# Topological entropy of the geodesic flow of non-positively curved metric
spaces

Nicola Cavallucci

###### Abstract

We extend the classical Otal-Peigné’s Theorem to the class of packed, Gromov-
hyperbolic spaces with a convex geodesic bicombing. Namely we prove that when
a group acts discretely and freely by isometries on one of these spaces then
its critical exponent equals the topological entropy of the geodesic flow of
the quotient metric space. An important tool will be a refinement of the
classical Bishop-Jones’ Theorem.

###### Contents

  1. 1 Introduction
  2. 2 Gromov-hyperbolic spaces
    1. 2.1 Gromov boundary
    2. 2.2 Visual metrics
  3. 3 Hausdorff and Packing dimensions
    1. 3.1 Definitions of Hausdorff and Packing dimensions
    2. 3.2 Local dimensions of measures
    3. 3.3 Visual dimensions
  4. 4 Bishop-Jones Theorem
    1. 4.1 Limit sets
    2. 4.2 Bishop-Jones’ Theorem
  5. 5 Dynamics on packed GCB spaces
    1. 5.1 The space of local geodesics
    2. 5.2 Packed GCB-spaces
  6. 6 Upper bound of the entropy
  7. 7 Lower bound of the entropy
  8. 8 Additional remarks

##  1 Introduction

The relation between the topological entropy of the geodesic flow of a
Riemannian manifold $(M,g)$ with non-positive sectional curvature and the
critical exponent of the discrete group of isometries $\Gamma\cong\pi_{1}(M)$
acting freely on the universal covering $\tilde{M}$ of $M$ has been
intensively studied during the years. A.Manning proved that in case of compact
manifolds with non-positive sectional curvature the topological entropy of the
geodesic flow equals the volume entropy of the universal covering (cp.
[Man79]) and so the critical exponent of the group $\Gamma$, whose definition
will be recalled later. This result has been generalized to compact quotients
of geodesically complete, CAT$(0)$-spaces by R.Ricks [Ric19]. Its proof works
in case of Busemann convex metric spaces, but it is actually enough to
consider metric spaces satisfying a weaker notion of non-positive curvature:
that is spaces supporting a convex geodesic bicombing that is geodesically
complete. We recall that a geodesic bicombing is a map $\sigma\colon X\times
X\times[0,1]$ such that for all $x,y\in X$ the map
$t\mapsto\sigma(x,y,\cdot)=\sigma_{xy}(\cdot)$ is a geodesic between $x$ and
$y$ parametrized proportionally to arc-length. Every geodesic $\sigma_{xy}$ is
said a $\sigma$-geodesic. The bicombing $\sigma$ is convex if the map
$t\mapsto d(\sigma_{xy}(t),\sigma_{x^{\prime}y^{\prime}}(t))$ is convex on
$[0,1]$ for every $x,y,x^{\prime},y^{\prime}\in X$, while it is geodesically
complete if every $\sigma$-geodesic can be extended to a bigger
$\sigma$-geodesic. Examples of these spaces are geodesically complete CAT$(0)$
and Busemann convex metric spaces, but there are also examples that are not
uniquely geodesic, like all Banach spaces.

However we are more interested in the much more complicated case of non-
cocompact actions. For Riemannian manifolds the following is true:

######  Theorem (Otal-Peigné, [OP04]).

Let $M=\Gamma\backslash\tilde{M}$ be a Riemannian manifold with pinched,
negative sectional curvature, i.e. $-b^{2}\leq\textup{Sec}_{g}\leq-a^{2}<0$.
Then the topological entropy of the geodesic flow on $M$ equals the critical
exponent of $\Gamma$.

The price to pay in order to consider any possible group acting discretely and
freely on $\tilde{M}$ is the condition on the sectional curvature. While the
lower bound is quite natural, since every compact manifold has such a bound,
the negative upper bound marks a difference among the cocompact case and the
general one. The proof of Otal-Peigné’s Theorem uses local estimates that are
true only in the strict negative curvature setting.

The purpose of this paper is to extend Otal-Peigné’s Theorem to a wider class
of metric spaces. As in [Cav21] we will consider Gromov-hyperbolic, packed,
GCB-spaces $(X,\sigma)$. The packing condition, that is a uniform upper bound
on the cardinality of $2r_{0}$-nets inside any ball of radius $3r_{0}$ for
some $r_{0}>0$, can be considered as a weak lower bound on the curvature (cp.
[CS20b], [CS20a], [Cav21]), while Gromov-hyperbolicity is a condition of
negative curvature at large scales. Finally the GCB-condition, that is the
existence of a geodesically complete, convex geodesic bicombing, gives some
control on the local geometry, but it is only a very weak notion of non-
positive curvature. Before stating our generalization of Otal-Peigné’s Theorem
we need to introduce some terminology, due to the difference from the manifold
case to ours.
If a group $\Gamma$ acts discretely and freely by $\sigma$-isometries on $X$
(this means that $g\sigma_{xy}=\sigma_{g(x)g(y)}$ for every $g\in\Gamma$ and
every $x,y\in X$) then it acts properly discountinously by homeomorphisms on
the space of $\sigma$-geodesic lines Geod${}_{\sigma}(X)$, that is the space
of geodesic lines whose finite subsegments are $\sigma$-geodesics. The
quotient by this action is the space Loc-Geod${}_{\sigma}(\Gamma\backslash
X)$, called the space of $\sigma$-local geodesic lines of $\Gamma\backslash
X$. The space Loc-Geod${}_{\sigma}(\Gamma\backslash X)$ has a natural
reparametrization flow $\\{\Phi_{t}\\}_{t\in\mathbb{R}}$ defined by
$\Phi_{t}\gamma=\gamma(\cdot+t)$. We will be interested to the dynamical
system $(\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$. We remark
that in case of geodesically complete Busemann convex or CAT$(0)$-spaces every
$\sigma$-geodesic line is a geodesic line and $\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$ is exactly the space of all local
geodesics of the quotient space $\Gamma\backslash X$, as in case of manifolds.

The topological entropy of the dynamical system $(\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$ is by definition:

$h_{\text{top}}=\sup_{\mu\in\mathcal{E}_{1}}h_{\mu},$

where $\mathcal{E}_{1}$ is the set of ergodic invariant measures of the
dynamical system and $h_{\mu}$ is the Kolmogorov-Sinai entropy of $\mu$. Other
versions of entropy, inspired by the one introduced by Brin-Katok ([BK83]),
are the lower and upper local entropies, namely:

$\underline{h}^{\text{loc}}=\sup_{\mu\in\mathcal{E}_{1}}\inf_{\text{\texthtd}}\underline{h}^{\text{loc}}_{\mu,\text{\texthtd}},\qquad\overline{h}^{\text{loc}}=\sup_{\mu\in\mathcal{E}_{1}}\inf_{\text{\texthtd}}\overline{h}^{\text{loc}}_{\mu,\text{\texthtd}},$

where the infimum is among all distances on $\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X,\Phi_{1})$ inducing its topology,

$\underline{h}^{\text{loc}}_{\mu,\text{\texthtd}}=\underset{\mu}{\text{ess}\inf}\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{\text{\texthtd}^{n}}(\gamma,r))$

and

$\overline{h}^{\text{loc}}_{\mu,\text{\texthtd}}=\underset{\mu}{\text{ess}\sup}\lim_{r\to
0}\limsup_{n\to+\infty}-\frac{1}{n}\log\mu(B_{\text{\texthtd}^{n}}(\gamma,r)).$

Here $B_{\text{\texthtd}^{n}}(\gamma,r)$ is the classical $n$-dynamical ball
of center $\gamma$ and radius $r$.
The topological entropy $h_{\text{top}}$ has an interpretation à la Bowen in
consequence of the variational principle (see [HKR95]):

$h_{\text{top}}=\inf_{\textup{\texthtd}}\sup_{K}\lim_{r\to
0}\limsup_{n\to+\infty}\frac{1}{n}\log\text{Cov}_{\text{\texthtd}^{n}}(K,r),$
(1)

where the infimum is again among all distances inducing the topology of
$\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$, the supremum is among every
possible compact subsets of $\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$
and $\text{Cov}_{\text{\texthtd}^{n}}(K,r)$ is the covering number at scale
$r$ of the set $K$ with respect to the dynamical distance
$\text{\texthtd}^{n}$.
If in (1) we consider the supremum only among compact $\Phi_{1}$-invariant
subsets we obtain another quantity which we call the invariant-topological
entropy and that will be denoted by $h_{\text{inv-top}}$. For a generic
dynamical system it holds:

$h_{\text{inv-top}}\leq
h_{\text{top}}\leq\underline{h}^{\text{loc}}\leq\overline{h}^{\text{loc}}.$

The main result of the paper is that in our case these inequalities are
actually equalities and that the common value equals the critical exponent of
$\Gamma$.

######  Theorem A.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $\Gamma$ be a group of $\sigma$-isometries of $X$ acting
discretely and freely. Then

$h_{\textup{inv-
top}}=h_{\textup{top}}=\underline{h}^{\textup{loc}}=\overline{h}^{\textup{loc}}=h_{\Gamma},$

where $h_{\Gamma}$ is the critical exponent of $\Gamma$.

This result clearly generalizes Otal-Peigné’s Theorem. The critical exponent
of $\Gamma$ is defined as

$h_{\Gamma}=\limsup_{T\to+\infty}\frac{1}{T}\log\\#\Gamma x\cap B(x,T).$

Using Bishop-Jones’ Theorem we will conclude that the limit superior in the
definition of $h_{\Gamma}$ is a true limit, generalizing Roblin’s Theorem
[Rob02] holding for CAT$(-1)$-spaces to Gromov-hyperbolic metric spaces.
Indeed we have:

######  Theorem B.

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $\Gamma$ be a
discrete group of isometries of $X$. Then

$\limsup_{T\to+\infty}\frac{1}{T}\log\\#\Gamma x\cap
B(x,T)=\liminf_{T\to+\infty}\frac{1}{T}\log\\#\Gamma x\cap B(x,T)=h_{\Gamma}.$

The proof of Theorem A is divided in two parts: $h_{\text{inv-top}}\geq
h_{\Gamma}$ and $\overline{h}^{\text{loc}}\leq h_{\Gamma}$.
The first inequality is based on the estimate of the topological entropy of
the compact $\Phi_{1}$-invariant subsets made by local geodesics never
excaping a fixed compact set of $\Gamma\backslash X$. Namely we will fix a
basepoint $x_{0}\in\Gamma\backslash X$ and for every $\tau\geq 0$ we will
consider the set $K_{\tau}$ of $\sigma$-local geodesic lines completely
contained in the compact ball of radius $\tau$ around $x_{0}$. Since the sets
$K_{\tau}$ are compact and $\Phi_{1}$-invariant we will use them to estimate
from below the invariant-topological entropy. Using special distances on
$K_{\tau}$ we will be able to prove that the topological entropy of the system
$(K_{\tau},\Phi_{1})$ is at least the lower Lipschitz-topological entropy (see
[Cav21] for more properties of this invariant) of the set
Geod${}_{\sigma}(\Lambda_{\tau})$ of $\sigma$-geodesic lines of the universal
cover $X$ with both endpoints in the $\tau$-uniform radial set
$\Lambda_{\tau}$. The $\tau$-uniform radial set is the subset of the limit set
of $\Gamma$ made by geodesic rays $[\tilde{x}_{0},z]$ staying at distance at
most $\tau$ from the orbit $\Gamma\tilde{x}_{0}$, where $\tilde{x}_{0}$ is a
covering point of $x_{0}$. Observe that any such geodesic ray defines a local
geodesic in the quotient that is entirely contained in the compact ball of
radius $\tau$ around $x_{0}$, showing why the topological entropy of
$K_{\tau}$ is related to dynamical properties of the set $\Lambda_{\tau}$. The
lower Lipschitz-topological entropy of the set
Geod${}_{\sigma}(\Lambda_{\tau})$ (whose definition will be recalled in
Section 7) equals the lower Minkowski dimension of $\Lambda_{\tau}$,
$\underline{\text{MD}}(\Lambda_{\tau})$, by a result of [Cav21]. Finally by
Bishop-Jones Theorem we will conclude

$h_{\text{inv-top}}\geq\sup_{\tau\geq
0}h_{\text{top}}(K_{\tau},\Phi_{1})\geq\sup_{\tau\geq
0}\underline{\text{MD}}(\Lambda_{\tau})\geq h_{\Gamma}.$

The proof of the inequality $\overline{h}^{\text{loc}}\leq h_{\Gamma}$ is
inspired to Ledrappier’s work [Led13]. We sketch here the main ideas: to every
ergodic measure $\mu\in\mathcal{E}_{1}$ we associate a (non-canonical) measure
$\nu$ on $\partial X$ with the following properties:

  * -

for every distance it holds

$h_{\mu,\text{\texthtd}}^{\text{loc}}\leq\underset{\nu}{\textup{ess}\sup}\limsup_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho};$ (2)

  * -

$\nu$ gives full measure to a special subset of the limit set of $\Gamma$,
that we will call ergodic limit set and we will denote as
$\Lambda_{\text{erg}}(\Gamma)$.

The intuition behind the definition of the ergodic limit set is the following.
For every geodesic $\gamma\in\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ and
for every compact $K\subseteq\Gamma\backslash X$ of positive $\mu$-measure we
can consider the sequence of returning integer times of $\gamma$ in $K$. By
Birkhoff’s Ergodic Theorem we deduce that this sequence has a very nice
controlled behaviour for $\mu$-a.e.$\gamma$, that is if
$\vartheta_{i}(\gamma)$ is the $i$-th integer time such that
$\gamma(\vartheta_{i}(\gamma))\in K$ then for $\mu$-a.e.$\gamma$
$\exists\lim_{i\to+\infty}\frac{\vartheta_{i}(\gamma)}{i}=\frac{1}{\mu(K)}<+\infty$.
Correspondingly the set of ergodic limit points is defined in a similar way:
we say that a point $z\in\partial X$ belongs to $\Lambda_{\text{erg}}(\Gamma)$
if one (hence every) geodesic ray $[x,z]$, where $x$ is a fixed basepoint of
$X$, satisfies this condition: there are points $y_{i}\in[x,z]$,
$i\in\mathbb{N}$ and a constant $\tau\geq 0$, such that

  * (a)

$d(y_{i},\Gamma x)\leq\tau$ for every $i$,

  * (b)

$d(x,y_{i})\to+\infty$,

  * (c)

$\exists\lim\frac{d(x,y_{i})}{i}<+\infty$.

Observe that conditions (a) and (b) alone define the well known radial limit
points, while (c) captures the idea of well-behaved returning times.
Now, the right hand side of (2) is the classical definition of the upper
packing dimension of the measure $\nu$ (see Section 3) and it is always
bounded from above by the packing dimension of any set of full measure.
Therefore, indicating by PD$(\Lambda_{\text{erg}})$ the packing dimension of
the ergodic limit set, we conclude:

$\overline{h}^{\text{loc}}\leq\text{PD}(\Lambda_{\text{erg}}).$

Finally the end of the proof of this inequality is based on the following
refinement of the easy part of Bishop-Jones’ Theorem:

######  Theorem C.

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $\Gamma$ be a
discrete group of isometries of $X$. Then:

$\textup{PD}(\Lambda_{\textup{erg}})=h_{\Gamma}.$

The packing dimension was introduced by Tricot ([Tri82]) with a dual
construction with respect to the classical Hausdorff dimension (denoted HD).
For a subset $B$ of a general metric space $Z$ it holds

$\text{HD}(B)\leq\text{PD}(B)\leq\underline{\text{MD}}(B)\leq\overline{\text{MD}}(B),$

where the last two quantities are respectively the lower and upper Minkowski
dimension of $B$, where inequalities can well be strict.
An interesting question is the following: is there an example of a proper
$\delta$-hyperbolic space with a discrete group of isometries $\Gamma$ such
that PD$(\Lambda_{\text{rad}}(\Gamma))>h_{\Gamma}$? Here
$\Lambda_{\text{rad}}(\Gamma)$ denotes the radial limit set of $\Gamma$. The
author thinks the answer should be affirmative.

##  2 Gromov-hyperbolic spaces

Let $(X,d)$ be a metric space. The open (resp.closed) ball of radius $r$ and
center $x\in X$ is denoted by $B(x,r)$ (resp. $\overline{B}(x,r)$). If we need
to specify the metric we will write $B_{d}(x,r)$ (resp. $\overline{B}(x,r))$.
A geodesic segment is an isometry $\gamma\colon I\to X$ where $I=[a,b]$ is a a
bounded interval of $\mathbb{R}$. The points $\gamma(a),\gamma(b)$ are called
the endpoints of $\gamma$. A metric space $X$ is said geodesic if for all
couple of points $x,y\in X$ there exists a geodesic segment whose endpoints
are $x$ and $y$. When we will not need to consider a specific geodesic between
$x$ and $y$ we will denote any geodesic segment between them, with an abuse of
notation, by $[x,y]$. A geodesic ray is an isometry $\xi\colon[0,+\infty)\to
X$ while a geodesic line is an isometry $\gamma\colon\mathbb{R}\to X$. A map
$\gamma\colon I\to X$, where $I$ is an interval of $\mathbb{R}$ is a local
geodesic if for every $t\in I$ there exists $\varepsilon>0$ such that
$\gamma|_{[t-\varepsilon,t+\varepsilon]}$ is a geodesic segment.

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

$(y,z)_{x}=\frac{1}{2}\big{(}d(x,y)+d(x,z)-d(y,z)\big{)}.$

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

$(x,z)_{w}\geq\min\\{(x,y)_{w},(y,z)_{w}\\}-\delta$ (3)

or, equivalently,

$d(x,y)+d(z,w)\leq\max\\{d(x,z)+d(y,w),d(x,w)+d(y,z)\\}+2\delta.$ (4)

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 any CAT$(\kappa)$ metric
space, with $\kappa<0$ 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 [LN19], [CS20b] and [CS20a]).

###  2.1 Gromov boundary

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

$\partial X=\\{(z_{n})_{n\in\mathbb{N}}\subseteq X\hskip 2.84526pt|\hskip
2.84526pt\lim_{n,m\to+\infty}(z_{n},z_{m})_{x}=+\infty\\}\hskip
2.84526pt/_{\sim},$

where $(z_{n})_{n\in\mathbb{N}}$ is a sequence of points in $X$ and $\sim$ is
the equivalence relation defined by
$(z_{n})_{n\in\mathbb{N}}\sim(z_{n}^{\prime})_{n\in\mathbb{N}}$ if and only if
$\lim_{n,m\to+\infty}(z_{n},z_{m}^{\prime})_{x}=+\infty$. 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 $[x,z_{n}]$ converges to a
geodesic ray $\xi$ which has the properties above (cp. Lemma III.3.13 of
[BH13]). A geodesic ray joining $x$ to $z\in\partial X$ will be denoted by
$\xi_{xz}$ or simply $[x,z]$. The relation between Gromov product and geodesic
ray is highlighted in the following lemma.

######  Lemma 2.1 ([Cav21], Lemma 5.4).

Let $X$ be a proper, $\delta$-hyperbolic metric space,
$z,z^{\prime}\in\partial X$ and $x\in X$. Then

  * (i)

if $(z,z^{\prime})_{x}\geq T$ then
$d(\xi_{xz}(T-\delta),\xi_{xz^{\prime}}(T-\delta))\leq 4\delta$;

  * (ii)

for all $b>0$, if $d(\xi_{xz}(T),\xi_{xz^{\prime}}(T))<2b$ then
$(z,z^{\prime})_{x}>T-b$.

The quasiconvex hull of a subset $C$ of $\partial X$ is the union of all the
geodesic lines joining two points of $C$ and it is denoted by QC-Hull$(C)$.
The following is a standard computation, see [BCGS17] for instance.

######  Lemma 2.2.

Let $X$ be a proper, $\delta$-hyperbolic metric space. Then every two geodesic
rays $\xi,\xi^{\prime}$ with same endpoints at infinity are at distance at
most $8\delta$, i.e. there exist $t_{1},t_{2}\geq 0$ such that
$t_{1}+t_{2}=d(\xi(0),\xi^{\prime}(0))$ and
$d(\xi(t+t_{1}),\xi^{\prime}(t+t_{2}))\leq 8\delta$ 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 parameter
$a\in\left(0,\frac{1}{2\delta\cdot\log_{2}e}\right)$ and center $x\in X$ if
there exists $V>0$ such that for all $z,z^{\prime}\in\partial X$ it holds

$\frac{1}{V}e^{-a(z,z^{\prime})_{x}}\leq D_{x,a}(z,z^{\prime})\leq
Ve^{-a(z,z^{\prime})_{x}}.$ (5)

For all $a$ as before and $x\in X$ there exists always a visual metric of
parameter $a$ and center $x$, see [Pau96]. As in [Pau96] and [Cav21] we define
the generalized visual ball of center $z\in\partial X$ and radius $\rho\geq 0$
as

$B(z,\rho)=\bigg{\\{}z^{\prime}\in\partial X\text{ s.t.
}(z,z^{\prime})_{x}>\log\frac{1}{\rho}\bigg{\\}}.$

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

######  Lemma 2.3.

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

$B_{D_{x,a}}\left(z,\frac{1}{V}\rho^{a}\right)\subseteq B(z,\rho)\subseteq
B_{D_{x,a}}(z,V\rho^{a}).$

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
$r>0$ casted by a point $y\in X$ with center $x$ is the set:

$\text{Shad}_{x}(y,r)=\\{z\in\partial X\text{ s.t. }[x,z]\cap
B(y,r)\neq\emptyset\text{ for all rays }[x,z]\\}.$

For our purposes we just need:

######  Lemma 2.4.

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

$\textup{Shad}_{x}\left(\xi_{z}\left(T\right),r\right)\subseteq
B(z,e^{-T+r}).$

##  3 Hausdorff and Packing dimensions

In the first part we recall briefly the definitions of Hausdorff and packing
dimensions of a subset of a metric space. In the second part we will recall
four definitions of local dimensions of measures and we will relate them to
Hasudorff and packing dimensions of suitable subsets. Finally 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 literature.

###  3.1 Definitions of Hausdorff and Packing dimensions

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

$\mathcal{H}^{\alpha}_{d}(B)=\lim_{\eta\to
0}\inf\left\\{\sum_{i\in\mathbb{N}}r_{i}^{\alpha}\text{ s.t.
}B\subseteq\bigcup_{i\in\mathbb{N}}B(x_{i},r_{i})\text{ and
}r_{i}\leq\eta\right\\}.$

The argument of the limit is increasing when $\eta$ tends to $0$, so the limit
exists. This formula actually defines a measure on $X$. The Hausdorff
dimension of a Borelian subset $B$ of $X$, denoted HD${}_{d}(B)$ is the unique
real number $\alpha\geq 0$ such that $\mathcal{H}^{\alpha^{\prime}}_{d}(B)=0$
for all $\alpha^{\prime}>\alpha$ and
$\mathcal{H}^{\alpha^{\prime}}_{d}(B)=+\infty$ for all
$\alpha^{\prime}<\alpha$.

The packing dimension is defined in a similar way, but using disjoint balls
inside $B$ instead of coverings. To be precise we define, for all $\alpha\geq
0$ and for all Borelian subsets $B$ of $X$,

$\mathcal{P}^{\alpha}_{d}(B)=\lim_{\eta\to
0}\sup\left\\{\sum_{i\in\mathbb{N}}r_{i}^{\alpha}\text{ s.t.
}B(x_{i},r_{i})\text{ are disjoint, }x_{i}\in B\text{ and
}r_{i}\leq\eta\right\\}.$

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

$\hat{\mathcal{P}}_{d}^{\alpha}(B)=\inf\left\\{\sum_{k=1}^{\infty}\mathcal{P}_{d}^{\alpha}(B_{k})\text{
s.t. }B\subseteq\bigcup_{k=1}^{\infty}B_{k}\right\\}.$

The packing dimension of a Borelian subset $B\subseteq X$, denoted
PD${}_{d}(B)$, is the unique real number $\alpha\geq 0$ such that
$\hat{\mathcal{P}}^{\alpha^{\prime}}_{d}(B)=0$ for all
$\alpha^{\prime}>\alpha$ and
$\hat{\mathcal{P}}^{\alpha^{\prime}}_{d}(B)=+\infty$ for all
$\alpha^{\prime}<\alpha$.
The packing dimension has another useful interpretation (see [Fal04],
Proposition 3.8): indeed for all Borelian subsets $B\subseteq X$ we have

$\text{PD}_{d}(B)=\inf\left\\{\sup_{k}\overline{\text{MD}}_{d}(B_{k})\text{
s.t. }B\subseteq\bigcup_{k=1}^{\infty}B_{k}\right\\}.$ (6)

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

$\overline{\text{MD}}_{d}(B)=\limsup_{r\to
0}\frac{\log\text{Cov}_{d}(B,r)}{\log\frac{1}{r}},$ (7)

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 (7) one defines the lower Minkowski
dimension of $B$, denoted $\underline{\text{MD}}_{d}(B)$.

###  3.2 Local dimensions of measures

Let $(X,d)$ be a metric space, let $\mathscr{B}$ be the $\sigma$-algebra
generated by the Borelian subsets of $X$ and let $\nu$ be a probability
measure on the measure space $(X,\mathscr{B})$. There are several notions of
dimension of the measure $\nu$, here we recall four of them. The _lower and
upper Hausdorff dimensions_ of $\nu$ are respectively:

$\underline{\text{HD}}_{d}(\nu)=\underset{\nu}{\textup{ess}\inf}\liminf_{r\to
0}\frac{\log\nu(B(y,r))}{\log r}$
$\overline{\text{HD}}_{d}(\nu)=\underset{\nu}{\textup{ess}\sup}\liminf_{r\to
0}\frac{\log\nu(B(y,r))}{\log r}$

while the _lower and upper packing dimension_ of $\nu$ are respectively:

$\underline{\text{PD}}_{d}(\nu)=\underset{\nu}{\textup{ess}\inf}\limsup_{r\to
0}\frac{\log\nu(B(y,r))}{\log r},$
$\overline{\text{PD}}_{d}(\nu)=\underset{\nu}{\textup{ess}\sup}\limsup_{r\to
0}\frac{\log\nu(B(y,r))}{\log r}.$

The name Hausdorff and packing dimension is justified by the following facts,
that are well-known at least in the Euclidean spaces.

######  Proposition 3.1.

Let $(X,d)$ be a separable metric space and let $\nu$ be a probability measure
on the measure space $(X,\mathscr{B})$. Then

$\displaystyle\overline{\textup{PD}}_{d}(\nu)$
$\displaystyle=\inf\\{\textup{PD}_{d}(B)\textup{ s.t. }\nu(B)=1\\}$
$\displaystyle\underline{\textup{PD}}_{d}(\nu)$
$\displaystyle=\inf\\{\textup{PD}_{d}(B)\textup{ s.t. }\nu(B)>0\\}$
$\displaystyle\overline{\textup{HD}}_{d}(\nu)$
$\displaystyle=\inf\\{\textup{HD}_{d}(B)\textup{ s.t. }\nu(B)=1\\}$
$\displaystyle\underline{\textup{HD}}_{d}(\nu)$
$\displaystyle=\inf\\{\textup{HD}_{d}(B)\textup{ s.t. }\nu(B)>0\\}.$

###### Proof.

The last two equalities are proved in [Led13], Proposition 2.5 and Section 7.
We remark that the properness assumption in these references can be dropped to
separability.
Let us now show the first equality, the proof of the second equality is
similar and it will be omitted. Let $\alpha=\overline{\textup{PD}}_{d}(\nu)$,
that is for all $\varepsilon>0$ it exists a set $A_{\varepsilon}\subseteq X$
of positive measure such that $\limsup_{r\to 0}\frac{\log\nu(B(x,r))}{\log
r}\geq\alpha-\varepsilon$ for all $x\in A_{\varepsilon}$. Now we apply
Egoroff’s Theorem as in [Led13]. For all $r>0$ and all $x\in X$ we set

$f_{r}(x)=\frac{\log\nu(B(x,r))}{\log r},\qquad
g_{r}(x)=\sup_{0<r^{\prime}<r}f_{r^{\prime}}(x),\qquad
h_{n}(x)=g_{\frac{1}{n}}(x).$

By definition $\limsup_{r\to 0}f_{r}(x)=\lim_{r\to
0}g_{r}(x)=\lim_{n\to+\infty}h_{n}(x)=:\alpha(x)$. By Egoroff’s Theorem there
exists a set of positive measure $A_{\varepsilon}^{\prime}$ such that
$|h_{n}(x)-\alpha(x)|<\varepsilon$ for all $x\in
A_{\varepsilon}^{\prime}\subseteq A_{\varepsilon}$ and all $n\geq
n_{\varepsilon}$. This means that for all
$r<r_{\varepsilon}=\frac{1}{n_{\varepsilon}}$ and for all $x\in
A_{\varepsilon}^{\prime}$ it holds $\frac{\log\nu(B(x,r))}{\log
r}\geq\alpha(x)-\varepsilon\geq\alpha-2\varepsilon$. In other words for all
$r\leq r_{\varepsilon}$ and for all $x\in A_{\varepsilon}^{\prime}$ it holds
$\nu(B(x,r))\leq r^{\alpha-2\varepsilon}$.
Let now $B$ be a borelian subset of $X$ with $\nu(B)=1$. By (6) we know that

$\text{PD}_{d}(B)=\inf\left\\{\sup\overline{\text{MD}}_{d}(B_{k})\text{ s.t.
}\bigcup_{k}B_{k}\supseteq B\right\\}.$

Let $B_{k}$ be subsets of $X$ such that $\bigcup_{k}B_{k}\supseteq B$. Since
$\nu(B)=1$ there must be some index $k$ such that $\nu(B_{k}\cap
A_{\varepsilon}^{\prime})>0$. Observe that from the estimates we proved before
we get

$\text{Cov}_{d}(B_{k}\cap A_{\varepsilon}^{\prime},r)\geq\nu(B_{k}\cap
A_{\varepsilon}^{\prime})\cdot\left(\frac{1}{r}\right)^{\alpha-2\varepsilon}$

for every $r<r_{\varepsilon}$. This directly implies

$\overline{\text{MD}}_{d}(B_{k})\geq\overline{\text{MD}}_{d}(B_{k}\cap
A_{\varepsilon}^{\prime})\geq\alpha-2\varepsilon,$

and so $\text{PD}_{d}(B)\geq\alpha-2\varepsilon$. By the arbitrariness of
$\varepsilon$ we conclude that

$\overline{\textup{PD}}_{d}(\nu)\leq\inf\\{\textup{PD}_{d}(B)\textup{ s.t.
}\nu(B)=1\\}.$

For the other inequality for every $\varepsilon>0$ we take the set

$B_{\varepsilon}=\left\\{x\in X\text{ s.t. }\limsup_{r\to
0}\frac{\log\nu(B(x,r))}{\log r}\leq\alpha+\varepsilon\right\\}.$

Clearly it satisfies $\nu(B_{\varepsilon})=1$. Moreover it can be covered by
the union of the sets

$B_{\varepsilon,k}=\left\\{x\in X\text{ s.t. }\frac{\log\nu(B(x,r))}{\log
r}\leq\alpha+2\varepsilon\text{ for all }r\leq\frac{1}{k}\right\\},$

where $k\in\mathbb{N}$. By similar estimates as before we can conclude that
$\overline{\text{MD}}(B_{\varepsilon,k})\leq\alpha+2\varepsilon$ for every
$k$, and so PD${}_{d}(B_{\varepsilon})\leq\alpha+2\varepsilon$. By the
arbitrariness of $\varepsilon$ we get the desired equality.
∎

###  3.3 Visual dimensions

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $x\in X$. We
know that $\partial X$ 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 $a$. For a Borelian subset $B$ of $\partial X$ and for
all $\alpha\geq 0$ we set, following [Pau96],

$\mathcal{H}^{\alpha}(B)=\lim_{\eta\to
0}\inf\left\\{\sum_{i\in\mathbb{N}}\rho_{i}^{\alpha}\text{ s.t.
}B\subseteq\bigcup_{i\in\mathbb{N}}B(z_{i},\rho_{i})\text{ and
}\rho_{i}\leq\eta\right\\},$

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 $\mathcal{H}^{\alpha^{\prime}}(B)=0$ for all
$\alpha^{\prime}>\alpha$ and $\mathcal{H}^{\alpha^{\prime}}(B)=+\infty$ for
all $\alpha^{\prime}<\alpha$. The visual Hausdorff dimension of the borelian
subset $B$ is denoted by HD$(B)$. By Lemma 2.3, see also [Pau96], we have
HD$(B)=a\cdot\text{HD}_{D_{x,a}}(B)$ for all visual metrics $D_{x,a}$ of
center $x$ and parameter $a$.

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

$\mathcal{P}^{\alpha}(B)=\lim_{\eta\to
0}\sup\left\\{\sum_{i\in\mathbb{N}}\rho_{i}^{\alpha}\text{ s.t.
}B(z_{i},\rho_{i})\text{ are disjoint, }x_{i}\in B\text{ and
}\rho_{i}\leq\eta\right\\},$

and also in this case $B(z_{i},\rho_{i})$ are generalized visual balls. As
usual we can define the visual $\alpha$-packing measure by

$\hat{\mathcal{P}}^{\alpha}(B)=\inf\left\\{\sum_{k=1}^{\infty}\mathcal{P}^{\alpha}(B_{k})\text{
s.t. }B\subseteq\bigcup_{k=1}^{\infty}B_{k}\right\\}.$

Consequently it is defined the visual packing dimension of a Borelian set $B$,
denoted by PD$(B)$. Using Lemma 2.3 as in the case of the Hausdorff measure
(see [Pau96]) it is easy to check that for every visual metric $D_{x,a}$ of
center $x$ and parameter $a$ it holds:

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

for all $\alpha\geq 0$ and all Borelian sets $B\subseteq\partial X$. Therefore
for every Borelian set $B$ it holds 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 also the visual upper and lower Minkowski
dimension of a subset $B\subseteq\partial X$, respectively:

$\overline{\text{MD}}(B)=\limsup_{\rho\to
0}\frac{\log\text{Cov}(B,\rho)}{\log\rho},\qquad\underline{\text{MD}}(B)=\liminf_{\rho\to
0}\frac{\log\text{Cov}(B,\rho)}{\log\rho},$

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 (see also
[Cav21]) one has
$\overline{\text{MD}}(B)=a\cdot\overline{\text{MD}}_{D_{x,a}}(B)$ for every
Borelian set $B$ and every visual metric of center $x$ and parameter $a$. The
same of course holds for the lower Minkowski dimensions.

Moreover it is easy to check that for every Borelian set $B$ of $\partial X$
the numbers HD$(B)$, PD$(B)$, $\underline{\text{MD}}(B)$,
$\overline{\text{MD}}(B)$ do not depend on $x$ too, see Proposition 6.4 of
[Pau96]. Clearly, using their comparisons with the classical dimensions
defined by a visual metric, we get

$\text{HD}(B)\leq\text{PD}(B)\leq\underline{\text{MD}}(B)\leq\overline{\text{MD}}(B)$
(8)

and

$\text{PD}(B)=\inf\left\\{\sup_{k}\overline{\text{MD}}(B_{k})\text{ s.t.
}B\subseteq\bigcup_{k=1}^{\infty}B_{k}\right\\}.$ (9)

for all Borelian subsets $B$ of $\partial X$.

Let $\mathscr{B}$ be the $\sigma$-algebra generated by the Borelian subsets of
$\partial X$ and let $\nu$ be a probability measure on the measure space
$(\partial X,\mathscr{B})$. We define the _visual lower and upper Hausdorff
dimensions_ of $\nu$ as:

$\underline{\text{HD}}(\nu)=\underset{\nu}{\textup{ess}\inf}\liminf_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho}$
$\overline{\text{HD}}(\nu)=\underset{\nu}{\textup{ess}\sup}\liminf_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho}$

while the _visual lower and upper packing dimension_ of $\nu$ are
respectively:

$\underline{\text{PD}}(\nu)=\underset{\nu}{\textup{ess}\inf}\limsup_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho},$
$\overline{\text{PD}}(\nu)=\underset{\nu}{\textup{ess}\sup}\limsup_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho}.$

Here the difference is that $B(z,\rho)$ denotes the generalized visual ball
instead of metric balls with respect to a visual distance $D_{x,a}$. Using
again Lemma 2.3 it is straightforward to prove that

$\underline{\text{HD}}(\nu)=a\cdot\underline{\text{HD}}_{D_{x,a}}(\nu)$

for every visual metric $D_{x,a}$, and similarly for the other dimensions.
Therefore

$\displaystyle\overline{\text{PD}}(\nu)$
$\displaystyle=\inf\\{\text{PD}(B)\text{ s.t. }\nu(B)=1\\}$ (10)
$\displaystyle\underline{\text{PD}}(\nu)$
$\displaystyle=\inf\\{\text{PD}(B)\text{ s.t. }\nu(B)>0\\}$
$\displaystyle\overline{\text{HD}}(\nu)$
$\displaystyle=\inf\\{\text{HD}(B)\text{ s.t. }\nu(B)=1\\}$
$\displaystyle\underline{\text{HD}}(\nu)$
$\displaystyle=\inf\\{\text{HD}(B)\text{ s.t. }\nu(B)>0\\}.$

##  4 Bishop-Jones Theorem

If $X$ is a proper metric space we denote by Isom$(X)$ its group of
isometries, endowed with the uniform convergence on compact subsets of $X$. A
subgroup $\Gamma$ of Isom$(X)$ is called discrete if the following equivalent
conditions (see [BCGS17]) hold:

  * (a)

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

  * (b)

$\forall x\in X$ and $R\geq 0$ the set $\Sigma_{R}(x)=\\{g\in\Gamma\text{ s.t.
}gx\in\overline{B}(x,R)\\}$ 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 ([Cav21], [Coo93]), as

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

where $x$ is a fixed point of $X$. Clearly 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 [Rob02] it is proved that if $\Gamma$ is a
discrete group of isometries of a CAT$(-1)$ space then
$\overline{h_{\Gamma}}(X)=\underline{h_{\Gamma}}(X)$. We will generalize this
result to proper, $\delta$-hyperbolic spaces (see Corollary 4.5).

###  4.1 Limit sets

We specialize the general situation above 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 (cp. [Coo93], 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 $\Gamma\backslash X$, as we will see in
the second part of the paper. In order to recall their definiton we need to
introduce a class of subsets of $\partial X$.
We fix a basepoint $x\in X$. Let $\tau$ and
$\Theta=\\{\vartheta_{i}\\}_{i\in\mathbb{N}}$ 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 $[x,z]$.

######  Lemma 4.1.

In the situation above it holds:

  * (i)

$\Lambda_{\tau,\Theta}(\Gamma)\subseteq\Lambda(\Gamma)$;

  * (ii)

the set $\Lambda_{\tau,\Theta}(\Gamma)$ is closed.

###### Proof.

The first statement is obvious, so we focus on (ii). Let
$z^{k}\in\Lambda_{\tau,\Theta}(\Gamma)$ converging to $z^{\infty}$. For every
$k$ let $\xi^{k}=[x,z^{k}]$ be a geodesic ray as in the definition of
$\Lambda_{\tau,\Theta}(\Gamma)$. We know that, up to a subsequence, the
sequence $\xi^{k}$ converges uniformly on compact sets of $[0,+\infty)$ to a
geodesic ray $\xi^{\infty}=[x,z^{\infty}]$. We fix $i\in\mathbb{N}$ and we
take points $y_{i}^{k}$ with
$d(x,y_{i}^{k})\in[\vartheta_{i},\vartheta_{i+1}]$ and $d(y_{i}^{k},\Gamma
x)\leq\tau$. The sequence $y_{i}^{k}$ converges to a point
$y_{i}^{\infty}\in[x,z^{\infty}]$ with
$d(x,y_{i}^{\infty})\in[\vartheta_{i},\vartheta_{i+1}]$. Moreover clearly
$d(y_{i}^{\infty},\Gamma x)\leq\tau$. Since this is true for every
$i\in\mathbb{N}$ we conclude that
$z^{\infty}\in\Lambda_{\tau,\Theta}(\Gamma)$. ∎

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

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

The _uniform radial limit set_ is defined (see [DSU17]) as

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

where $\Lambda_{\tau}(\Gamma)=\Lambda_{\tau,\\{i\tau\\}}(\Gamma)$.
Another interesting set that will play an important role in the paper is the
_ergodic limit set_ , defined as:

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

where a sequence $\Theta=\\{\vartheta_{i}\\}$ belongs to $\Theta_{\text{erg}}$
if $\exists\lim_{i\to+\infty}\frac{\vartheta_{i}}{i}<+\infty$. The name of
this set will be justified in the second part of the paper: for the moment we
just say it is related to properties of ergodic measures of the geodesic flow
in the quotient space.
When $\Gamma$ is clear in the context we will simply write
$\Lambda_{\tau,\Theta},\Lambda_{\text{rad}},\Lambda_{\text{u-rad}},\Lambda_{\textup{erg}},\Lambda$,
omitting $\Gamma$.

######  Lemma 4.2.

In the situation above the sets
$\Lambda_{\textup{rad}},\Lambda_{\textup{u-rad}}$ and $\Lambda_{\textup{erg}}$
are $\Gamma$-invariant and do not depend on $x$.

###### Proof.

Let $y$ be another point of $X$ and let $z\in\partial X$. By Lemma 2.2 for
every geodesic rays $\xi=[y,z]$, $\xi^{\prime}=[x,z]$ there are
$t_{1},t_{2}\geq 0$ such that $t_{1}+t_{2}\leq d(x,y)$ and
$d(\xi(t+t_{1}),\xi^{\prime}(t+t_{2}))\leq 8\delta$. This means that
$d(\xi(t),\xi^{\prime}(t))\leq d(x,y)+8\delta$ for every $t\geq 0$. It is then
straightforward to see that if $z\in\Lambda_{\tau,\Theta}$ (as defined with
respect to $x$) then it belongs to $\Lambda_{\tau+d(x,y)+8\delta,\Theta}$ as
defined with respect to $y$. This shows the thesis. ∎

###  4.2 Bishop-Jones’ Theorem

The celebrated Bishop-Jones’ Theorem, in the generic version of [DSU17],
states the following:

######  Theorem 4.3 ([BJ97], [DSU17]).

Let $\Gamma$ be a discrete group of isometries of a proper,
$\delta$-hyperbolic metric space $X$. Then:

$h_{\Gamma}=\textup{HD}(\Lambda_{\textup{rad}})=\textup{HD}(\Lambda_{\textup{u-rad}})=\sup_{\tau\geq
0}\textup{HD}(\Lambda_{\tau}).$

Our contribution to Bishop-Jones’ Theorem is the following:

######  Theorem 4.4.

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $\Gamma$ be a
discrete group of isometries of $X$. Then:

$\textup{PD}(\Lambda_{\textup{erg}})=h_{\Gamma}$

In order to introduce the techniques we will use in the proof of this theorem
we start with another easier result that generalizes Roblin’s Theorem,
[Rob02]. We remark that the proof is completely different.

######  Proposition 4.5.

Let $X$ be a proper, $\delta$-hyperbolic metric space and let $\Gamma$ be a
discrete group of isometries. Then the critical exponent is a true limit, i.e.

$h_{\Gamma}=\lim_{T\to+\infty}\frac{1}{T}\log\\#\Gamma x\cap B(x,T).$

###### Proof.

By Bishop-Jones’ Theorem we have

$h_{\Gamma}=\sup_{\tau\geq 0}\text{HD}(\Lambda_{\tau})\leq\sup_{\tau\geq
0}\underline{\text{MD}}(\Lambda_{\tau}).$

So it would be enough to show that

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

We fix $\tau\geq 0$. For every $\varepsilon>0$ we take a subsequence
$T_{j}\to+\infty$ such that for all $j$ it holds

$\frac{1}{T_{j}}\log\\#\Gamma
x\cap\overline{B}(x,T_{j})\leq\underline{h_{\Gamma}}(X)+\varepsilon.$

We define $\rho_{j}=e^{-T_{j}}$: notice that $\rho_{j}\to 0$. Let
$k_{j}\in\mathbb{N}$ such that $(k_{j}-1)\tau<T_{j}\leq k_{j}\tau$. For each
$g\in\Gamma$ such that $(k_{j}-1)\tau\leq d(x,gx)\leq(k_{j}+1)\tau$ we
consider the shadow Shad${}_{x}(gx,2\tau)$. First of all if
$z\in\Lambda_{\tau}$ then we know there exists $g\in\Gamma$ as before such
that $d([x,z],gx)\leq\tau$, so $z\in\text{Shad}_{x}(gx,2\tau)$. In other words
these shadows cover $\Lambda_{\tau}$, and their cardinality is at most
$e^{(\underline{h_{\Gamma}}(X)+\varepsilon)(k_{j}+1)\tau}$. So, by Lemma 2.4,
$\Lambda_{\tau}$ is covered by at most
$e^{(\underline{h_{\Gamma}}(X)+\varepsilon)(k_{j}+1)\tau}$ generalized visual
balls of radius $e^{-T_{j}+3\tau}=e^{3\tau}\rho_{j}$. Therefore

$\displaystyle\underline{\text{MD}}(\Lambda_{\tau})$
$\displaystyle\leq\liminf_{j\to+\infty}\frac{\log\text{Cov}(\Lambda_{\tau},e^{2\tau}\rho_{j})}{\log\frac{1}{e^{3\tau}\rho_{j}}}$
$\displaystyle\leq\liminf_{j\to+\infty}\frac{(\underline{h_{\Gamma}}(X)+\varepsilon)(k_{j}+1)\tau}{-3\tau+(k_{j}-1)\tau}=\underline{h_{\Gamma}}(X)+\varepsilon.$

By the arbitrariness of $\varepsilon$ we can conclude the proof. ∎

There are several remarks we can do about this proof:

  * (a)

The proof is still valid for every sequence $T_{j}\to+\infty$, so it implies
also that $\sup_{\tau\geq 0}\overline{\textup{MD}}(\Lambda_{\tau})\leq
h_{\Gamma}$. Therefore we have another improvement of Bishop-Jones Theorem,
namely:

$\sup_{\tau\geq 0}{\textup{HD}}(\Lambda_{\tau})=\sup_{\tau\geq
0}\underline{\textup{MD}}(\Lambda_{\tau})=\sup_{\tau\geq
0}\overline{\textup{MD}}(\Lambda_{\tau})=h_{\Gamma}.$ (11)

  * (b)

$\Lambda_{\textup{u-rad}}=\bigcup_{\tau\in\mathbb{N}}\Lambda_{\tau}$, so by
(a) and (9) we deduce that $\textup{PD}(\Lambda_{\textup{u-rad}})=h_{\Gamma}$.

  * (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
$\limsup_{i\to+\infty}\frac{\vartheta_{i+1}}{\vartheta_{i}}=1$. Then we can
cover this set by the $2\tau$-shadows casted 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

$\underline{\textup{MD}}(\Lambda_{\tau,\Theta})\leq\liminf_{j\to+\infty}\frac{(\underline{h_{\Gamma}}(X)+\varepsilon)\vartheta_{i_{j}+1}}{\vartheta_{i_{j}-1}}\leq\underline{h_{\Gamma}}(X)+\varepsilon,$

where the last step follows by the asymptotic behaviour of the sequence
$\Theta$. A similar estimate holds for the upper Minkowski dimension.

  * (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
$\limsup_{i\to+\infty}\frac{\vartheta_{i+1}}{\vartheta_{i}}=1$, is $\leq
h_{\Gamma}$. But this is not necessarily true since in (9) it is required a
countable covering and not an arbitrary covering. That is why the estimate of
the packing dimension of the ergodic limit set $\Lambda_{\textup{erg}}$ in
Theorem 4.4 is not so easy. However as we will see in a moment the ideas
behind the proof are similar to the ones used in the proposition above.

###### Proof of Theorem 4.4.

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 $\\{B_{k}\\}_{k\in\mathbb{N}}$ 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 (9):

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

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

$B_{m,l,n}=\bigcup_{\Theta}\Lambda_{m,\Theta},$

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

$l-\eta_{l}\leq\frac{\vartheta_{i}}{i}\leq l+\eta_{l},$

where $\eta_{l}=\frac{\varepsilon}{2+\varepsilon}\cdot l$.
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
$\lim_{i\to+\infty}\frac{\vartheta_{i}}{i}=L<+\infty$, in particular it exists
$n\in\mathbb{N}$ such that for all $i\geq n$ it holds
$L-\beta\leq\frac{\vartheta_{i}}{i}\leq L+\beta$, where
$\beta=\frac{2+\varepsilon}{4+3\varepsilon}\cdot\eta_{L}$. Now we take
$l\in\mathbb{Q}_{>0}$ such that $|L-l|<\beta$. Then it is easy to see that
$[L-\beta,L+\beta]\subseteq[l-2\beta,l+2\beta]$ and
$\eta_{l}\geq\eta_{L}-\frac{\varepsilon}{2+\varepsilon}\beta\geq 2\beta$. 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 for every $T\geq T_{0}$ it holds

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

Let us fix any $\rho\leq e^{-\max\\{T_{0},n(l-\eta_{l})\\}}$. We consider
$j\in\mathbb{N}$ with the following property:
$(j-1)(l-\eta_{l})<\log\frac{1}{\rho}\leq j(l-\eta_{l})$. We observe that the
condition on $\rho$ gives $\log\frac{1}{\rho}\geq n(l-\eta_{l})$, implying
$j\geq n$.
We consider the set of elements $g\in\Gamma$ such that

$j(l-\eta_{l})-m\leq d(x,gx)\leq(j+1)(l+\eta_{l})+m.$ (12)

For any such $g$ we consider the shadow $\text{Shad}_{x}(gx,2m)$. 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
$l-\eta_{l}\leq\frac{\vartheta_{i}}{i}\leq l+\eta_{l}$ for all $i\geq n$. In
particular this holds for $i=j$, and so $j(l-\eta_{l})\leq\vartheta_{j}\leq
j(l+\eta_{l})$, so there exists a point $y$ along a geodesic ray $[x,z]$
satisfying:

$j(l-\eta_{l})\leq\vartheta_{j}\leq
d(x,y)\leq\vartheta_{j+1}\leq(j+1)(l+\eta_{l}),\qquad d(y,\Gamma x)\leq m.$

So there is $g\in\Gamma$ satisfying (12) such that
$z\in\text{Shad}_{x}(gx,2m)$. Moreover these shadows are casted by points at
distance at least $j(l-\eta_{l})-m$ from $x$, so at distance at least
$\log\frac{1}{e^{m}\rho}$ from $x$. We need to estimate the number of such
$g$’s. 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 $e^{3m}\rho$. Thus

$\displaystyle\overline{\text{MD}}(B_{m,l,n})$ $\displaystyle=\limsup_{\rho\to
0}\frac{\text{Cov}(B_{m,l,n},e^{3m}\rho)}{\log\frac{1}{e^{3m}\rho}}$
$\displaystyle\leq\limsup_{j\to+\infty}\frac{(h_{\Gamma}+\varepsilon)[(j+1)(l+\eta_{l})+m]}{-3m+(j-1)(l-\eta_{l})}$
$\displaystyle\leq(h_{\Gamma}+\varepsilon)(1+\varepsilon),$

where the last inequality follows from the choice of $\eta_{l}$. ∎

##  5 Dynamics on packed GCB spaces

In this section we study quotients of proper metric spaces in general. In the
second part we will apply these results to the special case of packed, GCB-
spaces.

Let $X$ be a proper metric space and $\Gamma$ 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 it is
defined a standard pseudometric 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 $d(x,g_{n}y)\leq\frac{1}{n}$. In particular
$d(x,g_{n}x)\leq d(x,g_{n}y)+d(g_{n}y,g_{n}x)\leq d(x,y)+1$ for every $n$. So
the cardinality of these $g_{n}$’s is finite. Thus there must be one of these
$g_{n}$’s such that $d(x,g_{n}y)=0$, i.e. $x=g_{n}y$, and so $\pi x=\pi y$.
The map $\pi$ is $1$-Lipschitz and moreover if $\Gamma$ acts freely (i.e.
$gx=x$ for some $x\in X$, $g\in\Gamma$ implies $g=\text{id}$) then it is a
local isometry. In this case for every $x\in X$ it is defined the injectivity
radius at $x$ by

$\iota(x)=\sup\\{r>0\text{ s.t. }\pi_{|_{B(x,r)}}\colon
B(x,r)\to\pi(B(x,r))\text{ is an isometry}\\}.$

This defines a map $\iota\colon X\to(0,+\infty]$ with the following
properties:

  * -

$\iota(x)>0$ for every $x\in X$;

  * -

$\iota$ is lower-semicontinuous;

  * -

$\iota$ is $\Gamma$-equivariant.

As a consequence it induces a function $\iota\colon\Gamma\backslash
X\to(0,+\infty]$ and for every compact subset $K\subseteq\Gamma\backslash X$
there exists the (strictly positive) minimum of $\iota$ on $K$, which is
called the injectivity radius of $K$. The following lemma is standard.

######  Lemma 5.1.

Let $X$ be a proper metric space and let $\Gamma$ be a group of isometries of
$X$ acting discretely and freely. Then

  * (i)

for all $x,y\in X$ such that $d(\pi x,\pi y)<\iota(x)$ there exists a unique
$g\in\Gamma$ such that $d(x,gy)<\iota(x)$. In particular $d(x,gy)=d(\pi x,\pi
y)$;

  * (ii)

if $X$ is a length space then $\pi$ is a locally isometric covering.

###  5.1 The space of local geodesics

Let $X$ be a proper metric space and let Loc-Geod$(X)$ be its space of local
geodesic lines, thus a subset of the set of maps $\gamma\colon\mathbb{R}\to
X$, endowed with the topology of uniform convergence on compact subsets of
$\mathbb{R}$. It is metrizable, indeed we consider, as in [Cav21], the class
$\mathcal{F}$ of continuous functions $f\colon\mathbb{R}\to\mathbb{R}$
satisfying

  * (a)

$f(s)>0$ for all $s\in\mathbb{R}$;

  * (b)

$f(s)=f(-s)$ for all $s\in\mathbb{R}$;

  * (c)

$\int_{-\infty}^{+\infty}f(s)ds=1$;

  * (d)

$\int_{-\infty}^{+\infty}2|s|f(s)ds=C(f)<+\infty$

and we define the quantity

$f(\gamma,\gamma^{\prime}):=\int_{-\infty}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))f(s)ds.$

######  Lemma 5.2.

Let $X$ be a proper, length metric space and let Loc-Geod$(X)$ be its space of
local geodesics. Then

  * (i)

for all $\gamma\in\textup{Loc-Geod}(X)$ and for all $t<s$ it holds
$d(\gamma(t),\gamma(s))\leq\ell(\gamma|_{[t,s]})=|s-t|$, where $\ell(\cdot)$
denotes the length of a curve;

  * (ii)

for all $\gamma,\gamma^{\prime}\in\textup{Loc-Geod}(X)$ and for all
$f\in\mathcal{F}$ it holds

$f(\gamma,\gamma^{\prime})\leq d(\gamma(0),\gamma^{\prime}(0))+C(f),$

and this expression defines a distance on $\textup{Loc-Geod}(X)$ which induces
its topology. Moreover

$d(\gamma(0),\gamma^{\prime}(0))\leq f(\gamma,\gamma^{\prime})+C(f).$

###### Proof.

The proof of (i) is easy. The first inequality in (ii) can be proved exactly
as in Lemma 2.2 of [Cav21], using (i) to show that the integral is finite. The
second inequality in (ii) is easy: for every $s\in\mathbb{R}$ it holds
$d(\gamma(s),\gamma^{\prime}(s))\geq d(\gamma(0),\gamma^{\prime}(0))-2|s|$ by
(i) and this implies the estimate. The fact that the distance $f$ induces the
topology of Loc-Geod$(X)$ can be proved in the same way of Lemma 2.2 of
[Cav21]. ∎

Every isometry $g$ of $X$ acts on Loc-Geod$(X)$ by
$(g\gamma)(\cdot)=g\gamma(\cdot)$ for all $\gamma\in\text{Loc-Geod}(X)$. The
action is clearly continuous, so $g$ defines a homeomorphism of Loc-Geod$(X)$.
Thus if $\Gamma$ is a group of isometries of $X$ then $\Gamma$ acts by
homeomorphisms on Loc-Geod$(X)$.

######  Lemma 5.3.

Let $\Gamma$ be a group of isometries of a proper metric space $X$ acting
discretely and freely. Then the action of $\Gamma$ on Loc-Geod$(X)$ is
properly discontinuous.

###### Proof.

We fix $\gamma\in\text{Loc-Geod}(X)$, we choose $\varepsilon>0$ such that if
$d(g\gamma(0),\gamma(0))<\varepsilon$ then $g=\text{id}$ and we consider the
set

$U=\left\\{\gamma^{\prime}\in\text{Loc-Geod}(X)\text{ s.t.
}\gamma^{\prime}(0)\in
B\left(\gamma(0),\frac{\varepsilon}{2}\right)\right\\}.$

The set $U$ is clearly an open neighbourhood of $\gamma$. Now if $g\in\Gamma$
is such that $g\gamma^{\prime}\in U$ for some $\gamma^{\prime}\in U$ then both
$d(\gamma^{\prime}(0),\gamma(0))$ and $d(g\gamma^{\prime}(0),\gamma(0))$ are
$<\frac{\varepsilon}{2}$, so $d(\gamma(0),g\gamma(0))<\varepsilon$ and
$g=\text{id}$. ∎

We fix a group of isometries $\Gamma$ of a proper metric space $X$ acting
discretely and freely. There are two natural objects we can consider: the
quotient space $\Gamma\backslash\text{Loc-Geod}(X)$ endowed with the quotient
topology and the space of parametrized local geodesic lines of the metric
space $\Gamma\backslash X$, namely Loc-Geod$(\Gamma\backslash X)$, endowed as
usual with the topology of uniform convergence on compact subsets of
$\mathbb{R}$.

######  Lemma 5.4.

Let $X$ be a proper, length metric space and let Loc-Geod$(X)$ be its space of
local geodesics. Let $\Gamma$ be a group of isometries of $X$ acting
discretely and freely and let $f\in\mathcal{F}$. Then

  * (i)

the natural action of $\Gamma$ on $(\textup{Loc-Geod}(X),f)$ is by isometries
and discrete;

  * (ii)

the space $\Gamma\backslash\textup{Loc-Geod}(X)$ is metrizable.

###### Proof.

The action of $\Gamma$ on $(\textup{Loc-Geod}(X),f)$ is clearly by isometries.
Moreover for every $\gamma\in\textup{Loc-Geod}(X)$ and every $R\geq 0$ we
have, by Lemma 5.2,

$\\#\\{g\in\Gamma\text{ s.t. }f(g\gamma,\gamma)\leq
R\\}\leq\\#\\{g\in\Gamma\text{ s.t. }d(g\gamma(0),\gamma(0))\leq
R+C(f)\\}<+\infty.$

By the same argument at the beginning of Section 5 we conclude that the
quotient pseudometric induced by $f$ is actually a metric, which implies (ii)
since the quotient metric induces the quotient topology. ∎

######  Proposition 5.5.

Let $\Gamma$ be a group of isometries of a proper, length metric space $X$
acting discretely and freely. Then the map

$\Pi\colon\textup{Loc-Geod}(X)\mapsto\textup{Loc-Geod}(\Gamma\backslash
X),\quad\Pi(\gamma)(\cdot)=\pi\gamma(\cdot)$

is continuous, surjective and $\Gamma$-equivariant. So it induces a map

$\bar{\Pi}\colon\Gamma\backslash\textup{Loc-Geod}(X)\mapsto\textup{Loc-
Geod}(\Gamma\backslash X)$

which is a homeomorphism.

###### Proof.

First of all if $\gamma=g\eta$ for some $g\in\Gamma$ and
$\gamma,\eta\in\text{Loc-Geod}(X)$ then clearly
$\pi\gamma(\cdot)=\pi\eta(\cdot)$, so $\Pi$ is $\Gamma$-equivariant. Moreover
every map $\gamma\colon\mathbb{R}\to\Gamma\backslash X$ can be lifted to a map
$\tilde{\gamma}\colon\mathbb{R}\to X$ since $\pi\colon X\to\Gamma\backslash X$
is a covering map by Lemma 5.1. Clearly if $\gamma$ is a local geodesic then
$\tilde{\gamma}$ is a local geodesic of $X$, again by Lemma 5.1. This shows
that $\Pi$ is surjective. Furthermore since $\pi$ is $1$-Lipschitz then $\Pi$
is continuous.
It remains to show that the map $\bar{\Pi}$ is a homeomorphism. Let us show it
is injective: suppose $\Pi(\gamma)=\Pi(\eta)$, i.e. $\pi\gamma(t)=\pi\eta(t)$
for every $t\in\mathbb{R}$. Therefore for every $t\in\mathbb{R}$ there is a
unique $g_{t}\in\Gamma$ such that $g_{t}\eta(t)=\gamma(t)$, Lemma 5.1. We
consider the set $A=\\{t\in\mathbb{R}\text{ s.t. }g_{t}=g_{0}\\}$. It is
closed: indeed if $t_{k}\to t_{\infty}$ with $t_{k}\in A$ then we have

$\displaystyle d(g_{t_{\infty}}\eta(t_{\infty}),g_{0}\eta(t_{\infty}))$
$\displaystyle\leq
d(g_{t_{\infty}}\eta(t_{\infty}),g_{t_{k}}\eta(t_{k}))+d(g_{t_{k}}\eta(t_{k}),g_{0}\eta(t_{\infty}))$
$\displaystyle\leq
d(\gamma(t_{\infty}),\gamma(t_{k}))+d(\eta(t_{k}),\eta(t_{\infty}))$

and the last quantity is as small as we want. So $g_{t_{\infty}}=g_{0}$, i.e.
$t_{\infty}\in A$. Moreover $A$ is open: let $t\in A$ and suppose there is a
sequence $t_{k}\to t$ such that $g_{t_{k}}\neq g_{0}$. Then

$d(\gamma(t),g_{t_{k}}\eta(t))\leq
d(\gamma(t),\gamma(t_{k}))+d(g_{t_{k}}\eta(t_{k}),g_{t_{k}}\eta(t))\leq
2|t-t_{k}|.$

When $2|t-t_{k}|<\iota(\eta(t))$ we get
$d(g_{0}^{-1}g_{t_{k}}\eta(t),\eta(t))<\iota(\eta(t))$ and by Lemma 5.1 this
implies $g_{0}^{-1}g_{t_{k}}=\text{id}$, i.e. $g_{t_{k}}=g_{0}$ for every
$t_{k}$ sufficiently close to $t$. Since $A$ is non empty we conclude that
$A=\mathbb{R}$, i.e. $\gamma(t)=g_{0}\eta(t)$ for every $t\in\mathbb{R}$. So
$\gamma=g_{0}\eta$, that is $\bar{\Pi}$ is injective.
The continuity of $\bar{\Pi}^{-1}$ can be checked on sequences since both
spaces are metrizable by Lemma 5.2 and 5.4. Let $p\colon\text{Loc-
Geod}(X)\to\Gamma\backslash\text{Loc-Geod}(X)$ be the standard projection map.
We take $\gamma_{n},\gamma_{\infty}\in\text{Loc-Geod}(\Gamma\backslash X)$
such that $\gamma_{k}\to\gamma_{\infty}$ uniformly on compact subsets of
$\mathbb{R}$. Let $T\geq 0$, let $\varepsilon>0$ be any real number which is
less than the injectivity radius of the compact set $\gamma_{\infty}([-T,T])$
and let $\tilde{\gamma}_{k},\tilde{\gamma}_{\infty}$ be any covering local
geodesics of $\gamma_{k},\gamma_{\infty}$ respectively. We know that for $k$
big enough it holds $d(\gamma_{k}(t),\gamma_{\infty}(t))<\varepsilon$ for
every $t\in[-T,T]$, then there is a unique $g_{k}(t)\in\Gamma$ such that
$d(g_{k}(t)\tilde{\gamma}_{k}(t),\tilde{\gamma}_{\infty}(t))<\varepsilon$ by
Lemma 5.1. Arguing as before we conclude that for every such $k$ it holds
$g_{k}(t)=g_{k}(0)=:g_{k}$ for every $t\in[-T,T]$. This implies that
$p(\tilde{\gamma_{k}})$ converges uniformly on $[-T,T]$ to
$p(\tilde{\gamma_{\infty}})$. Since this is true for every $T\geq 0$ we get
the continuity of $\bar{\Pi}^{-1}$. ∎

There is a natural action of $\mathbb{R}$ on $\text{Loc-Geod}(X)$ defined by
reparametrization:

$\Phi_{t}\gamma(\cdot)=\gamma(\cdot+t)$

for every $t\in\mathbb{R}$. It is easy to see it is a continuous action, i.e.
$\Phi_{t}\circ\Phi_{s}=\Phi_{t+s}$ for all $t,s\in\mathbb{R}$ and for every
$t\in\mathbb{R}$ the map $\Phi_{t}$ is a homeomorphism of $\text{Loc-
Geod}(X)$. This action is called the _local geodesic flow_ on $X$.

###  5.2 Packed GCB-spaces

From now on we will add some other assumptions on our metric space $X$, in
terms of weak upper and lower bounds on the curvature. As a lower bound we
take a bounded packing condition at a fixed scale.
Let $Y$ be any subset of a metric space $X$:
– a subset $S$ of $Y$ is called $r$-dense if $\forall y\in Y$ $\exists z\in S$
such that $d(y,z)\leq r$;
– a subset $S$ of $Y$ is called $r$-separated if $\forall y,z\in S$ it holds
$d(y,z)>r$.
The packing number of $Y$ at scale $r$ is the maximal cardinality of a
$2r$-separated subset of $Y$ and it is denoted by $\text{Pack}(Y,r)$. The
covering number of $Y$ is the minimal cardinality of a $r$-dense subset of $Y$
and it is denoted by $\text{Cov}(Y,r)$. The packing and the covering functions
of $X$ are respectively

$\text{Pack}(R,r)=\sup_{x\in
X}\text{Pack}(\overline{B}(x,R),r),\qquad\text{Cov}(R,r)=\sup_{x\in
X}\text{Cov}(\overline{B}(x,R),r).$

We say that a metric space $X$ is $P_{0}$-packed at scale $r_{0}$ if
Pack$(3r_{0},r_{0})\leq P_{0}$, that is every ball of radius $3r_{0}$ contains
no more than $P_{0}$ points that are $2r_{0}$-separated.

As upper curvature bound we take a very weak notion of non-positive curvature.
Let $X$ be a metric space. A geodesic bicombing is a map $\sigma\colon X\times
X\times[0,1]\to X$ with the property that for all $(x,y)\in X\times X$ the map
$\sigma_{xy}\colon t\mapsto\sigma(x,y,t)$ is a geodesic from $x$ to $y$
parametrized proportionally to arc-length, i.e.
$d(\sigma_{xy}(t),\sigma_{xy}(t^{\prime}))=|t-t^{\prime}|d(x,y)$ for all
$t,t^{\prime}\in[0,1]$ and $\sigma_{xy}(0)=x,\sigma_{xy}(1)=y$.
_When $X$ is equipped with a geodesic bicombing then for all $x,y\in X$ we
will denote by $[x,y]$ the geodesic $\sigma_{xy}$ parametrized by arc-length._
A geodesic bicombing is:

  * •

_convex_ if the map $t\mapsto
d(\sigma_{xy}(t),\sigma_{x^{\prime}y^{\prime}}(t))$ is convex on $[0,1]$ for
all $x,y,x^{\prime},y^{\prime}\in X$;

  * •

_consistent_ if for all $x,y\in X$, for all $0\leq s\leq t\leq 1$ and for all
$\lambda\in[0,1]$ it holds
$\sigma_{pq}(\lambda)=\sigma_{xy}((1-\lambda)s+\lambda t)$, where
$p:=\sigma_{xy}(s)$ and $q:=\sigma_{xy}(t)$;

  * •

_reversible_ if $\sigma_{xy}(t)=\sigma_{yx}(1-t)$ for all $t\in[0,1]$.

For instance every convex metric space in the sense of Busemann (so also every
CAT$(0)$ metric space) admits a unique convex, consistent, reversible geodesic
bicombing.
Given a geodesic bicombing $\sigma$ we say that a geodesic (segment, ray,
line) $\gamma$ is a $\sigma$-geodesic (segment, ray, line) if for all
$x,y\in\gamma$ we have that $[x,y]$ coincides with the subsegment of $\gamma$
between $x$ and $y$.
A geodesic bicombing is _geodesically complete_ if every $\sigma$-geodesic
segment is contained in a $\sigma$-geodesic line. A couple $(X,\sigma)$ is
said a GCB-space if $\sigma$ is a convex, consistent, reversible, geodesically
complete geodesic bicombing on the complete metric space $X$. The packing
condition has a controlled behaviour in GCB-spaces.

######  Proposition 5.6 (Proposition 3.2 of [CS20a]).

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. Then:

  * (i)

for all $r\leq r_{0}$, the space $X$ is $P_{0}$-packed at scale $r$ and it is
proper;

  * (ii)

for every $0<r\leq R$ and every $x\in X$ it holds:

$\textup{Pack}(R,r)\leq P_{0}(1+P_{0})^{\frac{R}{r}-1}\text{, if }r\leq
r_{0};$ $\textup{Pack}(R,r)\leq P_{0}(1+P_{0})^{\frac{R}{r_{0}}-1}\text{, if
}r>r_{0}.$

Basic examples of GCB-spaces that are $P_{0}$-packed at scale $r_{0}$ are:

  * i)

complete and simply connected Riemannian manifolds with sectional curvature
pinched between two nonpositive constants $\kappa^{\prime}\leq\kappa<0$;

  * ii)

simply connected $M^{\kappa}$-complexes, with $\kappa\leq 0$, without free
faces and bounded geometry (i.e., with valency at most $V_{0}$, size at most
$S_{0}$ and positive injectivity radius);

  * iii)

complete, geodesically complete, CAT$(0)$ metric spaces $X$ with dimension at
most $n$ and volume of balls of radius $R_{0}$ bounded above by $V$.

For further details on the second and the third class of examples we refer to
[CS20b].

Let $(X,\sigma)$ be a proper GCB-space. The space of parametrized
$\sigma$-geodesic lines of $X$ is

$\text{Geod}_{\sigma}(X)=\\{\gamma\colon\mathbb{R}\to X\text{ isometry s.t.
}\gamma(\mathbb{R})\text{ is a }\sigma\text{-geodesic line}\\},$

considered as a subset of Loc-Geod$(X)$. By the continuity of $\sigma$ (see
[Des15], [Cav21]) we have that $\text{Geod}_{\sigma}(X)$ is closed in
$\text{Loc-Geod}(X)$. Moreover the local geodesic flow on Loc-Geod$(X)$
restricts as an action on $\text{Geod}_{\sigma}(X)$, called the
$\sigma$-geodesic flow on $X$. The evaluation map
$E\colon\text{Geod}_{\sigma}(X)\to X$, which is defined as
$E(\gamma)=\gamma(0)$, is continuous and proper ([BL12], Lemma 1.10 and
[Cav21]). Moreover this restriction is surjective since $\sigma$ is assumed
geodesically complete. The topology on $\text{Geod}_{\sigma}(X)$ is metrizable
by Lemma 5.2.

Let $(X,\sigma)$ be a GCB-space. An isometry $g$ of $X$ is a $\sigma$-isometry
if for all $x,y\in X$ it holds $\sigma_{g(x)g(y)}=g(\sigma_{xy})$. We say that
a group of isometries of $X$ is a group of $\sigma$-isometries if every
element of the group is a $\sigma$-isometry. If $\Gamma$ is a group of
$\sigma$-isometries of $X$ acting discretely and freely then
Geod${}_{\sigma}(X)$ is $\Gamma$-invariant and we set Loc-
Geod${}_{\sigma}(\Gamma\backslash X):=\Pi(\text{Geod}_{\sigma}(X))$, which is
homeomorphic to $\Gamma\backslash\text{Geod}_{\sigma}(X)$ by Proposition 5.5.
Since in Busemann convex (so also CAT$(0)$) metric spaces $X$ every local
geodesic is a geodesic then it holds Loc-Geod$(X)=$Geod${}_{\sigma}(X)$, where
$\sigma$ is the unique bicombing, so Loc-Geod${}_{\sigma}(\Gamma\backslash
X)=$ Loc-Geod$(\Gamma\backslash X)$.
We end this section with two easy results and a remark.

######  Lemma 5.7.

Let $(X,\sigma)$ be a proper GCB-space. Then the natural map
$\Pi\colon(\textup{Geod}_{\sigma}(X),f)\to(\textup{Loc-Geod}_{\sigma}(X),f)$
is $1$-Lipschitz for all $f\in\mathcal{F}$.

###### Proof.

For all $\gamma,\gamma^{\prime}\in\text{Geod}_{\sigma}(X)$ we have

$\displaystyle f(\gamma,\gamma^{\prime})$
$\displaystyle=\int_{-\infty}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))f(s)ds$
$\displaystyle\geq\int_{-\infty}^{+\infty}d(\pi\gamma(s),\pi\gamma^{\prime}(s))f(s)ds=f(\Pi(\gamma),\Pi(\gamma^{\prime})).$

∎

######  Lemma 5.8.

Let $(X,\sigma)$ be a GCB-space and let $\Gamma$ be a group of
$\sigma$-isometries of $X$ acting discretely and freely. Then the metric on
$\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ induced by every
$f\in\mathcal{F}$ is complete.

###### Proof.

We observe the result is not completely trivial because the metric $f$ on
$\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ does not coincide with the
quotient metric. However given a Cauchy sequence $\\{\gamma_{k}\\}$ we know
that for every $t\in\mathbb{R}$ the sequence $\\{\gamma_{k}(t)\\}$ is bounded,
so we can apply Ascoli-Arzela’s Theorem to find a limit map
$\gamma_{\infty}\colon\mathbb{R}\to\Gamma\backslash X$. We need to show
$\gamma_{\infty}$ is a local geodesic. Let $t\in\mathbb{R}$ and let $2\iota>0$
be smaller than the injectivity radius at $\gamma_{\infty}(t)$. Then by Lemma
5.1 for every $k$ big enough there is a subsegment of length $2\iota$ of
$\gamma_{k}$ which converges to $\gamma_{\infty}([t-\iota,t+\iota])$ and that
is a true geodesic. Therefore $\gamma_{\infty}([t-\iota,t+\iota])$ is a true
geodesic, that is $\gamma_{\infty}$ is a local geodesic.
Let $\tilde{\gamma}_{k},\tilde{\gamma}_{\infty}$ be any covering local
geodesic of $\gamma_{k},\gamma_{\infty}$ respectively. Arguing as in the proof
of Proposition 5.5 we know that for every $T\in\mathbb{R}$ there exists
$g_{T}\in\Gamma$ such that $g_{T}\tilde{\gamma}_{k}$ converges uniformly to
$\tilde{\gamma}_{\infty}$ on $[-T,T]$. It is then clear by Lemma 5.1 that if
$T^{\prime}\geq T$ then $g_{T^{\prime}}=g_{T}$. In particular we can find
$g\in\Gamma$ such that $g\tilde{\gamma}_{k}$ converges uniformly on every
compact subset of $\mathbb{R}$ to $\tilde{\gamma}_{\infty}$. Observe that each
$g\tilde{\gamma}_{k}$ is a $\sigma$-geodesic line, so it is
$\tilde{\gamma}_{\infty}$, i.e. $\gamma_{\infty}\in\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$. ∎

######  Remark 5.9.

We observe that the proof shows also that Loc-
Geod${}_{\sigma}(\Gamma\backslash X)$ is closed in Loc-Geod$(\Gamma\backslash
X)$.

In our situation the space $\Gamma\backslash X$ is naturally a locally GCB-
space with local convex bicombing $\sigma$ as defined in [Mie15]. A
$\sigma$-local geodesic can be defined as a local geodesic $\gamma$ such that
for all close enough $t<t^{\prime}$ it holds
$\gamma|_{[t,t^{\prime}]}=\sigma_{\gamma(t),\gamma(t^{\prime})}$. Of course
this equality has a meaning only locally, where $\sigma$-geodesics are
defined. Clearly every $\sigma$-local geodesic can be lifted to a
$\sigma$-local geodesic of $X$ and by convexity it is easy to check that every
$\sigma$-local geodesic of $X$ is actually a $\sigma$-geodesic. So
$\textup{Loc-Geod}_{\sigma}(X)$ can be seen exactly as the space of
$\sigma$-local geodesics of $\Gamma\backslash X$.
This picture can be generalized: if $X$ is a complete, connected, locally GCB-
space then by Theorem 1.1 of [Mie15] it admits a universal cover $\tilde{X}$
that can be endowed with a unique structure of GCB-space $(\tilde{X},\sigma)$
and $X=\Gamma\backslash\tilde{X}$, where $\Gamma$ is a group of
$\sigma$-isometries of $\tilde{X}$ acting discretely and freely. In this
setting the natural space of $\sigma$-local geodesics of $X$ coincides with
the space $\textup{Loc-Geod}_{\sigma}(\Gamma\backslash\tilde{X})$ defined
above.

##  6 Upper bound of the entropy

We consider the dynamical system $(\text{Loc-Geod}_{\sigma}(\Gamma\backslash
X),\Phi_{1})$, where $(X,\sigma)$ is a $\delta$-hyperbolic GCB-space that is
$P_{0}$-packed at scale $r_{0}$, $\Gamma$ is a group of $\sigma$-isometries of
$X$ acting discretely and freely and $\Phi_{1}$ is the classical time one
reparametrization flow, i.e. $\Phi_{1}(\gamma)(\cdot)=\gamma(\cdot+1)$ for all
$\gamma\in\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$.
In this section we prove that the topological entropy of the dynamical system
above is less than or equal to the critical exponent of $\Gamma$.
Before we recall the definition of the topological entropy and the local
entropies of a dynamical system. For us a dynamical system is a couple $(Y,T)$
where $Y$ is a topological space and $T\colon Y\to Y$ is a continuous map. We
will always denote by $\mathscr{B}$ the $\sigma$-algebra of Borelian subsets
of $Y$. We denote by $\mathcal{M}_{1}(Y,T)$ the set of $T$-invariant
probability measure on $(Y,\mathscr{B})$. We recall that a measure
$\mu\in\mathcal{M}_{1}(Y,T)$ is ergodic if every subset $A$ of $Y$ such that
$T^{-1}A\subseteq A$ has $\mu$-measure $0$ or $1$. We denote the set of
ergodic measures by $\mathcal{E}_{1}(Y,T)$.
_For the rest of the section we will always assume that $Y$ is a locally
compact, metrizable, topological space._

The topological entropy of the dynamical system $(Y,T)$ is by definition:

$h_{\text{top}}(Y,T):=\sup_{\mu\in\mathcal{M}_{1}(Y,T)}h_{\mu}(Y,T)=\sup_{\mu\in\mathcal{E}_{1}(Y,T)}h_{\mu}(Y,T),$
(13)

where $h_{\mu}(Y,T)$ denotes the Kolmogorov-Sinai entropy of the measure
$\mu$. For our purposes we will not need to recall its definition. The second
equality is classical and follows from the convexity of the entropy
functional.
We introduce two other possible definitions of entropy of a measure
$\mu\in\mathcal{E}_{1}(Y,T)$. We restrict to ergodic measures as we have seen
that it is enough to consider them in order to compute the topological entropy
of $(Y,T)$. Let $\mu\in\mathcal{E}_{1}(Y,T)$ and be a metric on $Y$ inducing
its topology. The upper local entropy of $\mu$ with respect to is defined as

$\overline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T)=\underset{\mu}{\text{ess}\sup}\lim_{r\to
0}\limsup_{n\to+\infty}-\frac{1}{n}\log\mu(B_{\text{\texthtd}^{n}}(y,r)),$

while the lower local entropy is

$\underline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T)=\underset{\mu}{\text{ess}\inf}\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{\text{\texthtd}^{n}}(y,r)).$

Here $\text{\texthtd}^{n}$ is the dynamical distance defined as usual as:

$\text{\texthtd}^{n}(y,y^{\prime})=\max_{i=0,\ldots,n-1}\text{\texthtd}(T^{i}y,T^{i}y^{\prime}),\qquad\forall
y,y^{\prime}\in Y.$

Observe that this definition of the lower local entropy agrees with the one
given in [Led13] while it is a bit different from the one of [Riq16]. However
it is certainly bigger than or equal to this last one. So we conclude:

######  Lemma 6.1 ([Riq16], Theorem 1.28).

Let $\mu\in\mathcal{E}_{1}(Y,T)$ and let  be a complete metric on $Y$ inducing
its topology. Then
$h_{\mu}(Y,T)\leq\underline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}(Y,T).$

It is natural to define the _upper and lower local entropy_ of the dynamical
system $(Y,T)$, respectively, as

$\overline{h}^{\text{loc}}(Y,T)=\sup_{\mu\in\mathcal{E}_{1}(Y,T)}\inf_{\text{\texthtd}}\overline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T),\qquad\underline{h}^{\text{loc}}(Y,T)=\sup_{\mu\in\mathcal{E}_{1}(Y,T)}\inf_{\text{\texthtd}}\underline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T),$

where the infimum are among all complete metrics on $Y$ inducing its topology.
Clearly by Lemma 6.1 we have

$h_{\text{top}}(Y,T)\leq\underline{h}^{\text{loc}}(Y,T)\leq\overline{h}^{\text{loc}}(Y,T).$
(14)

As said at the beginning of the section we consider the dynamical system
$(\text{Loc-Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$. To simplify the
notations we denote by $\mathcal{M}_{1}$, $\mathcal{E}_{1}$ the sets
$\mathcal{M}_{1}(\text{Loc-Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$ and
$\mathcal{E}_{1}(\text{Loc-Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$. In
the same way the topological entropy, the lower and the upper local entropy of
$(\text{Loc-Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$ will be simply
denoted by, respectively, $h_{\text{top}}$, $\underline{h}^{\text{loc}}$ and
$\overline{h}^{\text{loc}}$. In order to prove $h_{\text{top}}\leq h_{\Gamma}$
the strategy is to show $\overline{h}^{\textup{loc}}\leq h_{\Gamma}$. This
would be enough by (14). Before proceeding in this direction we need to
simplify the expression of the upper local entropy.

By Lemma 5.8 every $f\in\mathcal{F}$ defines a complete metric on Loc-
Geod${}_{\sigma}(\Gamma\backslash X)$ which induces its topology, so by
definition it is clear that

$\overline{h}^{\text{loc}}\leq\sup_{\mu\in\mathcal{E}_{1}}\overline{h}^{\text{loc}}_{\mu,f},\qquad\underline{h}^{\text{loc}}\leq\sup_{\mu\in\mathcal{E}_{1}}\underline{h}^{\text{loc}}_{\mu,f}.$

Our aim is to simplify the computation of $\overline{h}^{\text{loc}}_{\mu,f}$
and $\underline{h}^{\text{loc}}_{\mu,f}$. In order to do that we need the key
lemma of [Cav21].

######  Lemma 6.2 (Key lemma, [Cav21]).

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. Let
$f\in\mathcal{F}$, $\gamma\in\textup{Geod}_{\sigma}(X)$ and $0<r\leq R$. Then

$\lim_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{f^{T}}(B_{f^{T}}(\gamma,R),r)=0.$

We are now ready to prove:

######  Lemma 6.3.

Let $\mu\in\mathcal{E}_{1}$ anf $f\in\mathcal{F}$. Then for every $R>0$ and
for $\mu$-a.e.$\gamma\in\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ it
holds:

$\underline{h}_{\mu,f}^{\textup{loc}}=\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R)).$

The same corresponding conclusion is true for the upper local entropy.

###### Proof.

We fix $R>0$ and we take $0<r\leq R$. Clearly $B_{f^{n}}(\gamma,R)\supseteq
B_{f^{n}}(\gamma,r)$ for all $\gamma\in\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$, so

$\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))\leq\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,r))$

for all $\gamma\in\textup{Loc-Geod}_{\sigma}(\Gamma\backslash X)$. In the
other direction we notice that for all $\gamma\in\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$ it holds

$\mu(B_{f^{n}}(\gamma,R))\leq\text{Cov}_{f^{n}}(B_{f^{n}}(\gamma,R),r)\cdot\underset{\mu}{\textup{ess}\sup}\mu(B_{f^{n}}(\gamma^{\prime},r)).$

So we deduce

$\displaystyle\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))$
$\displaystyle\geq\liminf_{n\to+\infty}-\frac{1}{n}\log\text{Cov}_{f^{n}}(B_{f^{n}}(\gamma,R),r)\cdot\underset{\mu}{\textup{ess}\sup}\mu(B_{f^{n}}(\gamma^{\prime},r))$
$\displaystyle=\liminf_{n\to+\infty}-\frac{1}{n}\log\text{Cov}_{f^{n}}(B_{f^{n}}(\gamma,R),r)$
$\displaystyle\quad+\liminf_{n\to+\infty}-\frac{1}{n}\log\underset{\mu}{\textup{ess}\sup}\mu(B_{f^{n}}(\gamma^{\prime},r))$
$\displaystyle=\liminf_{n\to+\infty}-\frac{1}{n}\log\underset{\mu}{\textup{ess}\sup}\mu(B_{f^{n}}(\gamma^{\prime},r))$
$\displaystyle\geq\underset{\mu}{\textup{ess}\sup}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma^{\prime},r)),$

where the second inequality follows directly by Lemma 6.2. By the
arbitrariness of $r$ we conclude:

$\displaystyle\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))$
$\displaystyle\geq\lim_{r\to
0}\underset{\mu}{\textup{ess}\sup}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma^{\prime},r))$
$\displaystyle\geq\underset{\mu}{\textup{ess}\sup}\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma^{\prime},r)).$

Since this is true for all $\gamma\in\textup{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$ we obtain:

$\displaystyle\underset{\mu}{\textup{ess}\inf}\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,r))$
$\displaystyle\geq\underset{\mu}{\textup{ess}\inf}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))$
$\displaystyle\geq\underset{\mu}{\textup{ess}\sup}\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma^{\prime},r)).$

This implies that for $\mu$-a.e.$\gamma$ the function

$\gamma\mapsto\lim_{r\to
0}\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma^{\prime},r))$

is constantly equal to $\underline{h}_{\mu,f}^{\text{loc}}$, by definition of
lower local entropy. So for $\mu$-a.e.$\gamma$ we conclude

$\underline{h}_{\mu,f}^{\text{loc}}=\liminf_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R)).$

The proof for the upper entropy is the same. ∎

We are ready to prove the first part of Theorem A, namely

######  Theorem 6.4.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $\Gamma$ be a group of $\sigma$-isometries of $X$ acting
discretely and freely. Then

$\overline{h}^{\textup{loc}}\leq h_{\Gamma}.$

The proof is inspired by [Led13]. Essentially to every measure
$\mu\in\mathcal{E}_{1}$ we associate a measure $\nu$ on $\partial X$ with the
following properties:

  * (i)

$\overline{h}^{\textup{loc}}_{\mu,f}\leq\overline{\text{PD}}(\nu)$, where
$f\in\mathcal{F}$ is arbitrary;

  * (ii)

$\nu(\Lambda_{\text{erg}})=1$.

Then we will use (10) and Theorem 4.4 to conclude the estimate.

###### Proof.

It is enough to show that for all $\mu\in\mathcal{E}_{1}$ it holds
$\overline{h}^{\textup{loc}}_{\mu,f}\leq h_{\Gamma}$ for some
$f\in\mathcal{F}$. So we fix $f\in\mathcal{F}$ and $\mu\in\mathcal{E}_{1}$.
For every $x\in X$ we consider the ball $B(x,\iota(x))$, where $\iota(x)$ is
the injectivity radius at $x$. Since $X$ is proper, so separable, we can find
a countable set $\\{x_{i}\\}_{i\in\mathbb{N}}\subseteq X$ such that
$X=\bigcup_{i\in\mathbb{N}}B(x_{i},\iota(x_{i}))$. We consider the sets

$U_{i}=\\{\gamma\in\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)\text{ s.t.
}\gamma(0)\in B(\pi x_{i},\iota(x_{i}))\\}.$

Clearly $\bigcup_{i\in\mathbb{N}}U_{i}=\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$, so there must be some index $i_{0}$ such
that $\mu(U_{i_{0}})=c>0$. By Lemma 5.1 for all $\gamma\in U_{i_{0}}$ there
exists a unique $\tilde{\gamma}\in\text{Geod}_{\sigma}(X)$ such that
$\Pi\tilde{\gamma}=\gamma$ and
$d(\tilde{\gamma}(0),x_{i_{0}})<\iota(x_{i_{0}})$.
Therefore we can define the map $+\colon U_{i_{0}}\to\partial X$ by
$+\gamma=\tilde{\gamma}^{+}$, where $\tilde{\gamma}$ is the unique covering
geodesic of $\gamma$ with $d(\tilde{\gamma}(0),x_{i_{0}})<\iota(x_{i_{0}})$.
Let us show it is continuous: let $\gamma_{k}\in U_{i_{0}}$ converging
uniformly on compact subsets to $\gamma_{\infty}\in U_{i_{0}}$. We showed in
Lemma 5.8 that there exist covering $\sigma$-geodesics of $\gamma_{k}$ that
converge uniformly on compact subsets to $\tilde{\gamma}_{\infty}$. It is then
clear that these covering $\sigma$-geodesics must be the $\sigma$-geodesics
$\tilde{\gamma}_{k}$. So $\tilde{\gamma}_{k}^{+}$ converges to
$\tilde{\gamma}_{\infty}^{+}$. Hence we can define the measure $\nu$ on
$(\partial X,\mathscr{B})$ by $\nu=+_{\ast}\mu$, i.e. $\nu(A)=\mu(+^{-1}A)$
for every Borelian subset $A$ of $\partial X$.

Step 1: $\overline{h}^{\textup{loc}}_{\mu,f}\leq\overline{\text{PD}}(\nu)$.
We fix $R=28\delta+3\iota(x_{i_{0}})+2C(f)$. By Lemma 6.3 we know that

$\overline{h}_{\mu,f}^{\textup{loc}}=\limsup_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))$
(15)

for $\mu$-a.e.$\gamma\in U_{i_{0}}$. Clearly the measure $\nu$ is concentrated
on the image of the map $+$, so in order to compute its upper packing
dimension we can just consider the points of this set. So for all $\gamma\in
U_{i_{0}}$ we consider the point $z=\tilde{\gamma}^{+}\in\partial X$ and the
generalized visual ball $B(z,e^{-n})$, where we choose $x_{i_{0}}$ as
basepoint of $X$. We want to estimate
$\nu(B(z,e^{-n}))=\mu(+^{-1}B(z,e^{-n}))$. We observe that every
$\eta\in+^{-1}B(z,e^{-n})$ satisfies:

$d(\tilde{\eta}(0),\tilde{\gamma}(0))\leq 2\iota(x_{i_{0}}),\qquad
d(\tilde{\eta}(n),\tilde{\gamma}(n))\leq 14\delta+\iota(x_{i_{0}}).$

Indeed the first inequality is by definition of $\tilde{\gamma}$ and
$\tilde{\eta}$, while the second one follows by Lemma 2.1 and Lemma 2.2. By
standard arguments, see [Cav21], this implies

$f^{n}(\tilde{\eta},\tilde{\gamma})\leq 28\delta+3\iota(x_{i_{0}})+2C(f)=R$

and so, by Lemma 5.7, $f^{n}(\eta,\gamma)\leq R$. In other words

$+^{-1}B(+\gamma,e^{-n})\subseteq B_{f^{n}}(\gamma,R)$

for all $\gamma\in U_{i_{0}}$. We conclude that

$\displaystyle\overline{\text{PD}}(\nu)$
$\displaystyle=\underset{\nu}{\text{ess}\sup}\limsup_{\rho\to
0}\frac{\log\nu(B(z,\rho))}{\log\rho}$
$\displaystyle\geq\underset{\nu}{\text{ess}\sup}\limsup_{n\to+\infty}-\frac{1}{n}\log\nu(B(z,e^{-n}))$
$\displaystyle\geq\underset{\mu}{\text{ess}\sup}\limsup_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))=\overline{h}_{\mu,f}^{\textup{loc}}.$

The first inequality is clear choosing $\rho=e^{-n}$ and using the property
that the limit superior along a selected subsequence is smaller than or equal
to the limit superior along any subsequence. The second follows from what we
said before: indeed $\nu$ gives full measure to the set of points
$z\in\partial X$ that are in the image of the map $+$. This, together with the
fact that
$\nu(B(+\gamma,e^{-n}))=\mu(+^{-1}B(+\gamma,e^{-n}))\leq\mu(B_{f^{n}}(\gamma,R))$,
implies the second inequality. The last equality is direct consequence of (15)
and the fact that $U_{i_{0}}$ has positive $\mu$-measure.

Step 2: $\nu$ gives full measure to the set $\Lambda_{\text{erg}}$.
In this step we will use for the first (and last) time the ergodicity of the
measure $\mu$. We associate to $\gamma\in U_{i_{0}}$ the set of integers
$\Theta(\gamma)=\\{\vartheta_{i}(\gamma)\\}$ defined recursively by

$\vartheta_{0}(\gamma)=0,\qquad\vartheta_{i+1}(\gamma)=\inf\\{n>\vartheta_{i}(\gamma)\text{
s.t. }\Phi_{n}(\gamma)\in U_{i_{0}}\\}.$

We notice that an integer $n$ satisfies

$d(\Phi_{n}\gamma(0),\pi x_{i_{0}})=d(\gamma(n),\pi
x_{i_{0}})<\iota({x_{i_{0}}})$

if and only if $n\in\Theta(\gamma)$, i.e. $\Theta(\gamma)$ is exactly the set
of (integer) returning times of $\gamma$ to the ball $B(\pi
x_{i_{0}},\iota(x_{i_{0}}))$.
We apply Birkhoff’s Ergodic Theorem to the indicator function of the set
$U_{i_{0}}$, namely $\chi_{U_{i_{0}}}$, obtaining that for
$\mu$-a.e.$\gamma\in\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ it holds

$\exists\lim_{N\to+\infty}\frac{1}{N}\sum_{j=0}^{N-1}\chi_{U_{i_{0}}}\circ\Phi_{j}(\gamma)=\mu(U_{i_{0}})=c>0.$

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

$\lim_{N\to+\infty}\frac{1}{N}\sum_{j=0}^{N-1}\chi_{U_{i_{0}}}\circ\Phi_{j}(\gamma)=\lim_{N\to+\infty}\frac{\\#\Theta(\gamma)\cap[0,N-1]}{N},$

and the right hand side is by definition the density of the set
$\Theta(\gamma)$. It is classical that, given the standard increasing
enumeration $\\{\vartheta_{0}(\gamma),\vartheta_{1}(\gamma),\ldots\\}$ of
$\Theta(\gamma)$, it holds

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

Putting all together we conclude that for $\mu$-a.e.$\gamma\in\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$ the following is true

$\exists\lim_{N\to+\infty}\frac{\vartheta_{N}(\gamma)}{N}=\frac{1}{c}<+\infty.$
(16)

Now we claim that for every $\gamma\in U_{i_{0}}$ satisfying (16) it holds
$+\gamma\in\Lambda_{\text{erg}}$. For every such $\gamma$ we take its covering
geodesic $\tilde{\gamma}$ whose limit point $\tilde{\gamma}^{+}$ defines
$+\gamma=z$. We recall that our basepoint of $X$ is $x_{i_{0}}$. We fix any
geodesic ray $\xi=[x_{i_{0}},z]$. First of all we know that for all $t\geq 0$
it holds $d(\xi(t),\tilde{\gamma}(t))\leq 8\delta+\iota(x_{i_{0}})$ by Lemma
2.2. Moreover we know that for every $N\in\mathbb{N}$ the geodesic $\gamma$
returns to $B(\pi x_{i_{0}},\iota(x_{i_{0}}))$ at time
$\vartheta_{N}(\gamma)$. This implies that
$d(\tilde{\gamma}(\vartheta_{N}(\gamma)),\Gamma
x_{i_{0}})<\iota({x_{i_{0}}})$, and so $d(\xi(\vartheta_{N}(\gamma)),\Gamma
x_{i_{0}})<8\delta+2\iota(x_{i_{0}})$. By definition this means that
$z\in\Lambda_{\tau,\Theta(\gamma)}$, where $\tau=8\delta+2\iota(x_{i_{0}})$.
Finally we observe that the sequence
$\Theta(\gamma)=\\{\vartheta_{N}(\gamma)\\}$ satisfies (16), that is exactly
the condition that defines a sequence involved in the definition of
$\Lambda_{\text{erg}}$. This shows the claim and so that
$\nu(\Lambda_{\text{erg}})=1$.

Step 3. By Step 1, Step 2, (10) and Theorem 4.4 we conclude that

$h_{\Gamma}=\text{PD}(\Lambda_{\text{erg}})\geq\overline{\text{PD}}(\nu)\geq\overline{h}_{\mu,f}^{\textup{loc}}\geq\overline{h}_{\mu}^{\textup{loc}}.$

Finally, from the arbitrariness of $\mu$ we get the thesis. ∎

##  7 Lower bound of the entropy

We consider again the dynamical system $(\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$, where $(X,\sigma)$ is a
$\delta$-hyperbolic GCB-space that is $P_{0}$-packed at scale $r_{0}$ and
$\Gamma$ is a group of $\sigma$-isometries of $X$ acting discretely and
freely. By Remark 5.9 and Lemma 5.4 the space $\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$ is locally compact and metrizable, so the
topological entropy $h_{\text{top}}$ can be computed via the variational
principle, see [HKR95], as

$h_{\textup{top}}=\inf_{\text{\texthtd}}\sup_{K}\lim_{r\to
0}\limsup_{n\to+\infty}\frac{1}{n}\log\textup{Cov}_{\text{\texthtd}^{n}}(K,r),$
(17)

where the infimum is among all metrics inducing the natural topology of
$\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$ and the supremum is among all
compact subsets of $\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$. Observe
that in the right hand side of (17) the compact subsets $K$ are not required
to be $T$-invariant, where by $T$-invariant we mean $T(K)=K$. For a general
dynamical system the restriction to $T$-invariant compact subsets gives a
strictly smaller quantity, see the remark below Lemma 1.6 of [HKR95]. We
remark that when $K$ is a compact $T$-invariant subset then the dynamical
system $(K,T)$ is supported on a compact metrizable space, so the quantity

$\lim_{r\to
0}\limsup_{n\to+\infty}\frac{1}{n}\log\textup{Cov}_{\text{\texthtd}^{n}}(K,r)$

does not depend on the choice of the metric and it coincides with the
topological entropy of the dynamical system $(K,T)$ by the variational
principle for compact dynamical systems. We define

$h_{\textup{inv-top}}=\sup_{K\,\,\,T\text{-inv.}}\lim_{r\to
0}\limsup_{n\to+\infty}\frac{1}{n}\log\textup{Cov}_{\text{\texthtd}^{n}}(K,r)=\sup_{K\,\,\,T\text{-inv.}}h_{\text{top}}(K,T),$
(18)

called the _invariant topological entropy_ of $(\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X),\Phi_{1})$. Here the supremum is among all
compact $T$-invariant subsets $K$ and is any metric on the compact subset $K$
inducing its topology. Clearly $h_{\textup{inv-top}}\leq h_{\textup{top}}$.
The strategy to conclude the proof of Theorem A is to show $h_{\textup{inv-
top}}\geq h_{\Gamma}$. First of all we need to identify the invariant compact
subsets of $\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)$. We fix a basepoint
$x\in X$ and we call $\pi x$ the projection of $x$ in $\Gamma\backslash X$, as
usual. Then the following are the prototype of compact $\Phi_{1}$-invariant
subsets:

$K_{\tau}=\\{\gamma\in\text{Loc-Geod}_{\sigma}(\Gamma\backslash X)\text{ s.t.
}\gamma(n)\in\overline{B}(\pi x,\tau)\text{ for all }n\in\mathbb{Z}\\},$

where $\tau\geq 0$ is a real number. The set $K_{\tau}$ is essentially the set
of local geodesic lines whose image is completely contained in a fixed compact
set of the quotient $\Gamma\backslash X$.

######  Lemma 7.1.

$K_{\tau}$ is a compact $\Phi_{1}$-invariant subset for all $\tau\geq 0$.
Moreover any compact $\Phi_{1}$-invariant subset is contained in $K_{\tau}$
for some $\tau\geq 0$.

###### Proof.

For all $\tau\geq 0$ the set $K_{\tau}$ is clearly $\Phi_{1}$-invariant and
closed by Remark 5.9. Let now $\gamma_{k}\colon\mathbb{R}\to\overline{B}(\pi
x,\tau)$ be a sequence of elements of $K_{\tau}$. By Ascoli-Arzela’s Theorem
there exists a map $\gamma_{\infty}\colon\mathbb{R}\to\overline{B}(\pi
x,\tau)$ such that $\gamma_{k}\to\gamma_{\infty}$ uniformly on compact subsets
of $\mathbb{R}$ and since $K_{\tau}$ is closed clearly $\gamma_{\infty}\in
K_{\tau}$. This completes the proof of the first statement.
Now let $K$ be a compact $\Phi_{1}$-invariant subset of $\text{Loc-
Geod}_{\sigma}(\Gamma\backslash X)$. We need to show it exists $\tau\geq 0$
such that $K$ is contained in $K_{\tau}$. Suppose it is not true. Then for
every $k\geq 0$ there exists $\gamma_{k}\in K$ such that
$\gamma_{k}(n_{k})\notin\overline{B}(\pi x,k)$ for some $n_{k}\in\mathbb{Z}$.
For every $k$ we reparametrize $\gamma_{k}$ so that
$\gamma_{k}(0)\notin\overline{B}(\pi x,k)$. Since $K$ is $\Phi_{1}$-invariant
then the reparametrized geodesics belong again to $K$. Moreover $K$ is
compact, then there exists a subsequence, denoted again by $\gamma_{k}$, that
converges to $\gamma_{\infty}\in K$. In particular the sequence
$\gamma_{k}(0)$ converges to $\gamma_{\infty}(0)$, but $d(\gamma_{k}(0),\pi
x)>k$ for every $k$, which is a contradiction. ∎

As a corollary we directly have: $h_{\text{inv-top}}=\sup_{\tau\geq
0}h_{\text{top}}(K_{\tau},\Phi_{1}).$ Now the idea is to relate the
topological entropy of the dynamical system $(K_{\tau},\Phi_{1})$ to the
Minkowski dimension of the set $\Lambda_{\tau}$, computed with respect to the
basepoint $x$. The tool we need is the Lipschitz-topological entropy of a
closed subset of the boundary at infinity, studied in [Cav21]. We recall
briefly its definition. Let $(X,\sigma)$ be a $\delta$-hyperbolic, GCB-space
that is $P_{0}$-packed at scale $r_{0}$. For a subset $C$ of $\partial X$ and
$Y\subseteq X$ we set

$\text{Geod}_{\sigma}(Y,C)=\\{\gamma\in\text{Geod}_{\sigma}(X)\text{ s.t.
}\gamma^{\pm}\in C\text{ and }\gamma(0)\in Y\\}.$

If $Y=X$ we simply write $\text{Geod}_{\sigma}(C)$. Clearly
$\text{Geod}_{\sigma}(C)$ is a $\Phi$-invariant subset of
$\text{Geod}_{\sigma}(X)$, so the geodesic flow is well defined on it. The
lower Lipschitz-topological entropy of $\text{Geod}_{\sigma}(C)$ is defined as

$\underline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(C))=\inf_{\text{\texthtd}}\sup_{K}\lim_{r\to
0}\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(K,r),$

where the infimum is taken among all geometric metrics on
$\text{Geod}_{\sigma}(C)$, the supremum is among all compact subsets of
$\text{Geod}_{\sigma}(C)$, $\text{\texthtd}^{T}$ denotes the metric
$\text{\texthtd}^{T}(\gamma,\gamma^{\prime})=\sup_{t\in[0,T]}\text{\texthtd}(\Phi_{t}\gamma,\Phi_{t}\gamma^{\prime})$
and $\text{Cov}_{\text{\texthtd}^{T}}(K,r)$ denotes the covering number of the
set $K$ at scale $r$ with respect to the metric $\text{\texthtd}^{T}$. Taking
the limit superior instead of the limit inferior we define the so-called upper
Lipschitz-topological entropy of Geod${}_{\sigma}(C)$, denoted
$\overline{h_{\text{Lip-top}}}(\text{Geod}_{\sigma}(C))$. The computation of
these invariants can be really simplified.

######  Proposition 7.2 ([Cav21]).

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $C$ be a subset of $\partial X$ and $x\in X$. Then there
exists a constant $L$ depending only on $x$ such that

$\underline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(C))=\sup_{C^{\prime}\subseteq
C}\liminf_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}),r)$

for every $f\in\mathcal{F}$ and every $r>0$, where the supremum is among all
closed subsets $C^{\prime}$ of $C$. Moreover if $C$ is closed then
$\underline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(C))=\underline{\textup{MD}}(C).$
In particular for every subset $C$ of $\partial X$ it holds

$\underline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(C))=\sup_{C^{\prime}\subseteq
C}\underline{\textup{MD}}(C^{\prime}),$

where the supremum is among all closed subsets of $C$. The corresponding
results are true for the upper Lipschitz-topological entropy.

We are now ready to show:

######  Theorem 7.3.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $\Gamma$ be a group of $\sigma$-isometries of $X$ acting
discretely and freely. Then

$h_{\textup{inv-top}}\geq h_{\Gamma}.$

###### Proof.

We already know that $h_{\text{inv-top}}=\sup_{\tau\geq
0}h_{\text{top}}(K_{\tau},\Phi_{1}).$ Moreover $K_{\tau}$ is compact, so we
can use every metric on $K_{\tau}$ inducing its topology in order to compute
its topological entropy. We are going to select a specific metric in the
following way: first of all we call $\iota_{\tau}$ the positive injectivity
radius of the compact set $\overline{B}(x,\tau)$. Secondly we consider

$f_{\tau}(s)=\frac{1}{2}\frac{4}{\iota_{\tau}}e^{-\frac{4}{\iota_{\tau}}|s|}.$

It is easy to check that $f_{\tau}\in\mathcal{F}$ and

$C(f_{\tau})=\int_{-\infty}^{+\infty}2|s|f_{\tau}(s)ds=\frac{\iota_{\tau}}{2}.$

Moreover we have:

$\displaystyle h_{\text{top}}(K_{\tau},\Phi_{1})$ $\displaystyle=\lim_{r\to
0}\limsup_{n\to+\infty}\frac{1}{n}\log\text{Cov}_{f^{n}_{\tau}}(K_{\tau},r)\geq\lim_{r\to
0}\liminf_{n\to+\infty}\frac{1}{n}\log\text{Cov}_{f^{n}_{\tau}}(K_{\tau},r)$
$\displaystyle\geq\liminf_{n\to+\infty}\frac{1}{n}\log\text{Cov}_{f^{n}_{\tau}}\left(K_{\tau},\frac{\iota_{\tau}}{2}\right)\geq\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}_{\tau}}\left(K_{\tau},\frac{\iota_{\tau}}{2}\right).$

Our aim is to show

$\small\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}_{\tau}}\left(K_{\tau},\frac{\iota_{\tau}}{2}\right)\geq\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f_{\tau}^{T}}\left(\text{Geod}_{\sigma}(\overline{B}(x,L),\Lambda_{\tau-8\delta-L}),\frac{3}{2}\iota_{\tau}\right),$
(19)

where $L$ is the constant of Proposition 7.2 that depends only on $x$ and not
on $\tau$. Indeed the right hand side equals $\underline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(\Lambda_{\tau-8\delta-L}))$ by Proposition 7.2, so
we would have

$h_{\text{inv-top}}=\sup_{\tau\geq
0}h_{\text{top}}(K_{\tau},\Phi_{1})\geq\sup_{\tau\geq
0}\underline{h_{\text{Lip-top}}}(\Lambda_{\tau-8\delta-L})=\sup_{\tau\geq
0}\underline{\text{MD}}(\Lambda_{\tau})=h_{\Gamma}$

again by Proposition 7.2 and (11), concluding the proof.
It remains to show (19). For every $T\geq 0$ let
$\gamma_{1},\ldots,\gamma_{N}$ be a set of $\sigma$-local geodesics realizing
$\text{Cov}_{f_{\tau}^{T}}(K_{\tau},\frac{\iota_{\tau}}{2}).$ For every
$i=1,\ldots,N$ we consider the set

$A_{i}=\\{\tilde{\gamma}\in\text{Geod}_{\sigma}(X)\text{ s.t.
}\Pi\tilde{\gamma}=\gamma_{i}\text{ and
}\tilde{\gamma}(0)\in\overline{B}(x,L+\iota_{\tau})\\}.$

Observe that for every two elements $\tilde{\gamma},\tilde{\gamma}^{\prime}\in
A_{i}$ we have $\Pi\tilde{\gamma}(0)=\Pi\tilde{\gamma}^{\prime}(0)$, so by
Lemma 5.1 either $\tilde{\gamma}=\tilde{\gamma}^{\prime}$ or
$d(\tilde{\gamma}(0),\tilde{\gamma}^{\prime}(0))\geq\iota_{\tau}$ since
$\iota_{\tau}$ is smaller than or equal to the injectivity radius at
$\gamma_{i}(0)$. We conclude that the set
$\\{\tilde{\gamma}(0)\\}_{\tilde{\gamma}\in A_{i}}$ is a
$\frac{\iota_{\tau}}{2}$-separated subset of $\overline{B}(x,L+\iota_{\tau})$,
implying $\\#A_{i}\leq\text{Pack}(L+\iota_{\tau},\frac{\iota_{\tau}}{2})$.
We consider the set $A_{T}=\bigcup_{i=1}^{N}A_{i}$ whose cardinality satisfies

$\\#A_{T}\leq\text{Cov}_{f_{\tau}^{T}}(K_{\tau},r)\cdot\text{Pack}\left(L+\iota_{\tau},\frac{\iota_{\tau}}{2}\right).$

We claim that $A_{T}$ covers the set
$\text{Geod}_{\sigma}(\overline{B}(x,L),\Lambda_{\tau-8\delta-L})$ at scale
$\frac{3}{2}\iota_{\tau}$ with respect to the metric $f_{\tau}^{T}$. Indeed
let
$\tilde{\gamma}\in\text{Geod}_{\sigma}(\overline{B}(x,L),\Lambda_{\tau-8\delta-L})$.
This means that a geodesic ray $\xi=[x,\tilde{\gamma}^{+}]$ is contained in
the $(\tau-8\delta-L)$-neighbourhood of the orbit $\Gamma x$. Moreover by
Lemma 2.2 we know that $d(\tilde{\gamma}(t),\xi(t))\leq 8\delta+L$ for every
$t\geq 0$. The same consideration holds for the negative ray
$\tilde{\gamma}|_{(-\infty,0]}$, so $d(\tilde{\gamma}(t),\Gamma x)\leq\tau$
for all $t\in\mathbb{R}$: this implies that $\Pi\tilde{\gamma}$ belongs to
$K_{\tau}$. By definition there exists $i\in\\{1,\ldots,N\\}$ such that
$f_{\tau}^{T}(\gamma_{i},\Pi\tilde{\gamma})\leq\frac{\iota_{\sigma}}{2}.$
Therefore for every $t\in[0,T]$ we know that

$\int_{-\infty}^{+\infty}d(\gamma_{i}(t+s),\Pi\tilde{\gamma}(t+s))f_{\tau}(s)ds\leq\frac{\iota_{\tau}}{2}.$

Moreover from the choice of $f_{\tau}$ we have

$\displaystyle\frac{\iota_{\tau}}{2}$
$\displaystyle\geq\int_{-\infty}^{+\infty}d(\gamma_{i}(t+s),\Pi\tilde{\gamma}(t+s))f_{\tau}(s)ds$
$\displaystyle\geq\int_{-\infty}^{+\infty}(d(\gamma_{i}(t),\Pi\tilde{\gamma}(t))-2|s|)f_{\tau}(s)ds\geq
d(\gamma_{i}(t),\Pi\tilde{\gamma}(t))-\frac{\iota_{\tau}}{2}.$

In conclusion $d(\gamma_{i}(t),\Pi\tilde{\gamma}(t))\leq\iota_{\tau}$ for
every $t\in[0,T]$. By definition of injectivity radius, since
$\Pi\tilde{\gamma}(t)\in\overline{B}(x,\tau)$, we conclude that for every
$t\in[0,T]$ there exists a covering geodesic $\tilde{\gamma}_{t}$ of
$\gamma_{i}$ such that

$d(\tilde{\gamma}_{t}(t),\tilde{\gamma}(t))=d(\gamma_{i}(t),\Pi\tilde{\gamma}(t))\leq\iota_{\tau}.$

For $t=0$ we have the covering geodesic $\tilde{\gamma}_{0}$ of $\gamma_{i}$
that satisfies

$d(\tilde{\gamma}_{0}(0),x)\leq
d(\tilde{\gamma}_{0}(0),\tilde{\gamma}(0))+d(\tilde{\gamma}(0),x)\leq
L+\iota_{\tau},$

so $\tilde{\gamma}_{0}\in A_{i}$. Moreover for every $t\in[0,T]$ there exists
an element $g_{t}\in\Gamma$ such that
$\tilde{\gamma}_{t}=g_{t}\tilde{\gamma}_{0}$. But arguing as in the proof of
Proposition 5.5 we conclude that $g_{t}=\text{id}$ for every $t\in[0,T]$ and
so $d(\tilde{\gamma}_{0}(t),\tilde{\gamma}(t))\leq\iota_{\tau}$ for every
$t\in[0,T]$.
We can now estimate the distance between $\tilde{\gamma}_{0}$ and
$\tilde{\gamma}$ with respect to $f_{\tau}^{T}$. For every $t\in[0,T]$ we have

$\displaystyle f^{t}(\tilde{\gamma}_{0},\tilde{\gamma})$
$\displaystyle=\int_{-\infty}^{+\infty}d(\tilde{\gamma}_{0}(t+s),\tilde{\gamma}(t+s))f_{\tau}(s)ds$
$\displaystyle\leq\int_{-\infty}^{+\infty}(d(\tilde{\gamma}_{0}(t),\tilde{\gamma}(t))+2|s|)f_{\tau}(s)ds\leq\iota_{\tau}+\frac{\iota_{\tau}}{2}=\frac{3}{2}\iota_{\tau}.$

Since $\tilde{\gamma}_{0}\in A_{T}$ we conclude that the set $A_{T}$ covers
Geod${}_{\sigma}(\overline{B}(x,L),\Lambda_{\tau-8\delta-L})$ at scale
$\frac{3}{2}\iota_{\tau}$ with respect to the metric $f_{\tau}^{T}$. We can
finish the proof of the theorem since

$\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f_{\tau}^{T}}\left(\text{Geod}_{\sigma}(\overline{B}(x_{0},L),\Lambda_{\tau-8\delta-L}),\frac{3}{2}\iota_{\tau}\right)$

is less than or equal to

$\displaystyle\liminf_{T\to+\infty}\frac{1}{T}\log\\#A_{T}$
$\displaystyle\leq\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f_{\tau}^{T}}\left(K_{\tau},\frac{\iota_{\tau}}{2}\right)\cdot\text{Pack}\left(L+\iota_{\tau},\frac{\iota_{\tau}}{2}\right)$
$\displaystyle=\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f_{\tau}^{T}}\left(K_{\tau},\frac{\iota_{\tau}}{2}\right),$

where the last equality follows from Proposition 5.6. ∎

##  8 Additional remarks

The definition of upper and lower local entropy of a dynamical system $(Y,T)$
as

$\overline{h}^{\text{loc}}(Y,T)=\sup_{\mu\in\mathcal{E}_{1}(Y,T)}\inf_{\text{\texthtd}}\overline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T),\quad\underline{h}^{\text{loc}}(Y,T)=\sup_{\mu\in\mathcal{E}_{1}(Y,T)}\inf_{\text{\texthtd}}\underline{h}_{\mu,\text{\texthtd}}^{\text{loc}}(Y,T)$

could seem arbitrary, in the sense that the order of the supremum and the
infimum could be switched. In our case there is no difference.

######  Corollary 8.1.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $\Gamma$ be a group of $\sigma$-isometries of $X$ acting
discretely and freely. Then

$\inf_{\textup{\texthtd}}\sup_{\mu\in\mathcal{E}_{1}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}=\inf_{\textup{\texthtd}}\sup_{\mu\in\mathcal{E}_{1}}\underline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}=h_{\Gamma}.$

###### Proof.

Clearly it always holds

$\inf_{\textup{\texthtd}}\sup_{\mu\in\mathcal{E}_{1}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}\geq\sup_{\mu\in\mathcal{E}_{1}}\inf_{\textup{\texthtd}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}.$

But, for $f\in\mathcal{F}$, we have also

$h_{\Gamma}=\sup_{\mu\in\mathcal{E}_{1}}h_{\mu}\leq\sup_{\mu\in\mathcal{E}_{1}}\inf_{\textup{\texthtd}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}\leq\sup_{\mu\in\mathcal{E}_{1}}\overline{h}_{\mu,f}^{\textup{loc}}\leq
h_{\Gamma}.$

So

$h_{\Gamma}\leq\sup_{\mu\in\mathcal{E}_{1}}\inf_{\textup{\texthtd}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}\leq\inf_{\textup{\texthtd}}\sup_{\mu\in\mathcal{E}_{1}}\overline{h}_{\mu,\textup{\texthtd}}^{\textup{loc}}\leq\sup_{\mu\in\mathcal{E}_{1}}\overline{h}_{\mu,f}^{\textup{loc}}\leq
h_{\Gamma},$

implying the equalities. The same proof holds for the lower entropies. ∎

The second observation is about measure of maximal entropy.

######  Corollary 8.2.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. Let $\Gamma$ be a group of $\sigma$-isometries of $X$ acting
discretely and freely. If there exists a measure $\mu\in\mathcal{E}_{1}$ of
maximal entropy, that is $h_{\mu}=h_{\textup{top}}$, then for all
$f\in\mathcal{F}$ it holds:

$h_{\Gamma}=h_{\mu}=\underline{h}_{\mu}^{\textup{loc}}=\underline{h}_{\mu,f}^{\textup{loc}}=\overline{h}_{\mu}^{\textup{loc}}=\overline{h}_{\mu,f}^{\textup{loc}}.$

Moreover for $\mu$-a.e.$\gamma\in\textup{Loc-Geod}_{\sigma}(\Gamma\backslash
X)$ it holds

$h_{\Gamma}=\lim_{n\to+\infty}-\frac{1}{n}\log\mu(B_{f^{n}}(\gamma,R))$

for every $R>0$.

We recall that it can happen that a measure of maximal entropy does not exist.
For instance if $M$ is a Riemannian manifold with pinched negative curvature
it is known by Otal-Peigné’s Theorem (cp. [OP04]) that such a measure exists
if and only if the Bowen-Margulis measure is finite, and it that case this
measure, once normalized, is the unique measure of maximal entropy. Probably
the same is true in the setting of Gromov-hyperbolic, packed, GCB-spaces but
the construction of the Bowen-Margulis entropy is less direct. However using
the tecnhiques of [Ric14] and [Lin19] it is plausible that Otal-Peigné’s
result can be generalized to our setting.

We conclude with a corollary and an example.

######  Corollary 8.3.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$, $\Gamma$ be a discrete group of $\sigma$-isometries of $X$ and
$x$ be a fixed basepoint of $X$. Then

$h_{\Gamma}=\overline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(\Lambda_{\textup{u-rad}}))=\underline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(\Lambda_{\textup{u-rad}})).$

###### Proof.

The set $\Lambda_{\text{u-rad}}$ is the increasing union of the closed subsets
$\Lambda_{\tau}$, $\tau\geq 0$. Let $C^{\prime}$ be a closed subset of
$\Lambda_{\text{u-rad}}$. There are two possibilities: either
$C^{\prime}=\Lambda_{\text{u-rad}}$ or it is contained in $\Lambda_{\tau}$ for
some $\tau\geq 0$. In the first case by Theorem 12.2.7 of [DSU17] we get
$\Lambda_{\text{u-rad}}=\Lambda_{\tau}$ for some $\tau\geq 0$. Therefore in
both cases
$\overline{\text{MD}}(C^{\prime})\leq\overline{\text{MD}}(\Lambda_{\tau})$,
and the same holds for the lower Minkowski dimensions. This implies the thesis
by Proposition 7.2. ∎

Now we show an example of subset $C$ of $\partial X$ such that

$\overline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(C))<\overline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(\overline{C})),$ (20)

where $\overline{C}$ is the closure of $C$. In [DPPS09] there is an example of
a pinched negatively curved Riemannian manifold $(M,g)$ admitting a non-
uniform lattice $\Gamma$ (i.e. the volume of $\Gamma\backslash M$ is finite)
such that $h_{\Gamma}<h_{\text{vol}}(M)$, where $h_{\text{vol}}(M)$ is the
volume entropy of $M$. By [Cav21] we know that

$h_{\text{vol}}(M)=\overline{h_{\textup{Lip-top}}}(\textup{Geod}(\partial
M))=\overline{h_{\textup{Lip-top}}}(\textup{Geod}(\Lambda(\Gamma))),$

while

$h_{\Gamma}=\overline{h_{\textup{Lip-
top}}}(\textup{Geod}(\Lambda_{\textup{u-rad}}(\Gamma)))$

by the corollary above. Since the closure of $\Lambda_{\text{u-rad}}(\Gamma)$
is $\Lambda(\Gamma)$ then (20) holds.

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