamma _k)d(p^{-1}x,x)}(d(x,px)+C")$ . Thus, by applying [Reference McMullenMcM99, Theorem 6.2] for the sequence of exponents $\delta (\Gamma _k)-c$ , the right-hand side of (6) will be arbitrarily small for $\epsilon '$ sufficiently small and k sufficiently large.

Proof. Denote $M^{a,b}=M^{>a}\cap M^{<b}$ . Take $U_1,\ldots U_m\subset M$ balls with compact closure, whose union covers $C(M)^{\epsilon ,r} = C(M) \cap M^{\epsilon ,r}$ . Take $\bar {\varphi }_1,\ldots ,\bar {\varphi }_m$ partition of unity subordinated to $U_1,\ldots U_m$ , in the sense that $\bar {\varphi }=\sum _{i=1}^m \bar {\varphi }_i$ has support contained in $M^{\epsilon -\eta ,r+\eta }$ and is identically equal to $1$ in $C(M)^{\epsilon ,r}$ , for some arbitrarily small $\eta>0$ . Let $\tilde {U_i}$ be a lift of $U_i$ in $\mathbb {H}^n$ , such that the union covers a fundamental domain of M. We denote $\varphi _i$ a compactly supported function subordinated to $\tilde {U}_i$ , such that $\varphi _i=\bar {\varphi }_{i}\circ \text {Proj}$ .

Then since the Patterson-Sullivan measures $\mu ^k_{x_0}$ converge weakly to $\mu _{x_0}$ , the critical exponents $\delta _k=\delta (\Gamma _k)$ converge to $\delta =\delta (\Gamma )$ , and we can express the Bowen-Margulis measures as $d\tilde {m}^k_{\mathrm {{BM}}}(v)=e^{-\delta _k(\beta _{v_{-}}(\pi (v), x_{0})+\beta _{v^{+}}(\pi (v), x_{0}))}d\mu ^k_{x_{0}}(v_{-})d\mu ^k_{x_{0}}(v_{+})dt$ , then, for k sufficiently large, we have

(7) $$ \begin{align} \bigg|\int_{T^1\tilde{U_i}}\varphi_id\tilde{m}^k_{\mathrm{BM}} - \int_{T^1\tilde{U_i}}\varphi_i d\tilde{m}_{\mathrm{BM}}\bigg| <\alpha, \end{align} $$

for some small $\alpha>0$ .

By Proposition 4.1, we have that $\int _{T^1 M^{<\epsilon }_k} dm^k_{\mathrm {BM}}, \int _{T^1 M^{<\epsilon }} dm_{\mathrm {BM}}<\alpha $ , and by construction, we have that

(8) $$ \begin{align} \bigg|\int_{T^1M^{<r}}dm_{\mathrm{BM}} - \sum_{i=1}^m\int_{T^1\tilde{U_i}}\varphi_id\tilde{m}_{\mathrm{BM}}\bigg| < \int_{T^1M^{<\epsilon}} dm_{\mathrm{BM}} + \int_{T^1M^{r,r+\eta}} dm_{\mathrm{BM}}. \end{align} $$

Now, since $M_k$ converges strongly to M, for k large, we have that $\sum _{i=1}^m\int _{T^1\tilde {U_i}}\varphi _id\tilde {m}^{k}_{\mathrm {BM}}$ is bounded between $(1-\alpha )\int _{T^1M_k^{\epsilon +\eta , r-\eta }}dm^k_{\mathrm { BM}}$ and $(1+\alpha )\int _{T^1M_k^{\epsilon -\eta , r+\eta }}dm^k_{\mathrm {BM}}$ . Hence

(9) $$ \begin{align} \bigg|\int_{T^1M_k^{<r}}dm^k_{\mathrm{BM}} - \sum_{i=1}^m\int_{T^1\tilde{U_i}}\varphi_id\tilde{m}^k_{\mathrm{BM}}\bigg| < \int_{T^1M_k^{<\epsilon}} dm^k_{\mathrm{BM}} + \alpha\int_{T^1M_k^{\epsilon,r}} dm^k_{\mathrm{BM}} + \int_{T^1M_k^{r,r+\eta}} dm^k_{\mathrm{BM}}. \end{align} $$

By a similar partition of unity argument, we can show that for any $0<a<b$ , there exists $\eta _0>0$ sufficiently small so that for any $k\gg 1$ sufficiently large, we have that

(10) $$ \begin{align} \int_{T^1M_k^{a,b}}dm^k_{\mathrm{BM}} < 2 \int_{T^1M^{a-\eta_0,b+\eta_0}}dm_{\mathrm{BM}}. \end{align} $$

Finally, we have to see that the function $(a,b)\mapsto \int _{T^1M^{a,b}}dm^k_{\mathrm {BM}}$ is continuous. Because of monotonicity, this reduces to prove that for any $r>0$ , $\int _{T^1\partial M^{<r}}dm^k_{\mathrm {BM}}=0$ . Indeed, the lift $\partial \tilde {M}^{<r}\subseteq \mathbb {H}^n$ is contained in the union of tubes around closed geodesics of length $\leq r$ (considering parabolic cusps corresponding to $0$ length geodesics). For core geodesics of length strictly less than r, these tubes are strictly convex, and hence, the boundaries intersect any geodesic in a discrete set. If we happen to have a geodesic of length r, then the intersection of $\partial \tilde {M}^{<r}$ with any geodesic is a discrete set, unless the geodesic is equal to the geodesic axis. In either case, the set $\partial \tilde {M}^{<r}\subseteq \mathbb {H}^n$ has zero measure for the Bowen-Margulis measure $d\tilde {m}_{\mathrm {{BM}}}(v)= e^{-2\delta (v_-|v_+)_{x_0}}d\mu _{x_{0}}(v_{-})d\mu _{x_{0}}(v_{+})dt$ , as for almost every geodesic line $\ell $ , the intersection $\partial \tilde {M}^{<r}\cap \ell $ has length $0$ .

Applying the triangular inequality, replacing equations (7), (8), (9), and then using Proposition 4.1, (10) (for sufficiently large k and $\eta $ sufficiently small), we have that

(11) $$ \begin{align} \begin{aligned} \bigg| \int_{T^1M_k^{<r}}dm^k_{\mathrm{BM}} - \int_{T^1M^{<r}} dm_{\mathrm{BM}} \bigg| &< \bigg|\int_{T^1 M_k^{<r}}dm^k_{\mathrm{BM}} - \sum_{i=1}^m\int_{T^1\tilde{U_i}}\varphi_id\tilde{m}^k_{\mathrm{BM}}\bigg| \\&+ \sum_{i=1}^m \bigg|\int_{T^1\tilde{U_i}}\varphi_{i}d\tilde{m}^k_{\mathrm{BM}} - \int_{T^1\tilde{U_i}}\varphi_i d\tilde{m}_{\mathrm{BM}}\bigg| \\&+ \bigg|\int_{T^1M^{<r}}dm_{\mathrm{BM}} - \sum_{i=1}^m\int_{T^1\tilde{U_i}}\varphi_id\tilde{m}_{\mathrm{BM}}\bigg|\\ &<\int_{T^1M_k^{<\epsilon}} dm^k_{\mathrm{BM}} + \alpha\int_{T^1M_k^{\epsilon,r}} dm^k_{\mathrm{BM}} + \int_{T^1M_k^{r,r+\eta}} dm^k_{\mathrm{BM}} \\&+ m\alpha + \int_{T^1M^{<\epsilon}} dm_{\mathrm{BM}} + \int_{T^1M^{r,r+\eta}} dm_{\mathrm{BM}}\\ &<(m+2)\alpha + 2\alpha\int_{T^1M^{<r+\eta}} dm_{\mathrm{BM}} + 3 \int_{T^1M^{r-\eta,r+2\eta}} dm_{\mathrm{BM}} \end{aligned} \end{align} $$

which goes to $0$ as $k\rightarrow +\infty $ , and $\alpha ,\eta \rightarrow 0$ .

Proof of Corollary 1.8.

Observe that since we have strong convergence for well-positioned convex sets $D_k\rightarrow D$ , we can take lifts $\widetilde {D_k}, \tilde {D}\subset \mathbb {H}^n$ and compact sets $E_k\subset \partial \widetilde {D_k}, E \subset \partial \tilde {D}$ so that $E_k$ , E are fundamental domains for the support of $\sigma ^{\pm }_{\partial D_k}, \sigma ^{\pm }_{\partial D}$ (respectively) and $E_k$ converges strongly to E. We can further assume there exists a set $F\subset \mathbb {S}^{n-1}$ so that $P_{D_k}(F)$ , $P_D(F)$ cover $E_k$ and E (respectively) on its interior, see (1) for the definition of the maps $P_{D_{k}}$ and $P_{D}$ . Under these assumptions, we have

$$\begin{align*}\Vert\sigma^{\pm}_{\partial D_k}\Vert \leq \tilde{\sigma}^{\pm}_{\widetilde{D_k}}(P_{D_k}(F)) \rightarrow \tilde{\sigma}^{\pm}_{\tilde{D}}(P_D(F)). \end{align*}$$

Reducing the set F so that $\tilde {\sigma }^{\pm }_{\widetilde {D_k}}(P_{D_k}(F)\setminus E_k), \tilde {\sigma }^{\pm }_{\tilde {D}}(P_D(F)\setminus E)$ are arbitrarily small, we then have

$$\begin{align*}\Vert\sigma^{\pm}_{\partial D_k}\Vert=\tilde{\sigma}^{\pm}_{\widetilde{D_k}}(E_k) \rightarrow \tilde{\sigma}^{\pm}_{\tilde{D}}(E) =\Vert\sigma^{\pm}_{\partial D}\Vert, \end{align*}$$

which proves the first statement

The relative result for subsets $\Omega _k, \Omega $ is proved by taking the fundamental domains $E^{\prime }_k\subset \partial \widetilde {D_k}$ , $E'\subset \partial \tilde {D}$ for the support of $\sigma ^{\pm }_{\partial \Omega _{k}}, \sigma ^{\pm }_{\partial \Omega }$ (respectively) and arguing as above.

Proof. Let’s prove that case for $\Omega ^{+}$ , and the proof for $\Omega ^-$ is similar. As done in the proof of [Reference RoblinRob03, Proposition 6.2] (using [Reference RoblinRob00, Section 3.1]), the function $(v,R)\mapsto \mu _{W^\pm (v)}(B^\pm (v,R))$ is continuous for $v\in T^1\mathbb {H}^n, R>0$ , as well as $\Gamma $ -invariant. Moreover, since $\Omega ^+$ is compact, there exists $R>0$ sufficiently large so that the function $v\mapsto \mu _{W^-(v)}(B^-(v,R))$ is a uniformly continuous positive function in some neighborhood of $\Omega ^-$ . It suffices then to prove the statement for sufficiently large k.

Denote by $A(v,R,r) = B^-(v,(1+r)R)\setminus B^-(v,R) \subset W^-(v)$ the annulus in $W^-(v)$ with center v between radius $R, (1+r)R$ . We will show that there exists $m>0$ and function $\eta (\epsilon )>0$ so that for k large, $v\in \mathcal {V}_{\eta ,\eta } (\Omega ^{+})$ and $0<r<\eta $ the following two statements hold

1. 1. $ \mu ^k_{W^-(v)}(B^-(v,R))\geq m$ ,

2. 2. $ \mu ^k_{W^-(v)}(A(v,R,r)) < \epsilon $ .

Then it is clear that the statement follows from Items (1) and (2) by making $\epsilon $ arbitrarily small. Now we prove Items (1) and (2) respectively.

1. 1. For a vector $u\in T^{1}\mathbb {H}^{n}$ , we define a function $P_{u}: W^{-}(u)\rightarrow \partial _{\infty }\mathbb {H}^{n}$ , where $P_{u}(v)$ is the endpoint of the bi-infinite geodesic $u_{-}\pi (v)$ different from $u_{-}$ , as shown in Figure 1. Since $\mathcal {V}_{\eta ,\eta } (\Omega ^+)$ has compact closure, we can take finitely many $v_i \in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ so that for any $u\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ , there exists $v_i$ , such that

   $$\begin{align*}P_u^{-1}P_{v_i}(B^-(v_i,R/2))\subseteq B^-(u,R). \end{align*}$$
   Moreover, we can assume that the conformal factor between $\mu ^k_{W^-(v_i)}$ and $\mu ^k_{W^-(u)}$ at the sets $B^-(v_i,R/2)$ , $P_u^{-1}P_{v_i}(B^-(v_i,R/2))$ is between $\frac 12$ and $2$ . This can be done uniformly for all k by following [Reference RoblinRob03, Section 1.H]. By taking $\eta $ small, we can assume that $\mu _{W^-(v_i)}(B^-(v_i,R/2))> 2m$ for some fixed $m>0$ and for any $v_i\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ . Then, by weak-convergence of measures, we have that for any $v_i$ (and large k) $\mu ^k_{W^-(v_i)}(B^-(v_i,R/2))> 2m$ . Then it follows that
   $$\begin{align*}\mu^k_{W^-(u)}(B^-(u,R)) \geq \mu^k_{W^-(u)}(P_u^{-1}P_{v_i}(B^-(v_i,R/2))) \geq \frac12 \mu^k_{W^-(v_i)}(B^-(v_i,R/2))> m. \end{align*}$$

2. 2. Since $\mathcal {V}_{\eta ,\eta } (\Omega ^+)$ has compact closure and $\delta (\Gamma )>(n-1)/2$ , given $\epsilon>0$ , we can take $\eta $ small enough so that for $v\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ , we have that $\mu _{W^-(v)}(A(v,R,5\eta ))<\epsilon $ . We will take again a finite collection of vectors $v_i$ , although now they need to satisfy the following list of properties.

   • ○ The finite collection of $v_i$ is taken so that $B^-(v_i,4\eta )\subset A(v,R,5\eta )$ for some $v\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ . Denote their total number by $C_2$ ,

   • ○ For any $v\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ and any $B^-(u,2\eta )\subset A(v,R,5\eta )$ , we have that

     $$\begin{align*}P_u^{-1}P_{v_i}(B^-(v_i,4\eta))\supseteq B^-(u,2\eta) \end{align*}$$
     with conformal factor bounded between $\frac 12$ and $2$ .

     Figure 1

     

   Take sufficiently large k so that $\mu ^k_{W^-(v_i)}(B^-(v_i,4\eta )) \leq \mu _{W^-(v_i)}(B^-(v_i,4\eta )) + \zeta $ for $\zeta $ small still to be determined.Let $v\in \mathcal {V}_{\eta ,\eta } (\Omega ^+)$ . Cover $A(v,R,\eta )$ by finitely many disjoint measurable sets $B_j$ , so that each $B_j$ is contained in a ball $B^-(u_j,2\eta )$ inside of $A(v,R,5\eta )$ . Then by the second bullet point, for each $u_j$ , we choose $v_i$ so that

   $$\begin{align*}P_{u_j}^{-1}P_{v_i}(B^-(v_i,4\eta))\supseteq B^-({u_j},2\eta). \end{align*}$$
   Observe that each $v_i$ can only be repeatedly selected less than $C_3$ times, for some constant $C_3$ depending only on the dimension n. Then we have the following chain of inequalities, which follow from the covering $\{B_j\}$ of $A(v,R,r)$ , the inclusion $P_{u_j}^{-1}P_{v_i}(B^-(v_i,4\eta ))\supseteq B^-({u_j},2\eta ) \supseteq B_j$ , the bound on the conformal factor of $P_u^{-1}P_{v_i}$ , the convergence $\mu ^{k}_{W^-(v)} \rightarrow \mu _{W^-(v)}$ , the inclusion $B^-(v_i,4\eta )\subset A(v,R,5\eta )$ , and the bound on the cardinality of the finite set of $v_i$ ’s
   (12) $$ \begin{align} \mu^k_{W^-(v)}(A(v,R,r)) &\leq \sum_j \mu^k_{W^-(v)}(B_j) \leq C_3\sum_i \mu^k_{W^-(v)}(P_{u_j}^{-1}P_{v_i}(B^-(v_i,4\eta)))\nonumber\\ &\leq 2 C_3\sum_i \mu^k_{W^-(v)}(B^-(v_i,4\eta)) \leq 2C_3 \sum_i \left(\mu_{W^-(v)}(B^-(v_i,4\eta)) + \zeta\right)\nonumber\\ &\leq 4C_3\sum_i \left(\mu_{W^-(v)}(A(v,R,5\eta)) + \zeta\right) \leq 4C_2 C_3 (\epsilon + \zeta) \end{align} $$
   which is arbitrarily small for $\eta $ small and k large.

Proof. There is an explicit counting formula of $\mathcal N_{D^{-}, D^{+}}(t)$ for orthogeodesic arcs between two convex sets $D^{\pm }$ given in [Reference Parkkonen and PaulinPP17, Theorem 3]:

$$ \begin{align*}\mathcal N_{D^{-}, D^{+}}(t)=\dfrac{||\sigma^{+}_{D^{-}}||\cdot || \sigma^{-}_{D^{+}}||}{\delta ||m_{\mathrm{{BM}}}||} e^{\delta(\Gamma) t}(1+O(e^{-\kappa t})).\end{align*} $$

This formula holds under the assumption that $(\mathbb {H}^{n}, \Gamma )$ has radius-continuous strong stable/unstable masses. The constant $O(\cdot )$ and the parameter $\kappa $ depend on $\Gamma $ , the convex sets $D^{\pm }$ , the speed of mixing, and the property of radius-continuous strong stable/unstable masses.

By Proposition 4.2 and Corollary 1.8, the Bowen-Margulis measure and the skinning measures converge to the ones of the limit manifold M weakly. The critical exponent $\delta (\Gamma _{k})$ converges to $\delta (\Gamma )$ [Reference McMullenMcM99, Theorem 1.5]. The convergence of the speed of mixing is controlled by the spectral gap [Reference Edwards and OhEO21]. Hence, this quantity also converges to the one of the limit manifold by Theorem 1.1. Therefore, it suffices to prove the sequence $\Gamma _k$ and the limit $\Gamma $ have uniform radius-continuous strong stable/unstable ball masses property, which follows from Proposition 5.2.

Proof of Theorem 1.3.

By Example 2.12, connected components $D_{k}^{\pm }$ in the thin part of $M_{k}$ are well-positioned convex sets that are strongly convergent to the well-positioned convex sets $D^{\pm }$ (respectively). By Theorem 5.3, there is a uniform counting formula for orthogeodesics between $D^{-}_{k}$ to $D^{+}_{k}$ along the sequence. This proves Item (1). Similarly, for small $r>0$ , the radius r embedded balls centered at $x_{k}$ are well-positioned convex subsets which are strongly convergent to the embedded r-ball centered at x. In that case, let $D^{+}_{k}=D^{-}_{k}$ be the radius r ball at $x_k$ , and $D^+=D^-$ be the radius r ball at x. Observe that if we change the radius $r>0$ to a radius $s>0, s<r$ , we have a one-to-one correspondence between the set of orthogeodesics by extending/shortening the geodesic arcs. Such correspondence takes an orthogeodesic of length $\ell $ to its extension of length $\ell +2(r-s)$ . Hence, applying Theorem 5.3 again and making s arbitrarily small (or equivalently, translating by $2r$ the counting function for the balls of radius r), we obtain the uniform counting for geodesic loops based at $x_k$ along the sequence.

Proof of Corollary 1.5.

As explained for instance by Roblin in [Reference RoblinRob03, Chapter 5], one can deduce an asymptotic counting of closed primitive geodesics in manifolds with negative pinched curvature from the asymptotic counting of orbit distance (i.e., geodesic loops), which only depends on the geometry of the universal cover. Namely, if $\mathcal {G}_{M}(\ell )$ is the set of closed primitive geodesics in M of length less than $\ell>0$ , then [Reference RoblinRob03, Corollary 5.3]

$$\begin{align*}\#\mathcal{G}_{M}(\ell) \approx \frac{e^{\delta\ell}}{\delta\ell} \text{ as } \ell\rightarrow+\infty. \end{align*}$$

Combining with the uniform counting of geodesic loops (Theorem 1.3), we obtain the uniform counting of closed primitive geodesics along a strongly convergent sequence of hyperbolic manifolds.

Proof. Since both terms in (13) are bilinear in $\psi ^{\pm }_{k}$ , we can assume without loss of generality that, by using a partition of unity, the support of $\psi ^{\pm }_{k}$ is contained in a small relatively compact open set $U^{\pm }_{k}$ in $T^{1} M_k$ , and there is a small relatively compact open set $\widetilde {U^{\pm }_{k}}$ in $T^{1}\mathbb {H}^{n}$ , such that the restriction of the quotient map $q_k: T^{1}\mathbb {H}^{n}\rightarrow T^{1} M_{k}$ to $\widetilde {U^{\pm }_{k}}$ is a diffeomorphism to $U^{\pm }_{k}$ . Define $\widetilde {\psi ^\pm _k}\in C^\infty _0(T^1\mathbb {H})$ with support in $\widetilde {U^{\pm }_{k}}$ and coinciding with $\psi ^\pm \circ q_k$ on $\widetilde {U^{\pm }_{k}}$ . Similarly, we can define a compactly supported function $\tilde {\psi }\in C^\infty _0(T^1\mathbb {H})$ corresponding to $\psi $ . Observe that we can choose the lifts $\widetilde {U^{\pm }_{k}}$ and $\widetilde {\psi ^{\pm }_{k}}$ appropriately, such that $\widetilde {\psi ^\pm _k}$ converges strongly to $\widetilde {\psi ^\pm }$ and

$$\begin{align*}\int_{\partial^1_\pm \widetilde{D^\mp_k}} \widetilde{\psi^\mp_k} d\sigma^\pm_{\widetilde{D^\mp_k}} = \int_{\partial^1_\pm D^\mp_k} \psi^\mp_k d\sigma^\pm_{\partial D^\mp_k}, \end{align*}$$

where $\widetilde {D^{\pm }_k}$ are lifts of $D^{\pm }_k$ . From now on, we will distinguish components of $\widetilde {D^\pm _k}$ . By abuse of notation, we still denote by $D^\pm _k$ a connected component of $\widetilde {D^\pm _k}$ , which we assume is the only connected component of $\widetilde {D^{\pm }_{k}}$ so that the intersection of $\partial ^1_\mp D^\pm _k$ with $\widetilde {U^{\pm }_{k}}$ is nonempty by using partition of unity. Observe that then we can label other components by $\Gamma _k$ left action $\gamma \mapsto \gamma D^\pm _k$ . These labels are redundant (i.e., label the same set) if and only if $\gamma _1^{-1}\gamma _2$ belongs to the stabilizer of $D^\pm _k$ .

Take $\eta , R>0$ and k sufficiently large so that the statement of Proposition 5.2 applies for the sequence of convergent precompact subsets $(\Omega ^\pm _k:=\partial ^1_\mp \widetilde {D^\pm _k} \cap supp(\widetilde {\psi ^\pm _k}))_{k\in \mathbb {N}}$ . We will fix $R>0$ from now on, but will keep taking smaller (independent of k) $\eta $ . Observe that for small, fixed $\tau>0$ , we have the inclusion

$$\begin{align*}\mathcal{V}_{\eta e^{-\tau},Re^{-\tau}} (\Omega^\pm_k) \subset \mathcal{V}_{\eta,R} (\Omega^\pm_k) \end{align*}$$

is precompact in each slice $V^{\pm }_{w,\eta ,R}$ , and converges as a whole to $\mathcal {V}_{\eta e^{-\tau },Re^{-\tau }} (\Omega ^\pm ) \subset \mathcal {V}_{\eta ,R} (\Omega ^\pm )$ in the usual sense. If by we denote the characteristic function of a set A, then we can construct smooth functions $\chi ^\pm _k\in C^\infty (T^1 \mathbb {H}^n)$ so that the following items hold

1. 1. For $w\in \Omega ^\pm _k, v \in W^{0\mp }(w)$

   

2. 2. The Sobolev norms $\Vert \chi ^\pm _k \Vert _\beta $ are uniformly bounded (i.e., independent of k), where $\beta $ is the Sobolev norm appearing in the statement of [Reference Edwards and OhEO21, Theorem 1.1].

3. 3. For any $w\in \Omega ^\pm _k$ , we have that

   $$\begin{align*}e^{-\epsilon} \nu^\pm_w (V^\mp_{w,\eta,R}) \leq \int_{V^\mp_{w,\eta,R}} \chi^\pm_k d\nu^\pm_w \leq \nu^\pm_w (V^\mp_{w,\eta,R}) \end{align*}$$
   for $\epsilon>0$ independent of k, where $d\nu ^\pm _w := dsd\mu _{W^\mp (w)}$ .

In order to define the test functions to apply exponential mixing, we start with the functions $H^\pm _k:\partial ^1_\mp \widetilde { D^\pm _k} \rightarrow \mathbb {R}$ defined by

$$\begin{align*}H^\pm_k(w) = \frac{1}{\int_{V^\mp_{w,\eta,R}} \chi^\pm_k d\nu^\pm_w}. \end{align*}$$

Let $\Phi ^\pm _k:T^1 \mathbb {H}^n\rightarrow \mathbb {R}$ defined by

$$\begin{align*}\Phi^\pm_k = (H^\pm_k \widetilde{\psi^\pm_k})\circ f^\mp_{D^\pm_k}\chi^\pm_k. \end{align*}$$

By construction, we have that $\Vert \Phi ^\pm _k \Vert _\beta $ are uniformly bounded and have support in $\mathcal {V}_{\eta ,R} (\Omega ^\pm _k)$ . Moreover, $\Phi ^\pm _k$ are nonnegative, measurable functions satisfying

(14) $$ \begin{align} \int_{T^1\mathbb{H}^n}\Phi^\pm_k d\tilde{m}^k_{\mathrm{{BM}}} = \int_{\partial^1_\mp D^\pm_k} \psi^\pm_k d\sigma_{k}^\mp .\end{align} $$

Following [Reference Parkkonen and PaulinPP17], we will estimate in two ways the quantity

(15) $$ \begin{align} I_k(T) := \int^T_0 e^{\delta(\Gamma_k) t} \sum_{\gamma\in\Gamma_k} \int_{T^1\mathbb{H}^n} (\Phi^-_k\circ g^{-t/2})(\Phi^+_k\circ g^{t/2}\circ \gamma^{-1}) d\tilde{m}^k_{\mathrm{{BM}}}dt .\end{align} $$

By [Reference Edwards and OhEO21, Theorem 1.1] and Theorem 1.1, there exist uniform $\kappa>0, O(.)$ , such that

(16) $$ \begin{align} I_k(T) &= \int^T_0 e^{\delta(\Gamma_k) t} \left(\frac{1}{\Vert m^k_{\mathrm{{BM}}}\Vert} \int_{T^1\mathbb{H}^n}\Phi^-_k d\tilde{m}^k_{\mathrm{{BM}}} \int_{T^1\mathbb{H}^n}\Phi^+_k d\tilde{m}^k_{\mathrm{{BM}}} + O(e^{-\kappa t}\Vert \Phi^-_k \Vert_\beta\Vert \Phi^+_k \Vert_\beta)\right)dt\nonumber\\ &=\frac{e^{\delta(\Gamma_k) T}}{\delta(\Gamma_k)\Vert m^k_{\mathrm{{BM}}}\Vert} \int_{\partial^1_- D^-_k} \psi^- d\sigma_{k}^- \int_{\partial^1_+ D^+_k} \psi^+ d\sigma_{k}^+ + \int^T_{0} e^{\delta(\Gamma_k) t} O(e^{-\kappa t}\Vert \Phi^-_k \Vert_\beta\Vert \Phi^+_k \Vert_\beta)dt\nonumber\\ &=e^{\delta(\Gamma_k) T} \left( \frac{\sigma^{+}_{D^{-}_k}(\psi^-_k)\cdot \sigma^{-}_{D^{+}_k}(\psi^+_k)}{\delta(\Gamma_k) \Vert m^k_{\mathrm{{BM}}}\Vert} + e^{-\delta(\Gamma_k) T}\int^T_{0} e^{\delta(\Gamma_k) t} O(e^{-\kappa t}\Vert \Phi^-_k \Vert_\beta\Vert \Phi^+_k \Vert_\beta)dt\right),\end{align} $$

where we used (14) for the second equality. Observe in the final line that we can make the error term $e^{-\delta (\Gamma _k) T}\int ^T_{0} e^{\delta (\Gamma _k) t} O(e^{-\kappa t}\Vert \Phi ^-_k \Vert _\beta \Vert \Phi ^+_k \Vert _\beta )dt$ arbitrarily small for any $T>T_0$ , where $T_0$ is sufficiently large and independent of k. Now we use a second way to compute this integral $I_{k}(T)$ . Let $\delta _k=\delta (\Gamma _k)$ . We interchange the integral over t and the summation over $\gamma $ . Then

$$ \begin{align*}I_{k}(T)=\sum_{\gamma\in \Gamma_k} \int_{0}^{T} e^{\delta_{k} t} \int_{T^{1}\mathbb{H}^{n}}(\Phi^{-}_{k} \circ g^{-t/2})(\Phi^{+}_{k}\circ g^{t/2}\circ \gamma^{-1})d \tilde{m}^{k}_{\mathrm{{BM}}} dt.\end{align*} $$

Suppose that if $v\in T^{1}\mathbb {H}^{n}$ belongs to the support of $(\Phi ^{-}_{k} \circ g^{-t/2})(\Phi ^{+}_{k}\circ g^{t/2}\circ \gamma ^{-1})$ , then

$$ \begin{align*}v\in g^{t/2} \mathcal{V}_{\eta, R}(\partial^{1}_{+}D_{k}^{-})\cap g^{-t/2} \mathcal{V}_{\eta, R}(\gamma \partial^{1}_{-}D_{k}^{+}).\end{align*} $$

Then by [Reference Parkkonen and PaulinPP17, Lemma 7], which is proved by using hyperbolic geometry in $\mathbb {H}^{n}$ , we have the following

(17) $$ \begin{align} d(w_{k}^{\pm}, v_{\gamma}^{\pm})=O(\eta+e^{-\ell_{\gamma}/2}) ,\end{align} $$

where $w_{k}^{-}=f^{+}_{D_{k}}(v), w_{k}^{+}=f^{-}_{\gamma D_{k}^{+}}(v)$ , $v_{\gamma }^{\pm }$ are endpoints of the common perpendicular between $D_{k}^{-}$ and $\gamma D_{k}^{+}$ , and $\ell _{\gamma }$ is the length of the common perpendicular. Since the Lipschitz norm of $\widetilde {\psi ^\pm _k}$ are uniformly bounded, and in particular bounded by the $\beta $ Sobolev norm of $\psi ^\pm _k$ , we have

$$ \begin{align*}|\widetilde{\psi^{\pm}_{k}}(w^{\pm}_{k})-\widetilde{\psi^{\pm}_{k}}(v^{\pm}_{\gamma})|=O((\eta+e^{-\ell_{\gamma}/2})|| \psi_{k}^{\pm}||_{\beta}).\end{align*} $$

If we define $\hat {\Phi }^{\pm }_{k}=H^{\pm }_{k}\circ f^{\mp }_{D^{\pm }_{k}}\chi ^{\pm }_{k}$ so that $\Phi ^\pm _k= (\widetilde {\psi ^\pm _k}\circ f^\mp _{D_{k}^{\pm }})\hat {\Phi }^\pm _k$ , by applying the previous equation, we obtain

(18) $$ \begin{align} I_{k}(T)=\sum_{\gamma \in \Gamma_{k}} [\psi_{k}^{\pm}(v_{\gamma}^{-}) \psi_{k}^{\pm}(v^{+}_{\gamma})&+O((\eta+e^{-\ell_{\gamma}/2})|| \psi^{-}_{k}||_{\beta} || \psi^{+}_{k}||_{\beta})]\times \nonumber\\ &\int_{0}^{T} e^{\delta_{k}t} \int_{v\in T^{1}\mathbb{H}^{n}} \hat{\Phi}^{-}_{k}(g^{-t/2}v) \hat{\Phi}^{+}_{k}(\gamma^{-1} g^{t/2}v) d\tilde{m}_{\mathrm{{BM}}}^{k}(v)dt, \end{align} $$

for $O(.)$ independent of k.

We now related another test function to $\hat {\Phi }^{\pm }_{k}$ following [Reference Parkkonen and PaulinPP17, Lemma 8]. Let $h^{\pm }_{k}: T^{1}\mathbb {H}^{n}\rightarrow \left [ 0, \infty \right ] $ be the $\Gamma _{k}$ -invariant measurable map defined by

(19) $$ \begin{align} h^{\mp}_{k}(w)=\dfrac{1}{2\eta \mu_{W^{\pm}_{w}}(B^{\pm}(w, R))} \end{align} $$

if $\mu _{W^{\pm }(w)}(B^{\pm }(w, R))>0$ , and $h^{\pm }_{k}(w)=0$ otherwise. We define the test function $\phi ^{\mp }_{k}=\phi ^{\mp }_{\eta , R, \Omega ^{\pm }_{k}}: T^{1}\mathbb {H}^{n}\rightarrow \left [ 0, \infty \right ] $ by

By the properties of $\chi _{k}^{\pm }$ , we have

$$ \begin{align*}\phi^{\pm}_{\eta e^{-\tau}, R e^{-\tau}, \partial^{1}_{\mp} \Omega^{\pm}_{k}} e^{-\epsilon}\leq \hat{\Phi}^{\pm}_{k}\leq \phi^{\pm}_{k}.\end{align*} $$

Hence, it suffices to consider the integral

(20) $$ \begin{align} i_{k}(T)=\sum_{\gamma \in \Gamma_{k}} [\psi_{k}^{\pm}(v_{\gamma}^{-}) \psi_{k}^{\pm}(v^{+}_{\gamma})&+O((\eta+e^{-\ell_{\gamma}/2})|| \psi^{-}_{k}||_{\beta} || \psi^{+}_{k}||_{\beta})]\times \nonumber\\ &\int^{T}_{0} e^{\delta_{k} t} \int_{T^{1} \mathbb{H}^{n}} (\phi^{-}_{k}\circ g^{-t/2})(\phi^{+}_{k}\circ g^{t/2}\circ \gamma^{-1})d \tilde{m}^{k}_{\mathrm{{BM}}} dt. \end{align} $$

By the definition of $\phi _{k}^{\pm }$ , the right-hand side of (20) is equal to

(21)

By the $\Gamma $ -invariance of $h^{\pm }_{k}$ , one has

$$ \begin{align*}h^{-}_{k} \circ f^{+}_{D^{-}_{k}}(g^{-t/2}v)&=e^{-\delta_{k}(t/2)} h^{-}_{k, e^{-t/2}R}(g^{t/2}w^{-}_{k}),\\h^{+}_{k}\circ f^{-}_{D^{+}_{k}}(\gamma^{-1} g^{t/2}v)&=e^{-\delta_{k}(t/2)}h^{+}_{k, e^{-t/2}R}(g^{-t/2}w^{+}_{k}), \end{align*} $$

where $w^{-}_{k}=f^{+}_{D^{-}_{k}}(v)$ , $w^{+}_{k}=f^{-}_{\gamma D^{+}_{k}}(v)=\gamma f^{-}_{D^{+}_{k}} (\gamma ^{-1}v)$ , and $h^{-}_{k, e^{-t/2}R}$ is defined the same as in (19), except we replace R by $e^{-t/2}R$ . Therefore,

$$ \begin{align*}h^{-}_{k}\circ f^{+}_{D_{k}^{-}}(g^{-t/2}v)h^{+}_{k}\circ f^{-}_{D_{k}^{+}}(\gamma^{-1} g^{t/2}v)=e^{-\delta_{k}t} h^{-}_{k, e^{-t/2}R}(g^{t/2}w^{-}_{k})h^{+}_{k, e^{-t/2}R}(g^{-t/2}w^{+}_{k}).\end{align*} $$

The remaining part is nonzero if and only if

$$ \begin{align*}v\in g^{t/2} \mathcal{V}_{\eta, R}(\Omega_{k}^{-})\cap \gamma g^{-t/2} \mathcal{V}_{\eta, R}(\Omega_{k}^{+})=\mathcal{V}_{\eta, e^{-t/2}R}(g^{t/2}\Omega^{-}_{k})\cap \mathcal{V}_{\eta, e^{-t/2}R}(\gamma g^{-t/2}\Omega^{+}_{k}).\end{align*} $$

By [Reference Parkkonen and PaulinPP17, Lemma 7], there exist constants $t_{0}>0$ and $c_{0}$ (independent of k), such that if $t\geq t_{0}$ , the following holds: there exists a common perpendicular $\alpha _{\gamma }$ from $D^{-}_{k}$ to $\gamma (D^{+}_{k})$ with

1. 1. $\vert \ell _{\gamma }-t \vert \leq 2\eta +c_{0} e^{-t/2}$ ,

2. 2. $d(\pi (v^{\pm }_{\gamma }), \pi (w^{\pm }_{k}))\leq c_{0}e^{-t/2},$

3. 3. $d(\pi (g^{\pm t/2}w^{\mp }_{k}), \pi (v))\leq \eta +c_{0}e^{-t/2}$ .

For all $\gamma \in \Gamma _{k}$ and $T\geq t_{0}$ , we define

$$ \begin{align*}\mathcal{A}_{k, \gamma}(T)=\{ (t, v)\in [t_{0}, T]\times T^{1}\mathbb{H}^{n}: v\in \mathcal{V}_{\eta, e^{-t/2}R}(g^{t/2}\Omega^{-}_{k})\cap \mathcal{V}_{\eta, e^{-t/2}R}(\gamma g^{-t/2}\Omega^{+}_{k})\},\end{align*} $$

and the integral

(22) $$ \begin{align} j_{k, \gamma}(T)& =\int\int_{(t, v)\in \mathcal{A}_{k, \gamma}(T)} h^{-}_{k, e^{-t/2}R}(g^{t/2}w^{-}_{k})h^{+}_{k, e^{-t/2}R} (g^{-t/2} w^{+}_{k}) dt d \tilde{m}_{BM}^{k}(v)\nonumber\\ &=\dfrac{1}{(2\eta)^{2}} \int \int_{(t, v)\in \mathcal{A}_{k, \gamma}(T)}\dfrac{dt d\tilde{m}^{k}_{BM}(v)}{\mu_{W^{+}(w^{-}_{t})}(B^{+}(w^{-}_{t}, r_{t})) \mu_{W^{-}(w^{+}_{t})} (B^{-}(w^{+}_{t}, r_{t}))} ,\end{align} $$

where

$$ \begin{align*}r_{t}=e^{-t/2}R, \quad w^{-}_{t}=g^{t/2}w^{-}_{k}, \quad w^{+}_{t}=g^{-t/2} w^{+}_{k}.\end{align*} $$

There exists then a constant $c^{\prime \prime }_{0}>0$ (independent of k), such that for $T\geq t_{0}$ , one has

(23) $$ \begin{align} -c^{\prime\prime}_{0}+\sum_{\gamma\in \Gamma_{T-O(\eta+e^{-\ell_{\gamma}/2}), -O(\eta+e^{-\ell_{\gamma}/2}), k}} [\psi_{k}^{\pm}(v_{\gamma}^{-}) \psi_{k}^{\pm}(v^{+}_{\gamma})&+O((\eta+e^{-\ell_{\gamma}/2})|| \psi^{-}_{k}||_{\beta} || \psi^{+}_{k}||_{\beta})] j_{k, \gamma}(T)\nonumber\\&\leq i_{k}(T)\leq\nonumber\\ c^{\prime\prime}_{0}+\sum_{\gamma\in \Gamma_{T+O(\eta+e^{-\ell_{\gamma}/2}), O(\eta+e^{-\ell_{\gamma}/2}), k}} [\psi_{k}^{\pm}(v_{\gamma}^{-}) \psi_{k}^{\pm}(v^{+}_{\gamma})&+O((\eta+e^{-\ell_{\gamma}/2})|| \psi^{-}_{k}||_{\beta} || \psi^{+}_{k}||_{\beta})] j_{k, \gamma}(T+O(\eta+e^{-\ell_{\gamma}/2})), \end{align} $$

where $\Gamma _{s, r, k}=\{\gamma \in \Gamma _k| t_0+2+c_0\leq \ell _\gamma \leq s, v^{\pm }_{\gamma }\in N_{r}\Omega ^{\pm } \}$ for all $s, r\in \mathbb {R}$ .

Claim 5.6. For any $\epsilon>0$ , if $\eta $ is small enough and $\ell _{\gamma }$ is large enough, then

$$ \begin{align*}j_{k, \gamma}(T)=e^{O(\eta+e^{-\ell_{\gamma}/2})}e^{O(\epsilon^{c'})}\dfrac{(2\eta+O(e^{-\ell_{\gamma}/2}))^{2}}{(2\eta)^{2}},\end{align*} $$

for $c'>0$ independent of k.

Proof. Since $(\mathbb {H}^{n}, \Gamma _{k})$ has radius-continous strong stable/unstable ball masses. By [Reference Parkkonen and PaulinPP17, Lemma 11], for every $\epsilon>0$ and every $(t, v)\in \mathcal {A}_{k, \gamma }(T)$ , one has

$$ \begin{align*}\mu_{W^{\pm}(w_{t}^{\mp})}(B^{\pm}(w^{\mp}_{t}, r_{t}))=e^{O(\epsilon)} \mu_{W^{\pm}(v_{\gamma})} (B^{\pm}(v_{\gamma}, r_{\ell_{\gamma}}))\end{align*} $$

if $\eta $ is small enough and $\ell _{\gamma }$ is large enough, independent of k. Here, $v_{\gamma }$ denote the midpoint of the common perpendicular from $D^{-}_{k}$ to $\gamma (D^{+}_{k})$ . Hence,

$$ \begin{align*}j_{k, \gamma}(T)=\dfrac{e^{O(\epsilon)}\int\int_{(t, v)\in \mathcal{A}_{k, \gamma}(T)} dt d\tilde{m}_{\mathrm{{BM}}}^{k}(v)}{\mu_{W^{+}(v_{\gamma})} (B^{+}(v_{\gamma}, r_{\ell_{\gamma}}))\mu_{W^{-}(v_{\gamma})} (B^{-}(v_{\gamma}, r_{\ell_{\gamma}}))}.\end{align*} $$

By [Reference Parkkonen and PaulinPP17, Lemma 10], for every $(t, v)\in \mathcal {A}_{k, \gamma }(T)$ , one has

$$ \begin{align*}dt d\tilde{m}_{\mathrm{{BM}}}^{k}(v)=e^{O(\eta+e^{-\ell_{\gamma}/2})}dt ds d\mu_{W^{-}(v_{\gamma})}(v') d\mu_{W^{+}(v_{\gamma})}(v")dt,\end{align*} $$

where $v'=f^{+}_{HB_{-}(v_{\gamma })}(v)$ and $v"=f^{-}_{HB_{+}(v_{\gamma })}(v)$ . By [Reference Parkkonen and PaulinPP17, Lemma 9], the distances $d(v, v_{\gamma }), d(v', v_{\gamma })$ , and $d(v", v_{\gamma })$ are $O(\eta +e^{-t/2})$ . Combining these equations together, the claim follows.

Applying then Claim 5.6 in equation (23), we get

(24) $$ \begin{align} I_k(t) = \sum_{\gamma\in \Gamma_{k}} &[\psi_{k}^{\pm}(v_{\gamma}^{-}) \psi_{k}^{\pm}(v^{+}_{\gamma})] + O(\eta e^{\delta_k t}), \end{align} $$

for t sufficiently large independent of k, and $O(.)$ independent of $k,\eta $ . Then by multiplying $e^{-\delta _k t}$ to equations (16), (24), we get that for fixed $\eta>0$

(25) $$ \begin{align} \frac{\sigma^{+}_{D^{-}_k}(\psi^-_k)\cdot \sigma^{-}_{D^{+}_k}(\psi^+_k)}{\delta(\Gamma_k) ||m^k_{\mathrm{{BM}}}||} - O(\eta) \leq \frac{N_{\psi^{-}_k, \psi^{+}_k}(t)}{e^{\delta(\Gamma_k) t}} \leq \frac{\sigma^{+}_{D^{-}_k}(\psi^-_k)\cdot \sigma^{-}_{D^{+}_k}(\psi^+_k)}{\delta(\Gamma_k) ||m^k_{\mathrm{{BM}}}||} + O(\eta) \end{align} $$

for t sufficiently large independent of k and $O(.)$ independent of $k,\eta $ , from where the result follows.

More about the proof of Theorem 5.3.

As $(D^\pm _k)_{k\in \mathbb {N}}$ is a sequence of well-positioned convex sets in $M_k$ which converges strongly to a well-positioned set $D^\pm $ in M, we can select $\psi ^\pm _k \in C^\infty _0(T^1M_k)$ , $\psi ^\pm \in C^\infty _0(T^1M)$ be compactly supported functions so that $(\psi ^\pm _k)_{k\in \mathbb {N}}$ converges strongly to $\psi ^\pm $ and $\psi ^\pm _k \equiv 1$ in $supp(\sigma ^\mp _{\partial D^\pm _k})$ . Hence, in the notation of Theorem 5.5

$$\begin{align*}\mathcal N_{\psi^-_k, \psi^+_k}(t) =\mathcal N_{D^{-}_k, D^{+}_k}(t),\quad \sigma^\mp_{D^\pm_k}(\psi^\pm_k)= \Vert \sigma^\mp_{D^\pm_k}\Vert, \end{align*}$$

and in particular $\frac {\sigma ^{+}_{D^{-}_k}(\psi ^-_k)\cdot \sigma ^{-}_{D^{+}_k}(\psi ^+_k)}{\delta (\Gamma _k) ||m^k_{\mathrm {{BM}}}||}\neq 0$ . Then we can restate the conclusion of Theorem 5.3 by a multiplicative error uniformly close to 1 along the sequence as t gets larger.