2. Gromov-hyperbolic spaces

Let $(X,d)$ be a metric space. The open (respectively closed) ball of radius r and center $x \in X$ is denoted by $B(x,r)$ (respectively $\overline {B}(x,r)$ ). If we need to specify the metric, we will write $B_d(x,r)$ (respectively $\overline {B}_d(x,r))$ . A geodesic segment is an isometric embedding $\gamma \colon I \to X$ , where $I=[a,b] \subseteq \mathbb {R}$ is a bounded interval. The points $\gamma (a), \gamma (b)$ are called the endpoints of $\gamma $ . A metric space X is called geodesic if for every couple of points $x,y\in X$ , there exists a geodesic segment whose endpoints are x and y. Every such geodesic segment will be denoted, with an abuse of notation, by $[x,y]$ . A geodesic ray is an isometric embedding $\xi \colon [0,+\infty )\to X$ while a geodesic line is an isometric embedding $\gamma \colon \mathbb {R}\to X$ .

Let X be a geodesic metric space and let $x,y,z \in X$ . The Gromov product of y and z with respect to x is defined as

$$ \begin{align*} (y,z)_x = \tfrac{1}{2}( d(x,y) + d(x,z) - d(y,z)). \end{align*} $$

The space X is called $\delta $ -hyperbolic if for every $x,y,z,w \in X$ , the following 4-points condition holds:

(2) $$ \begin{align} (x,z)_w \geq \min\lbrace (x,y)_w, (y,z)_w \rbrace - \delta \end{align} $$

or, equivalently,

(3) $$ \begin{align} d(x,y) + d(z,w) \leq \max \lbrace d(x,z) + d(y,w), d(x,w) + d(y,z) \rbrace + 2\delta. \end{align} $$

The space X is Gromov hyperbolic if it is $\delta $ -hyperbolic for some $\delta \geq 0$ .

We recall that Gromov-hyperbolicity should be considered as a negative-curvature condition at large scale: for instance, every CAT $(\kappa )$ metric space, where $\kappa <0$ is $\delta $ -hyperbolic for a constant $\delta $ depending only on $\kappa $ . The converse is false, essentially because the CAT $(\kappa )$ condition controls the local geometry much better than the Gromov-hyperbolicity due to the convexity of the distance functions in such spaces (see for instance [Reference Cavallucci and SambusettiCS21, Reference Cavallucci and SambusettiCS24, Reference Lytchak and NaganoLN19]).

2.1. Gromov boundary

Let X be a proper, $\delta $ -hyperbolic metric space and let x be a point of X. The Gromov boundary of X is defined as the quotient

$$ \begin{align*} \partial X = \Big\{ (z_n)_{n \in \mathbb{N}} \subseteq X \hspace{1mm} | \hspace{1mm} \lim_{n,m \to +\infty} (z_n,z_m)_{x} = + \infty \Big\} \hspace{1mm} /_\sim, \end{align*} $$

where $(z_n)_{n \in \mathbb {N}}$ is a sequence of points in X and $\sim $ is the equivalence relation defined by $(z_n)_{n \in \mathbb {N}} \sim (z_n')_{n \in \mathbb {N}}$ if and only if $\lim _{n,m \to +\infty } (z_n,z_m')_{x} = + \infty $ . We will write $ z = [(z_n)] \in \partial X$ for short, and we say that $(z_n)$ converges to z. This definition does not depend on the basepoint x. There is a natural topology on $X\cup \partial X$ that extends the metric topology of X.

Every geodesic ray $\xi $ defines a point $\xi ^+=[(\xi (n))_{n \in \mathbb {N}}]$ of the Gromov boundary $ \partial X$ : we say that $\xi $ joins $\xi (0) = y$ to $\xi ^+ = z$ . Moreover, for every $z\in \partial X$ and every $x\in X$ , it is possible to find a geodesic ray $\xi $ such that $\xi (0)=x$ and $\xi ^+ = z$ . Indeed, if $(z_n)$ is a sequence of points converging to z then, by properness of X, the sequence of geodesics $[x,z_n]$ subconverges to a geodesic ray $\xi $ which has the properties above (cf. [Reference Bridson and HaefligerBH13, Lemma III.3.13]). A geodesic ray joining x to $z\in \partial X$ will be denoted by $\xi _{x,z}$ or simply $[x,z]$ . The relation between Gromov product and geodesic ray is highlighted in the following lemma.

Lemma 2.1. [Reference CavallucciCav23, Lemma 4.2]

Let X be a proper, $\delta $ -hyperbolic metric space, $z, z' \in \partial X$ , $x\in X$ , $b>0$ . Then:

1. (i) if $(z,z')_{x} \geq T$ , then $d(\xi _{x,z}(T - \delta ),\xi _{x,z'}(T - \delta )) \leq 4\delta $ ;

2. (ii) if $d(\xi _{x,z}(T),\xi _{x,z'}(T)) < 2b$ , then $(z,z')_{x}> T - b$ .

The following is a standard computation, see [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Proposition 8.10] for instance.

Lemma 2.2. Let X be a proper, $\delta $ -hyperbolic metric space. Then every two geodesic rays $\xi , \xi '$ with same endpoints at infinity are at distance at most $8\delta $ , that is, there exist $t_1,t_2\geq 0$ such that $t_1+t_2=d(\xi (0),\xi '(0))$ and $d(\xi (t + t_1),\xi '(t+t_2)) \leq 8\delta $ for all $t\in \mathbb {R}$ .

2.2. Visual metrics

When X is a proper, $\delta $ -hyperbolic metric space, it is known that the boundary $\partial X$ is metrizable. A metric $D_{x,a}$ on $\partial X$ is called a visual metric of center $x \in X$ and parameter $a\in (0,{1}/({2\delta \cdot \log _2e}))$ if there exists $V> 0$ such that for all $z,z' \in \partial X$ , it holds that

(4) $$ \begin{align} \frac{1}{V}e^{-a(z,z')_{x}}\leq D_{x,a}(z,z')\leq V e^{-a(z,z')_{x}}. \end{align} $$

For every a as before and every $x\in X$ , there exists a visual metric of parameter a and center x, see [Reference PaulinPau96]. As in [Reference CavallucciCav23, Reference PaulinPau96], we define the generalized visual ball of center $z \in \partial X$ and radius $\rho \geq 0$ as

$$ \begin{align*} B(z,\rho) = \bigg\lbrace z' \in \partial X \text{ s.t. } (z,z')_{x}> \log \frac{1}{\rho} \bigg\rbrace. \end{align*} $$

It is comparable to the metric balls of the visual metrics on $\partial X$ .

Lemma 2.3. Let $D_{x,a}$ be a visual metric of center x and parameter a on $\partial X$ . Then for every $z\in \partial X$ and for every $\rho>0$ , it holds that

$$ \begin{align*}B_{D_{x,a}}\bigg(z, \frac{1}{V}\rho^a\bigg) \subseteq B(z,\rho)\subseteq B_{D_{x,a}}(z, V\rho^a ).\end{align*} $$

It is classical that generalized visual balls are related to shadows, whose definition is the following. Let $x\in X$ be a basepoint. The shadow of radius $r>0$ caste by a point $y\in X$ with center x is the set:

$$ \begin{align*} \text{Shad}_x(y,r) = \lbrace z\in \partial X \text{ s.t. } [x,z]\cap B(y,r) \neq \emptyset \text{ for every ray } [x,z]\rbrace. \end{align*} $$

For our purposes, we just need the following lemma.

Lemma 2.4. [Reference CavallucciCav23, Lemma 4.8]

Let X be a proper, $\delta $ -hyperbolic metric space. Let $z\in \partial X$ , $x\in X$ , and $T\geq 0$ . Then for every $r>0$ , it holds that

$$ \begin{align*} \mathrm{Shad}_{x} (\xi_{x,z}(T), r) \subseteq B(z, e^{-T + r}). \end{align*} $$

3. Hausdorff and packing dimensions

In this section, we recall briefly the definitions of Hausdorff and packing dimensions of a subset of a metric space. Then we will adapt these constructions and results to the case of the boundary at infinity of a $\delta $ -hyperbolic metric space. The facts presented here are classical and can be found easily in the literature.

3.1. Definitions of Hausdorff and packing dimensions

Let $(X,d)$ be a metric space and $\alpha \geq 0$ . The $\alpha $ -Hausdorff measure of a Borel subset $B\subset X$ is defined as

$$ \begin{align*}\mathcal{H}^\alpha_d(B) = \lim_{\eta \to 0}\inf \bigg\lbrace \sum_{i\in \mathbb{N}} r_i^\alpha \text{ such that } B\subseteq \bigcup_{i\in \mathbb{N}}B(x_i,r_i) \text{ and } r_i\leq \eta\bigg\rbrace.\end{align*} $$

The argument of the limit is increasing when $\eta $ tends to $0$ , so the limit exists. This formula actually defines a Borel measure on X. To be precise, what we introduced is the definition of the spherical Hausdorff measure. It is comparable to the classical Hausdorff measure. The Hausdorff dimension of a Borel subset B of X, denoted HD $_d(B)$ , is the unique real number $\alpha \geq 0$ such that $\mathcal {H}^{\alpha '}_d(B) = 0$ for every $\alpha '> \alpha $ and $\mathcal {H}^{\alpha '}_d(B) = +\infty $ for every $\alpha ' < \alpha $ .

The packing dimension is defined in a similar way, but using disjoint balls inside B instead of coverings. For every $\alpha \geq 0$ and for every Borel subset B of X, we define

$$ \begin{align*} \mathcal{P}^\alpha_d(B) = \lim_{\eta \to 0} \sup\bigg\lbrace \sum_{i\in \mathbb{N}} r_i^\alpha \text{ such that } B(x_i,r_i) \text{ are disjoint, }x_i\in B \text{, and } r_i\leq \eta\bigg\rbrace. \end{align*} $$

This is not a measure on X but only a pre-measure. By a standard procedure, one can define the $\alpha $ -packing measure as

$$ \begin{align*} \hat{\mathcal{P}}_d^\alpha(B) = \inf\bigg\lbrace \sum_{k=1}^\infty \mathcal{P}_d^\alpha(B_k) \text{ such that } B\subseteq \bigcup_{k=1}^\infty B_k, \, B_k \text{ Borel}\bigg\rbrace. \end{align*} $$

The packing dimension of a Borel subset $B\subseteq X$ , denoted PD $_d(B)$ , is the unique real number $\alpha \geq 0$ such that $\hat {\mathcal {P}}^{\alpha '}_d(B) = 0$ for every $\alpha '> \alpha $ and $\hat {\mathcal {P}}^{\alpha '}_d(B) = +\infty $ for every $\alpha ' < \alpha $ . The packing dimension has another useful interpretation (cf. [Reference FalconerFal04, Proposition 3.8]): for every Borel subset $B\subseteq X$ , we have

(5) $$ \begin{align} \text{PD}_d(B) = \inf\bigg\lbrace \sup_k \overline{\text{MD}}_d(B_k) \text{ such that } B\subseteq \bigcup_{k=1}^\infty B_k,\, B_k \text{ Borel}\bigg\rbrace. \end{align} $$

The quantity $\overline {\text {MD}}_d$ denotes the upper Minkowski dimension, namely:

(6) $$ \begin{align} \overline{\text{MD}}_d(B) = \limsup_{r \to 0}\frac{\log\text{Cov}_d(B,r)}{\log({1}/{r})}, \end{align} $$

where B is any subset of X and $\text {Cov}_d(B,r)$ denotes the minimal number of d-balls of radius r needed to cover B. Taking the limit inferior in place of the limit superior in equation (6), one defines the lower Minkowski dimension of B, denoted $\underline {\text {MD}}_d(B)$ .

3.2. Visual dimensions

Let X be a proper, $\delta $ -hyperbolic metric space and let $x\in X$ . The boundary at infinity $\partial X$ supports several visual metrics $D_{x,a}$ , so the Hausdorff dimension, the packing dimension, and the Minkowski dimension of subsets of $\partial X$ are well defined with respect to $D_{x,a}$ . There is a way to define universal versions of these quantities that do not depend neither on x nor on a. Fix $\alpha \geq 0$ . For a Borel subset B of $\partial X$ , we set, following [Reference PaulinPau96],

$$ \begin{align*} \mathcal{H}^\alpha(B) = \lim_{\eta \to 0}\inf \bigg\lbrace \sum_{i\in \mathbb{N}} \rho_i^\alpha \text{ such that } B\subseteq \bigcup_{i\in \mathbb{N}}B(z_i,\rho_i) \text{ and } \rho_i\leq \eta\bigg\rbrace, \end{align*} $$

where $B(z_i,\rho _i)$ are generalized visual balls. As in the classical case, the visual Hausdorff dimension of B is defined as the unique $\alpha \geq 0$ such that $\mathcal {H}^{\alpha '}(B) = 0$ for every $\alpha '> \alpha $ and $\mathcal {H}^{\alpha '}(B) = +\infty $ for every $\alpha '<\alpha $ . The visual Hausdorff dimension of the Borel subset B is denoted by HD $(B)$ . By Lemma 2.3, see also [Reference PaulinPau96], we have HD $(B) = a\cdot \text {HD}_{D_{x,a}}(B)$ for every visual metric $D_{x,a}$ of center x and parameter a.

In the same way, we can define the visual $\alpha $ -packing pre-measure of a Borel subset B of $\partial X$ by

$$ \begin{align*}\mathcal{P}^\alpha(B) = \lim_{\eta \to 0} \sup\bigg\lbrace \sum_{i\in \mathbb{N}} \rho_i^\alpha \text{ such that } B(z_i,\rho_i) \text{ are disjoint, }x_i\in B \text{, and } \rho_i\leq \eta\bigg\rbrace,\end{align*} $$

where $B(z_i,\rho _i)$ are again generalized visual balls. As usual, we can define the visual $\alpha $ -packing measure by

$$ \begin{align*}\hat{\mathcal{P}}^\alpha(B) = \inf\bigg\lbrace \sum_{k=1}^\infty \mathcal{P}^\alpha(B_k) \text{ such that } B\subseteq \bigcup_{k=1}^\infty B_k,\, B_k \text{ Borel}\bigg\rbrace.\end{align*} $$

Consequently, the visual packing dimension of a Borel set B is defined, denoted by PD $(B)$ . Using Lemma 2.3, as in the case of the Hausdorff measure (see [Reference PaulinPau96]), one can check that for every visual metric $D_{x,a}$ of center x and parameter a, it holds that

$$ \begin{align*} \frac{1}{V^a} \hat{\mathcal{P}}_{D_{x,a}}^{{\alpha}/{a}}(B) \leq \hat{\mathcal{P}}^\alpha(B) \leq V^a \hat{\mathcal{P}}_{D_{x,a}}^{{\alpha}/{a}}(B) \end{align*} $$

for every $\alpha \geq 0$ and every Borel subset $B\subseteq \partial X$ . Therefore, for every Borel set B, it holds that PD $(B)=a\cdot \text {PD}_{D_{x,a}}(B)$ .

Using generalized visual balls, instead of metric balls with respect to a visual metric, one can define the visual upper and lower Minkowski dimension of a subset $B\subseteq \partial X$ :

$$ \begin{align*} \overline{\text{MD}}(B) = \limsup_{\rho \to 0}\frac{\log \text{Cov}(B,\rho)}{\log \rho},\quad \underline{\text{MD}}(B) = \liminf_{\rho \to 0}\frac{\log \text{Cov}(B,\rho)}{\log \rho}, \end{align*} $$

where $\text {Cov}(B,\rho )$ denotes the minimal number of generalized visual balls of radius $\rho $ needed to cover B. Using again Lemma 2.3, one has $\overline {\text {MD}}(B) = a\cdot \overline {\text {MD}}_{D_{x,a}}(B)$ for every Borel set B, and every visual metric of center x and parameter a. The same holds for the lower Minkowski dimension.

It is easy to check that for every Borel set B of $\partial X$ , the numbers HD $(B)$ , PD $(B)$ , $\underline {\text {MD}}(B)$ , $\overline {\text {MD}}(B)$ do not depend on x, see [Reference PaulinPau96, Proposition 6.4], and their definition is independent also on a. Using the classical facts holding for metric spaces, we get

(7) $$ \begin{align} \text{HD}(B) \leq \text{PD}(B) \leq \underline{\text{MD}}(B) \leq \overline{\text{MD}}(B) \end{align} $$

and

(8) $$ \begin{align} \text{PD}(B) = \inf\bigg\lbrace \sup_k \overline{\text{MD}}(B_k) \text{ such that } B\subseteq \bigcup_{k=1}^\infty B_k,\,B_k \text{ Borel}\bigg\rbrace \end{align} $$

for every Borel subset B of $\partial X$ .

4. Limit sets of discrete groups of isometries

If X is a proper metric space, we denote its group of isometries by Isom $(X)$ and we endow it with the uniform convergence on compact subsets of X. A subgroup $\Gamma $ of Isom $(X)$ is called discrete if the following equivalent conditions hold:

1. (a) $\Gamma $ is discrete as a subspace of Isom $(X)$ ;

2. (b) $\text {for all } x\in X$ and $R\geq 0$ , the set $\Sigma _R(x) = \lbrace g \in \Gamma \text { such that } g x\in \overline {B}(x,R)\rbrace $ is finite.

The critical exponent of a discrete group of isometries $\Gamma $ acting on a proper metric space X can be defined using the Poincaré series, or alternatively [Reference CavallucciCav23, Reference CoornaertCoo93], as

$$ \begin{align*} \overline{h_\Gamma}(X) = \limsup_{T \to +\infty}\frac{1}{T}\log \# (\Gamma x \cap B(x,T)), \end{align*} $$

where x is a fixed point of X. This quantity does not depend on the choice of x. In the following, we will often write $\overline {h_\Gamma }(X)=:h_\Gamma $ . Taking the limit inferior instead of the limit superior, we define the lower critical exponent, denoted by $\underline {h_\Gamma }(X)$ . In [Reference RoblinRob02], it is proved that if $\Gamma $ is a discrete, non-elementary group of isometries of a CAT $(-1)$ space, then $\overline {h_\Gamma }(X) = \underline {h_\Gamma }(X)$ . Theorem B generalizes this result to proper, $\delta $ -hyperbolic spaces.

We specialize the situation to the case of a proper, $\delta $ -hyperbolic metric space X. Every isometry of X acts naturally on $\partial X$ and the resulting map on $X\cup \partial X$ is a homeomorphism. The limit set $\Lambda (\Gamma )$ of a discrete group of isometries $\Gamma $ is the set of accumulation points of the orbit $\Gamma x$ on $\partial X$ , where x is any point of X; it is the smallest $\Gamma $ -invariant closed set of the Gromov boundary (cf. [Reference CoornaertCoo93, Theorem 5.1]) and it does not depend on x.

There are several interesting subsets of the limit set: the radial limit set, the uniformly radial limit set, etc. They are related to important sets of the geodesic flow on the quotient space $\Gamma \backslash X$ . We will see an instance in the second part of the paper. To recall their definition, we need to introduce a more general class of subsets of $\partial X$ .

We fix a basepoint $x\in X$ . Let $\tau $ and $\Theta = \lbrace \vartheta _i \rbrace _{i\in \mathbb {N}}$ be respectively a positive real number and an increasing sequence of real numbers with $\lim _{i \to +\infty }\vartheta _i = +\infty $ . We define $\Lambda _{\tau , \Theta }(\Gamma )$ as the set of points $z\in \partial X$ such that there exists a geodesic ray $[x,z]$ satisfying the following: for every $i\in \mathbb {N}$ , there exists a point $y_i \in [x,z]$ with $d(x,y_i) \in [\vartheta _i, \vartheta _{i+1}]$ such that $d(y_i,\Gamma x) \leq \tau $ . We observe that up to change $\tau $ with $\tau + 8\delta $ , the definition above does not depend on the choice of the geodesic ray $[x,z]$ , by Lemma 2.2.

We can now introduce some interesting subsets of the limit set of $\Gamma $ . Let $\Theta _{\text {rad}}$ be the set of increasing, unbounded sequences of real numbers. The radial limit set is classically defined as

$$ \begin{align*} \Lambda_{\text{rad}}(\Gamma) = \bigcup_{\tau \geq 0}\bigcup_{\Theta \in \Theta_{\text{rad}}}\Lambda_{\tau, \Theta}(\Gamma). \end{align*} $$

The uniform radial limit set is defined (see [Reference Das, Simmons and UrbańskiDSU17]) as

$$ \begin{align*} \Lambda_{\text{u-rad}}(\Gamma) = \bigcup_{\tau \geq 0}\Lambda_{\tau}(\Gamma), \end{align*} $$

where $\Lambda _\tau (\Gamma )=\Lambda _{\tau , \lbrace i\tau \rbrace }(\Gamma )$ .

Another interesting set is what we call the ergodic limit set, defined as

$$ \begin{align*} \Lambda_{\text{erg}}(\Gamma) = \bigcup_{\tau \geq 0}\bigcup_{\Theta \in \Theta_{\text{erg}}}\Lambda_{\tau, \Theta}(\Gamma), \end{align*} $$

where a sequence $\Theta = \lbrace \vartheta _i\rbrace $ belongs to $\Theta _{\text {erg}}$ if $\text {there exists }\lim _{i\to +\infty } ({\vartheta _{i}}/{i}) \in (0,+\infty )$ . The name is justified by Theorem C stating that every ergodic measure which is invariant by the geodesic flow on $\Gamma \backslash X$ is concentrated on geodesics whose endpoints belong to $\Lambda _{\text {erg}}$ .

When $\Gamma $ is clear in the context, we will simply write $\Lambda _{\tau , \Theta }, \Lambda _{\text {rad}}, \Lambda _{\text {u-rad}}, \Lambda _{\mathrm{erg}}, \Lambda $ , omitting $\Gamma $ .

5. Bishop–Jones’ theorem

The celebrated Bishop–Jones theorem, in the general version of [Reference Das, Simmons and UrbańskiDSU17], states the following.

Theorem 5.1. [Reference Bishop and JonesBJ97, Reference Das, Simmons and UrbańskiDSU17, Reference PaulinPau97]

Let X be a proper, $\delta $ -hyperbolic metric space and let $\Gamma < \mathrm{Isom}(X)$ be discrete and non-elementary. Then

$$ \begin{align*}h_\Gamma = \mathrm{HD}(\Lambda_{\mathrm{rad}}) = \mathrm{HD}(\Lambda_{\mathrm{u-rad}}) = \sup_{\tau \geq 0} \mathrm{HD}(\Lambda_{\tau}).\end{align*} $$

To introduce the techniques we will use in the proof of Theorem A, we start with the following proof.

Proof of Theorem B

By Theorem 5.1, we have

$$ \begin{align*}\overline{h_{\Gamma}}(X) = h_\Gamma = \sup_{\tau \geq 0} \text{HD}(\Lambda_\tau) \leq \sup_{\tau \geq 0} \underline{\text{MD}}(\Lambda_\tau).\end{align*} $$

So it would be enough to show that

$$ \begin{align*}\sup_{\tau \geq 0} \underline{\text{MD}}(\Lambda_\tau) \leq \underline{h_\Gamma}(X).\end{align*} $$

We fix $\tau \geq 0$ . For every $\varepsilon> 0$ , we take a subsequence $T_j \to +\infty $ such that

$$ \begin{align*}\frac{1}{T_j}\log \#(\Gamma x \cap \overline{B}(x,T_j)) \leq \underline{h_\Gamma}(X) + \varepsilon\end{align*} $$

for every j. We define $\rho _j = e^{-T_j}$ : notice that $\rho _j \to 0$ . Let $k_j\in \mathbb {N}$ be such that $(k_j-1)\tau \le T_j < k_j\tau $ . If $z\in \Lambda _\tau $ , then there exists a geodesic ray $[x,z]$ and a point $y_j \in [x,z]$ with $d(x,y_j) \in [(k_j-3)\tau , (k_j-2)\tau ]$ and $d(y_j, gx) \le \tau $ for some $g\in \Gamma $ . This g satisfies $d(x,gx)\le (k_j-1)\tau \le T_j$ . Moreover, $z\in \text {Shad}_x(gx, \tau + 8\delta )$ , since $d(gx,[x,z]) \le \tau $ and since every two parallel geodesic rays are $8\delta $ apart by Lemma 2.2. We showed that the set of shadows $\{\text {Shad}_x(gx, \tau + 8\delta )\}$ with $g\in \Gamma $ such that $(k_j-4)\tau \le d(x,gx)\le (k_j-1)\tau \le T_j$ cover $\Lambda _\tau $ . The cardinality of this set of shadows is at most $e^{(\underline {h_\Gamma }(X) + \varepsilon )T_j} \le e^{(\underline {h_\Gamma }(X) + \varepsilon )k_j\tau }$ . Among these shadows indexed by these elements $g\in \Gamma $ , we select those that intersect $\Lambda _\tau $ . For these, the construction above gives a point $z_g \in \Lambda _\tau $ , a point $y_g$ along $[x,z_g]$ such that $(k_j-3)\tau \le d(x,y_g) \le (k_j -2)\tau $ and $d(y_g,gx)\le \tau $ . Therefore,

(9) $$ \begin{align} \text{Shad}_{x}(gx, \tau + 8\delta)\subseteq \text{Shad}_{x}(y_g, 2\tau + 8\delta) &\subseteq B(z_g, e^{2\tau+8\delta}e^{-d(x,y_g)})\nonumber\\ &\subseteq B(z_g, e^{5\tau+8\delta}\rho_j), \end{align} $$

by Lemma 2.4. This shows that $\Lambda _\tau $ is covered by at most $e^{(\underline {h_\Gamma }(X) + \varepsilon )k_j\tau }$ generalized visual balls of radius $e^{5\tau +8\delta }\rho _j$ . Therefore,

$$ \begin{align*} \begin{aligned} \underline{\text{MD}}(\Lambda_{\tau})&\leq \liminf_{j \to +\infty}\frac{\log\text{Cov}(\Lambda_{\tau}, e^{5\tau + 8\delta}\rho_j)}{\log ({1}/{e^{5\tau + 8\delta}\rho_j})} \\ &\leq \liminf_{j \to +\infty}\frac{(\underline{h_\Gamma}(X) + \varepsilon)k_j\tau}{-5\tau - 8\delta + (k_j-1)\tau} = \underline{h_\Gamma}(X)+\varepsilon. \end{aligned} \end{align*} $$

By the arbitrariness of $\varepsilon $ , we conclude the proof.

There are several remarks we can do about this proof.

1. (a) The proof is still valid for every sequence $T_j \to + \infty $ , so it implies also that $\sup _{\tau \geq 0} \overline {\mathrm{MD}}(\Lambda _\tau ) \leq h_\Gamma $ . Therefore, we have another improvement of the Bishop–Jones theorem, namely:

   (10) $$ \begin{align} \sup_{\tau \geq 0} {\mathrm{HD}}(\Lambda_\tau) = \sup_{\tau \geq 0} \underline{\mathrm{MD}}(\Lambda_\tau) = \sup_{\tau \geq 0} \overline{\mathrm{MD}}(\Lambda_\tau)=h_\Gamma. \end{align} $$

2. (b) $\Lambda _{\mathrm{u-rad}}\hspace{-1pt} =\hspace{-1pt} \bigcup _{\tau \in \mathbb {N}}\Lambda _\tau $ , so by item (a) and equation (8), we deduce that $\mathrm{PD}(\Lambda _{\mathrm{u-rad}})\hspace{-1pt}=\hspace{-1pt}h_\Gamma $ .

3. (c) We can get the same estimate of the Minkowski dimensions from above, weakening the assumptions on the sets $\Lambda _\tau $ . Indeed, take a set $\Lambda _{\tau , \Theta }$ such that $\limsup _{i\to +\infty } ({\vartheta _{i+1}}/{\vartheta _i}) = 1$ . Then we can cover this set by shadows caste by points of the orbit $\Gamma x$ whose distance from x is between $\vartheta _{i_j}$ and $\vartheta _{i_j + 1}$ , with $i_j \to +\infty $ when $j \to + \infty $ . Therefore, arguing as before, we obtain

   $$ \begin{align*}\underline{\mathrm{MD}}(\Lambda_{\tau, \Theta}) \leq \liminf_{j \to +\infty}\frac{(\underline{h_\Gamma}(X) + \varepsilon)\vartheta_{i_j + 1}}{\vartheta_{i_j - 1}} \leq \underline{h_\Gamma}(X)+\varepsilon,\end{align*} $$
   where the last step follows by the asymptotic behavior of the sequence $\Theta $ . A similar estimate holds for the upper Minkowski dimension.

4. (d) One could be tempted to conclude that the packing dimension of the set $\bigcup _{\tau \geq 0}\bigcup _{\Theta } \Lambda _{\tau , \Theta }$ , where $\Theta $ is a sequence such that $\limsup _{i\to +\infty } ({\vartheta _{i+1}}/{\vartheta _i}) = 1$ , is $\leq h_\Gamma $ . However, this is not necessarily true since in equation (8), a countable covering is required and not an arbitrary covering. That is why the estimate of the packing dimension of the ergodic limit set $\Lambda _{\mathrm{erg}}$ in Theorem A is not so easy. However, as we will see in a moment, the ideas behind the proof are similar to those used in the proof of Theorem B.

Proof of Theorem A

We notice it is enough to prove that PD $(\Lambda _{\text {erg}}) \leq h_\Gamma $ . The strategy is the following: for every $\varepsilon> 0$ , we want to find a countable family of sets $\lbrace B_k\rbrace _{k\in \mathbb {N}}$ of $\partial X$ such that $\Lambda _{\text {erg}} \subseteq \bigcup _{k=1}^\infty B_k$ and $\sup _{k\in \mathbb {N}}\overline {\text {MD}}(B_k)\leq (h_\Gamma + \varepsilon )(1+\varepsilon )$ . Indeed, if this is true, then by equation (8):

$$ \begin{align*}\text{PD}(\Lambda_{\text{erg}}) \leq \sup_{k\in \mathbb{N}}\overline{\text{MD}}(B_k)\leq (h_\Gamma + \varepsilon)(1+\varepsilon),\end{align*} $$

and by the arbitrariness of $\varepsilon $ , the thesis is true.

So we fix $\varepsilon> 0$ and we proceed to define the countable family. For $m,n\in \mathbb {N}$ and $l\in \mathbb {Q}_{> 0}$ , we define

$$ \begin{align*}B_{m,l,n} = \bigcup_{\Theta} \Lambda_{m,\Theta},\end{align*} $$

where $\Theta $ is taken among all sequences such that for every $i\geq n$ , it holds that

$$ \begin{align*} l-\eta_l \leq \frac{\vartheta_{i}}{i} \leq l + \eta_l, \end{align*} $$

where $\eta _l = ({\varepsilon }/({2+\varepsilon })) \cdot l$ .

First of all, if $z\in \Lambda _{\text {erg}}$ , we know that $z\in \Lambda _{m,\Theta }$ for some $m \in \mathbb {N}$ and $\Theta $ satisfying $\lim _{i\to +\infty } ({\vartheta _{i}}/{i})\hspace{-1pt} =\hspace{-1pt} L\hspace{-1pt} \in\hspace{-1pt} (0,\infty )$ , in particular, there exists $n\hspace{-1pt}\in\hspace{-1pt} \mathbb {N}$ such that $L\hspace{-1pt} -\hspace{-1pt} \beta\hspace{-1pt} \leq\hspace{-1pt} {\vartheta _{i}}/{i}\hspace{-1pt} \leq L + \beta $ for every $i\geq n$ , where $\beta = (({2+\varepsilon })/({4+3\varepsilon }))\cdot \eta _L$ . Now we take $l \in \mathbb {Q}_{>0}$ such that $\vert L - l \vert < \beta $ . Then it is easy to see that $[L-\beta ,L + \beta ]\subseteq [l-2\beta , l+ 2\beta ]$ and $\eta _l \geq \eta _L -( {\varepsilon }/({2+\varepsilon }))\beta \geq 2\beta $ . So by definition, $z\in B_{m,l,n}$ ; therefore, $\Lambda _{\text {erg}}\subseteq \bigcup _{m,l,n} B_{m,l,n}.$

Now we need to estimate the upper Minkowski dimension of each set $B_{m,l,n}$ . We take $T_0$ big enough such that

$$ \begin{align*} \frac{1}{T}\log \#(\Gamma x \cap \overline{B}(x,T)) \leq h_\Gamma + \varepsilon \end{align*} $$

for every $T\geq T_0$ . Let us fix $\rho \leq e^{-\max \lbrace T_0, n(l-\eta _l) \rbrace }$ . We consider $j\in \mathbb {N}$ with the following property: $(j-1)(l-\eta _l) < \log ({1}/{\rho })\leq j(l-\eta _l)$ . We observe that the condition on $\rho $ gives $\log ({1}/{\rho }) \geq n(l-\eta _l)$ , implying $j\geq n$ .

We consider the set of elements $g\in \Gamma $ such that

(11) $$ \begin{align} j(l-\eta_l) - m \leq d(x,gx) \leq (j+1)(l+\eta_l) + m. \end{align} $$

For any such g, we consider the shadow $\text {Shad}_{x}(gx,2m + 8\delta )$ . We claim that this set of shadows covers $B_{m,l,n}$ . Indeed, every point z of $B_{m,l,n}$ belongs to some $\Lambda _{m,\Theta }$ with $l-\eta _l \leq {\vartheta _{i}}/{i} \leq l + \eta _l$ for every $i\geq n$ . In particular, this holds for $i=j$ , and so $j(l-\eta _l) \leq \vartheta _j \leq j(l+\eta _l)$ . Hence, there exists a point y along a geodesic ray $[x,z]$ satisfying:

$$ \begin{align*}j(l-\eta_l)\leq \vartheta_j \leq d(x,y)\leq \vartheta_{j+1}\leq (j+1)(l+\eta_l), \quad d(y,\Gamma x) \leq m.\end{align*} $$

So there is $g\in \Gamma $ satisfying equation (11) such that $z\in \text {Shad}_{x}(gx,2m + 8\delta )$ , by Lemma 2.2. Moreover, these shadows are caste by points at distance at least $j(l-\eta _l) - m$ from x, so at distance at least $\log ({1}/{e^m\rho })$ from x. We need to estimate the number of such g elements. By the assumption on $\rho $ , we get that this number is less than or equal to $e^{(h_\Gamma +\varepsilon )[(j+1)(l+\eta _l) + m]}.$ Hence, using again Lemma 2.4, we conclude that $B_{m,l,n}$ is covered by at most $e^{(h_\Gamma +\varepsilon )[(j+1)(l+\eta _l) + m]}$ generalized visual balls of radius $e^{5m+8\delta }\rho $ . Thus,

$$ \begin{align*} \begin{aligned} \overline{\text{MD}}(B_{m,l,n})&=\limsup_{\rho \to 0}\frac{\log\text{Cov}(B_{m,l,n}, e^{5m+8\delta}\rho)}{\log ({1}/{e^{5m+8\delta}\rho})}\\ &\leq \limsup_{j\to +\infty}\frac{(h_\Gamma+\varepsilon)[(j+1)(l+\eta_l) + m]}{-5m -8\delta + (j-1)(l-\eta_l)} \\ &\leq (h_\Gamma+\varepsilon)(1+\varepsilon), \end{aligned} \end{align*} $$

where the last inequality follows from the choice of $\eta _l$ .

6. An interpretation of the ergodic limit set

Let X be a proper metric space. The space of parameterized geodesic lines of X is

$$ \begin{align*} \text{Geod}(X) = \lbrace \gamma\colon \mathbb{R} \to X \text{ isometric embedding}\rbrace, \end{align*} $$

considered as a subset of $C^0(\mathbb {R},X)$ , the space of continuous maps from $\mathbb {R}$ to X endowed with the uniform convergence on compact subsets of $\mathbb {R}$ . By lower semicontinuity of the length under uniform convergence (cf. [Reference Bridson and HaefligerBH13, Proposition I.1.20]), we have that $\text {Geod}(X)$ is closed in $C^0(\mathbb {R},X)$ . There is a natural action of $\mathbb {R}$ on $\text {Geod}(X)$ defined by reparameterization:

$$ \begin{align*} \Phi_t\gamma (\cdot) = \gamma(\cdot + t) \end{align*} $$

for every $t\in \mathbb {R}$ . It is a continuous action, that is, the map $\Phi _t$ is a homeomorphism of $\text {Geod}(X)$ for every $t\in \mathbb {R}$ and $\Phi _t \circ \Phi _s = \Phi _{t+s}$ for every $t,s\in \mathbb {R}$ . This action is called the geodesic flow on X.

Let $\Gamma $ be a discrete group of isometries of X. We consider the quotient space $\Gamma \backslash X$ and the standard projection $\pi \colon X \to \Gamma \backslash X$ . On the quotient, a standard pseudometric is defined by $d(\pi x, \pi y) = \inf _{g\in \Gamma }d(x, gy)$ . Since the action is discrete, then this pseudometric is actually a metric. Indeed, if $d(\pi x, \pi y)= 0$ , then for every $n> 0$ , there exists $g_n\in \Gamma $ such that $d(x,g_ny)\hspace{-1pt}\leq\hspace{-1pt} {1}/{n}$ . In particular, $d(x,g_nx)\hspace{-1pt}\leq\hspace{-1pt} d(x,g_ny)\hspace{-1pt} +\hspace{-1pt} d(g_ny,g_nx) \leq d(x,y)\hspace{-1pt} +\hspace{-1pt} 1$ for every n. The cardinality of these $g_n$ is finite, and thus there must be one of these $g_n$ such that $d(x, g_ny) = 0$ , that is, $x= g_ny$ , and so $\pi x = \pi y$ .

The group $\Gamma $ acts on $\mathrm{Geod}(X)$ by $(g\gamma )(\cdot ) = g(\gamma (\cdot ))$ . This action is by homeomorphisms and we define the space

$$ \begin{align*} \mathrm{Proj-Geod}(\Gamma\backslash X) := \Gamma \backslash \mathrm{Geod}(X), \end{align*} $$

endowed with the quotient topology. The elements of $\text {Proj-Geod}(\Gamma \backslash X)$ will be denoted by $[\gamma ]$ , where $\gamma \in \text {Geod}(X)$ is a representative. The action of $\Gamma $ commutes with the flow $\Phi _t$ in the sense that $g\circ \Phi _t = \Phi _t \circ g$ for every $g\in \Gamma $ and $t\in \mathbb {R}$ . Therefore, the flow $\Phi _t$ defines a flow on $\mathrm{Proj-Geod}(\Gamma \backslash X)$ , that is, an action of $\mathbb {R}$ by homeomorphisms. This flow, still denoted $\Phi _t$ , is called the geodesic flow on $\Gamma \backslash X$ .

The couple $(\mathrm{Proj-Geod}(\Gamma \backslash X), \Phi _1)$ , where $\Phi _1$ is the geodesic flow of $\Gamma \backslash X$ at time  $1$ , is a dynamical system. An important role in its study is played by $\Phi _1$ -invariant probability measures, that is, Borel measures $\mu $ on $\mathrm{Proj-Geod}(\Gamma \backslash X)$ with total mass $1$ and such that $(\Phi _1)_\#\mu = \mu $ , where $(\Phi _1)_\#$ denotes the pushforward. The set of $\Phi _1$ -invariant probability measures is a closed, convex subset of all Borel measures on $\mathrm{Proj-Geod}(\Gamma \backslash X)$ , whose extremal points are ergodic. We recall that a $\Phi _1$ -invariant probability measure is ergodic if for every $\Phi _1$ -invariant subset $A\subseteq \mathrm{Proj-Geod}(\Gamma \backslash X)$ , that is, such that ${\Phi _1^{-1}(A) = \Phi _{-1}(A) \subseteq A}$ , we have $\mu (A) \in \{0,1\}$ . Ergodic measures satisfy the famous Birkhoff ergodic theorem that we now state in our specific situation.

Proposition 6.2. Let X be a proper metric space, let $\Gamma < \mathrm{Isom}(X)$ be discrete. Let $(\mathrm{Proj-Geod}(\Gamma \backslash X), \Phi _1)$ be the geodesic flow on $\Gamma \backslash X$ as defined above. Let $\mu $ be an ergodic, $\Phi _1$ -invariant probability measure. For every $f \in L^1(\mu )$ , it holds that

(12) $$ \begin{align} \lim_{N\to +\infty}\frac{1}{N}\sum_{j=0}^{N-1} (f \circ \Phi_j)([\gamma]) = \int f\,d\mu \end{align} $$

for $\mu $ -almost every (a.e.) $[\gamma ] \in \mathrm{Proj-Geod}(\Gamma \backslash X)$ . In other words, the limit in equation (12) exists for $\mu $ -a.e. $[\gamma ] \in \mathrm{Proj-Geod}(\Gamma \backslash X)$ and equals the right-hand side.

The next result, which is a reformulation of Theorem C, motivates the name of the ergodic limit set.

Theorem 6.3. Let X be a proper, $\delta $ -hyperbolic space. Let $\Gamma < \mathrm{Isom}(X)$ be discrete and non-elementary. Let $\mu $ be an ergodic, $\Phi _1$ -invariant, probability measure on $\mathrm{Proj-Geod}(\Gamma \backslash X)$ . Then $\mu $ is concentrated on the set

$$ \begin{align*}\{[\gamma] \in \mathrm{Proj-Geod}(\Gamma\backslash X) \, : \, \gamma^\pm \in \Lambda_{\mathrm{erg}}\}.\end{align*} $$

Notice that the property $\gamma ^\pm \in \Lambda _{\mathrm{erg}}$ is well defined, that is, it does not depend on the representative of the class $[\gamma ]$ . This follows by the $\Gamma $ -invariance of $\Lambda _{\mathrm{erg}}$ , see Lemma 4.2.

Proof. Since X is proper, we can find a countable set $\{x_i\}_{i\in \mathbb {N}}\subseteq X$ such that $X = \bigcup _{i\in \mathbb {N}} B(x_i,1)$ . For every i, we define the sets

$$ \begin{align*} V_i := \{ \gamma \in \text{Geod}(X)\,:\, \gamma(0) \in B(x_i,1) \}\end{align*} $$

and

$$ \begin{align*} U_i := \Gamma \backslash V_i \subseteq \text{Proj-Geod}(\Gamma\backslash X).\end{align*} $$

Since $\{V_i\}_{i\in \mathbb {N}}$ is a covering of $\text {Geod}(X)$ , then also $\{U_i\}_{i\in \mathbb {N}}$ is a covering of $\text {Proj-Geod}(\Gamma \backslash X)$ . In particular, there must be some $i_0\in \mathbb {N}$ such that $\mu (U_{i_0}) = c> 0$ . To every $[\gamma ] \in U_{i_0}$ , we associate the set of integers $\Theta ([\gamma ]) = \lbrace \vartheta _i([\gamma ])\rbrace $ defined recursively by

$$ \begin{align*} \vartheta_0([\gamma])=0, \quad \vartheta_{i+1}([\gamma]) = \min \lbrace n\in\mathbb{N}, n> \vartheta_i([\gamma]) \text{ such that } \Phi_n([\gamma]) \in U_{i_0}\rbrace. \end{align*} $$

We apply Proposition 6.2 to the indicator function of the set $U_{i_0}$ , namely $\chi _{U_{i_0}}$ , obtaining that for $\mu $ -a.e. $[\gamma ] \in \text {Proj-Geod}(\Gamma \backslash X)$ , it holds that

$$ \begin{align*} \text{there exists } \lim_{N\to +\infty}\frac{1}{N}\sum_{j=0}^{N-1} (\chi_{U_{i_0}} \circ \Phi_j)([\gamma]) = \mu(U_{i_0}) = c \in (0,1]. \end{align*} $$

We remark that $(\chi _{U_{i_0}} \circ \Phi _j)([\gamma ]) = 1$ if and only if $j\in \Theta ([\gamma ])$ and it is $0$ otherwise. So

$$ \begin{align*}\lim_{N\to +\infty}\frac{1}{N}\sum_{j=0}^{N-1} (\chi_{U_{i_0}} \circ \Phi_j)([\gamma]) = \lim_{N\to +\infty}\frac{\#\Theta([\gamma]) \cap [0,N-1]}{N},\end{align*} $$

and the right-hand side is by definition the density of the set $\Theta ([\gamma ])$ . It is classical that, given the standard increasing enumeration $\lbrace \vartheta _0([\gamma ]), \vartheta _1([\gamma ]),\ldots \rbrace $ of $\Theta ([\gamma ])$ , it holds that

$$ \begin{align*}\lim_{N\to +\infty}\frac{\#\Theta([\gamma]) \cap [0,N-1]}{N} = \lim_{N\to +\infty}\frac{N}{\vartheta_N([\gamma])}.\end{align*} $$

Putting all together, we conclude that for $\mu $ -a.e. $[\gamma ] \in \text {Proj-Geod}(\Gamma \backslash X)$ , the following is true:

(13) $$ \begin{align} \text{there exists } \lim_{N\to +\infty}\frac{\vartheta_N([\gamma])}{N} = \frac{1}{c} \in [1,+\infty). \end{align} $$

In the same way, applying the same argument to the flow at time $-1$ , we get that for $\mu $ -a.e. $[\gamma ] \in \text {Proj-Geod}(\Gamma \backslash X)$ , we have

(14) $$ \begin{align} \text{there exists } \lim_{N\to +\infty}\frac{\vartheta_N([-\gamma])}{N} = \frac{1}{c} \in [1,+\infty). \end{align} $$

Here, $-\gamma $ denotes the curve $-\gamma (t) = \gamma (-t)$ . We deduce that equations (13) and (14) hold together for $\mu $ -a.e. $[\gamma ] \in \text {Proj-Geod}(\Gamma \backslash X)$ . Finally, we need to prove that for every $[\gamma ] \in \text {Proj-Geod}(\Gamma \backslash X)$ satisfying equations (13) and (14), we have $\gamma ^{\pm } \in \Lambda _{\text {erg}}$ . We show that $\gamma ^+ \in \Lambda _{\text {erg}}$ , the other being similar. We notice that an integer n satisfies $n \in \Theta ([\gamma ])$ if and only if there exists a representative $g\gamma $ of $[\gamma ]$ , with $g\in \Gamma $ , such that $\Phi _n(g\gamma ) \in V_{i_0}$ , that is, $g\gamma (n) \in B(x_{i_0},1)$ . In other words, $n \in \Theta ([\gamma ])$ if and only if

(15) $$ \begin{align} d(\gamma(n), \Gamma x_{i_0}) < 1. \end{align} $$

We choose $x_{i_0}$ as the basepoint of X. We fix a geodesic ray $\xi = [x_{i_0}, \gamma ^+]$ . By Lemma 2.2, we have that $d(\xi (t), \gamma (t)) \leq 8\delta + 1$ for every $t\geq 0$ . This, together with equation (15) says that $d(\xi (\vartheta _N([\gamma ])), \Gamma x_{i_0}) < 8\delta + 2$ . By definition, this means that $\gamma ^+ \in \Lambda _{\tau , \Theta ([\gamma ])}$ , where $\tau = 8\delta + 2$ . Finally, we observe that the sequence $\Theta ([\gamma ])=\lbrace \vartheta _N([\gamma ])\rbrace $ satisfies equation (13), which is exactly the condition that defines a sequence involved in the definition of $\Lambda _{\text {erg}}$ . Repeating the argument for $\gamma ^-$ , we get the thesis.