Proof Owing to Selberg’s lemma, $\Gamma $ has a finite index torsion-free subgroup $\Gamma _0<\Gamma $ . It follows that the order of every elliptic element of $\Gamma $ is at most $[\Gamma :\Gamma _0]$ .

We treat $\operatorname {Isom}^{+}\!{\mathcal {H}^{\mathtt {d}}}$ as the group of orientation-preserving Moebius transformations of $\mathbb {R}^{\mathtt {d}}$ which preserve the unit ball $\mathbb {B}^{\mathtt {d}}$ . When done so, it can be shown [Reference Gehring, Martin and Palka14, §3] that every elliptic element $\gamma \in \Gamma $ is conjugated to an orthogonal map T. We use the canonical form of orthogonal matrices as the (orthogonal) conjugate of a matrix of the form

$$ \begin{align*} \begin{pmatrix}R_{1} & & & & &\\ & \ddots & & & &\\ & & R_{k} & & &\\ & & & \pm1 & &\\ & & & & \ddots &\\ & & & & & \pm1 \end{pmatrix} \end{align*} $$

where $R_1,\ldots ,R_k$ are $2\times 2$ rotation matrices.

Because the order of T (as an element of the group of orthogonal matrices) is bounded, it is clear from the canonical form that there is some constant (depending only on the bound of the order) $c_0>0$ such that if $T\not =\operatorname {Id}$ , then $|\mathtt {d}-\operatorname {tr} T|>c_0$ , where $\operatorname {tr} T$ is the trace of T. Because the trace function is continuous, is invariant under conjugation, and of course satisfies $\mathtt {d}=\operatorname {tr}\operatorname {Id}$ , it follows that there is some constant $l_0>0$ such that $d(\gamma g,g)=d(g^{-1}\gamma g,e)>l_0$ , unless $\gamma =e$ .

Proof We use the upper half-space model and assume by conjugation of $\Gamma $ that $\xi \,{=}\,\infty $ . Recall that $\mathbb {R}^{\mathtt {d}-1}$ can be decomposed into a product $\mathbb {R}^{\mathtt {d}-1}=\mathbb {R}^{k}\times \mathbb {R}^{\mathtt {d}-k-1}$ , where $\operatorname {rank}(\infty )=k$ and $\mathbb {R}^{k}$ is a $\Gamma _{\infty }$ -invariant subspace such that $\Gamma _{\infty }\backslash \mathbb {R}^{k}$ is compact. Let $l_{\Gamma }>0$ be the minimal length of a translation in $\Gamma _{\infty }$ . Combining the definition of the thin part of $\Gamma \backslash \mathbb {H}^{\mathtt {d}}$ with the analysis of the possible types of parabolic isometries in $\Gamma _{\infty }$ , as in [Reference Apanasov3, §3.3], we conclude that

$$ \begin{align*} \Gamma_{\infty}\backslash\bigg\{x\in\mathbb{H}^{\mathtt{d}}:\:x_{\mathtt{d}}&\geq\frac{l_{\Gamma}\sqrt{1+4\|({1}/{l_{\Gamma}})(x_{k+1},\ldots,x_{\mathtt{d}-1})\|^{2}}}{2\sinh({\epsilon}/{2})}\bigg\}\subset\operatorname{cusp}_{\epsilon}(\infty)\\ &\subset\Gamma_{\infty}\backslash\bigg\{x\in\mathbb{H}^{n}:\:x_{\mathtt{d}}\geq\frac{l_{\Gamma}}{2\sinh({N_{\mathtt{d}}\epsilon}/{2})}\bigg\} \end{align*} $$

for all small enough $\epsilon>0$ and for some fixed $N_{\mathtt {d}}>0$ .

Because $\infty $ is a bounded parabolic fixed point (see §2.1), there is a constant $c_{\Gamma }$ such that

(3.1) $$ \begin{align} \Gamma_{\infty}\backslash\bigg\{x\in \operatorname{hull}(\Lambda(\Gamma)):\: x_{\mathtt{d}}&\geq\frac{c_{\Gamma}l_{\Gamma}}{2\sinh({\epsilon}/{2})}\bigg\}\subset \operatorname{cusp}_{\epsilon}(\infty)\cap\operatorname{core}(X)\nonumber\\&\subset\Gamma_{\infty}\backslash\bigg\{x\in \operatorname{hull}(\Lambda(\Gamma)):\:x_{\mathtt{d}}\geq\frac{l_{\Gamma}}{2\sinh({N_{\mathtt{d}}\epsilon}/{2})}\bigg\}. \end{align} $$

Item (1) follows directly from equation (3.1), together with the following.

1. (1) The action of $\Gamma _{\infty }$ by $\gamma x=\beta Ax+x^0$ , as described in the beginning of §3.2, where $\beta =1$ for parabolic and elliptic elements.

2. (2) The fact that discrete subgroups cannot contain both parabolic and loxodromic elements with a common fixed point [Reference Bowditch5], so $\Gamma _{\infty }$ contains only parabolic and elliptic elements.

For item (2), if $\Gamma z\not \in \mathcal {F}\operatorname {cusp}_{\tilde {\epsilon }_{\mathtt {d}}}(X)$ , there is nothing to show, so we assume that $\pi _{K}(z)\in T_{\tilde {\epsilon }_{\mathtt {d}}}(\Gamma _{\infty })$ . Either in the future, if $\Gamma z$ goes down in the cusp, or in the past, if z goes up, the trajectory $\{\Gamma za^{n}\}_{n\in \mathbb {Z}}$ visits $\Omega _{\mathrm {nc}}^{\tilde {\epsilon }_{\mathtt {d}}}$ , so there is an integer $n\in \mathbb {Z}$ with minimal absolute value such that $\Gamma za^{n}\in \Omega _{\mathrm {nc}}^{\tilde {\epsilon }_{\mathtt {d}}}$ . This is of course true if the non-wandering parts of the cusps of X are distant enough from each other (more than one unit), that is, if $\tilde {\epsilon }_{\mathtt {d}}$ is small enough. Without loss of generality, $y=za^{n}$ and z are on the same side of the geodesic semicircle defined by z, and $\pi _{K}(y)_{\mathtt {d}}<\pi _{K}(z)_{\mathtt {d}}$ .

Consider the following path in $\mathbb {H}^{\mathtt {d}}$ . First we connect by a straight line the points $\pi _{K}(y)$ and w, where w is the point right above $\pi _{K}(y)$ whose $\mathtt {d}$ th coordinate is equal to $\pi _{K}(z)_{\mathtt {d}}$ . Then we draw a straight horizontal line (with the same $\mathtt {d}$ th coordinate) from w to a point which is $(\pi _{K}(z))_{\mathtt {d}}$ far in the Euclidean direction $\overrightarrow {w\pi _{K}(z)}$ . This path starts at $\pi _K(y)$ and passes through $\pi _K(z)$ , and is of length $\log ({\pi _{K}(z)_{\mathtt {d}}}/{\pi _{K}(y)_{\mathtt {d}}})+1\geq n$ . To conclude item (2), use equation (3.1) again.

Item (3) follows similarly.

Proof We will give a covering of $\operatorname {core}_{\epsilon }(X)$ by X-balls $B(\Gamma z,\eta )$ , rather than a covering of $\Omega _{\mathrm {nc}}^{\epsilon }$ (which is a subset of $\mathcal {F}\operatorname {core}_{\epsilon }(X)$ ). To obtain a covering of $\Omega _{\mathrm {nc}}^{\epsilon }$ , one only needs to choose $\ll _{\eta }1$ many frames $\{\Gamma g_i\}_{i\in I}$ with $\pi _{K}(\Gamma g_i)\in B(\Gamma z,\eta )$ , such that

$$ \begin{align*}\mathcal{F}B(\Gamma z,\eta)\subset\bigcup_{i\in I}B(\Gamma g_i,\eta).\end{align*} $$

This is easily done using the compactness of K and the definition of the metrics as in §2.3. Because we only want to bound the number of balls up to a multiplicative constant, this will suffice.

Recall, once again, the structure of the cusps. For simplicity, assume that $\xi =\infty $ is a bounded parabolic fixed point in the upper half-space model. The boundary $\mathbb {R}^{\mathtt {d}-1}\subset \partial \mathbb {H}^{\mathtt {d}}$ decomposes into $\mathbb {R}^{\mathtt {d}-1}=\mathbb {R}^{k}\times \mathbb {R}^{\mathtt {d}-k-1}$ , where $\operatorname {rank}(\infty )=k$ and $\mathbb {R}^{k}$ is a $\Gamma _{\infty }$ -invariant subspace such that $\Gamma _{\infty }\backslash \mathbb {R}^{k}$ is compact.

As $\Gamma _{\infty }\backslash \mathbb {R}^{k}$ is compact, there is a subset $D_{\infty }\subset \mathbb {H}^{\mathtt {d}}$ , which contains representatives of all elements in $X=\Gamma \backslash \mathbb {H}^{\mathtt {d}}$ and is bounded in the directions defined by $\mathbb {R}^{k}$ . As $\infty $ is a bounded parabolic fixed point (see §2.1), the distance $d_{\operatorname {euc}}(\zeta ,\mathbb {R}^{k})$ is bounded for $\zeta \in \Lambda (\Gamma )\smallsetminus \{\infty \}$ , and so the part of $D_{\infty }$ corresponding to $\operatorname {core}(X)$ is bounded as well in the remaining $\mathtt {d}-k-1$ directions defined by $\mathbb {R}^{\mathtt {d}-k-1}$ .

In other words, we have simply created a $\mathtt {d}$ -dimensional box, bounded in $\mathtt {d}-1$ directions, which contains a fundamental domain for $\operatorname {core}(X)$ . For $\epsilon ,\epsilon ^\prime $ small enough, which satisfy $\epsilon <\tilde {\epsilon }_{\mathtt {d}}<\epsilon ^{\prime }<\epsilon _{\mathtt {d}}$ , the part of the fundamental domain corresponding to $T_{\epsilon ^\prime }(\Gamma _\infty )\smallsetminus T_{\epsilon }(\Gamma _\infty )$ is unbounded only in the upwards direction $e_{\mathtt {d}}$ , where it is of hyperbolic length $\ll |\!\log \epsilon |$ owing to Lemma 3.5. We make note that because the $\mathtt {d}$ th coordinate of a point in $T_{\epsilon ^\prime }(\Gamma _{\infty })$ is bounded from below by equation (3.1), boundedness also implies that the hyperbolic diameter in these directions is bounded from above (because $ds^2={\|dx\|^2}/{x_{\mathtt {d}}^2}$ ). Therefore, we can cover

$$ \begin{align*}\operatorname{hull}(\Lambda)\cap (T_{\epsilon^\prime}(\Gamma_\infty)\smallsetminus T_{\epsilon}(\Gamma_\infty))\end{align*} $$

by $\ll _{\eta }|\!\log \epsilon |$ many $\eta $ -balls. Each such ball, under the quotient by $\Gamma $ , is of the desired form.

As $\Gamma $ is geometrically finite, its bounded parabolic fixed points consist of a finite number of $\Gamma $ -orbits. So we may repeat this argument for each $\xi \in R$ , where R is a set of representatives for these orbits. Moreover, we can choose a cover of the compact set $\operatorname {core}_{\epsilon ^\prime }(X)$ by $\ll _{\eta } 1$ balls $\{\Gamma O_i\}_{i\in I}$ , where we choose $\{O_i\}_{i\in I}$ to be contained in some compact subset of $\mathbb {H}^{\mathtt {d}}$ . This would yield a cover of $\operatorname {core}_{\epsilon }(X)$ of the desired size.

Both requirements on $k_i$ , as in the statement of Lemma 3.9, are trivially satisfied, perhaps after making sure (in Lemma 3.5) that $\tilde {\epsilon }_{\mathtt {d}}$ is small enough with respect to $\epsilon ^\prime $ , and by choosing $\eta $ to be small enough so that these balls would not intersect $T_{\tilde {\epsilon }_{\mathtt {d}}}(\xi ^\prime )$ for $\xi ^{\prime }\not \in R$ .

Proof Let V be a function with $Z(V,\epsilon ,\epsilon ^\prime )\not =\emptyset $ and define the interval

$$ \begin{align*}J=[-\lceil|\!\log c_{1}\epsilon|\rceil,\lceil|\!\log c_{1}\epsilon|\rceil]\cap\mathbb{Z}.\end{align*} $$

Consider the restriction $V|_J$ . The main case of interest to us is $\lceil |\!\log c_{1}\epsilon |\rceil \leq N$ . In the other case, this restriction may be thought of as the trivial restriction to $[-N,N]$ . The following estimates might be much larger than required for the $\lceil |\!\log c_1\epsilon |\rceil>N$ case, yet it will not matter for the rest of the paper.

Assume that the restriction $V|_J$ is not identically equal to zero. Then, by Lemma 3.5, V is positive and constant (up to $\ll 1$ ‘fluctuations’ in value) on some subinterval of J in which the trajectory of any $x\in Z(v,\epsilon ,\epsilon ^\prime )$ visits $\Omega _{\mathrm {c}}^{\epsilon }$ , and V is zero outside of this subinterval. So the number of different restrictions $V|_J$ is bounded from above (up to a constant) by the number of different subintervals of J, that is, by $c\lceil |\!\log c_1\epsilon |\rceil ^2$ .

By taking images and preimages, there is the same number of possible restrictions to any $I\subset [-N,N]$ of length $2\lceil |\!\log c_{1}\epsilon |\rceil $ . As we can divide $[-N,N]$ into $\lceil ({2N+1})/({2\lceil |\!\log c_{1}\epsilon |\rceil })\rceil $ subintervals of length up to $2\lceil |\!\log c_{1}\epsilon |\rceil $ , we obtain up to

$$ \begin{align*} (c\lceil|\!\log c_{1}\epsilon|\rceil^{2})^{\lceil({2N+1})/({2\lceil|\!\log c_{1}\epsilon|\rceil})\rceil}&\leq|\!\log\epsilon|^{3(({N}/{|\!\log\epsilon|})+1)}\\&=|\!\log\epsilon|^{3}\cdot e^{({3\log(|\!\log\epsilon|)}/{|\!\log\epsilon|})N} \end{align*} $$

different functions V, for small enough $\epsilon $ .

Proof First note that the ball $\{\Gamma O_{i_{1}}\}$ is clearly a covering of $Z_{+}(O_{i_{1}},O_{i_{2}},V,\epsilon )$ of the correct form $\Gamma (\gamma _1 O_{l}a^{-0}\cap O_{i{}_{1}})$ . The size of this covering, $1$ , is of course bounded from above by

$$ \begin{align*} C^{s+t}\cdot e^{\delta{{\sum}_{j=1}^{t}}|J_{j}|+{{\sum}_{i=1}^{s}}({r_{i}}/{2})|I_{i}|} \end{align*} $$

for $t=s=0$ . For reasons of readability, the constant $C>0$ will be defined only later in the proof, independently of the following construction. It is important to note that this is consistent and does not raise any circular argument.

We prove the lemma by going through the time intervals by their order, assuming by induction that the lemma is true for all previous intervals. For both the first interval (base case) and the following intervals (induction step), we start with a covering of $Z_{+}(O_{i_{1}},O_{i_{2}},V,\epsilon )$ by up to

$$ \begin{align*} C^{s+t}\cdot e^{\delta{{\sum}_{j=1}^{t}}|J_{j}|+{{\sum}_{i=1}^{s}}({r_{i}}/{2})|I_{i}|} \end{align*} $$

sets of the form $\Gamma (\gamma _1 O_l a^{-K}\cap O_{i_{1}})$ , where the interval we consider is $[K,K^\prime ]\cap \mathbb {Z}$ , either of type $I_{s+1}$ or type $J_{t+1}$ . We write

$$ \begin{align*}K^{\prime}=\begin{cases}K+L, & K^{\prime}<N,\\ K+L+1, & K^{\prime}=N,\end{cases}\end{align*} $$

to simplify the notation in the proof.

The main idea is the following. Let $x\in Z_{+}(O_{i_1},O_{i_2},V,\epsilon )$ . Then $x=\Gamma g$ for some $g\in \gamma _1 O_l a^{-K}\cap O_{i_{1}}$ . By the construction of the intervals, $\Gamma ga^{K+L+1}\in \Omega _{\mathrm {nc}}^{\epsilon _{\mathtt {d}}^{\prime }}$ (even in the $K+L+1=N$ case, that is, the last interval, by the assumption $i_2\leq N_1$ ). Therefore, $ga^{K+L+1}=\gamma _{2}\tilde {g}_{2}$ for some $\gamma _{2}\in \Gamma $ and $\tilde {g}_{2}\in O_{j}$ , for $1\leq j\leq N_1$ . Then

$$ \begin{align*} g\in\gamma_{2}O_{j}a^{-(K+L+1)}\cap O_{i_{1}}, \end{align*} $$

which under the quotient by $\Gamma $ , is indeed of the desired form. We need to bound the number of possible sets of this form. This can be done by counting the number of possible 2-tuples $(O_{j},\gamma _{2})$ , or alternatively the number of 3-tuples $(O_{j},\gamma _{1},\gamma _{1}^{-1}\gamma _{2})$ .

The number of $O_j$ values is at most $N_1$ . The number of $\gamma _1$ values is at most the number of sets given in the previous step, which is bounded by

$$ \begin{align*} C^{s+t}\cdot e^{\delta\cdot{{\sum}_{j=1}^{t}}|J_{j}|+{{\sum}_{i=1}^{s}}({r_{i}}/{2})|I_{i}|}. \end{align*} $$

It only remains to count the elements of type $\gamma _{1}^{-1}\gamma _{2}$ . Let $\tilde {g}_1=\gamma _1^{-1}ga^{K}\in O_l$ . By construction, $\Gamma \tilde {g}_{1},\Gamma \tilde {g}_{2}\in \Omega _{\mathrm {nc}}^{\epsilon _{\mathtt {d}}^{\prime }}$ , and so by Lemma 3.9, we get $d(o,\tilde {g}_{1}),d(o,\tilde {g}_{2})<r_0$ . Moreover, because

$$ \begin{align*}\gamma_{1}^{-1}\gamma_{2}\tilde{g}_{2}=\tilde{g}_{1}a^{L+1},\end{align*} $$

we have

$$ \begin{align*}d(\gamma_{1}^{-1}\gamma_{2}\pi_{K}(o),\pi_{K}(o))\leq 2r_0+L+1.\end{align*} $$

Therefore, these maps $\gamma _{1}^{-1}\gamma _{2}$ are contained in

$$ \begin{align*} \Gamma_{J}=\{\gamma\in\Gamma:\:d(\gamma\pi_{K}(o),\pi_{K}(o))\leq 2r_0+L+1\}. \end{align*} $$

In the case where the next interval is of type $J_{t+1}$ , we cannot give a better restriction of what maps of $\Gamma $ can be of the former type, and we will have to bound $\Gamma _{J}$ itself using Proposition 2.17, to obtain

$$ \begin{align*} |\Gamma_J|\leq Be^{(2r_0+L+1)\delta}\leq\tilde{B}e^{\delta|J_{t+1}|}, \end{align*} $$

where $\tilde {B}=Be^{2r_0\delta }$ depends only on $\Gamma $ and o.

In the case where the next interval is of type $I_{t+1}$ , we shall be using the information about the trajectory spending time in the cusp to show that all the maps of the type $\gamma _{1}^{-1}\gamma _{2}$ are in fact contained in a (proper) subset of $\Gamma _{J}$ . This will allow us to cut down on the number of sets. Let R be a set of representatives for the $\Gamma $ -orbits of bounded parabolic fixed points, as in Lemma 3.9.

Indeed, in the $I_{t+1}$ case, $\Gamma \tilde {g}_{1},\Gamma \tilde {g}_{2}\in \Omega _{\mathrm {c}}^{\tilde {\epsilon }_{\mathtt {d}}}$ owing to Lemma 3.5. By the choice of the cover as in Lemma 3.9, there is a parabolic fixed point $\xi \in R$ such that $\tilde {g}_{1},\tilde {g_2}\in \mathcal {F}T_{\tilde {\epsilon }_{\mathtt {d}}}(\Gamma _{\xi })$ . Because $\{\Gamma \tilde {g}_1a^{m}\}_{m=0}^{L+1}\subset \mathcal {F}\operatorname {cusp}_{\tilde {\epsilon }_{\mathtt {d}}}(X)$ , that is, the trajectory stays in the cusp during the whole interval, we obtain from Lemma 3.5 that $\tilde {g}_{1}a^{L+1}\in \mathcal {F}T_{\tilde {\epsilon }_{\mathtt {d}}}(\Gamma _{\xi })$ as well. So we have obtained $\tilde {g}_{1}a^{L+1},\tilde {g}_{2}\in \mathcal {F}T_{\tilde {\epsilon }_{\mathtt {d}}}(\Gamma _{\xi })$ . Because $\gamma _{1}^{-1}\gamma _{2}\tilde {g}_{2}=\tilde {g}_{1}a^{L+1}$ , we get from the precise invariance of $T_{\tilde {\epsilon }_{\mathtt {d}}}(\Gamma _{\xi })$ that $\gamma _{1}^{-1}\gamma _{2}\in \Gamma _{\xi }$ (see §2.2).

Therefore, we have shown that in the $I_{t+1}$ case, it suffices to bound

$$ \begin{align*} \Gamma_{I}=\{\nu\in\Gamma_{\xi}:\:d(\gamma\pi_{K}(o),\pi_{K}(o))\leq 2r_0+L+1\}\subset\Gamma_{J}. \end{align*} $$

Let us do so. For simplicity, we may assume $\xi =\infty $ . Recall that there is a subgroup $H\triangleleft \Gamma _{\xi }$ of finite index $n(\Gamma )$ , isomorphic to $\mathbb {Z}^{r_{\xi }}$ , such that H acts cocompactly on $\mathbb {R}^{r_{\xi _{j}}}\subset \mathbb {R}^{\mathtt {d}-1}=\partial \mathbb {H}^{\mathtt {d}}\backslash \{\xi \}$ by translations. Say $\{\nu _{1},\ldots \nu _{n(\Gamma )}\}$ are representatives for $\Gamma _{\xi }/H$ , then every $\nu \in \Gamma _{I}$ is of the form $\nu =\gamma \nu _{k}$ for $\gamma \in H,\:1\leq k\leq n(\Gamma )$ . Therefore,

$$ \begin{align*} d(\gamma \pi_{K}(o),\pi_{K}(o))&\leq d(\nu\pi_{K}(o),\pi_{K}(o))+d(\nu_{k}\pi_{K}(o),\pi_{K}(o))\\& \leq 2r_0+L+1+\underset{1\leq k\leq n(\Gamma)}{\max}d(\nu_{k}\pi_{K}(o),\pi_{K}(o)). \end{align*} $$

We have obtained $|\Gamma _{I}|\leq n(\Gamma )|\Gamma _{I}^{\prime }|$ for

$$ \begin{align*} \Gamma_{I}^{\prime}=\{\gamma\in H:\:d(\gamma\pi_{K}(o),\pi_{K}(o))\leq 2r_0+\underset{1\leq k\leq n(\Gamma)}{\max}d(\upsilon_{k},o)+L+1\}. \end{align*} $$

A simple hyperbolic geometry calculation shows that if $\gamma \in H$ , then

$$ \begin{align*} \|\gamma z_{0}-z_{0}\|_{\mathbb{R}^{\mathtt{d}-1}}\ll e^{({1}/{2})d(\gamma\pi_{K}(o),\pi_{K}(o))} \end{align*} $$

for $z_{0}$ the point on $\partial \mathbb {H}^{\mathtt {d}}$ right below $\pi _{K}(o)$ , that is, with the $\mathtt {d}$ th coordinate equal to $0$ . Therefore, $|\Gamma _{I}^{\prime }|\leq |\Gamma _{\partial \mathbb {H}^{\mathtt {d}}}|$ for

$$ \begin{align*} \Gamma_{\partial\mathbb{H}^{\mathtt{d}}}=\{\gamma\in H:\:\|\gamma z_{0}-z_{0}\|_{\mathbb{R}^{\mathtt{d}-1}}\leq\tilde{F}e^{({1}/{2})L}\} \end{align*} $$

for some constant $\tilde {F}>0$ .

To bound the size of this set, note that $Hz_0-z_0$ is a lattice in $\mathbb {R}^{r_{\xi }}$ , and so $|\Gamma _{\partial \mathbb {H}^{\mathtt {d}}}|$ is roughly equal to the volume of a $r_{\xi }$ -dimensional ball of radius $\tilde {F}e^{(1/2)L}$ , divided by $d(H)$ , the volume of a fundamental polyhedron. That is, $|\Gamma _{\partial \mathbb {H}^{\mathtt {d}}}|\ll e^{({r_{\xi }}/{2})L}$ . Therefore, we obtain $|\Gamma _{I}|\leq \tilde {E} e^{(r_{s+1}/2)|I_{s+1}|}$ , where $\tilde {E}$ depends only on $\Gamma $ .

So we get that $Z_{+}(O_{i_{1}},O_{i_{2}},V,\epsilon )$ can be covered by

$$ \begin{align*} C^{s+t}\cdot e^{\delta\cdot{{\sum}_{j=1}^{t}}|J_{j}|+{{\sum}_{i=1}^{s}}({r_{i}}/{2})|I_{i}|}\cdot N_1\cdot\begin{cases}\tilde{E} e^{(r_{s+1}/2)|I_{s+1}|} & \mbox{the next interval is }I_{s+1}\\\tilde{B}e^{\delta|J_{t+1}|} & \mbox{the next interval is }J_{t+1}\end{cases} \end{align*} $$

sets of the desired form, which proves the claim for $C=N_1\cdot \max \{\tilde {E},\tilde {B}\}$ .

Proof Let $\epsilon ^{\prime }<\epsilon _{\mathtt {d}}^{\prime }$ be small enough with respect to all previous lemmas.

Recall that in Lemma 3.14, we assumed that $i_1,i_2\leq N_1$ , that is, $\Gamma O_{i_{1}},\Gamma O_{i_{2}}\subset \Omega _{\mathrm {nc}}^{\epsilon _{\mathtt {d}}^{\prime }}$ . Even if this is not the case, the main idea of the proof remains intact, and only minor adjustments are needed in the first and last intervals, as follows. Now, it is possible that in the first interval, $\tilde {g}_1$ is in $\Omega _{\mathrm {nc}}^{\epsilon ^{\prime }}$ rather than in $\Omega _{\mathrm {nc}}^{{\epsilon _{\mathtt {d}}^{\prime }}}$ , and similarly for $\tilde {g}_2$ in the last interval, and so

$$ \begin{align*} d(\tilde{g}_{i},o)<\tilde{r}_0+\lceil|\!\log c_{2}\epsilon^{\prime}|\rceil \end{align*} $$

rather than $<r_0$ . Therefore, $d(\gamma _1^{-1}\gamma _2\pi _{K}(o),\pi _{K}(o))$ would increase accordingly. This would yield an increase of $\ll e^{2|\!\log \epsilon ^{\prime }|\delta }$ in the number of element $\gamma _1^{-1}\gamma _2$ and so in the number of sets in the cover.

In any case, at the end of the $p+l$ iterations of Lemma 3.14, we emerge with a covering of $Z_{+}(O_{i_{1}},O_{i_{2}},V,\epsilon )$ by

$$ \begin{align*} f&\ll e^{2|\!\log\epsilon^{\prime}|\delta}\cdot C^{p+l}\cdot e^{\delta\cdot{{\sum}_{j=1}^{l}}|J_{j}|+{{\sum}_{i=1}^{p}}({r_{i}}/{2})|I_{i}|} \ll_{\epsilon^\prime} C^{p+l}\cdot e^{\delta N-{{\sum}_{i=1}^{\mathtt{d}-1}}(({2\delta-i})/{2})\cdot|V^{-1}(i)|} \end{align*} $$

sets, where we used the fact mentioned in §2.4 that $\delta>({r_{i}}/{2})$ for all cusps. Each of the sets is contained in the union of $\ll 1$ forward Bowen $(N,\eta $ )-balls. Note that

$$ \begin{align*} p+l\leq\bigg\lceil\frac{N}{({2\lceil|\!\log c_{1}\epsilon|\rceil+1})/{3}}\bigg\rceil\leq\bigg\lceil\frac{2N}{|\!\log\epsilon|}\bigg\rceil, \end{align*} $$

because each interval of type $I_{t}$ is of length at least $2\lceil |\!\log c_{1}\epsilon |\rceil -1$ . So we get

$$ \begin{align*} \ll_{\epsilon^{\prime}}C^{({2N}/{|\!\log\epsilon|})}\cdot e^{\delta N-{{\sum}_{i=1}^{\mathtt{d}-1}}(({2\delta-i})/{2})\cdot|V^{-1}(i)|} \end{align*} $$

forward Bowen $(N,\eta $ )-balls.

Let us now cover $Z(V,\epsilon ,\epsilon ^{\prime })$ by (non-forward) Bowen $(N,\eta $ )-balls. It is simply done by evoking Lemma 3.14 $N_0(\epsilon ^\prime )^2$ times (for all $i_1,i_2$ ) to obtain a covering of

$$ \begin{align*} T_{a}^{-N}Z(V,\epsilon,\epsilon^{\prime})&=\{x\in \Omega_{\mathrm{nc}}^{\epsilon^{\prime}}\cap T_{a}^{-2N}(\Omega_{\mathrm{nc}}^{\epsilon^{\prime}}):T_{a}^{m}x\in \Omega_{\mathrm{c},i}^{\epsilon}\iff V(m-N)=i,\\&\quad\qquad\:\text{ for all } m\in[0,2N]\cap\mathbb{Z},\ \text{for all } i\in[1,\mathtt{d}-1]\cap\mathbb{Z}\} \end{align*} $$

by

$$ \begin{align*} \ll_{\epsilon^\prime} N_0(\epsilon^\prime)^{2}\cdot C^{({4N}/{|\!\log\epsilon|})}\cdot e^{(2N+1)\delta-{{\sum}_{i=1}^{\mathtt{d}-1}}(({2\delta-i})/{2})\cdot|V^{-1}(i)|} \end{align*} $$

forward Bowen $(2N,\eta )$ -balls. Clearly $T_{a}^{N}(yB_{2N,\eta }^{+})\subset (T_{a}^{N}y)B_{N,\eta }$ and so this yields a covering of $Z(V,\epsilon ,\epsilon ^{\prime })$ by Bowen $(N,\eta )$ -balls.

To finish the proof, recall that $N_{0}(\epsilon ^{\prime })\ll _{\epsilon ^\prime } 1$ owing to Lemma 3.9 and Remark 3.10.

Proof Without loss of generality, by passing to a subsequence, assume $(\mu _i)_{i\in \mathbb {N}}$ converges to $\mu $ in the weak- $\star $ topology. For simplicity, for any $m\in \mathbb {Z}$ and $h>0$ , let $\mathrm {Compact}(m,h) = T_{a}^{-m}(\Omega _{\mathrm {nc}}^{h})$ and $\mathrm {Cusp}(m,h) = T_{a}^{-m}(\Omega _{\mathrm {c}}^{h})$ .

Let $\epsilon>0$ be small enough with respect to Lemma 3.15, and let

$$ \begin{align*} \kappa>\kappa_{\epsilon}:=\frac{1}{2\delta-r_{\max}}\cdot\frac{4\log(|\!\log\epsilon|)}{|\!\log\epsilon|}. \end{align*} $$

Take $\epsilon _{0}>0$ small enough such that

(3.2) $$ \begin{align} g(\epsilon,\epsilon_{0})&:=\frac{4\log(|\!\log\epsilon|)}{|\!\log\epsilon|}(1+3\epsilon_{0})+6(\delta+\alpha)\epsilon_{0}\nonumber\\&\quad-(2\delta-r_{\max})(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)<0, \end{align} $$

where $\alpha $ is as in the statement of Theorem 1.1. We may also assume that ${\epsilon _{0}}/({1+2\epsilon _{0}})<\kappa $ , and that $3\epsilon _{0}$ is small enough such that the assumptions of Theorem 1.1 and the results of Lemma 3.15 hold. This is possible because

$$ \begin{align*} \underset{\epsilon_{0}\to0^{+}}{\lim}g(\epsilon,\epsilon_{0})=\frac{4\log(|\!\log\epsilon|)}{|\!\log\epsilon|}-(2\delta-r_{\max})\kappa<0 \end{align*} $$

by the choice of $\kappa $ . The reason for this specific choice of $\epsilon _0$ will be clear at the end of the proof.

Let $(\lambda _i)_{i\in \mathbb {N}}$ be as in the statement of Theorem 1.1. Set the heights $H_{i}=\lambda _{i}^{\epsilon _{0}/4}$ , where $i\in \mathbb {N}$ is large enough such that $\lambda _j<1$ for all $j\geq i$ . Set $N_{i}=\lfloor -(1/2)\log \lambda _{i}\rfloor $ and $N_{i}^{\prime }=N_{i}+2\lfloor \epsilon _{0}N_{i}\rfloor $ . Set

$$ \begin{align*} E_{\kappa,i}=\bigg\{x\in \mathcal{F}X:\:\frac{1}{2N_{i}^{\prime}+1}\underset{n=-N_{i}^{\prime}}{\overset{N_{i}^{\prime}}{\sum}} 1_{\mathrm{Cusp}(n,\epsilon)}(x)>\kappa\bigg\} \end{align*} $$

and

$$ \begin{align*} X_{\kappa,i}=\mathrm{Compact}(-N_{i}^{\prime},H_i)\cap\mathrm{Compact}(N_{i}^{\prime},H_i)\cap E_{\kappa,i}. \end{align*} $$

In what follows, we will use Lemma 3.12 and Lemma 3.15 to cover $X_{\kappa ,i}$ by a relatively small number of Bowen balls, and deduce that $\mu _{i}(X_{\kappa ,i})$ is small. From this, the desired result will follow.

Note that once $H_{i}$ is small enough (that is, i is large enough), any trajectory of a point $x\in \mathrm {Compact}(0,H_i)$ visits $\mathrm {Compact}(0,\tilde {\epsilon }_{\mathtt {d}})$ in up to $\lfloor \epsilon _{0}N_{i}\rfloor $ steps, either in the past or the future. This is immediate from Lemma 3.5, because $\lceil |\!\log c_{2}H_{i}|\rceil \leq \lfloor \epsilon _{0}N_{i}\rfloor $ .

It follows that

$$ \begin{align*} \mathrm{Compact}(0,H_i) \subset\underset{k=-\lfloor\epsilon_{0}N_{i}\rfloor}{\overset{\lfloor\epsilon_{0}N_{i}\rfloor}{\bigcup}} \mathrm{Compact}(-k,\tilde{\epsilon}_{\mathtt{d}}). \end{align*} $$

Therefore,

$$ \begin{align*} X_{\kappa,i}\subset\underset{(k_{1},k_{2})\in\{-\lfloor\epsilon_{0}N_{i}\rfloor,\ldots,\lfloor\epsilon_{0}N_{i}\rfloor\}^{2}}{\bigcup}F_{k_{1},k_{2}} \end{align*} $$

for

$$ \begin{align*} F_{k_{1},k_{2}}=\mathrm{Compact}(-N_{i}^{\prime}-k_{1},\tilde{\epsilon}_{\mathtt{d}})\cap \mathrm{Compact}(N_{i}^{\prime}-k_{2},\tilde{\epsilon}_{\mathtt{d}})\cap E_{\kappa,i}. \end{align*} $$

For any $(k_{1},k_{2})$ , set $c=\lfloor -({k_{1}+k_{2}})/{2}\rfloor ,\:d=\lfloor ({k_{1}-k_{2}})/{2}\rfloor $ and $N=N_{i}^{\prime }+d$ . Note that

$$ \begin{align*} N_{i}+\lfloor\epsilon_{0}N_{i}\rfloor\leq N\leq N_{i}+3\lfloor\epsilon_{0}N_{i}\rfloor. \end{align*} $$

Then

$$ \begin{align*} T_{a}^{c}F_{k_{1},k_{2}}\subset\mathrm{Compact}(-N,\tilde{\epsilon}_{\mathtt{d}})\cap\mathrm{Compact}(N,\tilde{\epsilon}_{\mathtt{d}}) \end{align*} $$

or

$$ \begin{align*} T_{a}^{c}F_{k_{1},k_{2}}\subset \mathrm{Compact}(-N,\tilde{\epsilon}_{\mathtt{d}})\cap\mathrm{Compact}(N+1,\tilde{\epsilon}_{\mathtt{d}}), \end{align*} $$

depending on whether $k_{1}$ and $k_{2}$ have the same parity or not. In any case,

$$ \begin{align*} T_{a}^{c}F_{k_{1},k_{2}}\subset \mathrm{Compact}(-N,\epsilon)\cap \mathrm{Compact}(N,\epsilon), \end{align*} $$

assuming $\epsilon $ was initially chosen to be small enough.

Now, note that

$$ \begin{align*} \frac{2N_{i}^{\prime}+1}{2N+1}\geq\frac{2N_{i}}{2(N_{i}+3\lfloor\epsilon_{0}N_{i}\rfloor)+1}\geq\frac{2N_{i}}{2(1+4\epsilon_{0})N_{i}}=(1+4\epsilon_{0})^{-1} \end{align*} $$

for large enough i, and that

$$ \begin{align*} -\frac{|c-d|+|c+d|}{2N_{i}^{\prime}+1}\geq-\frac{2\lfloor\epsilon_{0}N_{i}\rfloor}{2(N_{i}+2\lfloor\epsilon_{0}N_{i}\rfloor)+1}\geq-\frac{\epsilon_{0}}{1+2\epsilon_{0}}. \end{align*} $$

Therefore, for all $x=T_{a}^{c}y\in T_{a}^{c}F_{k_{1},k_{2}}$ ,

$$ \begin{align*} \frac{1}{2N+1}\underset{n=-N}{\overset{N}{\sum}}1_{\mathrm{Cusp}(n,\epsilon)}(x)&= \frac{1}{2N+1}\underset{n=-N}{\overset{N}{\sum}}1_{\mathrm{Cusp}(n+c,\epsilon)}(y)\\&= \frac{1}{2N+1}\underset{n=-N_{i}^{\prime}+(c-d)}{\overset{N_{i}^{\prime}+(c+d)}{\sum}}1_{\mathrm{Cusp}(n,\epsilon)}(y)\\&\geq \frac{2N_{i}^{\prime}+1}{2N+1}\frac{1}{2N_{i}^{\prime}+1}\bigg(\underset{n=-N_{i}^{\prime}}{\overset{N_{i}^{\prime}}{\sum}}1_{\mathrm{Cusp}(n,\epsilon)}(y)-|c-d|-|c+d|\bigg)\\&\geq (1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg). \end{align*} $$

We have obtained

$$ \begin{align*} &T_{a}^{c}F_{k_{1},k_{2}}\subset\bigg\{x\in\mathrm{Compact}(-N,\epsilon)\cap\mathrm{Compact}(N,\epsilon):\:\\& \quad\qquad|\{n\in[-N,N]\cap\mathbb{Z}:\:x\in\mathrm{Cusp}(n,\epsilon)\}|\geq(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)(2N+1)\bigg\}. \end{align*} $$

By Lemma 3.12, $T_a^{c}F_{k_{1},k_{2}}$ can be covered by

$$ \begin{align*} \ll_{\epsilon}e^{({3\log(|\!\log\epsilon|)}/{|\!\log\epsilon|})N}\leq e^{({3\log(|\!\log\epsilon|)}/{|\!\log\epsilon|})(1+3\epsilon_{0})N_{i}} \end{align*} $$

sets $Z_{+}(V,\epsilon )$ with

$$ \begin{align*} |V^{-1}(\{1,\ldots,\mathtt{d}-1\})|&\geq (2N+1)(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)\\&\geq 2N_{i}(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg). \end{align*} $$

Each set is covered, owing to Lemma 3.15, by

$$ \begin{align*} f&\ll_{\epsilon} C^{({4N}/{|\!\log\epsilon|})}\cdot e^{2N\delta-((2\delta-r_{\max})/2)\cdot|V^{-1}(\{1,\ldots,\mathtt{d}-1\})|}\\& \ll_{\epsilon} \exp\bigg(\bigg[\frac{4(1+3\epsilon_0)\log C}{|\!\log\epsilon|}+2(1+3\epsilon_0)\delta\\&\hspace{1.7cm}-(2\delta-r_{\max})(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)\bigg]N_{i}\bigg) \end{align*} $$

Bowen $(N,\eta )$ -balls.

Therefore, $F_{k_{1},k_{2}}$ can be covered by that many $T_{a}^{-c}$ images of Bowen $(N,\eta )$ -balls. Each such image is contained in a Bowen $(N-|c|,\eta )$ -ball. Note that these balls are in fact subsets of Bowen $(N_{i},\eta )$ -balls because

$$ \begin{align*} N-|c|=(N_{i}+2\lfloor\epsilon_{0}N_{i}\rfloor)+d-|c|\geq N_{i}. \end{align*} $$

Therefore, $X_{\kappa ,i}$ can be covered by

$$ \begin{align*} l&\ll_{\epsilon}(\epsilon_{0}N_{i})^{2}\cdot e^{({3\log(|\!\log\epsilon|)}/{|\!\log\epsilon|})(1+3\epsilon_{0})N_{i}}\cdot f\\& \ll_{\epsilon,\epsilon_0}N_{i}^{2}\exp\bigg(\bigg[\frac{4\log(|\!\log\epsilon|)}{|\!\log\epsilon|}(1+3\epsilon_{0})+2(1+3\epsilon_0)\delta\\ &\hspace{2.5cm}-(2\delta-r_{\max})(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)\bigg]N_{i}\bigg) \end{align*} $$

Bowen $(N_{i},\eta )$ -balls, denoted $S_{1},\ldots S_{l}$ . We will use this cover to bound $\mu _i(X_{\kappa ,i})$ .

Note that these balls satisfy

$$ \begin{align*}T_{a}^{N_i}S_j\subset\mathrm{Compact}(-N_i+N+c,\epsilon).\end{align*} $$

Moreover,

$$ \begin{align*}-N_i+N+c=2\lfloor\epsilon_0 N_i\rfloor+(d+c)\end{align*} $$

and so $|-N_i+N+c|\leq 4\lfloor \epsilon _0 N_i\rfloor $ . Using Lemma 3.5, it follows that $T_{a}^{N_i}S_j\subset \Omega _{\mathrm {nc}}^{\lambda _{i}^{3\epsilon _0}}$ once i is large enough.

Let us make the cover disjoint, that is, define $\tilde {S}_{j}=S_{j}\smallsetminus{\kern-1.3pt}\bigcup_{k=1}^{j-1} S_{k}$ for all $1\leq j\leq l$ . Note that $S_{j}$ is a Bowen $(N_{i},\eta )$ -ball and so $T_a^{N_i}S_j$ is a backward Bowen $(2N_i,\eta )$ -ball. As such, it has width at most

$$ \begin{align*} 2\eta e^{-2N_{i}}\leq\lambda_{i} \end{align*} $$

in the $N^{-}$ direction, and width at most $1$ in the $MA$ and $N^+$ directions. Therefore, we have obtained

$$ \begin{align*} (T_a^{N_i}\tilde{S}_{j})\times(T_a^{N_i}\tilde{S}_{j})\subset\{(x,y)\in \mathcal{F}\operatorname{core}_{\lambda_{i}^{3\epsilon_0}}(X)\times \mathcal{F}\operatorname{core}_{\lambda_{i}^{3\epsilon_0}}(X):\:x\in yB_{1}^{N^{+}}B_{1}^{MA}B_{\lambda_{i}}^{N^{-}}\} \end{align*} $$

for all j. So, by the assumption of Theorem 1.1,

$$ \begin{align*} \mu_{i}\times\mu_{i}\bigg(\underset{j=1}{\overset{l}{\bigcup}}(T_a^{N_i}\tilde{S}_{j})\times(T_a^{N_i}\tilde{S}_{j})\bigg)\ll_{\epsilon_0}\lambda_{i}^{\delta-3\alpha\epsilon_{0}} \leq e^{-2(\delta-3\alpha\epsilon_{0})N_{i}}. \end{align*} $$

As $\{\tilde {S}_{j}\times \tilde {S}_{j}\}_{j=1}^{l}$ is a mutually disjoint collection, and because $\mu _i$ is invariant under the frame flow,

$$ \begin{align*} \underset{j=1}{\overset{l}{\sum}}\mu_{i}(\tilde{S}_{j})^{2}&=\underset{j=1}{\overset{l}{\sum}}\mu_{i}(T_{a}^{N_i}\tilde{S}_{j})^{2}=\underset{j=1}{\overset{l}{\sum}}\mu_{i}\times\mu_{i}(T_{a}^{N_i}\tilde{S}_{j}\times T_{a}^{N_i}\tilde{S}_{j})\\&=\mu_{i}\times\mu_{i}\bigg(\underset{j=1}{\overset{l}{\bigcup}}T_{a}^{N_i}\tilde{S}_{j}\times T_{a}^{N_i}\tilde{S}_{j}\bigg)\ll_{\epsilon_0}e^{-2(\delta-3\alpha\epsilon_{0})N_{i}}. \end{align*} $$

As $\{\tilde {S}_{j}\}_{j=1}^{l}$ is a covering of $X_{\kappa ,i}$ ,

$$ \begin{align*} \mu_{i}(X_{\kappa,i})^{2}&\leq\bigg(\underset{j=1}{\overset{l}{\sum}}\mu_{i}(\tilde{S}_{j})\bigg)^{2} \leq l \cdot\underset{j=1}{\overset{l}{\sum}}\mu_{i}(\tilde{S}_{j})^{2}\\& \ll_{\epsilon,\epsilon_0}N_{i}^{2}\exp\bigg(\bigg[\frac{4\log(|\!\log\epsilon|)}{|\!\log\epsilon|}(1+3\epsilon_{0})+6(\delta+\alpha)\epsilon_0\\&\quad-(2\delta-r_{\max})(1+4\epsilon_{0})^{-1}\bigg(\kappa-\frac{\epsilon_{0}}{1+2\epsilon_{0}}\bigg)\bigg]N_{i}\bigg). \end{align*} $$

By the choice of $\epsilon _{0}$ in equation (3.2), the latter exponent is negative, and therefore (as ${\lim _{i\to \infty }}N_{i}=\infty $ ), ${\lim _{i\to \infty }}\mu _{i}(X_{\kappa ,i})=0$ .

Because $\mu _{i}$ is $T_{a}$ -invariant and $\operatorname {supp}(\mu _{i})\subset \Omega _{\mathcal {F}}$ , and using the fact that

$$ \begin{align*} X = X_{\kappa,i} \cup \mathrm{Compact}(-N_{i}^{\prime},H_i)^{c} \cup \mathrm{Compact}(N_{i}^{\prime},H_i)^{c} \cup E_{\kappa,i}^{c}, \end{align*} $$

we have for all $n\in \mathbb {N}$ ,

$$ \begin{align*} \mu_{i}(\Omega_{\mathrm{c}}^{\epsilon})&=\mu_i(\mathrm{Cusp}(n,\epsilon))=\int_{X} 1_{\mathrm{Cusp}(n,\epsilon)}d\mu_i\\& \leq \mu_i(X_{\kappa,i}) + \mu_i(\mathrm{Compact}(-N_{i}^{\prime},H_i)^{c}) +\mu_i(\mathrm{Compact}(N_{i}^{\prime},H_i)^{c}) \\&\quad+\int_{E_{\kappa,i}^{c}} 1_{\mathrm{Cusp}(n,\epsilon)}d\mu_i\\& =\mu_{i}(X_{\kappa,i})+2\mu_{i}(\Omega_{\mathrm{c}}^{H_i})+\int_{E_{\kappa,i}^{c}} 1_{\mathrm{Cusp}(n,\epsilon)}d\mu_i, \end{align*} $$

and so, by the definition of $E_{\kappa ,i}$ ,

(3.3) $$ \begin{align} \mu_{i} (\Omega_{\mathrm{c}}^{\epsilon}) &=\frac{1}{2N_i^{\prime}+1}\sum_{n=-N_i^\prime}^{N_i^\prime}\mu_{i}(\mathrm{Cusp}(n,\epsilon))\\ & \leq \mu_{i}(X_{\kappa,i})+ 2\mu_i(\Omega_{\mathrm{c}}^{H_i})+ \int_{E_{\kappa,i}^{c}} \frac{1}{2N_i^{\prime}+1}\sum_{n=-N_i^\prime}^{N_i^\prime} 1_{\mathrm{Cusp}(n,\epsilon)}d\mu_i \notag\\ & \leq\mu_i(X_{\kappa,i})+2\mu_i(\Omega_{\mathrm{c}}^{H_i})+\kappa.\notag \end{align} $$

Note that ${\lim _{\epsilon \to 0^{+}}}\kappa _{\epsilon }=0$ and so we can take $\kappa $ to be as small as desired, if $\epsilon $ is small enough. Moreover, the first two terms in equation (3.3) tend to zero as $i\to \infty $ . So, we see that for all $\upsilon>0$ , there is $\epsilon>0$ small enough and $j\in \mathbb {N}$ such that $\mu _i(\Omega _{c}^{\epsilon })<\upsilon $ for all $i>j$ . This shows that the measures are ‘almost’ supported on a compact set, namely that the sequence of measures is ‘tight’. It follows from Prokhorov’s theorem [Reference Prokhorov30] that the sequence $(\mu _i)_{i\in \mathbb {N}}$ converges in the weak topology, and its limit $\mu $ is a probability measure.

Assume that $\epsilon $ was chosen such that $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon })=0$ . Then, taking the limit of equation (3.3) as $i\to \infty $ , we get from weak convergence that $\mu (\Omega _{\mathrm {nc}}^{\epsilon })\geq 1-\kappa $ . Because it holds for all $\kappa>\kappa _\epsilon $ , it follows that

(3.4) $$ \begin{align} \mu(\Omega_{\mathrm{nc}}^{\epsilon})\geq 1-\kappa_\epsilon \end{align} $$

as well, as desired.

Note that because $\mu $ is a finite measure, there cannot be more than countably many values of $\epsilon ^\prime $ for which $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon ^\prime })>0$ . So for such an $\epsilon ^\prime $ , we may choose arbitrarily close $\epsilon _i>\epsilon ^\prime $ , for which equation (3.4) holds, and take the limit as $\epsilon _i\to \epsilon ^\prime $ to obtain the result for $\epsilon ^{\prime }$ as well.

Proof Let $\rho>0$ be small enough with respect to the injectivity radius of $\Omega _{\mathrm {nc}}^{\epsilon }$ , for instance, $\rho =\alpha \min \{\eta ,l_0,\epsilon \}$ for $l_0$ the constant from Lemma 3.2 and $\alpha \ll 1$ .

Take a finite partition $\{P_{1},\ldots ,P_{m}\}$ of $\Omega _{\mathrm {nc}}^{\epsilon }$ whose elements are contained in the quotients by $\Gamma $ of balls of radius $\rho $ , namely $\Gamma k_{i}B_{\rho }^{G}$ for $k_{i}\in G$ . Of course, we may assume that these sets are mutually disjoint. Moreover, if $\mu $ is given, we can choose this partition to satisfy $\mu (\partial P_{j})=0$ for all $1\leq j\leq m$ , because for all $x\in \Gamma \backslash G$ , we have $\mu (\partial B_{r}(x))=0$ for all but countably many $r>0$ . Assume, for now, that $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon })=0$ as well.

Define $\mathscr {P}=\{\Omega _{\mathrm {c}}^{\epsilon },P_{1},\ldots ,P_{k}\}$ and take some $S={{\bigcap }_{n=-N}^{N}}T_{a}^{n}Q_{n}\in \mathscr {P}_{N}$ . It is clear that for all $x\in S$ and $-N\leq n\leq N$ ,

$$ \begin{align*} T_{a}^{-n}x\in \Omega_{\mathrm{c}}^{\epsilon}\iff Q_{n}=\Omega_{\mathrm{c}}^{\epsilon}. \end{align*} $$

In particular, the function

$$ \begin{align*} f(x)=\frac{1}{2N+1}\underset{n=-N}{\overset{N}{\sum}}1_{\Omega_{\mathrm{c}}^{\epsilon}}(T_{a}^{n}x) \end{align*} $$

is constant on S. Define

$$ \begin{align*} X^{\prime}=\{x\in T_{a}^{-N}( \Omega_{\mathrm{nc}}^{\epsilon}):\:f(x)\leq\kappa\}. \end{align*} $$

Given $\mu $ ,

$$ \begin{align*} \mu(\Omega_{\mathrm{c}}^{\epsilon})&=\frac{1}{2N+1}\underset{n=-N}{\overset{N}{\sum}}\mu(T_{a}^{n}( \Omega_{\mathrm{c}}^{\epsilon}))=\int fd\mu=\int_{f(x)>\kappa}fd\mu+\int_{f(x)\leq\kappa}fd\mu\\&\geq\kappa\cdot\mu(\{x\in\mathcal{F}X:\:f(x)>\kappa\})+0. \end{align*} $$

Set $A=T_{a}^{-N}(\Omega _{\mathrm {nc}}^{\epsilon })$ and $B=\{x\in \mathcal {F}X:\:f(x)\leq \kappa \}$ (and so $X^{\prime }=A\cap B)$ . Then, as $\mu $ is a $T_a$ -invariant probability measure supported on $\Omega _{\mathcal {F}}$ ,

$$ \begin{align*} \mu(X^{\prime})\geq1-\mu(A^{c})-\mu(B^{c})\geq1-(1+\kappa^{-1})\mu(\Omega_{\mathrm{c}}^{\epsilon})\geq1-2\kappa^{-1}\mu(\Omega_{\mathrm{c}}^{\epsilon}). \end{align*} $$

Clearly $X^{\prime }$ satisfies conditions (1) and (2) of the lemma, as it is precisely the union of elements $S={{\bigcap }_{n=-N}^{N}}T_{a}^{n}Q_{n}$ such that $Q_{-N}\not =\Omega _{\mathrm {c}}^{\epsilon }$ , and $Q_{n}=\Omega _{\mathrm {c}}^{\epsilon }$ for up to $\kappa (2N+1)$ different $-N\leq n\leq N$ .

It remains only to cover each such element by $3^{(\mathtt {d}-1)\kappa (2N+1)}$ Bowen $(N,\eta )$ -balls. For simplicity, for such S, define

$$ \begin{align*} S^{\prime}=T_{a}^{N}S=\underset{n=-N}{\overset{N}{\bigcap}}T_{a}^{N+n}Q_{n}=\underset{n=0}{\overset{2N}{\bigcap}}T_{a}^{n}Q_{n-N} \end{align*} $$

and

$$ \begin{align*} W=\{0\leq n\leq 2N:\:Q_{n-N}=\Omega_{\mathrm{c}}^{\epsilon}\}. \end{align*} $$

We will cover $S^{\prime }$ by $3^{(\mathtt {d}-1)\kappa (2N+1)}$ backward Bowen $(2N,\rho )$ -balls. By taking images of these balls by $T_{a}^{-N}$ , we will get a cover of S by the same amount of (two-sided) Bowen $(N,\rho )$ -balls. As $\rho <\eta $ , each such Bowen $(N,\rho )$ -ball is contained in a Bowen $(N,\eta )$ -ball, which will give the desired result.

We prove the following claim by induction: for all $0\leq n\leq 2N$ , the set $S^{\prime }$ can be covered by $3^{(\mathtt {d}-1)|\{0,\ldots ,n\}\cap W|}$ many backward Bowen $(n,\rho )$ -balls.

First, for $n=0$ , as

$$ \begin{align*} S^{\prime}\subset Q_{-N}\in\{P_{1},\ldots,P_{k}\} \end{align*} $$

and $P_{j}\subset \Gamma k_{j}B_{\rho }^{G}$ , the result is clear for $3^{0}$ sets, namely $P_{j}$ .

Assume we covered $S^{\prime }$ by $3^{(\mathtt {d}-1)|\{0,\ldots ,n\}\cap W|}$ many sets for some $n\in [0,2N-1]$ . Let $\Gamma kB_{n,\rho }^{-}$ be one of those. If $n+1\not \in W$ , then $S^{\prime }\subset T_{a}^{n+1}P_{j}$ for some j. Therefore,

$$ \begin{align*}S^{\prime}\cap\Gamma kB_{n,\rho}^{-}\subset\Gamma kB_{n,\rho}^{-}\cap\Gamma k_{j}B_{\rho}^{G}a^{n+1}.\end{align*} $$

Because $S^\prime \subset \Omega _{\mathrm {nc}}^{\epsilon }$ , and $\rho $ was chosen to be small enough with respect to the injectivity radius, it follows that, in fact,

$$ \begin{align*}S^\prime\cap\Gamma kB_{n,\rho}^{-}\subset \Gamma k^{\prime}B_{n+1,\rho}^{-}\end{align*} $$

for some $k^\prime \in G$ . Therefore, by replacing $\Gamma kB_{n,\rho }^{-}$ with $\Gamma k^{\prime }B_{n+1,\rho }^{-}$ , for each set $\Gamma kB_{n,\rho }^{-}$ in the given cover, we obtain a new cover of $S^{\prime }$ by backward Bowen $(n+1,\rho )$ -balls, again of size $3^{(\mathtt {d}-1)|\{0,\ldots ,n\}\cap W|}=3^{(\mathtt {d}-1)|\{0,\ldots ,n+1\}\cap W|}$ .

Otherwise, $n+1\in W$ . In this case, we simply split each backward Bowen $(n,\rho )$ -ball into $3^{\mathtt {d}-1}$ backward Bowen $(n+1,\rho )$ -balls, each of them smaller in the $N^{-}$ directions by a factor of $e<3$ . We emerge with

$$ \begin{align*}3^{(\mathtt{d}-1)}\cdot3^{(\mathtt{d}-1)|\{0,\ldots,n\}\cap W|}=3^{(\mathtt{d}-1)|\{0,\ldots,n+1\}\cap W|}\end{align*} $$

many sets of the right form. This proves the claim.

To conclude, in the last step of the induction, as $|W|\leq \kappa (2N+1)$ (because $S\subset X^{\prime }$ ), we get up to $3^{(\mathtt {d}-1)\kappa (2N+1)}$ sets as desired.

To finish the proof, one only needs to treat the case where $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon })>0$ . In this case, choose $\epsilon ^{\prime }>\epsilon $ such that $\mu (\Omega _{\mathrm {c}}^{\epsilon ^\prime })<\tfrac 43\mu (\Omega _{\mathrm {c}}^{\epsilon })$ and $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon ^{\prime }})=0$ . This is possible because, as we mentioned, there are only countably many cusp regions whose boundary is of positive measure. Then the partition $X^{\prime }$ that was constructed for $\epsilon ^{\prime }$ works for $\epsilon $ as well.

Proof of Theorem 1.1

Let $\mu $ be a weak- $\star $ limit of a subsequence of $(\mu _{n})_{n\in \mathbb {N}}$ . Without loss of generality, by passing to a subsequence, assume $(\mu _n)_{n\in \mathbb {N}}$ converges to $\mu $ . Fix some $0<\epsilon <\epsilon _{\mathtt {d}}$ , small enough with respect to all previous lemmas, and for which $\kappa =\mu (\Omega _{\mathrm {c}}^\epsilon )^{1/2}<\tfrac 12$ and $\mu (\partial \Omega _{\mathrm {c}}^{\epsilon })=0$ . Let $\mathscr {P}$ be the partition from Lemma 3.17, whose elements all have $\mu $ -null boundaries.

Let $\epsilon _{0}>0$ be given, small enough so that the assumptions of Theorem 1.1 hold. Let $m\in \mathbb {N}$ be such that $H_{i}=\lambda _{i}^{\epsilon _{0}}<\epsilon $ for all $i\geq m$ , and fix some $i\geq m$ . Set $N_{i}=\lfloor -\tfrac 12\log \lambda _{i}\rfloor $ . Define $X_{i}$ to be the set guaranteed in Lemma 3.17, with respect to $N_{i}$ and $\kappa $ . Here we assumed $\kappa =\mu (\Omega _{\mathrm {c}}^{\epsilon })^{1/2}>0$ . Otherwise, this is essentially the convex cocompact case, and we may set instead any $0<\kappa <\tfrac 12$ to obtain, by Lemma 3.17, a set $X_i$ of full $\mu _i$ -measure, and continue the same way.

For all $P=S_{j}\in \mathscr {P}_{N_{i}}$ in the union ${{\bigcup }_{j=1}^{l}}S_{j}$ defining $X_{i}$ , there is a covering $P\subset {{\bigcup }_{k=1}^{3^{(\mathtt {d}-1)\kappa (2N_{i}+1)}}}A_{k}$ by Bowen $(N_{i},\eta )$ -balls. Define

$$ \begin{align*} \tilde{P}_{n}=(A_{n}\cap P)\smallsetminus\underset{k=1}{\overset{n-1}{\bigcup}}A_{k} \end{align*} $$

for all $n\geq 1$ . Then $\{\tilde {P}_{n}\}_{n=1}^{3^{(\mathtt {d}-1)\kappa (2N_{i}+1)}}$ is a collection of mutually disjoint sets, each contained in a Bowen $(N_{i},\eta )$ -ball, whose union is P. So $\{\tilde {P}_{n}\}_{n=1}^{3^{(\mathtt {d}-1)\kappa (2N_{i}+1)}}$ is a measurable partition of $P=S_{j}$ .

Define

$$ \begin{align*} \mathcal{Q}_{i}=\underset{P\in\mathscr{P}_{N_{i}}:\:P\cap X_{i}=\emptyset}{\bigcup}\{P\}\cup\underset{P\in\mathscr{P}_{N_{i}}:\:P\cap X_{i}\not=\emptyset}{\bigcup}\{\tilde{P}_{n}\}_{n=1}^{3^{(\mathtt{d}-1)\kappa(2N_{i}+1)}}. \end{align*} $$

Clearly, $Q_{i}$ is a measurable partition of X.

Take $A\in \mathcal {Q}_{i},\:B\in \mathscr {P}_{N_{i}}$ .

1. (1) Assume $A=P$ for $P\in \mathscr {P}_{N_{i}}$ such that $P\cap X_{i}=\emptyset $ . Then $A,B\in \mathscr {P}_{N_{i}}$ and so $A\cap B=\emptyset $ or $A=B$ , as the elements of $\mathscr {P}_{N_{i}}$ are mutually disjoint.

2. (2) Otherwise, $A=\tilde {P}_{n}$ for $P\in \mathscr {P}_{N_{i}}$ such that $P\subset X_{i}$ (that is, P is one of the sets defining $X_{i}$ ). Then $A\cap B=\emptyset $ unless $B=P$ , in which case $A\cap B=A$ .

In particular, for a given $B\in \mathscr {P}_{N_{i}}$ , we have

$$ \begin{align*} \log((\mu_{i})_{B}(A))\cdot(\mu_{i})_{B}(A)\not=0 \end{align*} $$

only for at most $3^{(\mathtt {d}-1)\kappa (2N_{i}+1)}$ elements $A\in \mathcal {Q}_{i}$ , where $(\mu _i)_B$ is the restriction of $\mu _i$ to B, normalized to be a probability measure. Therefore, the entropy satisfies

$$ \begin{align*} H_{(\mu_{i})_{B}}(\mathcal{Q}_{i})\leq\log3^{(\mathtt{d}-1)\kappa(2N_{i}+1)}=\log3^{\mathtt{d}-1}\cdot\kappa(2N_{i}+1). \end{align*} $$

We get that the conditional entropy satisfies

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i}|\mathscr{P}_{N_{i}})=\underset{B\in\mathscr{P}_{N_{i}}}{\sum}\mu_{i}(B)\cdot H_{(\mu_{i})_{B}}(\mathcal{Q}_{i})\leq\log3^{\mathtt{d}-1}\cdot\kappa(2N_{i}+1). \end{align*} $$

As $\mathcal {Q}_{i}$ is finer than $\mathscr {P}_{N_{i}}$ , we get $\mathcal {Q}_{i}=\mathcal {Q}_{i}\vee \mathscr {P}_{N_{i}}$ , so

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i})=H_{\mu_{i}}(\mathscr{P}_{N_{i}})+H_{\mu_{i}}(\mathcal{Q}_{i}|\mathscr{P}_{N_{i}}), \end{align*} $$

and so

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i})\leq H_{\mu_{i}}(\mathscr{P}_{N_{i}})+\log3^{\mathtt{d}-1}\cdot\kappa(2N_{i}+1). \end{align*} $$

Therefore, to get a bound on $H_{\mu _{i}}(\mathscr {P}_{N_{i}})$ , we are interested in bounding $H_{\mu _{i}}(\mathcal {Q}_{i})$ . Clearly,

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i})\geq H_{\mu_{i}}(\mathcal{Q}_{i}|\{X_{i},X_{i}^{c}\})\geq\mu_{i}(X_{i})\cdot H_{(\mu_{i})_{X_{i}}}(\mathcal{Q}_{i}). \end{align*} $$

Using the fact that entropy is controlled by an $L^{2}$ -norm, and the already mentioned fact that for $A\in \mathcal {Q}_{i}$ , either $A\subset X_{i}$ or $A\cap X_{i}=\emptyset $ , we obtain

$$ \begin{align*} H_{(\mu_{i})_{X_{i}}}(\mathcal{Q}_{i})\geq-\log\underset{A\in\mathcal{Q}_{i}}{\sum}((\mu_{i})_{X_{i}}(A))^{2}=-\log\underset{A\in\mathcal{Q}_{i},\:A\subset X_{i}}{\sum}\frac{\mu_{i}(A)^{2}}{\mu_{i}(X_{i})^{2}}. \end{align*} $$

Let us estimate this sum. Every $A\in \mathcal {Q}_{i}$ with $A\subset X_{i}$ is contained in a Bowen $(N_{i},\eta )$ -ball. In addition, $\{A\in \mathcal {Q}_{i}:\:A\subset X_{i}\}$ is a mutually disjoint family and

$$ \begin{align*}X_{i}\subset T_{a}^{-N_i}(\Omega_{\mathrm{nc}}^{\epsilon})\subset T_{a}^{-N_i}(\mathcal{F}\operatorname{core}_{H_{i}}(X)).\end{align*} $$

Therefore, by the assumptions of Theorem 1.1 (as explained in the proof of Lemma 3.16),

$$ \begin{align*} \underset{A\in\mathcal{Q}_{i},\:A\subset X_{i}}{\sum}\mu_{i}(A)^{2}&=\mu_{i}\times\mu_{i}\bigg(\underset{A\in\mathcal{Q}_{i},\:A\subset X_{i}}{\bigcup}T_{a}^{N_i}(A)\times T_{a}^{N_i}(A)\bigg)\\& \leq E(\epsilon_{0})\lambda_{i}^{\delta-\alpha\epsilon_{0}}\leq E(\epsilon_{0})e^{-2(\delta-\alpha\epsilon_{0})N_{i}}, \end{align*} $$

where $E(\epsilon _0)$ is the implicit constant in assumption (2) of Theorem 1.1. Therefore,

$$ \begin{align*} H_{(\mu_{i})_{X_{i}}}(\mathcal{Q}_{i})\geq-\log\frac{E(\epsilon_{0})e^{-2(\delta-\alpha\epsilon_{0})N_{i}}}{\mu_{i}(X_{i})^{2}}=2\log(\mu_{i}(X_{i}))-\log E(\epsilon_{0})+2(\delta-\alpha\epsilon_{0})N_{i}, \end{align*} $$

and so,

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i})\geq2\mu_{i}(X_{i})\log(\mu_{i}(X_{i}))-\mu_{i}(X_{i})\log E(\epsilon_{0})+2\mu_{i}(X_{i})(\delta-\alpha\epsilon_{0})N_{i}. \end{align*} $$

Note that the first two terms are bounded as i approaches $\infty $ . Yet, the third term approaches $\infty $ because

$$ \begin{align*}\underset{i\to\infty}{\liminf}(\mu_i(X_i))\geq\underset{i\to\infty}{\liminf}(1-2\kappa^{-1}\mu_i(\Omega_{c}^{\epsilon}))=1-2\kappa^{-1}\mu(\Omega_{c}^{\epsilon})\geq1-2\kappa>0\end{align*} $$

and of course $N_i\to \infty $ . The first equality arised from $\mu $ being a probability measure by Lemma 3.16 and $\Omega _{\mathrm {c}}^{\epsilon }$ being a continuity set.

Therefore, for all large enough i,

$$ \begin{align*} H_{\mu_{i}}(\mathcal{Q}_{i})&\geq2\mu_{i}(X_{i})(\delta-\alpha\epsilon_{0})N_{i}-\alpha\epsilon_{0}\cdot\mu_{i}(X_{i})N_{i}=\mu_{i}(X_{i})(2\delta-3\alpha\epsilon_{0})N_{i}\\&\geq(1-2\kappa^{-1}\mu_{i}(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})\cdot N_{i}. \end{align*} $$

Altogether, we get

$$ \begin{align*} (1-2\kappa^{-1}\mu_{i}(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})\cdot N_{i}\leq H_{\mu_{i}}(\mathcal{Q}_{i})\leq H_{\mu_{i}}(\mathscr{P}_{N_{i}})+\log3^{\mathtt{d}-1}\cdot\kappa(2N_{i}+1). \end{align*} $$

Fix some $N_{0}\in \mathbb {N}$ . By subadditivity of entropy, and $T_{a}$ -invariance of $\mu _{i}$ , we get for all $i>N_{0}$ (set $k_{i}=\lceil {N_{i}}/{N_{0}}\rceil $ ):

$$ \begin{align*} H_{\mu_{i}}(\mathscr{P}_{N_{i}}) &\leq H_{\mu_{i}}(\mathscr{P}_{k_{i}N_{0}}) = H_{\mu_{i}}\bigg(\underset{j=1}{\overset{k_{i}}{\bigvee}}\underset{n=(2(j-1)-k_{i})N_{0}}{\overset{(2j-k_{i})N_{0}}{\bigvee}} T_{a}^{-n}\mathscr{P}\bigg)\\ &\leq\underset{j=1}{\overset{k_{i}}{\sum}}H_{\mu_{i}}\bigg(\underset{n=(2(j-1)-k_{i})N_{0}}{\overset{(2j-k_{i})N_{0}}{\bigvee}}T_{a}^{-n}\mathscr{P}\bigg)\\& =\underset{j=1}{\overset{k_{i}}{\sum}}H_{\mu_{i}}\bigg(T_{a}^{(2j-1-k_{i})N_{0}}\underset{n=(2(j-1)-k_{i})N_{0}}{\overset{(2j-k_{i})N_{0}}{\bigvee}}T_{a}^{-n}\mathscr{P}\bigg)\\& =\underset{j=1}{\overset{k_{i}}{\sum}}H_{\mu_{i}}\bigg(\underset{n=-N_{0}}{\overset{N_{0}}{\bigvee}}T_{a}^{-n}\mathscr{P}\bigg)=k_{i}H_{\mu_{i}}(\mathscr{P}_{N_{0}}), \end{align*} $$

and so

$$ \begin{align*} (1-2\kappa^{-1}\mu_{i}(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})\cdot N_{i}\leq\bigg\lceil\frac{N_{i}}{N_{0}}\bigg\rceil H_{\mu_{i}}(\mathscr{P}_{N_{0}})+\log3^{\mathtt{d}-1}\cdot\kappa(2N_{i}+1), \end{align*} $$

that is,

(3.5) $$ \begin{align} \begin{aligned} H_{\mu_{i}}(\mathscr{P}_{N_{0}}) &\geq((1-2\kappa^{-1}\mu_{i}(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})-2 \log3^{\mathtt{d}-1}\cdot\kappa)\cdot\frac{N_{i}}{\lceil{N_{i}}/{N_{0}}\rceil} \\ &\quad - \log3^{\mathtt{d}-1}\cdot\kappa\cdot\frac{1}{\lceil{N_{i}}/{N_{0}}\rceil}. \end{aligned} \end{align} $$

Note that ${\lim _{i\to \infty }}\mu _{i}(\Omega _{\mathrm {c}}^{\epsilon })=\mu (\Omega _{\mathrm {c}}^{\epsilon })$ . Moreover, ${N_{i}}/{\lceil {N_{i}}/{N_{0}}\rceil }\to N_{0}$ and ${1}/{\lceil {N_{i}}/{N_{0}}\rceil }\to 0$ as $i\to \infty $ . Therefore, the right-hand side of equation (3.5) approaches

$$ \begin{align*} ((1-2\kappa^{-1}\mu(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})-2\log3^{\mathtt{d}-1}\cdot\kappa)\cdot N_{0}. \end{align*} $$

As for the left-hand side of equation (3.5), we claim that it converges to $H_{\mu }(\mathscr {P}_{N_{0}})$ . Recall that per our choice, $\mu (\partial A)=0$ for all $A\in \mathscr {P}$ . As $T_{a}$ is a measure-preserving homeomorphism, we get $\mu (\partial (T_{a}^{k}(A)))=0$ for all $A\in \mathscr {P}$ and $|k|\leq N_{0}$ . In general, $\partial ({\bigcap _{i\in I}}A_{i})\subset {\bigcup _{i\in I}}\partial A_{i}$ for any finite set of indices I. As every set $A\in \mathscr {P}_{N_{0}}$ is of the form $A={{\bigcap }_{i=-N_{0}}^{N_{0}}}T_{a}^{k}(A)$ , we obtain $\mu (\partial A)=0$ for all $A\in \mathscr {P}_{N_{0}}$ . Recall that by definition,

$$ \begin{align*} H_{\mu_{i}}(\mathscr{P}_{N_{0}})=-\underset{A\in\mathscr{P}_{N_{0}}}{\sum}\mu_{i}(A)\log(\mu_{i}(A)). \end{align*} $$

As it is just a finite sum of elements, each converging to $\mu (A)\log (\mu (A))$ (by the Portmanteau theorem), we obtain

$$ \begin{align*} \underset{i\to\infty}{\lim}H_{\mu_{i}}(\mathscr{P}_{N_{0}})=H_{\mu}(\mathscr{P}_{N_{0}}) \end{align*} $$

as desired.

Therefore, by taking the limit of equation (3.5) as $i\to \infty $ , we get

$$ \begin{align*} H_{\mu}(\mathscr{P}_{N_{0}})\geq((1-2\kappa^{-1}\mu(\Omega_{\mathrm{c}}^{\epsilon}))\cdot(2\delta-3\alpha\epsilon_{0})-2\log3^{\mathtt{d}-1}\cdot\kappa)N_{0}. \end{align*} $$

Therefore,

$$ \begin{align*} h_{\mu}(T)&=\underset{\mathscr{Q}}{\sup}\underset{n\to\infty}{\lim}\frac{{{\bigvee}_{i=-n}^{n}}T^{-n}\mathscr{Q}}{2n+1}\geq\underset{N_{0}\to\infty}{\lim}\frac{H_{\mu}(\mathscr{P}_{N_{0}})}{2N_{0}+1}\\&\geq\underset{N_{0}\to\infty}{\lim}((1-2\mu(\Omega_{\mathrm{c}}^{\epsilon})^{1/2})\cdot(2\delta-3\alpha\epsilon_{0})-2\log3^{\mathtt{d}-1}\cdot\mu(\Omega_{\mathrm{c}}^{\epsilon})^{1/2})\cdot\frac{N_{0}}{2N_{0}+1}\\&=((1-2\mu(\Omega_{\mathrm{c}}^{\epsilon})^{1/2})\cdot(2\delta-3\alpha\epsilon_{0})-2\log3^{\mathtt{d}-1}\cdot\mu(\Omega_{\mathrm{c}}^{\epsilon})^{1/2})\cdot\tfrac{1}{2}. \end{align*} $$

As ${\lim _{\epsilon \to 0^{+}}}\mu (\Omega _{\mathrm {c}}^{\epsilon })=0$ and as the calculation holds for all small enough $\epsilon ,\epsilon _{0}>0$ (apart for countably many $\epsilon $ ), we can take the limit as $\epsilon ,\epsilon _{0}\to 0^{+}$ and get $h_{\mu }(T)\geq \delta $ .

Proof Let $\mu $ be an A-invariant probability measure on $\Gamma \backslash G$ . First, note that $h_{\mu }(a_1)=h_{(\pi _M)_{\ast }\mu }(a_1)$ , where $(\pi _M)_{\ast }\mu $ is the pushforward of $\mu $ to $\Gamma \backslash G/M$ , because compact (isometric) extensions do not increase entropy. In particular, the maximal entropy of the frame flow is $\delta $ , which is realized by $m_{\operatorname {BM}}^{\mathcal {F}}$ .

Next, assume that $\mu $ is of maximal entropy. Then its pushforward measure is of entropy $\delta $ . The uniqueness of the measure of entropy $\delta $ on $\Gamma \backslash G/M$ is known in our setup (Theorem 2.16), and so $\mu $ is some lift of $m_{\operatorname {BM}}$ .

For all $m\in M$ , define a measure $\mu _{m}$ on $\Gamma \backslash G$ by $\mu _{m}(Y)=\mu (Ym)$ . These measures are all A-invariant probability measures. It is clear that by averaging these measures with respect to the Haar measure $\lambda $ on M, we obtain a lift of $m_{\operatorname {BM}}$ which is M-invariant from the right. This must be the lift using the Haar measure $\lambda $ , by uniqueness of the Haar measure. In other words, this lift is the Bowen–Margulis measure on $\Gamma \backslash G$ . So we obtain

(3.6) $$ \begin{align} m_{\operatorname{BM}}^{\mathcal{F}}=\underset{M}{\int}\mu_{m}d\lambda(m). \end{align} $$

By [Reference Winter39], $m_{\operatorname {BM}}^{\mathcal {F}}$ is mixing and in particular ergodic. Therefore, it is an extreme point in the space of invariant probability measures, and in particular the decomposition in equation (3.6) is trivial and so $\mu =m_{\operatorname {BM}}^{\mathcal {F}}$ .