2.1 Teichmüller compactification

Let ${\mathcal {MF}}(S)$ be the space of measured foliations on S and let ${\mathcal {PMF}}(S)$ be the space of projective measured foliations. The space ${\mathcal {MF}}(S)$ is a powerful tool for understanding $\operatorname {\mathrm {Teich}}(S)$ . In fact, to each $q \in QD(S)$ on a Riemann surface X, one can associate two measured foliations: the vertical measured foliation $\mathcal {V}(q)$ and the horizontal measured foliation $\mathcal {H}(q)$ . This assignment gives a homeomorphism from $QD(S)$ onto its image in ${\mathcal {MF}}(S) \times {\mathcal {MF}}(S)$ (cf. [Reference Hubbard and Masur12]). Fixing a base point, the compactification of $\operatorname {\mathrm {Teich}}(S)$ obtained by considering Teichmüller geodesic rays emanating from the base point is called the Teichmüller compactification of $\operatorname {\mathrm {Teich}}(S)$ and is denoted by $\overline {\operatorname {\mathrm {Teich}}(S)}^{T}$ . The above assignment then gives a homeomorphism between the boundary $\overline {\operatorname {\mathrm {Teich}}(S)}^{T} \setminus \operatorname {\mathrm {Teich}}(S)$ and ${\mathcal {PMF}}(S)$ .

The Teichmüller compactification is useful for many purposes. However, we require more than merely a compactification. The Teichmüller compactification depends on the choice of a base point X and, therefore, the action of $\operatorname {\mathrm {Mod}}(S)$ on $\overline {\operatorname {\mathrm {Teich}}(S)}^{T}$ is not continuous [Reference Kerckhoff15]. Nevertheless, this action remains well behaved in the following sense. Let $\mathcal {UE}(S) \subset {\mathcal {PMF}}(S)$ denote the set of projective classes of uniquely ergodic measured foliations on S. Since no ambiguity will arise, we write $\mathcal {UE}(S)$ simply as $\mathcal {UE}$ . Let $\operatorname {\mathrm {Teich}}(S) \bigcup _{T} \mathcal {UE}$ denote the image of $\operatorname {\mathrm {Teich}}(S) \cup \mathcal {UE}$ in $\overline {\operatorname {\mathrm {Teich}}(S)}^{T}$ , equipped with the induced topology. We then have the following theorem.

2.2 Gardiner–Masur compactification

For the study of counting problems for mapping class groups, another compactification of $\operatorname {\mathrm {Teich}}(S)$ , namely the Gardiner–Masur compactification, is often more suitable. We give a brief introduction to it and collect certain properties that will be used later. The reader is referred to [Reference Gardiner and Masur10, Reference Liu and Weixu20] for a detailed treatment.

Denote by $\mathcal {C}$ the set of (isotopy classes of) simple closed curves on S and $\mathrm {P}(\mathbb {R}_{+}^{\mathcal {C}})$ the projective space of $\mathbb {R}_{+}^{\mathcal {C}}$ , where $\mathbb {R}_{+}^{\mathcal {C}}$ is the set of maps from $\mathcal {C}$ to $\mathbb {R}_{+}$ . Denote the natural quotient by

$$ \begin{align*}\mathfrak{p}:\mathbb{R}_{+}^{\mathcal{C}} \to \mathrm{P}(\mathbb{R}_{+}^{\mathcal{C}}).\end{align*} $$

The map

$$ \begin{align*}\iota: X \mapsto (\operatorname{\mathrm{Ext}}_{X}(\alpha)^{{1}/{2}})_{\alpha \in \mathcal{C}},\end{align*} $$

where $\operatorname {\mathrm {Ext}}_X(\alpha )$ denotes the extremal length of $\alpha $ on X, defines an embedding of $\operatorname {\mathrm {Teich}}(S)$ into $\mathbb {R}_{+}^{\mathcal {C}}$ . Thus, the composition $\mathfrak {p} \circ \iota $ defines a map from $\operatorname {\mathrm {Teich}}(S)$ to $\mathrm {P}(\mathbb {R}_{+}^{\mathcal {C}})$ . Gardiner and Masur showed that the map $\mathfrak {p} \circ \iota $ is also an embedding and has a compact closure in $\mathrm {P}(\mathbb {R}_{+}^{\mathcal {C}})$ . We identify $\operatorname {\mathrm {Teich}}(S)$ with $\mathfrak {p} \circ \iota (\operatorname {\mathrm {Teich}}(S))$ . The closure ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ in $\mathrm {P}(\mathbb {R}_{+}^{\mathcal {C}})$ is called the Gardiner–Masur compactification. The boundary ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} = {\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} \setminus \operatorname {\mathrm {Teich}}(S)$ is called the Gardiner–Masur boundary of $\operatorname {\mathrm {Teich}}(S)$ .

The Gardiner–Masur compactification of $\operatorname {\mathrm {Teich}}(S)$ has the following properties.

1. (A1) The Gardiner–Masur compactification of $\operatorname {\mathrm {Teich}}(S)$ is homeomorphic to the horofunction compactification of $\operatorname {\mathrm {Teich}}(S)$ endowed with the Teichmüller metric [Reference Liu and Weixu20, Reference Walsh35]. The latter can be defined for any proper metric space and plays an important role in metric geometry.

2. (A2) The action of $\operatorname {\mathrm {Mod}}(S)$ on $\operatorname {\mathrm {Teich}}(S)$ extends continuously to an action on ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ [Reference Liu and Weixu20]. Therefore, this compactification is different from the Teichmüller compactification $\overline {\operatorname {\mathrm {Teich}}(S)}^{T}$ .

3. (A3) The topology of ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is complicated; in contrast, $\overline {\operatorname {\mathrm {Teich}}(S)}^{T}$ is a $(6g-6)$ -dimensional ball.

4. (A4) Points in the Gardiner–Masur boundary ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ can be described by homogeneous functions on ${\mathcal {MF}}(S)$ [Reference Miyachi28]. Notice that [Reference Gardiner and Masur10, Reference Miyachi29]

   $$ \begin{align*}\mathcal{UE} \varsubsetneqq {\mathcal{PMF}}(S) \varsubsetneqq {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}.\end{align*} $$

   The inclusion ${\mathcal {PMF}}(S) \varsubsetneqq {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is only set-theoretic, whereas the inclusion $\mathcal {UE} \varsubsetneqq {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is a topological embedding when $\mathcal {UE}$ is equipped with the topology induced from ${\mathcal {PMF}}(S)$ [Reference Miyachi29]. This also implies that the Gardiner–Masur compactification is distinct from the Thurston compactification.

5. (A5) The identity map on $\operatorname {\mathrm {Teich}}(S)$ extends to an $\operatorname {\mathrm {Mod}}(S)$ -equivariant homeomorphism [Reference Masur24, Reference Miyachi29]:

   $$ \begin{align*}\operatorname{\mathrm{Teich}}(S) \bigcup_{T} \mathcal{UE} \cong \operatorname{\mathrm{Teich}}(S) \bigcup_\mathrm{GM} \mathcal{UE}. \end{align*} $$

2.3 Teichmüller geodesics

Since Teichmüller geodesics play an important role in our work, we require more precise information about the homeomorphism $\tilde {\Psi }$ from $QD(S)$ to its image in ${\mathcal {MF}}(S) \times {\mathcal {MF}}(S)$ [Reference Lindenstrauss and Mirzakhani18]. The homeomorphism $\tilde {\Psi }$ is defined as follows. Let $\mathcal {C}$ be the set of isotopy classes of simple closed curves on S, $i: {\mathcal {MF}}(S) \times {\mathcal {MF}}(S) \to \mathbb {R}_{+}$ be the intersection function on ${\mathcal {MF}}(S)$ , and

$$ \begin{align*}\tilde{\Delta} = \{(\unicode{x3bb},\eta) \in ({\mathcal{MF}}(S))^2: \exists\gamma \in \mathcal{C}, \ i(\unicode{x3bb}, \gamma) = i(\eta, \gamma) = 0\}.\end{align*} $$

Then, $\tilde {\Psi }: QD(S) \to {\mathcal {MF}}(S) \times {\mathcal {MF}}(S) \setminus \tilde {\Delta }$ is given by setting $\tilde {\Psi }(q) = (\mathcal {H}(q),\mathcal {V}(q))$ . Let

$$ \begin{align*}\Delta = \{(\unicode{x3bb},[\eta]) \in {\mathcal{MF}}(S) \times {\mathcal{PMF}}(S): \exists\gamma \in \mathcal{C},\ i(\unicode{x3bb}, \gamma) = i(\eta, \gamma) = 0\},\end{align*} $$

then this map further gives a homeomorphism

(2.1) $$ \begin{align} \begin{aligned} \Psi: QD^{1}(S) &\to {\mathcal{MF}}(S) \times {\mathcal{PMF}}(S) \setminus \Delta, \\ q &\mapsto (\mathcal{H}(q), [\mathcal{V}(q)]). \end{aligned} \end{align} $$

The fact that the map $\Psi $ is a homeomorphism implies that, for $([\unicode{x3bb} ],[\eta ]) \in {\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S) \setminus \Delta $ , there is a unique oriented unparameterized Teichmüller geodesic with projective horizontal measured foliation $[\unicode{x3bb} ]$ and projective vertical measured foliation $[\eta ]$ . Since every oriented Teichmüller geodesic line determines a unique pair $([\xi ],[\eta ]) \in {\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S)$ , the map $\Psi $ yields a parameterization of the space $\mathfrak {G}$ of all Teichmüller geodesic lines in $\operatorname {\mathrm {Teich}}(S)$ by a subset of ${\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S)$ . This parameterization gives a topological structure, and hence, a Borel structure, on the set of geodesic lines.

The parameterization of $\mathfrak {G}$ given by $\Psi $ is useful for many purposes; however, it is not compatible with the Teichmüller compactification in the sense that geodesic rays may have many accumulation points on the boundary. Since our approach relies heavily on geodesics, the Gardiner–Masur compactification (and hence, by property (A1), the horofunction compactification) provides, in a certain sense, a more compatible (though not one-to-one) parameterization that suffices for our purposes.

It is convenient to use the horofunction compactification, which we now recall, to describe this new parameterization of $\mathfrak {G}$ . Fix a base point $o \in \operatorname {\mathrm {Teich}}(S)$ and consider the set $C(\operatorname {\mathrm {Teich}}(S),o)$ of all continuous functions on $\operatorname {\mathrm {Teich}}(S)$ that vanish at o, equipped with the topology of uniform convergence on compact subsets. For any point $x \in \operatorname {\mathrm {Teich}}(S)$ , denote by $d_x$ the function $y \mapsto d(x,y) -d(x,o)$ . Then, the assignment

$$ \begin{align*}x \mapsto d_x\end{align*} $$

gives an embedding of $\operatorname {\mathrm {Teich}}(S)$ into $C(\operatorname {\mathrm {Teich}}(S),o)$ with a relatively compact closure $\overline {\operatorname {\mathrm {Teich}}(S)}^{\mathrm {horo}}$ , which is called the horofunction compactification of $\operatorname {\mathrm {Teich}}(S)$ . The complement $\partial _{h}\overline {\operatorname {\mathrm {Teich}}(S)}$ of $\operatorname {\mathrm {Teich}}(S)$ in $\overline {\operatorname {\mathrm {Teich}}(S)}^{\mathrm {horo}}$ is called the horofunction boundary. As noted in property (A1), it is known (cf. [Reference Liu and Weixu20, Reference Walsh35]) that ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ can be identified with $\overline {\operatorname {\mathrm {Teich}}(S)}^{\mathrm {horo}}$ , under which ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ corresponds to $\partial _{h}\overline {\operatorname {\mathrm {Teich}}(S)}^{\mathrm {horo}}$ . Accordingly, we shall work exclusively with ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ and its points will be regarded as functions on $\operatorname {\mathrm {Teich}}(S)$ .

Given any geodesic ray $\boldsymbol {r} \subset \operatorname {\mathrm {Teich}}(S)$ , it is proved in [Reference Walsh35] that $\boldsymbol {r}$ converges in ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ to a unique point in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . In particular, if $\boldsymbol {r}$ is determined by $q \in QD^1(S)$ whose vertical measured foliation $\mathcal {V}(q)$ is uniquely ergodic, then its unique limit point is given by the image of $\mathcal {V}(q)$ in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ under the inclusion $\mathcal {UE} \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . Points in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ that are limit points of geodesic rays are called Busemann points. Thus, points in $\mathcal {UE} \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ are Busemann points. However, not every point in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is a Busemann point. In fact, there are non-Busemann points in ${\mathcal {PMF}}(S) \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ [Reference Miyachi28].

In the case of non-Busemann points, we employ optimal geodesics as asymptotic rays. Recall that, for a horofunction $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , a geodesic $\ell \in \mathfrak {G}$ is called an optimal geodesic for $\xi $ if, for all $t \in \mathbb {R}$ , one has $\xi (\ell (t))-\xi (\ell (0)) = -t$ . An optimal geodesic for $\xi $ passing through x may be regarded as the most efficient path from x to $\xi $ . One can check that if $\ell : t \mapsto \ell (t)$ is a parameterized optimal geodesic, then so is $\ell (t+t_0)$ for all $t_0 \in \mathbb {R}$ . It is proved in [Reference Azemar2, Proposition 3.4] that given $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ and $x \in \operatorname {\mathrm {Teich}}(S)$ , there exists a unique optimal geodesic for $\xi $ passing through x. Moreover, when $\xi $ is a Busemann point, this unique optimal geodesic is precisely the geodesic ray defining $\xi $ .

To give a slightly more complete picture, we need to extend geodesic rays to geodesic lines. It is shown in [Reference Miyachi30, Theorem 1] that the intersection number function i on ${\mathcal {MF}}(S)$ can be extended to the preimage of ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ in $\mathbb {R}_{+}^{\mathcal {C}}$ . Hence, for any two boundary points $\xi ,\eta \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , one can discuss whether $i(\xi ,\eta )$ vanishes or not. A pair $(\xi ,\eta ) \in ({\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}})^2 $ is called a filling pair if for all $\zeta \in {\mathcal {MF}}(S) \setminus \{0\}$ ,

$$ \begin{align*}i(\xi,\zeta) + i(\eta,\zeta)> 0.\end{align*} $$

A geodesic line $\ell $ is called optimal for $(\xi ,\eta )$ if $\ell (t)$ is optimal for $\xi $ and $\ell (-t)$ is optimal for $\eta $ .

Notice that if $\xi \in {\mathcal {PMF}}(S)$ is uniquely ergodic, then for any $\zeta \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} \setminus \{\xi \}$ , $(\xi ,\zeta )$ is a filling pair [Reference Miyachi30, Lemma 7].

If $y \in \operatorname {\mathrm {Teich}}(S)$ and $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , we denote by $\mathcal {L}(y,\xi )$ the unique optimal geodesic for $\xi $ passing through y. If $(\xi ,\eta ) \in ({\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}})^2$ is a filling pair, we denote the unique optimal geodesic for $(\xi ,\eta )$ by $\mathcal {L}(\xi ,\eta )$ . Denote by $\mathcal {N}$ the set of non-filling pairs. Then, there is a map

(2.2) $$ \begin{align} \begin{aligned} \Upsilon: ({\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}})^2 \setminus \mathcal{N} &\to \mathfrak{G},\\ (\xi,\eta) &\mapsto \mathcal{L}(\xi,\eta). \end{aligned} \end{align} $$

Moreover, the restriction

$$ \begin{align*}\Upsilon|_{\mathcal{UE} \times \mathcal{UE} \setminus \text{Diag}}\end{align*} $$

is the same as the parameterization given by (2.1). The following lemma shows that, compared with other compactifications, ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is more compatible with the topological structure of $\mathfrak {G}$ when the latter is equipped with the topology of uniform convergence on compact sets.

Convention. When $g = 2$ , $\operatorname {\mathrm {Mod}}(S_2)$ does not act faithfully on $\operatorname {\mathrm {Teich}}(S_2)$ . Instead, if we denote the center of $\operatorname {\mathrm {Mod}}(S_2)$ by Z, then the quotient group $\mathrm {PMod(S_2)}= \operatorname {\mathrm {Mod}}(S_2)/Z$ acts faithfully on $\operatorname {\mathrm {Teich}}(S_2)$ . Since we are concerned only with orbits of the action of $\operatorname {\mathrm {Mod}}(S_2)$ on $\operatorname {\mathrm {Teich}}(S_2)$ , we adopt the following convention: when we refer to a non-elementary subgroup of $\operatorname {\mathrm {Mod}}(S_2)$ , we mean a non-elementary subgroup of $\mathrm {PMod(S_2)}$ .

3.1 Conformal densities

Let G be a subgroup of $\operatorname {\mathrm {Mod}}(S)$ . The subgroup G is called non-elementary if it contains two independent pseudo-Anosov mapping classes, that is, two pseudo-Anosov mapping classes with disjoint fixed point sets in $\mathcal {UE} = \mathcal {UE}(S)$ , the subset of uniquely ergodic points in ${\mathcal {PMF}}(S)$ . Such a subgroup is also called sufficiently large in [Reference McCarthy and Papadopoulos25]. The limit set, denoted by $\Lambda G$ , is defined to be the closure in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ of the set of fixed points in $\mathcal {UE} \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ of pseudo-Anosov mapping classes contained in G. So, if G is non-elementary, then $|\Lambda G| = \infty $ . Throughout this paper, the subgroup G is always assumed to be non-elementary.

One can define a G-conformal density in a very general situation; see, for instance, [Reference Coulon7, Reference Yang36]. Here, we use only a simplified version, which is sufficient for our purposes. We first recall the Busemann cocycles with respect to the Teichmüller metric. Let $x, y \in \operatorname {\mathrm {Teich}}(S)$ and $\xi \in \mathcal {UE}$ . The Busemann cocycle $\beta _{\xi }(x,y)$ at $\xi $ is defined to be

$$ \begin{align*}\beta_{\xi}(x,y) = \lim_{z_n \to \xi}(d(x,z_n)-d(y,z_n)),\end{align*} $$

where $z_n \in \operatorname {\mathrm {Teich}}(S)$ and $z_n \to \xi $ in ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . By [Reference Miyachi28, Reference Walsh35], for $\xi \in \mathcal {UE}$ ,

(3.1) $$ \begin{align} \begin{aligned} \beta_{\xi}(x,y) = \frac{1}{2}\ln \frac{\operatorname{\mathrm{Ext}}_{x}(\xi)}{\operatorname{\mathrm{Ext}}_{y}(\xi)}. \end{aligned} \end{align} $$

Hence, $\beta _{\cdot }(x,y)$ is a continuous function on $\mathcal {UE}$ . We note that the Busemann cocycle $\beta _{\xi }(x,y)$ considered here is defined only for $\xi \in \mathcal {UE}$ .

We now recall some general results on conformal densities for subgroups of mapping class groups from [Reference Yang36]. Notice that ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is the horofunction compactification of $\operatorname {\mathrm {Teich}}(S)$ . Let $s \geq 0$ and $x,y \in \operatorname {\mathrm {Teich}}(S)$ . The Poincaré series associated to $G \curvearrowright \operatorname {\mathrm {Teich}}(S)$ is defined to be

$$ \begin{align*}P_G(s,x,y) = \sum_{g \in G}\exp{(-sd(gx,y))}.\end{align*} $$

Notice that since G is non-elementary, any conformal density has positive dimension.

3.2 The Patterson–Sullivan construction

The Patterson–Sullivan construction is a recipe for constructing conformal densities. We now briefly review the Patterson–Sullivan construction and the Bowen–Margulis measure in the setting of Teichmüller space. We refer the reader to [Reference Quint31] for an excellent exposition of the Patterson–Sullivan theory in the setting of negatively curved manifolds.

Let $\delta _G$ be the critical exponent of G with respect to the Teichmüller metric d. That is, let $o \in \operatorname {\mathrm {Teich}}(S)$ ,

$$ \begin{align*}\delta_G = \inf\bigg\{s: \sum_{g \in G}\exp(-sd(o,go)) < \infty\bigg\}.\end{align*} $$

Notice that $\delta _G$ is independent of the choice of o and $\delta _G> 0$ as G is non-elementary. Recall that G is said to be of divergence type if its Poincaré series $P_G(s,o,o)$ of G diverges at the critical exponent $s = \delta _G$ . Since G is non-elementary, it has contracting elements [Reference Minsky27]. According to [Reference Yang37], $\delta _G$ is the same as the growth rate of G with respect to the Teichmüller metric.

We review the Patterson–Sullivan construction by assuming that G is of divergence type. For subgroups of convergence type, one can modify the construction as explained in [Reference Quint31]. Let $x \in \operatorname {\mathrm {Teich}}(S)$ and $s> \delta _G$ . Denote the Dirac measure at $y \in \operatorname {\mathrm {Teich}}(S)$ by $\mathcal {D}_y$ . Consider the family of probability measures $\{\nu _{x,s}\}$ on $\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}$ given by

$$ \begin{align*} \nu_{x,s} = \frac{1}{P_G(s,x,x)}\sum_{g \in G}\exp(-sd(x,gx))\mathcal{D}_{gx}.\end{align*} $$

Since ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is compact, the sequence of measures $\{\nu _{x,s}: s> \delta _G\}$ weak- $^{*}$ converges to some probability measure $\nu _x$ on ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ as $s \to \delta _G$ . We now claim that the support of $\nu _x$ is a subset of ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . Indeed, since $P_G(s,x,x)$ diverges at $s = \delta _G$ , by assumption, and the action of G on $\operatorname {\mathrm {Teich}}(S)$ is proper, $\nu _{x}(K) = 0$ holds for any compact subset K of $\operatorname {\mathrm {Teich}}(S)$ . Hence, it is supported on a subset of ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . We now define $\nu _y$ for every $y \in \mathrm{Teich}(S)$ by

$$ \begin{align*} d\nu_y(\xi)= \exp(-\delta_G\beta_{\xi}(y,x))d\nu_x(\xi)\quad \text{for all}~\xi \in \mathcal{UE}.\end{align*} $$

Proof. Since G is non-elementary, G contains pseudo-Anosov mapping classes. By [Reference Minsky27], pseudo-Anosov mapping classes are contracting elements for the action of $\operatorname {\mathrm {Mod}}(S)$ on $\operatorname {\mathrm {Teich}}(S)$ . However, [Reference Yang36] shows that the Gardiner–Masur boundary, equipped with the finite difference relation, gives a non-trivial convergence boundary. One can then apply [Reference Yang36] to conclude that $\{\nu _x\}$ is a $\delta _G$ -conformal density for G. Note that $\{\nu _x\}$ is a conformal density rather than a quasi-conformal density.

By [Reference Yang36] and Theorem 3.2, it follows that $\nu _x$ is fully supported on $\Lambda G \cap \mathcal {UE}$ . To see that $\nu _x$ has no atoms, suppose that $\nu _x$ has an atom $\eta \in \Lambda G$ , say of mass r. By Theorem 3.2 again, $\eta $ must be a conical point, that is, there exists $D> 0$ such that the D-neighborhood of the geodesic $[x, \eta )$ intersects the orbit $Gx$ infinitely often. Let $\gamma _n \in G$ be such a sequence. Then, by the triangle inequality, $\beta _{\eta }(\gamma _n x,x) \to -\infty $ . So,

$$ \begin{align*}\nu_x(\gamma_n^{-1}\eta) = \nu_{\gamma_n x}(\eta) = \exp(-\delta_G\beta_{\eta}(\gamma_n x,x))\nu_x(\eta) \to \infty,\end{align*} $$

which contradicts the finiteness of $\nu _x$ .

Since each $\nu _x$ admits a unique completion, we henceforth assume that all such measures are complete. We now turn to the definition of the Bowen–Margulis measure on $QD^1(S)/G$ . For $x\in \operatorname {\mathrm {Teich}}(S)$ and $\xi , \eta \in \mathcal {UE}$ with $\xi \neq \eta $ , set

(3.2) $$ \begin{align} \begin{aligned} \rho_{x}(\xi,\eta) = \lim_{z \to [\xi], w \to [\eta]} (d(x,z) + d(x,w) - d(z,w)) &= \beta_{\xi}(x,u) + \beta_{\eta}(x,u), \end{aligned} \end{align} $$

where $u \in \operatorname {\mathrm {Teich}}(S)$ is any point in the geodesic $\mathcal {L}(\xi ,\eta )$ . Now, we define a measure $\hat {\mu }_{G}$ on ${\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S)$ by setting

$$ \begin{align*} d\hat{\mu}_{G}(\xi,\eta) = \exp{(\delta_G \rho_x(\xi,\eta))}d\nu_x(\xi)\,d\nu_x(\eta)\quad \text{for all}~\xi \neq \eta \in \mathcal{UE}.\end{align*} $$

Notice that since $\nu _x$ is supported on $\mathcal {UE}$ , $\hat {\mu }_G$ extends to ${\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S)$ . We claim that $\hat {\mu }_G$ is independent of x and it is G-invariant for the diagonal action of G on ${\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S)$ . In fact, this claim follows from the definition of the conformal density and the definition of $\rho _x$ .

The homeomorphism $\Psi $ defined in (2.1) provides a Hopf parameterization of the unit cotangent bundle $QD^1(S)$ of $\operatorname {\mathrm {Teich}}(S)$ as $({\mathcal {PMF}}(S) \times {\mathcal {PMF}}(S) \setminus \Delta ) \times \mathbb {R}$ , where

$$ \begin{align*}\Delta = \{([\unicode{x3bb}],[\eta]) \in {\mathcal{PMF}}(S) \times {\mathcal{PMF}}(S): \exists \gamma \in \mathcal{C}, \ i(\unicode{x3bb},\gamma)=i(\eta,\gamma)=0\}.\end{align*} $$

Consequently, each point in $QD^1(S)$ can be parameterized as $(\xi ,\eta ,s)$ . We define a measure $\tilde {\mu }_G$ on $QD^1(S)$ by setting

$$ \begin{align*}d\tilde{\mu}_G = dt\,d\hat{\mu}_G.\end{align*} $$

Notice that since $\nu _x$ is fully supported on $\mathcal {UE}$ , $\tilde {\mu }_G$ is fully supported on geodesic lines whose horizontal and vertical measured foliations are both uniquely ergodic. The measure $\mu _G$ on the quotient $QD^1(S)/G$ that descends from $\tilde {\mu }_G$ is called the Bowen–Margulis measure associated to G. Note that, a priori, the Bowen–Margulis measure $\mu _G$ need not be finite.

5.1 Shadows, cones, and corridors

For the reader’s convenience, we introduce the notation, following [Reference Link19, Reference Roblin33], that will be used in the following. Let $QD^1(S)$ be the space of unit-area quadratic differentials, which can be identified with the cotangent bundle of $\operatorname {\mathrm {Teich}}(S)$ , and let ${\mathcal {MF}}(S)$ be the space of measured foliations on S. We denote by $\mathcal {UE}$ the set of uniquely ergodic points in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . The natural projections involved here are given by

(5.1) $$ \begin{align} \begin{aligned} \pi: QD^1(S) &\to \operatorname{\mathrm{Teich}}(S),\\ \mathcal{V}: QD^1(S) &\to {\mathcal{MF}}(S), \quad q \mapsto \mathcal{V}(q),\\ \mathcal{H}: QD^1(S) &\to {\mathcal{MF}}(S), \quad q \mapsto \mathcal{H}(q),\\ [ \cdot ]: {\mathcal{MF}}(S) &\to {\mathcal{PMF}}(S), \quad \xi \mapsto [\xi]. \end{aligned} \end{align} $$

We use $\xi ,\eta ,\ldots $ to denote boundary points in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . When $\xi \in {\mathcal {PMF}}(S)$ , we sometimes also use $\xi $ to refer to one of its preimages under $[\cdot ]$ , when no confusion arises.

Set $\mathfrak {F} = ({\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}})^2 \setminus \Delta $ , where

$$ \begin{align*}\Delta & = \{([\unicode{x3bb}],[\eta]) \in ({\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}})^2: \exists~ \gamma \in {\mathcal{MF}}(S)\setminus \{0\}, i(\unicode{x3bb},\gamma) =i(\eta,\gamma)=0\}.\end{align*} $$

Note that every pair $(\zeta , \psi ) \in \mathfrak {F}$ determines a unique unparameterized Teichmüller geodesic $\mathcal {L}(\zeta ,\psi )$ , called the Teichmüller geodesic determined by the pair $(\zeta ,\psi )$ , by taking the optimal geodesic line for $(\zeta ,\psi )$ . When naturally oriented, it is referred to as the oriented geodesic from $\zeta $ to $\psi $ . Note that if $\zeta $ and $\psi $ belong to $\mathcal {UE}$ , then $\mathcal {L}(\zeta ,\psi )$ is exactly

$$ \begin{align*}\mathcal{L}(\zeta,\psi) = \{\pi(q): q \in QD^1(S), [\mathcal{H}(q)] = \zeta, [\mathcal{V}(q)] = \psi\}.\end{align*} $$

For any point $x \in \operatorname {\mathrm {Teich}}(S)$ , the unique unit holomorphic quadratic differential q on x with $([\mathcal {H}(q)],[\mathcal {V}(q)]) = (\xi ,\eta )$ will be denoted by $q(x;\xi ,\eta )$ .

For any subset $A \subset \operatorname {\mathrm {Teich}}(S)$ and $r> 0$ , the r-neighborhood $\mathcal {N}_r(A)$ of A is the subset consisting of points in $\operatorname {\mathrm {Teich}}(S)$ whose distance to A is at most r. If A is a single point, we denote by $B(x,r)$ the open ball of radius r centered at x.

5.1.1 Shadows

For $r> 0$ , define the (measurable) shadow of the ball $B(y,r)$ seen from x in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ by

$$ \begin{align*}\mathcal{O}_r(x,y) = \{\beta \in \mathcal{UE}: \mathcal{L}(x,\beta) \cap B(y,r) \neq \emptyset\},\end{align*} $$

where $\mathcal {L}(x,\beta )$ is the unique optimal geodesic for $\beta $ passing through x. Since $\beta \in \mathcal {UE}$ , the geodesic $\mathcal {L}(x,\beta )$ is the geodesic ray starting from x whose associated quadratic differential has projectivized vertical measured foliation $\beta $ . Although one could define the shadow as the set of boundary points satisfying the same condition, we adopt the above definition. Let $\zeta \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ and $y \in \operatorname {\mathrm {Teich}}(S)$ . Define

$$ \begin{align*} \mathcal{O}_{r}(\zeta,y) = \{\eta \in \mathcal{UE}: \mathcal{L}(\zeta,\eta) \cap B(y,r) \neq \emptyset\}.\end{align*} $$

For $r>0$ and $x,y \in \operatorname {\mathrm {Teich}}(S)$ with $d(x,y)> 2r$ , define

(5.2) $$ \begin{align} \begin{aligned} &\mathcal{O}_r^{+}(x,y) = \{\beta \in \mathcal{UE}: \exists~ z \in B(x,r) ~\text{such that}~ \mathcal{L}(z,\beta) \cap B(y,r) \neq \emptyset\},\\ &\mathcal{O}_r^{-}(x,y) = \{\beta \in \mathcal{UE}: \forall~ z \in B(x,r),~ \mathcal{L}(z,\beta) \cap B(y,r) \neq \emptyset\}. \end{aligned} \end{align} $$

The following lemma should be compared with [Reference Link19, Lemma 2.1].

Lemma 5.1. Let $\zeta ,\kappa \in \mathcal {UE}$ be two distinct projective measured foliations determining a Teichmüller geodesic and $o \in \operatorname {\mathrm {Teich}}(S)$ be any point. Let $r> 0$ , $\epsilon> 0$ , and the Teichmüller geodesic line determined by $\zeta ,\kappa $ be $\mathcal {L}(\zeta ,\kappa )$ . Suppose that $d(o,\mathcal {L}(\zeta ,\kappa )) = r$ . Then, there exists an open neighborhood $O_1 \times O_2 \subset (\mathcal {UE})^2$ of $(\zeta ,\kappa )$ such that

$$ \begin{align*} |d(o,\mathcal{L}(\mathcal{F}_1,\mathcal{F}_2))-r| \leq \epsilon\end{align*} $$

for every $(\mathcal {F}_1,\mathcal {F}_2) \in O_1 \times O_2 $ . Moreover, every pair $(a,b) \in O_1 \times O_2$ is a filling pair and, hence, determines a Teichmüller geodesic.

Proof. By [Reference Kaimanovich and Masur13, Lemma 1.4.3], the distance function on

$$ \begin{align*}{\mathcal{PMF}}(S) \times {\mathcal{PMF}}(S) \setminus \Delta\end{align*} $$

is continuous, which proves the first assertion of the lemma. Here, the set ${\mathcal {PMF}}(S)$ is equipped with its natural topology instead of the induced topology from ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . For the second assertion, if not, then there are measured foliations $\xi _n \to \zeta $ , $\eta _n \to \kappa $ , and simple closed curves $\gamma _n \in \mathcal {C}$ such that $i(\xi _n,\gamma _n)= i(\eta _n,\gamma _n)=0$ . We may assume that $\gamma _n \to \iota $ with $\iota \in {\mathcal {MF}}(S)$ . Then, $i(\iota ,\zeta ) = i(\iota ,\kappa ) = 0$ . However, $\iota $ cannot be a minimal foliation (that is, a foliation without simple closed leaves); otherwise, $\zeta $ is topologically equivalent to $\kappa $ [Reference Rees32] and, hence, has zero intersection with $\kappa $ , which contradicts the assumption. So, there is a simple closed curve $\gamma \in \mathcal {C}$ such that $i(\gamma ,\zeta ) = i(\gamma ,\kappa ) = 0$ , which also contradicts the assumption.

Corollary 5.2. Let $\mathcal {O}_r(\zeta ,y)$ be as defined above. Then, $\mathcal {O}_r(\zeta ,y)$ is open in $\mathcal {UE}$ and, hence, it is $\nu _x$ -measurable for all $x \in \operatorname {\mathrm {Teich}}(S)$ .

Proof. Since the measure $\nu _x$ is complete by assumption, $\mathcal {UE}$ is $\nu _x$ -measurable. So, if ${\zeta \in \mathcal {UE}}$ , the result follows from Lemma 5.1. Otherwise, we apply Lemma 2.4.

Remark 5.3.

1. (1) The set $\mathcal {O}_{r}(x,y)$ is open in $\mathcal {UE}$ and, hence, $\nu _x$ -measurable.

2. (2) In general, it might not be true that if $x_n$ converges to $\xi $ in ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , then $\lim _{n \to \infty } \mathcal {O}^{\pm }_{r}(x_n,y) = \mathcal {O}_r(\xi ,y)$ .

3. (3) For $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , we define $\mathcal {O}^{\pm }_r(\xi ,y) = \mathcal {O}_r(\xi ,y)$ .

We will also use the following modified boundary of $\mathcal {O}_r(\xi ,x)$ , as in [Reference Link19],

$$ \begin{align*}\tilde{\partial} \mathcal{O}_r(\xi,x) = \{\eta \in \mathcal{UE}\setminus \{\xi\}: d(x,\mathcal{L}(\xi,\eta)) = r\}.\end{align*} $$

Notice that by Corollary 5.2, $\tilde {\partial } \mathcal {O}_r(\xi ,x)$ is also $\nu _x$ -measurable.

5.1.2 Cones

Let $x \in \operatorname {\mathrm {Teich}}(S)$ and $U \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . We denote by $\operatorname {\mathrm {Sect}}_x(U)$ the set of all points y in $\operatorname {\mathrm {Teich}}(S)$ such that there exists $\xi \in U$ with $y \in \mathcal {L}(x,\xi )$ .

For $r> 0$ , following [Reference Link19], define the cones

$$ \begin{align*}C_r^{+}(x,U) = \{y \in \operatorname{\mathrm{Teich}}(S): \mathcal{O}_r^{+}(x,y) \cap U \neq \emptyset\}\end{align*} $$

and

$$ \begin{align*}C_r^{-}(x,U) = \{y \in \operatorname{\mathrm{Teich}}(S): \mathcal{O}_r^{+}(x,y) \subset U\}.\end{align*} $$

5.1.3 Certain useful Borel sets

The following sets will be useful. For $r> 0$ and ${x,y \in \operatorname {\mathrm {Teich}}(S)}$ with $d(x,y)> 6r$ , the corridor $L_r(x,y)$ is, intuitively, the set of pairs $(\xi ,\eta ) \in \mathcal {UE} \times \mathcal {UE}$ with $\xi \neq \eta $ such that the unique (oriented) Teichmüller geodesic $\mathcal {L}(\xi ,\eta )$ passes first through $B(x,r)$ and then through $B(y,r)$ .

To define $L_r(x,y)$ rigorously, we use the nearest point projection. Let $x \in \operatorname {\mathrm {Teich}}(S)$ and $\ell =\mathcal {L}(\xi ,\eta )$ be a Teichmüller geodesic line. Denote

$$ \begin{align*}\Pi_{\ell}(x) = \{z \in \ell: d(x,z) = d(x,\ell)\}.\end{align*} $$

Then, $\Pi _{\ell }$ is called the nearest point projection to $\ell $ . Notice that in [Reference Link19, Reference Roblin33], $\Pi _{\ell }(x)$ is always a single point because the distance function on ${\mathrm {CAT(0)}}$ -space is convex. However, this is no longer true since the Teichmüller distance is, in general, not convex (see [Reference Bourque and Rafi5]). Therefore, $\Pi _{\ell }(x)$ is a non-empty closed subset that may contain more than one point. Nevertheless, the diameter of $\Pi (x)$ is bounded by $2d(x,\ell )$ . Recall that each ordered pair $(\xi ,\eta ) \in \mathcal {UE} \times \mathcal {UE}$ defines not only a unique unparameterized geodesic line $\mathcal {L}(\xi ,\eta )$ , but also an orientation $\overrightarrow {\xi \eta }$ on it pointing to $\eta $ . For any two distinct points $p,q$ in $\mathcal {L}(\xi ,\eta )$ , we say p is on the left of q if the oriented geodesic line $\overrightarrow {pq}$ is positive with respect to the orientation. For $x \in \operatorname {\mathrm {Teich}}(S)$ and an oriented geodesic $\mathcal {L}(\xi ,\eta )$ , we define the leftmost nearest projection (to $\mathcal {L}(\xi ,\eta )$ ) $w(x;\xi ,\eta )$ to be the unique unit holomorphic quadratic differential on z such that $[\mathcal {H}(w(x;\xi ,\eta ))] = \xi , [\mathcal {V}(w(x;\xi ,\eta ))] = \eta $ , $z = \pi (w(x;\xi ,\eta )) \in \Pi _{\mathcal {L}(\xi ,\eta )}(x)$ , and z is leftmost in $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ . Note that we use the fact that $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ is a closed set, which ensures that the notion of the leftmost point is well defined.

The corridor $L_r(x,y)$ for $d(x,y)> 6r$ is then defined to be the subset of $\mathcal {UE} \times \mathcal {UE}$ consisting of $(\xi ,\eta )$ with $d(x,\mathcal {L}(\xi ,\eta )) < r, d(y,\mathcal {L}(\xi ,\eta )) < r$ and so that $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ is on the left of $\Pi _{\mathcal {L}(\xi ,\eta )}(y)$ with $md(\Pi _{\mathcal {L}(\xi ,\eta )}(x), \Pi _{\mathcal {L}(\xi ,\eta )}(y))> r$ , where $md(A,B)$ stands for the minimal distance (rather than the Hausdorff distance) between two sets $A,B$ . By Lemma 5.1, $L_r(x,y)$ is an open subset of $\mathcal {UE} \times \mathcal {UE}$ ; hence, $\nu _x \otimes \nu _x$ -measurable.

Let $g_t$ be the Teichmüller geodesic flow on $QD^1(S)$ . Let $x \in \operatorname {\mathrm {Teich}}(S)$ and $A, B \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . Set $\widehat {\mathfrak {F}} = \mathcal {UE} \times \mathcal {UE} \setminus \Delta $ and define

(5.3) $$ \begin{align} \begin{aligned} K_r(x) &= \bigg\{g_sw(z;\xi,\eta): (\xi,\eta) \in \widehat{\mathfrak{F}}, d(x,\mathcal{L}(\xi,\eta)) \leq r, s \in \bigg(\!\!-\frac{r}{2},\frac{r}{2}\bigg)\bigg\},\\ K_A^{+} &= K^{+}_r(x,A) = \{g_tw(z;\xi,\eta)\in K_r(x): \eta \in A \},\\ K_{B}^{-} &= K^{-}_r(y,B) = \{g_tw(z;\xi,\eta)\in K_r(y): \xi \in B \}. \end{aligned} \end{align} $$

It is straightforward to show the following lemma.

Lemma 5.4. If A is $\nu _x$ -measurable and B is $\nu _y$ -measurable, the sets $K_r(x)$ , $K_A^{+}$ , and $K^{-}_{B}$ defined above are $\tilde {\mu }_G$ -measurable.

Proof. This follows from the continuity of the distance function at boundary points in $\mathcal {UE}$ (cf. [Reference Kaimanovich and Masur13, Lemma 1.4.3]) and from our choice of the leftmost nearest projection.

Remark 5.5. Since $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ has bounded diameter depending on $d(x,\mathcal {L}(\xi ,\eta ))$ , the reader can check that whenever we use $w(x;\xi ,\eta )$ , the only relevant property is that $w(x,\xi ,\eta ) \in \Pi _{\mathcal {L}(\xi ,\eta )}(x)$ . Thus, one may define $w(x;\xi ,\eta )$ in another way. Namely, define

$$ \begin{align*}K^{\prime}_r(x) = \{g_sq(z;\xi,\eta): (\xi,\eta) \in \widehat{\mathfrak{F}}, d(x,\mathcal{L}(\xi,\eta)) \leq r, s \in \bigg(\!\!-\frac{r}{2},\frac{r}{2}\bigg), z \in \Pi_{\mathcal{L}(\xi,\eta)}(x)\}.\end{align*} $$

One can check that this set is measurable. Since for any $(\xi ,\eta ) \in \mathcal {UE} \times \mathcal {UE}$ , the set $K^{\prime }_{r}(x) \cap \mathcal {L}(\xi ,\eta )$ is a compact set and the projection map $\phi : K^{\prime }_{r}(x) \to \mathcal {UE} \times \mathcal {UE}$ is continuous, one can find a measurable section of $\phi $ . We define $w(x;\xi ,\eta )$ to be the value of the measurable section of $\phi $ at $(\xi ,\eta )$ and define $K_r(x)$ as before. In particular, $K_r(x)$ is measurable.

Remark 5.6. As pointed out in [Reference Bourque and Rafi5], the work of J. H. C. Whitehead implies that for each $x \in \operatorname {\mathrm {Teich}}(S)$ , one can choose $r_x$ , depending on x, such that for each geodesic $\mathcal {L}(\xi ,\eta )$ passing through $B(x,r_x)$ , $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ is, in fact, a connected segment or a point. However, this is more than we need, since in the following, we will only use the boundedness of $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ .

5.2 Properties of shadows and cones

We now list important properties of shadows and cones, following [Reference Link19]. We defer to the appendix those arguments that carry over to our setting with only minor modifications.

Concerning $\mathcal {O}_r(\xi ,x)$ and $\mathcal {O}^{\pm }_r(x,y)$ defined in §5.1, we have the following lemma.

Lemma 5.7. Let $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ and $o \in \operatorname {\mathrm {Teich}}(S)$ . Then, the set

$$ \begin{align*}\{ r> 0: \nu_o(\tilde{\partial} \mathcal{O}_r(\xi,o)) > 0\}\end{align*} $$

is at most countable.

Proof. We first prove part (b). Choose $\eta \in \mathcal {O}_r(\xi ,o) \subset \mathcal {UE}$ . Consider the unique geodesic $\mathcal {L}(\xi ,\eta )$ determined by $\xi ,\eta $ , and the leftmost nearest projection $w_{\xi ,\eta }$ of o to $\mathcal {L}(\xi ,\eta )$ . It follows that $d(o,w_{\xi ,\eta }) < r$ . Let $\epsilon> 0$ be such that $B(w_{\xi ,\eta },\epsilon ) \subset B(o,r)$ . By Lemma 2.4, there exists an open neighborhood U of $\xi $ in ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ such that, for any points $z \in U$ , there exists a geodesic $\mathcal {L}(z, \eta )$ with

$$ \begin{align*}d(w_{\xi,\eta}, \mathcal{L}(z,\eta)) < \epsilon.\end{align*} $$

Notice that z may lie either in $\operatorname {\mathrm {Teich}}(S)$ or in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . Now, choose N such that for all $n \geq N$ , we have $B(y_n,r) \in U$ when $y_n \in \operatorname {\mathrm {Teich}}(S)$ and $y_n \in U$ when $y_n \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ . It follows that $\eta \in \mathcal {O}_r^{-}(y_n,o) \subset \mathcal {O}_r^{+}(y_n,o) $ for $n \geq N$ .

We now prove part (a). It suffices to show that

$$ \begin{align*}\limsup_{n \to \infty}(\mathcal{O}^{+}_r(y_n,o)) \subset \mathcal{O}_r(\xi,o) \cup \tilde{\partial}\mathcal{O}_r(\xi,o).\end{align*} $$

Let $\iota \in \limsup _{n \to \infty }(\mathcal {O}^{+}_r(y_n,o))$ . Then, there exists an infinite subsequence, still denoted by $\{y_n\}$ , such that $\iota \in \mathcal {O}^{+}_r(y_n,o)$ for all n. Denote by $\ell = \mathcal {L}(\xi ,\iota )$ the geodesic determined by $\xi $ and $\iota $ . We may assume that $y_n \in \operatorname {\mathrm {Teich}}(S)$ . Indeed, if $y_n \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , then we can take a sequence $y^{\prime }_n \in \operatorname {\mathrm {Teich}}(S)$ converging to y. Now, consider oriented geodesics $\ell _n$ constructed as follows. For $y_n \in \operatorname {\mathrm {Teich}}(S)$ , let $\ell _n$ be the geodesic extending the oriented geodesic ray $-\mathcal {L}(y_n,\iota )$ . Observe that $d(o,\ell _n) < r$ . Furthermore, reparameterize $\ell _n$ so that $\beta _{\iota }(o,\ell (o))= \beta _{\iota }(o,\ell _n(0)) = 0$ . Let $s_n \in \mathbb {R}$ be such that $\ell _n(s_n)$ is the leftmost point in $\Pi _{\ell _n}(o)$ . Computations as in [Reference Link19, (26)] show that $s_n \in (-r,r)$ . On the one hand, $y_n \to \xi $ . On the other hand, passing to a subsequence if necessary, one may assume that $s_n \to s \in [-r,r]$ . By Lemma 2.4, we have $d(\ell _n(s_n),\ell (s_n)) \to 0$ as $n \to \infty $ . Therefore,

(5.4) $$ \begin{align} d(o,\ell(\mathbb{R}) &\leq d(o,\ell(s))\nonumber \\ &\leq \lim_{n \to \infty}(d(o,\ell_n(s_n))+d(\ell_n(s_n), \ell(s_n)) + d(\ell(s_n),\ell(s)))\nonumber\\ &\leq r. \end{align} $$

Lemma 5.9. Let $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , $o \in \operatorname {\mathrm {Teich}}(S)$ , and $r> 0$ be such that $\nu _o(\mathcal {O}_r(\xi ,o))> 0$ and $\nu _o(\tilde {\partial } \mathcal {O}_r(\xi ,o)) = 0$ . Then, for every $\epsilon> 0$ , there exists an open neighborhood $U_{\xi }$ of $\xi $ in ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ such that for every $a \in U_{\xi }$ ,

$$ \begin{align*} e^{-\epsilon}\nu_o(\mathcal{O}_r(\xi,o)) \leq \nu_{o}(\mathcal{O}_r^{\pm}(a,o)) \leq e^{\epsilon}\nu_o(\mathcal{O}_r(\xi,o)).\end{align*} $$

Concerning $C^{\pm }_r(x,U)$ defined in §5.1, it is straightforward to check the following lemma.

We record the following properties of $K_{A}^{+} = K^+_r(x,A), K^-_B = K^-_{r}(y,B)$ . Let

$$ \begin{align*}\Psi': QD^1(S) \to {\mathcal{PMF}}(S) \times {\mathcal{PMF}}(S), \quad q \mapsto ([\mathcal{H}(q)],[\mathcal{V}(q)])\end{align*} $$

be the map obtained by composing $\Psi $ in (2.1) with the natural projection ${\mathcal {MF}}(S) \to {\mathcal {PMF}}(S)$ in the first factor.

Lemma 5.12. For all $\gamma \in G$ such that $d(x, \gamma y) \geq 12r$ , we have

$$ \begin{align*} \Psi'\bigg(\bigcup_{t> 0}\{K_A^{+} \cap g_{-t}\gamma K_B^{-}\}\bigg) = L_r(x, \gamma y) \cap (\gamma B \times A).\end{align*} $$

We will also need estimates for the Bowen–Margulis measures of $K_A^{+}$ and $K_B^{-}$ . First, recall that (see (3.2)) for $\xi \neq \eta \in \mathcal {UE}, x\in \operatorname {\mathrm {Teich}}(S)$ , and $ u \in \mathcal {L}(\xi ,\eta )$ ,

$$ \begin{align*}\rho_x(\xi,\eta) = \beta_{\xi}(x,u) + \beta_{\eta}(x,u),\end{align*} $$

and (see (3.1))

$$ \begin{align*} \beta_{\xi}(x,y) = \dfrac{1}{2}\ln \frac{\operatorname{\mathrm{Ext}}_x(\xi)}{\operatorname{\mathrm{Ext}}_y(\xi)}.\end{align*} $$

By Kerckhoff’s formula [Reference Kerckhoff15],

$$ \begin{align*}\beta_{\xi}(x,y) \leq d(x,y).\end{align*} $$

Hence,

(5.6) $$ \begin{align} \begin{aligned} \eta \in \mathcal{O}_{r}(\xi,x) \Rightarrow \rho_x(\xi,\eta) \leq 2r,\\ (\xi,\eta) \in L_r(x,\gamma y) \Rightarrow \rho_x(\xi,\eta) \leq 2r. \end{aligned} \end{align} $$

By the definition of $\tilde {\mu }_G$ and the fact that $\nu _x$ is fully supported on $\mathcal {UE}$ , we have

(5.7) $$ \begin{align} \begin{aligned} \tilde{\mu}_G(K_A^{+}) &= \int_A\bigg(\int_{\mathcal{O}_r(\xi,x)}e^{\delta_G \rho_x(\xi,\eta)}d\nu_x(\eta)\bigg)\,d\nu_x(\xi)\int \mathbb{1}_{K_A^{+}}(g_{t}w(x;\xi,\eta))\,dt \\ &= r\int_A\bigg(\int_{\mathcal{O}_r(\xi,x)}e^{\delta_G \rho_x(\xi,\eta)}d\nu_x(\eta)\bigg)\,d\nu_x(\xi),\\ \tilde{\mu}_G(K_B^{-})&= \int_B\bigg(\int_{\mathcal{O}_r(\eta,y)}e^{\delta_G \rho_x(\xi,\eta)}\,d\nu_y(\eta)\bigg)\,d\nu_y(\xi)\int \mathbb{1}_{K_B^{-}}(g_{t}w(x;\xi,\eta))\,dt \\ &= r\int_B\bigg(\int_{\mathcal{O}_r(\eta,y)}e^{\delta_G \rho_x(\xi,\eta)}d\nu_y(\xi)\bigg)\,d\nu_y(\eta). \end{aligned} \end{align} $$

From (5.7) and (5.6), it follows that

(5.8) $$ \begin{align} \begin{aligned} r\int_A\nu_x(\mathcal{O}_r(\xi,x))\,d\nu_x(\xi) \leq \tilde{\mu}_G(K_A^{+}) \leq re^{2r\delta_G}\int_A\nu_x(\mathcal{O}_r(\xi,x))\,d\nu_x(\xi),\\ r\int_B\nu_y(\mathcal{O}_r(\eta,y))\,d\nu_y(\eta) \leq \tilde{\mu}_G(K_B^{-}) \leq re^{2r\delta_G}\int_B\nu_y(\mathcal{O}_r(\eta,y))\,d\nu_y(\eta). \end{aligned} \end{align} $$

For the following lemma, recall that for $x \in \operatorname {\mathrm {Teich}}(S)$ and a geodesic line $\mathcal {L}(\xi ,\eta )$ , we have defined the nearest point projection $\Pi _{\mathcal {L}(\xi ,\eta )}(x)$ and the leftmost nearest projection $w(x;\xi ,\eta ) \in QD^1(S)$ .

Proof. Consider $w(x;\xi ,\eta )$ and $w(\gamma y; \xi ,\eta )$ . There is $\tau> 0$ such that $w(\gamma y;\xi ,\eta ) = g_{\tau }w(x;\xi ,\eta )$ . Since $(\xi ,\eta ) \in L_r(x,\gamma y)$ , we have $|d(x,\gamma y) - \tau | < 2r$ . Note that

$$ \begin{align*}K_r(\gamma y)\cap \mathcal{L}(\xi,\eta) = \bigg\{g_sw(\gamma y;\xi,\eta): s \in \bigg(-\frac{r}{2},\frac{r}{2}\bigg) \bigg\},\end{align*} $$

and $g_{t+s}w(x;\xi ,\eta ) \in K_r(\gamma y)$ if and only if $t+s-\tau \in (-{r}/{2},{r}/{2})$ . Hence, if ${\tau -s-({r}/{2}) \geq T_0}$ and $ \tau -s + ({r}/{2}) \leq T + 3r$ , then

(5.12) $$ \begin{align} &\int^{T+3r}_{T_0}e^{\delta_G t}\mathbb{1}_{K_r(\gamma y;\xi,\eta)}(g_{t+s}w(x;\xi,\eta))\,dt \nonumber \\ &\quad\geq \int^{\tau -s + {r}/{2}}_{\tau-s-{r}/{2}} e^{\delta_G t}\,dt \nonumber\\ &\quad\geq re^{-3r\delta_G}e^{\delta_Gd(x,\gamma y)}. \end{align} $$

However, one checks that $d(x,\gamma y) \in (T_0+3r,T]$ implies that $\tau -s-({r}/{2}) \geq T_0$ and $ \tau -s + ({r}/{2}) \leq T + 3r$ , and hence, item $(1)$ holds. For item $(2)$ , if $g_{t+s}w(x;\xi ,\eta ) \in K_r(\gamma y)$ , then

$$ \begin{align*}t+s-\tau \leq \frac{r}{2}, \tau - (t+s) \leq \frac{r}{2}.\end{align*} $$

It follows that

(5.13) $$ \begin{align} &\int^{T-6r}_{T_0}e^{\delta_G t}\mathbb{1}_{K_r(\gamma y)}(g_{t+s}w(x;\xi,\eta))\,dt \nonumber\\ &\quad\leq \int^{\tau -s + {r}/{2}}_{\tau -s -{r}/{2}}e^{\delta_G t}\,dt \nonumber\\ &\quad\leq re^{\delta_G(\tau -s + ({r}/{2}))} \nonumber\\ &\quad\leq re^{3r\delta_G}e^{\delta_Gd(x,\gamma y)}. \end{align} $$

For item $(3)$ , it suffices to note that if $d(x,\gamma y) \leq T_0-3r$ , then $\tau -s+({r}/{2}) \leq T_0$ , and if $d(x,\gamma y) \geq T$ , then $\tau -s-3r \geq T-6r$ .

Finally, we record the following consequence of Lemma 5.12.

Lemma 5.14. Let $\gamma \in G$ and $d(x,\gamma y) \geq 12r$ . Let $r> 0$ , then

(5.14) $$ \begin{align} &\tilde{\mu}_{G}(K_A^{+} \cap g_{-t}\gamma K_B^{-}) \nonumber\\ &\quad =\int_{L_{r}(x,\gamma y) \cap (\gamma B \times A)}e^{\delta_G \rho_x(\xi,\eta)}\,d\nu_x(\xi)\,d\nu_y(\eta) \int^{{r}/{2}}_{-({r}/{2})}\mathbb{1}_{K_r(\gamma y)}(g_{t+s}w(x;\xi,\eta))\,ds. \end{align} $$

5.3 Subexponential growth of orbits in cones

In this section, we prove a technical counting lemma that will serve as a substitute for [Reference Link19, Lemma 6.6].

We remark that in the CAT(0) setting of [Reference Link19, Lemma 6.6], the corresponding quantities were proved to be finite and do not merely have slow exponential growth rates. Due to various intricacies involved in Teichmüller geometry, we were not able to prove this in our setting. Nevertheless, the exponential estimates we were able to establish turn out to be sufficient.

The proof of Proposition 5.15 will rely on following lemma.

Lemma 5.16 immediately implies the following corollary.

We will also need the following lemma.

Lemma 5.18. Assume that the action $G\curvearrowright \operatorname {\mathrm {Teich}}(S)$ has purely exponential growth. For $x\in \operatorname {\mathrm {Teich}}(S)$ , let $A_{K,R}(x)$ be the set of $y\in \operatorname {\mathrm {Teich}}(S)$ with $R-K\leq d(x,y)<R$ . Then, for any $x,y\in \operatorname {\mathrm {Teich}}(S)$ , for large enough $K>0$ , there is a $C>0$ such that for any Borel set $A\subset {\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , we have

$$ \begin{align*} \lim \sup_{R\to \infty}\frac{|A\cap A_{K,R}(x) \cap Gy|}{|A_{K,R}(x) \cap Gy|}\leq C\nu_x(\overline{A}). \end{align*} $$

Proof. Let $K>0$ be large enough so that $\nu _x(\mathcal {O}_K(x,gy))\asymp _\times e^{-\delta d(x,gy)}$ [Reference Yang36] and $|A_{K,R}(x) \cap Gy|\asymp _\times e^{\delta R}$ . Here, the notation $A(R) \asymp _\times B(R)$ is $A(R) \asymp _{x,y,K} B(R)$ , which means $\lim _{R \to \infty } ({A(R)}/{B(R)}) = c \neq 0$ , where c depends only on the choice of $x,y$ , and K. Since the number of orbit points in any ball of finite radius is finite, a point of ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ is in at most D shadows $\mathcal {O}_K(x,gy), gy\in A_{K,R}(x)$ .

Thus,

$$ \begin{align*}\sum_{\{g: gy\in A_{K,R}(x)\cap A\}}\nu_x(\mathcal{O}_K(x,gy))\leq D \nu_x\bigg(\bigcup_{\{g:gy\in A_{K,R}(x)\cap A\}}\mathcal{O}_K(x,gy)\bigg).\end{align*} $$

Moreover, by Lemma 2.4, any uniquely ergodic point of

$$ \begin{align*}\bigcap_{n\in \mathbb{N}z}\bigcup_{\{g:gy\in A_{K,R}(x)\cap A\}}\mathcal{O}_K(x,gy)\end{align*} $$

lies in $\overline {A}$ .

Thus, for large enough n, we have

$$ \begin{align*}\nu_x\bigg(\bigcup_{\{g:gy\in A_{K,R}(x)\cap A\}}\mathcal{O}_K(x,gy)\bigg)\leq 2\nu_x(\overline{A}),\end{align*} $$

so

$$ \begin{align*} \begin{aligned} \frac{|A\cap A_{K,R}(x) \cap Gy|}{|A_{K,R}(x) \cap Gy|}&\simeq e^{-hR}|A\cap A_{K,R}(x) \cap Gy|\simeq \sum_{\{g:gy\in A_{K,R}(x)\cap A\}}\nu_x(\mathcal{O}_K(x,gy))\\ &\leq D \nu_x\bigg(\bigcup_{\{g:gy\in A_{K,R}(x)\cap A\}}\mathcal{O}_K(x,gy)\bigg)\leq 2D\nu_x(\overline{A}), \end{aligned} \end{align*} $$

where the implied constants depend on $x,y,$ and K but not on R.

5.4 Asymptotic formulas

Having laid out the necessary technical results in the previous sections, our proof follows the strategy of [Reference Link19, Reference Roblin33]. A key observation is that the argument in [Reference Link19] remains valid even when the number of orbit points in a sector that are far from the boundary is not finite, but merely has slower exponential growth rate than G itself, a property we established in Proposition 5.15. This observation allows us to use the estimates established in §5.3 to replace [Reference Link19, Lemma 6.6].

5.4.1 Proof of item (3)

Item $(3)$ in Theorem 5.19 follows from the following more general theorem by taking f to be $\mathbb {1}_{{\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} \times {\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}}$ . For $x \in \operatorname {\mathrm {Teich}}(S)$ , the Dirac measure at x is denoted by $\mathcal {D}_x$ .

Theorem 5.20. Let G be a non-elementary subgroup of $\operatorname {\mathrm {Mod}}(S)$ with finite Bowen–Margulis measure $\mu _G$ . Let $x,y \in \operatorname {\mathrm {Teich}}(S)$ , and denote the Patterson–Sullivan measures by $\nu _x$ and $\nu _y$ , respectively. Set $\lVert \mu _G\lVert = \mu _G(QD^1(S)/G)$ . Then, the following weak convergence in $C({\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} \times {\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}})^{*}$ holds as $R \to \infty $ ,

$$ \begin{align*}\delta_G e^{-\delta_G R} \sum_{g \in G \atop d(x, gy) \leq R}\mathcal{D}_{gy} \otimes \mathcal{D}_{g^{-1}x} \rightharpoonup \frac{1}{\lVert \mu_G \lVert}\nu_x \otimes \nu_y.\end{align*} $$

The remainder of this section is devoted to the proof of Theorem 5.20. We follow the arguments in [Reference Link19, § 8], with the necessary technical modifications. The proof is therefore not self-contained; we indicate only the required modifications and refer the reader to [Reference Link19, §8] for the remaining details.

Throughout this subsection, we assume that G is a non-elementary subgroup of $\operatorname {\mathrm {Mod}}(S)$ with finite Bowen–Margulis measure $\mu _G$ . To simplify notation in the following, for ${x,y \in \operatorname {\mathrm {Teich}}(S)}$ , we set

$$ \begin{align*}\mu^{R}_{x,y} = \delta_G e^{-\delta_G R} \sum_{g \in G \atop d(x, gy) \leq R}\mathcal{D}_{gy} \otimes \mathcal{D}_{g^{-1}x},\end{align*} $$

and, whenever $r, A, B$ are fixed, we write (see §5.1)

$$ \begin{align*}K^+ = K_r^+(x,A), \quad K^{-}= K_r^-(y,B).\end{align*} $$

Recall that conical points are contained in $\mathcal {UE}$ and, hence, for any $\xi \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , the pair $(\xi ,\eta )$ is a filling pair for every conical point $\eta \neq \xi $ . In particular, $\mathcal {L}(\xi ,\eta )$ is non-empty.

Proposition 5.21. Let $\epsilon> 0$ and $(\xi _0,\eta _0) \in ({\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}})^2$ . Let $x, y \in \operatorname {\mathrm {Teich}}(S)$ have trivial stabilizers in G and satisfy $x \in \mathcal {L}(\xi _0,\xi ^+), y \in \mathcal {L}(\eta _0,\eta ^{+})$ for some conical points $\xi ^{+} \in \mathrm {supp}(\nu _x)$ and $\eta ^{+} \in \mathrm {supp}(\nu _y)$ . Then, there exist open neighborhoods $U,W \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ of $\xi _0$ and $\eta _0$ , respectively, such that for all Borel subsets $A \subset U, B \subset W$ ,

$$ \begin{align*} &\limsup_{R \to \infty} \mu^{R}_{x,y}(C^{-}_{1}(x,A) \times C^{-}_{1}(y,B)) \leq e^{\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert, \\ &\liminf_{R \to \infty} \mu^{R}_{x,y}(C^{+}_{1}(x,A) \times C^{+}_{1}(y,B)) \geq e^{-\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert. \end{align*} $$

Proof. Throughout the proof, we adopt the notation of §5.1. Let $\epsilon> 0$ and $\rho = \min \{d(x,\gamma x),d(y,\gamma y): \gamma \in G \setminus \{\mathrm {id}\}\}> 0$ . Choose

$$ \begin{align*}r \in \bigg(0, \min\bigg\{\frac{1}{10}, \frac{\rho}{6}, \frac{\epsilon}{60\delta_G}\bigg\}\bigg)\end{align*} $$

such that

$$ \begin{align*} \nu_x(\tilde{\partial} \mathcal{O}_r(\xi_0,x)) = \nu_y (\tilde{\partial} \mathcal{O}_r(\eta_0,y)) = 0.\end{align*} $$

The existence of such an r is guaranteed by Lemma 5.7.

Claim 1. We have

$$ \begin{align*} \nu_x(\mathcal{O}_r(\xi_0,x)) \nu_y (\mathcal{O}_r(\eta_0,y))> 0.\end{align*} $$

Proof of Claim 1.

Indeed, by assumption, $\xi ^{+} \in \mathcal {O}_r(\xi _0,x), ~ \eta ^{+} \in \mathcal {O}_r(\eta _0,y)$ , and $\xi ^{+},\eta ^{+}$ lie in the support of measures $\nu _x,\nu _y$ . It suffices to show that there exist open neighborhoods $ O_1, O_2$ of $\xi _0$ and $\eta _0$ contained in the corresponding sets $\mathcal {O}_r(\xi _0,x)$ and $\mathcal {O}_r(\eta _0,y)$ . This follows from Corollary 5.2.

Thanks to Lemma 5.9, there exist open neighborhoods $\widehat {U},\widehat {W}$ of $\xi _0$ and $\eta _0$ , respectively, in the compactification, such that whenever $(a,b) \in \widehat {U} \times \widehat {W}$ , one has

(5.15) $$ \begin{align} \begin{aligned} e^{-({\epsilon}/{30})}\nu_x(\mathcal{O}_r(\xi_0,x)) & \leq \nu_x(\mathcal{O}^{\pm}_r(a,x)) \leq e^{{\epsilon}/{30}}\nu_x(\mathcal{O}_r(\xi_0,x)),\\ e^{-({\epsilon}/{30})}\nu_y(\mathcal{O}_r(\eta_0,y)) & \leq \nu_y(\mathcal{O}^{\pm}_r(b,y)) \leq e^{{\epsilon}/{30}}\nu_x(\mathcal{O}_r(\eta_0,y)). \end{aligned} \end{align} $$

Let $U,W$ be open neighborhoods of $\xi _0,\eta _0$ in ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ such that their closures satisfy

$$ \begin{align*}\bar{U} \subset \widehat{U} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}},\quad \bar{W} \subset \widehat{W} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}.\end{align*} $$

Let $A \subset U$ and $B \subset W$ be arbitrary Borel subsets, and set

$$ \begin{align*}M = r^2\nu_x(\mathcal{O}_r(\xi_0,x))\nu_y(\mathcal{O}_r(\xi_0,y)).\end{align*} $$

We apply Roblin’s method, which consists of providing bounds for

$$ \begin{align*} \int^{T + 3r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \end{align*} $$

and

$$ \begin{align*} \int^{T -6r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \end{align*} $$

using two approaches: the mixing properties of the geodesic flow and direct computations.

Claim 2. Let $T_0> 60r$ , $T - 6r> T_0$ . Then, for $t \gg T_0$ , we have

(5.16) $$ \begin{align} \begin{aligned} & (e^{\delta_G(T-6r)}-e^{\delta_G T_0})M\nu_x(A)\nu_x(B) \\ & \quad \leq e^{({2\epsilon}/{5})}\lVert \mu_G \lVert \delta_G \int^{T-6r}_{T_0}e^{\delta_G t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt, \\ & (e^{\delta_G(T+3r)}-e^{\delta_G T_0})M\nu_x(A)\nu_x(B) \\ & \quad \geq e^{-({8\epsilon}/{15})}\lVert \mu_G \lVert \delta_G \int^{T+3r}_{T_0}e^{\delta_G t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt. \end{aligned}\end{align} $$

Proof of Claim 2.

By the choice of r and $\rho $ , one has

$$ \begin{align*}\text{ for all } \gamma \in G \setminus \{\mathrm{id}\},\quad K_r(x) \cap \gamma K_r(x) = \emptyset, \quad K_r(y) \cap \gamma K_r(y) = \emptyset.\end{align*} $$

Thus, under the projection $QD^1(S) \to QD^1(S)/G$ , the subsets $K^{\pm }$ inject into $QD^1(S)/G$ . It follows from Corollary 4.3 that

$$ \begin{align*}\lim_{t \to \infty}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+}\cap g_{-t}\gamma K^{-}) = \frac{\tilde{\mu}_G(K^{+})\tilde{\mu}_G(K^{-})}{\mu_G(QD^1(S)/G)}.\end{align*} $$

Let $T_0> 20r$ . For $t \gg T_0$ , the computations in the proof of [Reference Link19, Proposition 8.2, pp. 827–828], together with (5.8) and (5.15), conclude the proof of the claim (see also Note A.5 for details of the computation).

Claim 3. We have

(5.17) $$ \begin{align} \begin{aligned} \liminf_{R \to \infty} \mu^{R}_{x,y}(C^{+}_{1}(x,A) \times C^{+}_{1}(y,B)) \geq e^{-\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert. \end{aligned} \end{align} $$

Proof of Claim 3.

By (3.2),

$$ \begin{align*} \rho_{x}(\xi,\eta) = \lim_{z \to [\xi], w \to [\eta]} (d(x,z) + d(x,w) - d(z,w)).\end{align*} $$

It follows that $\rho _x(\xi ,\eta ) \leq 2r$ whenever $(\xi ,\eta ) \in L_r(x,\gamma y)$ . Furthermore, as $T_0> 60r$ , $K^{+} \cap g_{-t}\gamma K^{-} \neq \emptyset $ for some $t \geq T_0$ implies that $d(x,\gamma y) \geq 12r$ . By the conformality of the measure $\nu _x$ , we have

$$ \begin{align*}\int_{\mathcal{O}_r^{+}(x,\gamma y)}\,d\nu_x(\eta)e^{\delta_G d(x,\gamma y)} \leq e^{4r\delta_G}\nu_y(\mathcal{O}_r^{+}(\gamma^{-1}x,y)). \end{align*} $$

Here, we use the fact, which follows from the triangle inequality, that if $\eta \in \mathcal {O}_r^{+}(x,\gamma y)$ , then the Busemann cocycle satisfies $\beta _{\eta }(x,y) \geq d(x,\gamma y) -4r$ . With these remarks in mind, since $r < 1$ and $T_0> 60r$ , it follows from Lemma 5.14, Lemma 5.13 (2), Lemma 5.11, Lemma 5.10 (5), and the fact that $\nu _x$ and $\nu _y$ are fully supported on $\mathcal {UE}$ that

(5.18) $$ \begin{align} &\int^{T - 6r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \nonumber \\ &\quad \overset{(5.14)}{=}\sum_{\gamma \in G} \int^{T - 6r}_{T_0}\,dt\, \bigg(e^{\delta_{G} t} \int_{L_{r}(x,\gamma y) \cap (\gamma B \times A)}e^{\delta_G \rho_x(\xi,\eta)}\,d\nu_x(\xi)\,d\nu_x(\eta) \nonumber \\ &\qquad \quad \times \int^{{r}/{2}}_{-{r}/{2}}\mathbb{1}_{K_r(\gamma y)}(g_{t+s}w(x;\xi,\eta))\,ds\bigg)\nonumber \\ &\!\overset{(5.10),~(5.11)}{\leq} r^2e^{5r\delta_G }\sum_{\gamma \in G \atop d(x,\gamma y) \leq T }e^{\delta_Gd(x,\gamma y)}\int_{L_{r}(x,\gamma y) \cap (\gamma B \times A)}\,d\nu_x(\xi)\,d\nu_x(\eta)\nonumber \\ &\!\!\overset{\text{Lemma~5.11(1)}}{\leq} r^2e^{5r\delta_G }\sum_{\substack{\gamma \in G \\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{+}(x,A) \times C_1^{+}(y,B)}}e^{\delta_Gd(x,\gamma y)}\nonumber \\ & \qquad \quad\times \int_{\mathcal{O}^{+}_r(\gamma y,x) \times \mathcal{O}_r^{+}(x,\gamma y)} d\nu_x(\xi)\,d\nu_x(\eta)\nonumber \\ &\qquad\leq r^2e^{5r\delta_G }\sum_{\substack{\gamma \in G \\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{+}(x,A) \times C_1^{+}(y,B)}}e^{\delta_Gd(x,\gamma y)}\nu_x(\mathcal{O}^{+}_r(\gamma y,x))\nu_x(\mathcal{O}_{r}^{+}(x,\gamma y))\nonumber \\ &\qquad\leq r^2e^{9r\delta_G }\sum_{\substack{\gamma \in G\\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{+}(x,A) \times C_1^{+}(y,B)}}\nu_x(\mathcal{O}^{+}_r(\gamma y,x))\nu_y(\mathcal{O}_{r}^{+}(\gamma^{-1}x,y)). \end{align} $$

Set

$$ \begin{align*}\mathcal{X} = (C_{1}^{+}(x,A) \times C_{1}^{+}(y,B)) \cap (\widehat{U} \times \widehat{W})\end{align*} $$

and

$$ \begin{align*}\mathcal{Y} =(C_{1}^{+}(x,A) \times C_{1}^{+}(y,B)) \setminus (\widehat{U} \times \widehat{W}).\end{align*} $$

Then,

$$ \begin{align*}\mathcal{X} \cup \mathcal{Y} =C_{1}^{+}(x,A) \times C_{1}^{+}(y,B).\end{align*} $$

It follows that

(5.19) $$ \begin{align} &\int^{T - 6r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \nonumber \\ &\quad\leq \underbrace{r^2e^{9r\delta_G }\sum_{\substack{\gamma \in G \\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{X}}}\nu_x(\mathcal{O}^{+}_r(\gamma y,x))\nu_y(\mathcal{O}_{r}^{+}(\gamma^{-1}x,y))}_{\mathrm{I}} \nonumber\\ &\qquad+\underbrace{r^2e^{9r\delta_G }\sum_{\substack{\gamma \in G\\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{Y}}}\nu_x(\mathcal{O}^{+}_r(\gamma y,x))\nu_y(\mathcal{O}_{r}^{+}(\gamma^{-1}x,y))}_{\mathrm {II}} \nonumber\\ &\quad\leq \mathrm {I} + \mathrm{II}. \end{align} $$

Note that, on the one hand, by (5.15) and the choice $r < \min \{1,{\epsilon }/{60\delta _G}\}$ , we have

(5.20) $$ \begin{align} \mathrm {I} & \leq r^2e^{{11\epsilon}/{30}}\sum_{\substack{\gamma \in G\\ d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{X}}}\nu_x(\mathcal{O}_r(\xi_0,x))\nu_y(\mathcal{O}_{r}(\eta_0,y)) \nonumber \\ &\leq e^{{11\epsilon}/{30}}M\frac{e^{T\delta_G}}{\delta_G}\mu^{T}_{x,y}(C_{1}^{+}(x,A) \times C_{1}^{+}(y,B)). \end{align} $$

On the other hand, by Proposition 5.15(1) and the fact that $\|\nu _x\| =\|\nu _y\| =1$ , we have, as $T \to \infty $ ,

(5.21) $$ \begin{align} \mathrm {II} = o(e^{T\delta_G}). \end{align} $$

Hence,

(5.22) $$ \begin{align} &\int^{T - 6r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \nonumber\\ &\quad\leq e^{{11\epsilon}/{30}}M\frac{e^{T\delta_G}}{\delta_G}\mu^{T}_{x,y}(C_{1}^{+}(x,A) \times C_{1}^{+}(y,B)) + e^{T\delta_G}o(1). \end{align} $$

Combining Claim 2 with the above estimates, we complete the proof of Claim 3 after dividing by $Me^{\delta _G (T-6r)}\lVert \mu _G\lVert $ .

Claim 4. We have

(5.23) $$ \begin{align} \begin{aligned} \limsup_{R \to \infty} \mu^{R}_{x,y}(C^{-}_{1}(x,A) \times C^{-}_{1}(y,B)) \leq e^{\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert .\end{aligned} \end{align} $$

Proof of Claim 4.

The proof of Claim 4 follows the same line as that of Claim 3. We start by making several remarks. Observe that if $(\xi ,\eta ) \in (\mathcal {UE})^2$ , then $\rho _x(\xi ,\eta ) \geq 0$ . By Lemma 5.11(2), Lemma 5.10, and the assumption that $r < 1$ , whenever $ (\gamma y,\gamma ^{-1}x) \in C_1^{-}(x,A) \times C^{-}_1(y,B) \subset C_r^{-}(x,A) \times C^{-}_r(y,B)$ ,

$$ \begin{align*}\{(\zeta,\eta): \eta \in \mathcal{O}^{-}_r(x, \gamma y),~ \zeta \in \mathcal{O}_r(\eta, x)\}\subset L_r(x,\gamma y) \cap (\gamma B \times A).\end{align*} $$

Moreover, since $\gamma y \in C_1^{-}(x,A) \subset C_r^{-}(x,A)$ , we have

$$ \begin{align*}\mathcal{O}_r^{-}(x,\gamma y) \subset \mathcal{O}_r^{+}(x,\gamma y) \subset A \subset \widehat{U}.\end{align*} $$

As a consequence of (5.15) at the beginning of the proof, if $\xi \in \mathcal {O}^{-}_r(x,\gamma y)$ , then

(5.24) $$ \begin{align} \begin{aligned} &\nu_x(\mathcal{O}_r(\xi,x)) \geq e^{-({\epsilon}/{30})}\nu_x(\mathcal{O}_r(\xi_0,x)). \end{aligned} \end{align} $$

Additionally, if $\gamma ^{-1}x \in \widehat {W}$ , then by (5.15) again and conformality, we have

(5.25) $$ \begin{align} \int_{\mathcal{O}_r^{-}(x,\gamma y)}d\nu_x(\xi)e^{\delta_G d(x,\gamma y)} &\geq \nu_y(\mathcal{O}^{-}_r(\gamma^{-1}x,y))\nonumber\\ &\geq e^{-({\epsilon}/{30})}\nu_y(\mathcal{O}_r(\eta_0,y)). \end{align} $$

As in the proof of Claim 3, combining the above remarks with Lemma 5.14, Lemma 5.13 (1), and the fact that $\nu _x$ and $\nu _y$ are fully supported on $\mathcal {UE}$ , we obtain

(5.26) $$ \begin{align} &\int^{T+3r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \nonumber\\ &\quad \overset{(5.14)}{\geq} \sum_{\gamma \in G} \int^{T +3r}_{T_0}\, dt \bigg(e^{0} \int_{L_{r}(x,\gamma y) \cap (\gamma B \times A)}\,d\nu_x(\xi)\,d\nu_y(\eta)\nonumber\\ &\qquad \quad \times \int^{{r}/{2}}_{-({r}/{2})}e^{\delta_Gt}\mathbb{1}_{K_r(\gamma y)}(g_{t+s}w(x;\xi,\eta))\,ds\bigg) \nonumber\\ &\quad \overset{(5.9)}{\geq} r^2e^{-3r\delta_G }\sum_{\gamma \in G \atop T_0 + 3r < d(x,\gamma y) \leq T }e^{\delta_Gd(x,\gamma y)}\int_{L_{r}(x,\gamma y) \cap (\gamma B \times A)}\,d\nu_x(\xi)\,d\nu_x(\eta) \nonumber\\ &\quad {\geq}\ r^2e^{-({\epsilon}/{20})}\sum_{\substack{\gamma \in G\\ T_0 +3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{-}(x,A) \times C_1^{-}(y,B)}}e^{\delta_Gd(x,\gamma y)}\int_{\mathcal{O}_r^{-}(x,\gamma y)}\,d\nu_x(\xi)\,d\nu_x(\mathcal{O}_r(\xi,x)) \nonumber \\ &\quad {\geq}\ r^2e^{-({\epsilon}/{12})}\sum_{\substack{\gamma \in G\\ T_0 +3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{-}(x,A) \times C_1^{-}(y,B)}}e^{\delta_Gd(x,\gamma y)}\int_{\mathcal{O}_r^{-}(x,\gamma y)}\,d\nu_x(\xi)\,d\nu_x(\mathcal{O}_r(\xi_0,x)) \nonumber \\ &\quad {\geq}\ r^2e^{-({\epsilon}/{12})}\sum_{\substack{\gamma \in G\\ T_0 +3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in C_1^{-}(x,A) \times C_1^{-}(y,B)}}\nu_y(\mathcal{O}^{-}_r(\gamma^{-1}x,y))\nu_x(\mathcal{O}_r(\xi_0,x)). \end{align} $$

Set

$$ \begin{align*}\mathcal{X'} = (C_{1}^{-}(x,A) \times C_{1}^{-}(y,B)) \cap (\widehat{U} \times \widehat{W})\end{align*} $$

and

$$ \begin{align*}\mathcal{Y'} =(C_{1}^{-}(x,A) \times C_{1}^{-}(y,B)) \setminus (\widehat{U} \times \widehat{W}).\end{align*} $$

Then,

(5.27) $$ \begin{align} &\int^{T+3r}_{T_0}e^{\delta_{G} t}\sum_{\gamma \in G}\tilde{\mu}_G(K^{+} \cap g_{-t}\gamma K^{-})\,dt \nonumber\\ &\quad \geq \underbrace{r^2e^{-({\epsilon}/{12})}\sum_{\substack{\gamma \in G\\ T_0 + 3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{X'}}}\nu_y(\mathcal{O}^{-}_r(\gamma^{-1}x,y))\nu_x(\mathcal{O}_r(\xi_0,x))}_{\mathrm{III}} \nonumber \\ &\qquad + \underbrace{r^2e^{-({\epsilon}/{12})}\sum_{\substack{\gamma \in G\\ T_0+3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{Y'}}}\nu_y(\mathcal{O}^{-}_r(\gamma^{-1}x,y))\nu_x(\mathcal{O}_r(\xi_0,x))}_{\mathrm{IV}}.\nonumber\\[-17pt] \end{align} $$

It follows from the previous remarks that

(5.28) $$ \begin{align} \mathrm {III} &\geq r^2e^{-({7\epsilon}/{60})}\sum_{\substack{\gamma \in G\\ T_0+3r < d(x,\gamma y) \leq T \\ (\gamma y, \gamma^{-1}x) \in \mathcal{X'}}}\nu_y(\mathcal{O}_r(\eta_0,y))\nu_x(\mathcal{O}_r(\xi_0,x)) \nonumber \\ &\geq e^{-({7\epsilon}/{60})}M \frac{e^{T\delta_G}}{\delta_G}\mu^{T}_{x,y}(\mathcal{X'})-C_1 \nonumber \\ &\geq e^{-({7\epsilon}/{60})}M \frac{e^{T\delta_G}}{\delta_G}\mu^{T}_{x,y}(C^{-}_{1}(x,A) \times C^{-}_{1}(y,B)) -Me^{-({7\epsilon}/{60})}\|\mathcal{Y}_T'\| -C_1, \end{align} $$

where $\|\mathcal {Y}_T'\|$ denotes the cardinality of

$$ \begin{align*}\mathcal{Y}_T'= \{\gamma \in G: d(s,\gamma y) \leq T, (\gamma y,\gamma^{-1}x) \in \mathcal{Y}'\}\end{align*} $$

and C arises from the fact that, for fixed x and y, there are only finitely many elements in $\gamma \in G$ with $d(x,\gamma y) \leq T_0+3r$ . Note that by Proposition 5.15, we have

$$ \begin{align*}\|\mathcal{Y}^{\prime}_T\| = o(e^{T\delta_G}).\end{align*} $$

Since $\mathrm {IV} \geq 0$ , combining these estimates, we complete the proof of Claim 4 exactly as in the proof of Claim 3.

The proof of the proposition follows from Claims 3 and 4.

Proposition 5.22. Let $\epsilon> 0$ and $x, y \in \operatorname {\mathrm {Teich}}(S)$ . Then, for any two points $\xi _0, \eta _0 \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , there exist $r> 0$ and open neighborhoods

$$ \begin{align*}U,W \subset {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}\end{align*} $$

of $\xi _0$ and $\eta _0$ , respectively, such that for all Borel subsets $A \subset U, B \subset W,$

$$ \begin{align*} \begin{aligned} \limsup_{R \to \infty} \mu^{R}_{x,y}(C^{-}_{r}(x,A) \times C^{-}_{r}(y,B)) &\leq e^{\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert, \\ \liminf_{R \to \infty} \mu^{R}_{x,y}(C^{+}_{r}(x,A) \times C^{+}_{r}(y,B)) &\geq e^{-\epsilon}\nu_x(A)\nu_y(B)/\lVert\mu_G\lVert. \end{aligned} \end{align*} $$

Proof. The proof follows the same line as that of [Reference Link19, Proposition 8.3], combined with the strategy used in Proposition 5.21, where [Reference Link19, Lemma 6.6(a)] is replaced by Proposition 5.15(1). We refer the reader to Note A.6 for details as well.

Proof of Theorem 5.20.

The proof follows the same line as that of [Reference Link19, Theorem 8.1]. Let $x, y \in \operatorname {\mathrm {Teich}}(S)$ and $\epsilon> 0$ . For any two points $\xi _0,\eta _0 \in {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ , fix $r> 0$ and open neighborhoods $U, W \subset {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ of $\xi _0,\eta _0$ ensured by Proposition 5.22. Let $\widehat {U}, \widehat {W}$ be two open sets of ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ such that $\widehat {U} \cap {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} = U$ and $\widehat {W} \cap {\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}} = W$ . Let $\mathcal {A}, \mathcal {B}$ be Borel subsets of ${\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ with closures $\overline {\mathcal {A}} \subset \widehat {U}, \overline {\mathcal {B}} \subset \widehat {W}$ such that $\partial (\mathcal {A} \times \mathcal {B})$ has $\nu _x \otimes \nu _y$ -measure zero.

Let $\alpha> 0$ . Choose open subsets $A_{+}, B_{+}$ and compact subsets $A_{-}, B_{-}$ of ${\partial _\mathrm{GM}\overline {\operatorname {\mathrm {Teich}}(S)}^\mathrm{GM}}$ such that

(5.29) $$\begin{align}\begin{aligned} A_{-} \subset \mathcal{A}^{\circ} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \subset \overline{\mathcal{A}} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \subset A_{+} \subset U,\\B_{-} \subset \mathcal{B}^{\circ} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \subset \overline{\mathcal{B}} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \subset B_{+} \subset W,\\ \nu_x(\mathcal{A}^{\circ} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \setminus A_{-})< \alpha, \nu_x( A_{+}\setminus \overline{\mathcal{A}} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}) < \alpha,\\\nu_y(\mathcal{B}^{\circ} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \setminus B_{-}) < \alpha, \nu_y( B_{+}\setminus \overline{\mathcal{B}} \cap {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}) < \alpha.\end{aligned}\end{align}$$

Note that

(5.30) $$ \begin{align} \begin{aligned} &\overline{\mathcal{A}} \times \overline{\mathcal{B}} \\& \quad =((\overline{\mathcal{A}} {\kern-1pt}\times{\kern-1pt} \overline{\mathcal{B}}) \setminus (C^{-}_{r}(x,A_{+}) {\kern-1pt}\times{\kern-1pt} C_{r}^{-}(y,B_{+}))) {\kern-1pt}\cup{\kern-1pt} ((\overline{\mathcal{A}} \times \overline{\mathcal{B}}) \cap (C^{-}_{r}(x,A_{+}) \times C_{r}^{-}(y,B_{+}))), \\&C^{+}_{r}(x,A_{-}) \times C_{r}^{+}(y,B_{-}) \\&\quad = ((\mathcal{A} \times \mathcal{B}) \cap (C^{+}_{r}(x,A_{-}) \times C_{r}^{+}(y,B_{-}))) \cup (C^{+}_{r}(x,A_{-}) \times C_{r}^{+}(y,B_{-}) \setminus (\mathcal{A} \times \mathcal{B})). \end{aligned} \end{align} $$

By Propositions 5.15, 5.22, and the fact that

$$ \begin{align*}\nu_x \otimes \nu_y(\partial (\mathcal{A} \times \mathcal{B})) = 0,\end{align*} $$

we have

(5.31) $$ \begin{align} \limsup_{R \to \infty}\mu^{R}_{x,y}( \mathcal{A} \times \mathcal{B}) &\leq \limsup_{R \to \infty}\mu^{R}_{x,y}((C^{-}_{r}(x,A_{+}) \times C_{r}^{-}(y,B_{+})) \nonumber\\ &\leq \frac{e^{\epsilon}\nu_x(A_{+})\nu_y(B_{+})}{\lVert\mu_G(G)\lVert} \nonumber\\ &\leq \frac{e^{\epsilon}(\nu_x(\overline{\mathcal{A}})\nu_y(\overline{\mathcal{B}})+\alpha(\lVert \nu_x\lVert\hspace{2pt} +\hspace{2pt}\lVert \nu_y\lVert))}{\lVert\mu_G(G)\lVert} \nonumber\\ &\leq \frac{e^{\epsilon}(\nu_x(\mathcal{A})\nu_y(\mathcal{B})+\alpha(\lVert \nu_x\lVert\hspace{2pt} +\hspace{2pt}\lVert \nu_y\lVert))}{\lVert\mu_G(G)\lVert}. \end{align} $$

Take $\alpha $ small enough. One has

$$ \begin{align*}\limsup_{R \to \infty}\mu^{R}_{x,y}( \mathcal{A} \times \mathcal{B}) \leq \frac{e^{\epsilon}\nu_x(\mathcal{A})\nu_y(\mathcal{B})}{\lVert\mu_G(G)\lVert}.\end{align*} $$

Similarly, one obtains that

$$ \begin{align*}\limsup_{R \to \infty}\mu^{R}_{x,y}( \mathcal{A} \times \mathcal{B}) \geq \frac{e^{-\epsilon}\nu_x(\mathcal{A})\nu_y(\mathcal{B})}{\lVert\mu_G(G)\lVert}.\end{align*} $$

Hence, for a continuous function f compactly supported in $\widehat {U} \times \widehat {W}$ ,

(5.32) $$ \begin{align} &e^{-\epsilon}\int_{{\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \times {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}} f \,d(\nu_x\otimes \nu_y) \nonumber\\ &\quad \leq \liminf_{R \to \infty}\int_{{\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \times {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}} f\,d(\mu^{R}_{x,y}) \nonumber \\ &\quad \leq \limsup_{R \to \infty}\int_{{\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \times {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}} f\,d(\mu^{R}_{x,y}) \nonumber \\ &\quad \leq e^{\epsilon}\int_{{\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}} \times {\partial_\mathrm{GM}\overline{\operatorname{\mathrm{Teich}}(S)}^\mathrm{GM}}} f \,d(\nu_x\otimes \nu_y). \end{align} $$

The proof then follows by exactly the same argument as in the last paragraph of the proof of [Reference Link19, Theorem 8.1].

6.1 Mapping class groups

If G is the full mapping class group $\operatorname {\mathrm {Mod}}(S)$ , then one can use item $(3)$ in Theorem 5.19 to get an asymptotic growth formula for the mapping class group with respect to the Teicmüller metric, since in this case, $\delta _G = 6g-6$ is the entropy of the Techmüller geodesic flow $g_t$ on the unit tangent bundle $QD^1(S)$ of $\operatorname {\mathrm {Teich}}(S)$ and $\mu _{G}$ is, up to a scale, the Masur–Veech measure constructed by Masur [Reference Masur23] and Veech [Reference Veech34], which is finite. This is the main result of [Reference Athreya, Bufetov, Eskin and Mirzakhani1], which is inspired by the method in Margulis’s thesis [Reference Margulis22]. Our method here is, however, modeled on the ones given by Roblin [Reference Roblin33] and Link [Reference Link19], so we obtain another proof for the case $\operatorname {\mathrm {Mod}}(S)$ .

6.2 Convex cocompact subgroups

A subgroup $G \leq \operatorname {\mathrm {Mod}}(S)$ is called convex cocompact if some G-orbit in $\operatorname {\mathrm {Teich}}(S)$ is quasi-convex (see [Reference Farb and Mosher9, Reference Kent and Leininger14]. Convex cocompact subgroups of $\operatorname {\mathrm {Mod}}(S)$ are analogs of convex cocompact subgroups of Kleinian groups. It also has many interesting properties, e.g. $\Lambda G$ consists of uniquely ergodic measured foliations and it is virtually purely pseudo-Anosov. An important feature for a convex cocompact subgroup G of $\operatorname {\mathrm {Mod}}(S)$ is that there is a G-invariant subset $W_G$ of $\overline {\operatorname {\mathrm {Teich}}(S)}^{Th}$ such that G acts on $W_G$ cocompactly. This shows that the corresponding Bowen–Margulis measure $\mu _G$ is finite, so one can use item $(3)$ in Theorem 5.19 to get an asymptotic formula for orbit growth. In fact, under the assumption that at least one pseudo-Anosov element has an axis in the principal stratum, this formula is obtained in the first named author’s thesis [Reference Gekhtman11], where a large part of the proof is quite different from the current paper.

6.3 Other examples

Statistically convex cocompact (SCC) subgroups for general actions with contracting elements were introduced by Yang [Reference Yang37]. In the setting of mapping class group actions on Teichmüller space, they are defined as follows. Let $G<\operatorname {\mathrm {Mod}}(S)$ , $o\in \operatorname {\mathrm {Teich}}(S)$ , and a constant $M>0$ . Let $O_{M}$ denote the set of $g\in G$ such that some geodesic from $B(o, M)$ to $B(go,M)$ lies outside of the M-neighborhood of the orbit $Go$ . We say G is SCC if for some $M>0$ , the set $O_{M}$ has strictly smaller exponential growth rate than G itself (this definition can be shown to be independent of the basepoint $o\in \operatorname {\mathrm {Teich}}(S)$ ). Yang [Reference Yang37] showed that SCC actions have coarse purely exponential growth:

$$ \begin{align*}0<\lim \inf_{R\to \infty} |Go\cap B(o,R)|/e^{\delta_G R}\leq \lim \inf_{R\to \infty} |Go\cap B(o,R)|/e^{\delta_G R}<\infty.\end{align*} $$

This immediately implies that G is divergent and item (2) of our Theorem 5.19 then implies that G admits finite Bowen–Margulis measure. Therefore, item (3) of Theorem 5.19 applies and gives precise multiplicative asymptotics for orbit growth of $G.$

Examples of non-convex-cocompact SCC subgroups of mapping class groups can be obtained as free products of a free abelian group H generated by Dehn twists and a cyclic group generated by a single pseudo-Anosov $\gamma $ . See [Reference Yang37, Proposition 6.6].