# Entropies of non-positively curved metric spaces

Nicola Cavallucci

###### Abstract

We show the equivalences of several notions of entropy, like a version of the
topological entropy of the geodesic flow and the Minkowski dimension of the
boundary, in metric spaces with convex geodesic bicombings satisfying a
uniform packing condition. Similar estimates will be given in case of closed
subsets of the boundary of Gromov-hyperbolic metric spaces with convex
geodesic bicombings.

###### Contents

  1. 1 Introduction
    1. 1.1 Lipschitz-topological entropy of the geodesic flow
    2. 1.2 Volume and Covering entropy
    3. 1.3 Minkowski dimension of the boundary
    4. 1.4 Equality of the entropies
    5. 1.5 Entropies of the closed subsets of the boundary
    6. 1.6 Differences of the invariants for geometrically finite groups
  2. 2 GCB-spaces and space of geodesics
  3. 3 Covering and volume entropy
    1. 3.1 Covering entropy
    2. 3.2 Volume entropy of homogeneous measures
  4. 4 Lipschitz-topological entropy
    1. 4.1 Topological entropy for the distances induced by $f\in\mathcal{F}$
    2. 4.2 Proof of Theorem 4.2
  5. 5 Gromov-hyperbolic metric spaces
    1. 5.1 Minkowski dimension
  6. 6 Entropies of subsets of the boundary
    1. 6.1 Covering and volume entropy
    2. 6.2 Lipschitz topological entropy
    3. 6.3 Minkowski dimension

##  1 Introduction

This paper is devoted to the investigation of different asymptotic quantities
associated to a metric space, some of them classical and widely studied. As we
will see in a minute these invariants have different nature: dynamical,
measure-theoretic and combinatoric. The purpose is two-fold:

  * -

to show the relations between these invariants, especially to understand when
they are equal;

  * -

to develop interesting tools and techniques to simplify the computation of
these invariants.

The second goal is extremely useful for applications: it is important to have
flexible and easy to compute definitions in order to study these invariants in
concrete cases. In the forthcoming paper [Cav21b] we will use this flexibility
to extend Otal-Peigné’s Theorem ([OP04]) to a large class of metric spaces. A
simplified version of the techniques developed on this paper were used to show
the continuity of the critical exponent of fundamental groups of a large class
of compact metric spaces under Gromov-Hausdorff convergence, see [Cav21a].

We are mainly interested on metric spaces satisfying weak bounds on the
curvature. As upper bound we consider a very weak convexity condition: the
existence of a convex geodesic bicombing. A geodesic bicombing on a metric
space $X$ is a map $\sigma\colon X\times X\times[0,1]\to X$ such that for all
$x,y\in X$ the function $\sigma_{xy}(\cdot)=\sigma(x,y,\cdot)$ is a geodesic
(parametrized proportionally to arc-length) from $x$ to $y$. A bicombing
$\sigma$ is convex if for all $x,y,x^{\prime},y^{\prime}\in X$ the map
$t\mapsto d(\sigma_{xy}(t),\sigma_{x^{\prime}y^{\prime}}(t))$ is convex. Among
metric spaces admitting a convex bicombing there are CAT$(0)$-spaces, Busemann
convex spaces and all normed vector spaces. The interest in this condition is
given by its stability under limits ([Des15], [CS20a]), while it is not the
case for Busemann convex spaces. Given a bicombing $\sigma$, every curve
$\sigma_{xy}$ is called a $\sigma$-geodesic. The bicombing is geodesically
complete if any $\sigma$-geodesic can be extended to a bigger
$\sigma$-geodesic. This notion coincides with the usual geodesic completeness
in case of Busemann convex metric spaces. A GCB-space is a couple $(X,\sigma)$
where $X$ is a complete metric space and $\sigma$ is a geodesically complete,
convex, geodesic bicombing on $X$.
As a lower bound on the curvature we take a uniform packing condition: a
metric space $X$ is said to be $P_{0}$-packed at scale $r_{0}$ if for all
$x\in X$ the cardinality of a maximal $2r_{0}$-separated subset of
$B(x,3r_{0})$ is at most $P_{0}$. This uniform packing condition interacts
very well with the weak convexity property given by a geodesically complete,
convex, geodesic bicombing, implying a uniform control of the packing
condition _at every scale_ (see Proposition 2.1 and [CS20a]). Sometimes,
especially in the second part of the paper where relative versions of the
invariants will be studied, we will impose also a Gromov-hyperbolicity
condition on our metric space.

###  1.1 Lipschitz-topological entropy of the geodesic flow

The topological entropy of the geodesic flow has been intensively studied in
case of Riemannian manifolds, especially in the negatively curved setting. If
such a manifold is denoted by $\bar{M}=M/\Gamma$, where $M$ is its universal
cover and $\Gamma$ is its fundamental group, then the set of parametrized
geodesic lines is identified with the unit tangent bundle $S\bar{M}$. Probably
the most important invariant associated to the geodesic flow is its
topological entropy, denoted $h_{\text{top}}(\bar{M})$. It equals the
Hausdorff dimension of the limit set of $\Gamma$ and the critical exponent of
$\Gamma$ (see [Sul84], [OP04]). Moreover if $\bar{M}$ is compact then it
coincides also with the volume entropy of $M$ ([Man79]), while this is no more
true in general, even when $\bar{M}$ has finite volume (cp. [DPPS09]): we will
come back to these examples at the end of the introduction.

In case of GCB-metric spaces $(X,\sigma)$ we restrict the attention to
$\sigma$-geodesic lines. The topological entropy of the $\sigma$-geodesic flow
is defined as the topological entropy (in the sense of Bowen, cp. [Bow73],
[HKR95]) of the dynamical system $(\text{Geod}_{\sigma}(X),\Phi_{t})$, where
$\text{Geod}_{\sigma}(X)$ is the space of parametrized $\sigma$-geodesic
lines, endowed with the topology of uniform convergence on compact subsets,
and $\Phi_{t}$ is the reparametrization flow. It is:

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

where the infimum is taken among all metrics on Geod${}_{\sigma}(X)$ inducing
its topology, the supremum is taken among all compact subsets of
Geod${}_{\sigma}(X)$, T is the distance
$\text{\texthtd}^{T}(\gamma,\gamma^{\prime})=\max_{t\in[0,T]}\text{\texthtd}(\Phi_{t}(\gamma),\Phi_{t}(\gamma^{\prime}))$
and $\text{Cov}_{\text{\texthtd}^{T}}(K,r)$ is the minimal number of balls
(with respect to the metric T) of radius $r$ needed to cover $K$. We remark
that in case of Busemann convex (or CAT$(0)$) metric spaces the space of
$\sigma$-geodesic lines coincides with the set of geodesic lines. This flow
has no recurrent geodesics, so applying the variational principle (cp.
[HKR95]) it is straightforward to conclude that its topological entropy is
zero (Lemma 4.1). Looking carefully at the proof of the variational principle
it turns out that the metrics on $\text{Geod}_{\sigma}(X)$ almost realizing
the infimum in the definition of the topological entropy are restriction to
Geod${}_{\sigma}(X)$ of metrics on its one-point compactification. In
particular they are no the natural ones to consider in this setting. That is
why, in Section 4, we will restrict the attention to the class of geometric
metrics : those with the property that the evaluation map
$E\colon(\text{Geod}_{\sigma}(X),\text{\texthtd})\to(X,d)$ defined as
$E(\gamma)=\gamma(0)$ is Lipschitz. Notice that for a geometric metric two
geodesic lines are not close if they are distant at time $0$. Accordingly the
_Lipschitz-topological entropy_ of the geodesic flow is defined as

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

where _the infimum is taken only among the geometric metrics of_
Geod${}_{\sigma}(X)$. Although the definition of the Lipschitz-topological
entropy is quite complicated, its computation can be remarkably simplified.
Indeed one of the most used metric on Geod${}_{\sigma}(X)$ (see for instance
[BL12]) is:

$d_{\text{Geod}}(\gamma,\gamma^{\prime})=\int_{-\infty}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))\frac{1}{2e^{|s|}}ds$

that induces the topology of $\text{Geod}_{\sigma}(X)$ and is geometric, and
it turns out that it realizes the infimum in the definition of the Lipschitz-
topological entropy.

######  Theorem A (Extract from Theorem 4.2 & Proposition 4.3).

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

$h_{\textup{Lip-
top}}(\textup{Geod}_{\sigma}(X))=\lim_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{d_{\textup{Geod}}^{T}}(\textup{Geod}_{\sigma}(x),r_{0}),$

where $\textup{Geod}_{\sigma}(x)$ is the set of $\sigma$-geodesic lines
passing through $x$ at time $0$.

Therefore the infimum in the definition of the Lipschitz topological entropy
is actually realized by the metric $d_{\text{Geod}}$ and the supremum among
the compact sets can be replaced by a fixed (relatively small) compact set.
Moreover also the scale $r$ can be fixed to be $r_{0}$ (or any other positive
real number).
Actually the result of Theorem A is still valid for a whole family of metrics
on Geod${}_{\sigma}(X)$: this flexibility will be one of the fundamental
ingredient in the main result of [Cav21b].

###  1.2 Volume and Covering entropy

The second definition of entropy we consider (see Section 3.2) is the volume
entropy. If $X$ is a metric space equipped with a measure $\mu$ it is
classical to consider the exponential growth rate of the volume of balls,
namely:

$h_{\mu}(X):=\lim_{T\to+\infty}\frac{1}{T}\log\mu(\overline{B}(x,T)).$

It is called the volume entropy of $X$ with respect to the measure $\mu$ and
it does not depend on the choice of the basepoint $x\in X$ by triangular
inequality. This invariant has been studied intensively in case of complete
Riemannian manifolds with non positive sectional curvature, where $\mu$ is the
Riemannian volume on the universal cover. It is related to other interesting
invariants as the simplicial volume of the manifold (see [Gro82]), [BS20]), a
macroscopical condition on the scalar curvature (cp. [Sab17]) and the systole
in case of compact, non-geometric $3$-manifolds (cp. [CS19]). Moreover the
infimum of the volume entropy among all the possible Riemannian metrics of
volume $1$ on a fixed closed manifold is a subtle homotopic invariant (see
[Bab93], [Bru08] for general considerations and [BCG95], [Pie19] for the
computation of the minimal volume entropy in case of, respectively, closed
$n$-dimensional manifolds supporting a locally symmetric metric of negative
curvature and $3$-manifolds).

A measure $\mu$ is called $H$-homogeneous at scale $r$ if

$\frac{1}{H}\leq\mu(\overline{B}(x,r))\leq H$

for every $x\in X$. Among homogeneous measures there is a remarkable example:
the volume measure $\mu_{X}$ of a complete, geodesically complete, CAT$(0)$
metric space $X$ that is $P_{0}$-packed at scale $r_{0}$ (see [CS20b] and
[LN19] for a description of the measure). In case $X$ is a Riemannian manifold
of non-positive sectional curvature then $\mu_{X}$ coincides with the
Riemannian volume, up to a universal multiplicative constant.
A more combinatoric and intrinsic version of the volume entropy of a generic
metric space is the _covering entropy_ , defined as:

$h_{\text{Cov}}(X):=\lim_{T\to+\infty}\frac{1}{T}\log\text{Cov}(\overline{B}(x,T),r),$

where $x$ is a point of $X$ and $\text{Cov}(\overline{B}(x,T),r)$ is the
minimal number of balls of radius $r$ needed to cover $\overline{B}(x,T)$. It
does not depend on $x$ but it can depend on the choice of $r$. This is not the
case when $X$ is a GCB-space that is $P_{0}$-packed at scale $r_{0}$, as
follows by Proposition 3.1. Moreover it is always finite (cp.Lemma 3.3).

###  1.3 Minkowski dimension of the boundary

The expression of the Lipschitz-topological entropy given by Theorem A
suggests the possibility to relate that invariant to some property of the
boundary at infinity of $X$. For simplicity we suppose $X$ is also Gromov-
hyperbolic, so that the boundary at infinity is metrizable. If we denote by
$(\cdot,\cdot)_{x}$ the Gromov product based on $x$ then the generalized
visual ball of center $z\in\partial X$ and radius $\rho$ is
$B(z,\rho)=\\{z^{\prime}\in\partial X\text{ s.t.
}(z,z^{\prime})_{x}>\log\frac{1}{\rho}\\}$. The visual Minkowski dimension of
the Gromov boundary $\partial X$ is:

$\text{MD}(\partial X)=\lim_{T\to+\infty}\frac{1}{T}\log\text{Cov}(\partial
X,e^{-T}),$

where $\text{Cov}(\partial X,e^{-T})$ is the minimal number of generalized
visual balls of radius $e^{-T}$ needed to cover $\partial X$. If the
generalized visual balls are metric balls for some visual metric $D_{x,a}$
then we refind the usual definition of Minkowski dimension of the metric space
$(\partial X,D_{x,a})$, once the obvious change of variable $\rho=e^{-T}$ is
made. This invariant is presented in Section 5.1.

###  1.4 Equality of the entropies

One of our main results is:

######  Theorem B.

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

$h_{\textup{Lip-
top}}(\textup{Geod}_{\sigma}(X))=h_{\mu}(X)=h_{\textup{Cov}}(X),$

where $\mu$ is every homogeneous measure on $X$. Moreover if $X$ is also
$\delta$-hyperbolic then the quantities above coincide also with
$\textup{MD}(\partial X)$.

Actually something more is true but in order to state it we need to recall the
notion of equivalent asymptotic behaviour of two functions introduced in
[Cav21a]. Given $f,g\colon[0,+\infty)\to\mathbb{R}$ we say that $f$ and $g$
have the same asymptotic behaviour, and we write $f\asymp g$, if for all
$\varepsilon>0$ there exists $T_{\varepsilon}\geq 0$ such that if $T\geq
T_{\varepsilon}$ then $|f(T)-g(T)|\leq\varepsilon$. The function
$T_{\varepsilon}$ is called the _threshold function_. Usually we will write
$f\underset{P_{0},r_{0},\delta,\ldots}{\asymp}g$ meaning that the threshold
function can be expressed only in terms of $\varepsilon$ and
$P_{0},r_{0},\delta,\ldots$

######  Theorem C.

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. Then
the functions defining the quantities of Theorem B have the same asymptotic
behaviour and the threshold functions depend only on $P_{0},r_{0},\delta$ and
the homogeneous constants of $\mu$.

Therefore not only all the introduced quantities define the same number, but
all of them also have the same asymptotic behaviour. This means that if one
can control the rate of convergence to the limit of one of these quantities
then also the rate of convergence of all the other quantities is bounded. We
remark that, differently from many of the papers in the literature, we do not
require any group action on our metric spaces.
The control of the asymptotic behaviour of the function defining the Minkowski
dimension is the main ingredient of the continuity theorem proved in [Cav21a].
The same ideas can be used to show similar continuity statements in more
general settings, but we won’t explore these applications here.

###  1.5 Entropies of the closed subsets of the boundary

In case $X$ is $\delta$-hyperbolic it is possible to define the versions of
all the different notions of entropies relative to subsets of the boundary
$\partial X$. For every subset $C\subseteq\partial X$ we denote by
Geod${}_{\sigma}(C)$ the set of parametrized $\sigma$-geodesic lines with
endpoints belonging to $C$ and with QC-Hull$(C)$ the union of the points
belonging to the geodesics joining any two points of $C$. Actually the
hyperbolicity assumption (or at least a visibility assumption on $\partial X$)
is necessary since otherwise the sets Geod${}_{\sigma}(C)$ and QC-Hull$(C)$
could be empty. The numbers

$\displaystyle h_{\text{Cov}}(C)$
$\displaystyle=\lim_{T\to+\infty}\frac{1}{T}\log\text{Cov}(\overline{B}(x,T)\cap\text{QC-
Hull}(C),r_{0})$ $\displaystyle h_{\text{Lip-top}}(\text{Geod}_{\sigma}(C))$
$\displaystyle=\inf_{\text{\texthtd}}\sup_{K\subseteq\text{Geod}_{\sigma}(C)}\lim_{r\to+\infty}\lim_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(K,r)$
$\displaystyle\text{MD}(C)$
$\displaystyle=\lim_{T\to+\infty}\frac{1}{T}\log\text{Cov}(C,e^{-T})$

are called, respectively, covering entropy of $C$, Lipschitz-topological
entropy of Geod${}_{\sigma}(C)$ and visual Minkowski dimension of $C$. The
volume entropy of $C$ with respect to a measure $\mu$ is

$h_{\mu}(C)=\sup_{\tau\geq
0}\lim_{T\to+\infty}\frac{1}{T}\log\mu(\overline{B}(x,T)\cap\overline{B}(\text{QC-
Hull}(C),\tau)),$

where $\overline{B}(Y,\tau)$ is the closed $\tau$-neighbourhood of $Y\subseteq
X$. If $\mu$ is $H$-homogeneous at scale $r$ then the volume entropy can be
computed taking $\tau=r$ in place of the supremum (Proposition 6.3). For
instance when $X$ is a Riemannian manifold with pinched negative curvature
then the the Riemannian volume $\mu_{X}$ is $H(r)$-homogeneous at every scale
$r>0$, so the definition does not depend on $\tau$ at all. Most of the
relations of Theorem B remain true for subsets of the boundary, but the
asymptotic behaviour of the different functions involved in the definitions of
the entropies depend also on the choice of the basepoint $x\in X$. The best
possible choice, $x\in\text{QC-Hull}(C)$, allows us to give again uniform
asymptotic estimates.

######  Theorem D.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$ and let $C\subseteq\partial X$. Then

$h_{\textup{Cov}}(C)=\textup{MD}(C)=h_{\mu}(C)$

for every homogeneous measure $\mu$ on $X$. All the functions defining the
quantities above have the same asymptotic behaviour and the threshold
functions can be expressed only in terms of $P_{0},r_{0},\delta$ and the
homogeneous constants of $\mu$, if the basepoint $x$ belongs to QC-Hull$(C)$.

The proof of this result does not follow by the same arguments of Theorem B,
indeed it will be based heavily on the Gromov-hyperbolicity of $X$. The
relation between the Lipschitz-topological entropy of Geod${}_{\sigma}(C)$ and
the other definitions of entropy is more complicated. We have

######  Theorem E.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$ and let $C\subseteq\partial X$. Then

  * (i)

if $C$ is closed then $h_{\textup{Lip-
top}}(\textup{Geod}_{\sigma}(C))=h_{\textup{Cov}}(C)$ and the functions
defining these two quantities have the same asymptotic behaviour with
thresholds function depending only on $P_{0},r_{0},\delta$.
More precisely, if $x\in\textup{QC-Hull}(C)$ then

$h_{\textup{Lip-
top}}(\textup{Geod}_{\sigma}(C))=\lim_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{d_{\textup{Geod}}^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,22\delta),C),r_{0}),$

where $\textup{Geod}_{\sigma}(\overline{B}(x,22\delta),C)$ is the set of
geodesic lines with endpoints in $C$ and passing through
$\overline{B}(x,22\delta)$ at time $0$.

  * (ii)

if $C$ is not closed then

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

where the supremum is taken among the closed subsets of $C$.

The inequality in (ii) can be strict, as shown in [Cav21b].

###  1.6 Differences of the invariants for geometrically finite groups

In this last part of the introduction we restrict the attention to the case of
Riemannian manifolds $\bar{M}=M/\Gamma$ with pinched negative sectional
curvature. If $\Gamma$ is geometrically finite then the limit set of $\Gamma$
is the union of the radial limit set $\Lambda_{\text{r}}(\Gamma)$ and the
bounded parabolic points. The latter is a countable set, therefore the
Hausdorff dimension of the limit set coincides with the Hausdorff dimension of
the radial limit set. So by Bishop-Jones’ Theorem it holds (here HD denotes
the Hausdorff dimension):

$\text{HD}(\Lambda(\Gamma))=\text{HD}(\Lambda_{r}(\Gamma))=h_{\Gamma}.$ (1.1)

We remark that this is not true if $\Gamma$ is not geometrically finite, even
when $M$ is the hyperbolic space.

######  Example 1.1.

In general it can happen
$\text{HD}(\Lambda_{\text{r}}(\Gamma)))<\text{HD}(\Lambda(\Gamma))$. Indeed
let $\Gamma$ be a cocompact group of $\mathbb{H}^{2}$ and let
$\Gamma^{\prime}$ be a normal subgroup of $\Gamma$ such that
$\Gamma/\Gamma^{\prime}$ is not amenable. Let
$F\subseteq\Lambda(\Gamma^{\prime})$ be the subsets of points $z$ that are
fixed by some $g\in\Gamma^{\prime}$. For every $z\in F$ and every $h\in\Gamma$
we have $hz=hgz=g^{\prime}hz$ for some $g^{\prime}\in\Gamma^{\prime}$ since
$\Gamma^{\prime}$ is normal. Then $hz$ is fixed by $g^{\prime}$ and so it
belongs to $F$, i.e. $F$ is $\Gamma$-invariant. By minimality of
$\Lambda(\Gamma)$ we get $\Lambda(\Gamma^{\prime})=\Lambda(\Gamma)$, so
$\text{HD}(\Lambda(\Gamma^{\prime}))=\text{HD}(\Lambda(\Gamma))$. But by the
growth tightness of $\Gamma$ (cp. for instance [Sam02]) we have

$\text{HD}(\Lambda_{\text{r}}(\Gamma^{\prime}))=h_{\Gamma^{\prime}}<h_{\Gamma}=\text{HD}(\Lambda(\Gamma))=\text{HD}(\Lambda(\Gamma^{\prime})).$

However if $M$ is the hyperbolic space and $\Gamma$ is geometrically finite
then even something more is true, indeed by [SU96]:

$h_{\Gamma}=\text{HD}(\Lambda(\Gamma))=\text{MD}(\Lambda(\Gamma)).$ (1.2)

This equality fails to be true for geometrically finite (actually of finite
covolume) groups of manifolds with pinched, but variable, negative curvature.
Indeed we have:

######  Example 1.2.

In [DPPS09] it is presented an example of a smooth Riemannian manifold $M$
with pinched negative sectional curvature admitting a (non-uniform) lattice
(i.e. a group of isometries $\Gamma$ with $\text{Vol}(M/\Gamma)<+\infty)$ such
that $h_{\Gamma}<h_{\mu_{M}}(M)$ (recall that $\mu_{M}$ denotes the Riemannian
volume of $M$). We observe that since $\Gamma$ is a lattice then
$\Lambda(\Gamma)=\partial M$, so $h_{\mu_{M}}(M)=\text{MD}(\Lambda(\Gamma))$
by Theorem B, while
$h_{\Gamma}=\text{HD}(\Lambda_{\text{r}}(\Gamma))=\text{HD}(\Lambda(\Gamma))$
by (1.1).

The example above is due to a relevant variation of the curvature of $M$.
Indeed in [DPPS19] is shown that for non-uniform lattices $\Gamma$ of
asymptotically $1/4$-pinched manifolds with negative curvature $M$ it holds
$h_{\mu_{M}}(M)=h_{\Gamma}$. The general situation in the geometrically finite
case is:

$\displaystyle\text{HD}(\Lambda(\Gamma))=h_{\Gamma}=h_{\text{top}}(M/\Gamma)$
(1.3) $\displaystyle h_{\text{Lip-top}}(\text{Geod}(\Lambda(\Gamma$
$\displaystyle)))=h_{\text{Cov}}(\Lambda(\Gamma))=h_{\mu_{M}}(\Lambda(\Gamma))=\text{MD}(\Lambda(\Gamma)),$

where the equalities in the second line follow by Theorem D and Theorem E,
while the equalities in the first line are consequences of (1.1) and Otal-
Peigne’s variational principle [OP04]. Moreover it is clear that the first
line is always less than or equal to the second one, since the Hausdorff
dimension is always smaller than or equal to the Minkowski dimension.

The relations in (1.3) allow us to give new interpretations of the phenomena
occurring in Example 1.2, i.e. the possible difference between the critical
exponent of the group and the volume entropy of $\Lambda(\Gamma)$:

  * •

_measure-theoretic interpretation:_ it can be seen as the difference between
the Hausdorff and the Minkowski dimension of the limit set $\Lambda(\Gamma)$,
so it is related to the fractal structure of the limit set;

  * •

_dynamical interpretation:_ it can be seen as the difference between the
topological entropy of the geodesic flow of the quotient and the Lipschitz-
topological entropy of Geod$(\Lambda(\Gamma))$.

  * •

_combinatoric interpretation:_ it can be seen as the difference between
$h_{\Gamma}$ and $h_{\text{Cov}}(\Lambda(\Gamma))$, where the former counts
the exponential growth rate of an orbit while the latter counts the
exponential growth rate of the cardinality of $r$-nets, for some (any) $r>0$.
Here the difference arises in terms of sparsity of the orbit.

##  2 GCB-spaces and space of geodesics

Throughout the paper $X$ will denote a metric space with metric $d$. The open
(resp.closed) ball of radius $r$ and center $x$ is denoted by $B(x,r)$ (resp.
$\overline{B}(x,r)$), while the metric sphere of center $x$ and radius $R$ is
denoted by $S(x,R)$. We use the notation $A(x,r,r^{\prime})$ to denote the
closed annulus of center $x$ and radii $0<r<r^{\prime}$, i.e. the set of
points $y\in X$ such that $r\leq d(x,y)\leq r^{\prime}$. 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 $y$. We
will denote any geodesic segment between two points $x$ and $y$, with an abuse
of notation, as $[x,y]$. A geodesic ray is an isometry
$\gamma\colon[0,+\infty)\to X$ while a geodesic line is an isometry
$\gamma\colon\mathbb{R}\to X$.

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 following inequalities are
classical:

$\text{Pack}(Y,2r)\leq\text{Cov}(Y,2r)\leq\text{Pack}(Y,r).$ (2.1)

The packing and the covering functions of $X$ are respectively

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

They take values on $[0,+\infty]$. By (2.1) it holds

$\text{Pack}(R,2r)\leq\text{Cov}(R,2r)\leq\text{Pack}(R,r).$ (2.2)

Let $C_{0},P_{0},r_{0}>0$. 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. The
space $X$ is $C_{0}$-covered at scale $r_{0}$ if Cov$(3r_{0},r_{0})\leq
C_{0}$, that is every ball of radius $3r_{0}$ can be covered by at most
$C_{0}$ balls of radius $r_{0}$.

A geodesic bicombing on a metric space $X$ 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
$t,t^{\prime}\in[0,1]$, $\sigma_{xy}(0)=x,\sigma_{xy}(1)=y$.
_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 any
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 2.1 (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 is
proper;

  * (ii)

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

$\displaystyle\textup{Pack}(R,r)$ $\displaystyle\leq
P_{0}(1+P_{0})^{\frac{R}{r}-1}\text{, if }r\leq r_{0};$
$\displaystyle\textup{Pack}(R,r)$ $\displaystyle\leq
P_{0}(1+P_{0})^{\frac{R}{r_{0}}-1}\text{, if }r>r_{0};$
$\displaystyle\textup{Cov}(R,r)$ $\displaystyle\leq
P_{0}(1+P_{0})^{\frac{2R}{r}-1}\text{, if }r\leq 2r_{0};$
$\displaystyle\textup{Cov}(R,r)$ $\displaystyle\leq
P_{0}(1+P_{0})^{\frac{R}{r_{0}}-1}\text{, if }r>2r_{0}.$

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].

When $(X,\sigma)$ is a proper GCB-space we can consider the space of
parametrized geodesic lines of $X$,

$\text{Geod}(X)=\\{\gamma\colon\mathbb{R}\to X\text{ isometry}\\},$

endowed with the topology of uniform convergence on compact subsets of
$\mathbb{R}$, and its subset $\text{Geod}_{\sigma}(X)$ made of elements whose
image is a $\sigma$-geodesic line. By the continuity of $\sigma$ (due to its
convexity, see [Des15], [CS20a]) we have that $\text{Geod}_{\sigma}(X)$ is
closed in $\text{Geod}(X)$. There is a natural action of $\mathbb{R}$ on
$\text{Geod}(X)$ 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 $\text{Geod}(X)$.
Moreover the action restricts as an action on $\text{Geod}_{\sigma}(X)$. This
action on $\text{Geod}_{\sigma}(X)$ is called the $\sigma$-geodesic flow on
$X$. The evaluation map $E\colon\text{Geod}(X)\to X$, which is defined as
$E(\gamma)=\gamma(0)$, is continuous and proper ([BL12], Lemma 1.10), so its
restriction to $\text{Geod}_{\sigma}(X)$ has the same properties. Moreover
this restriction is surjective since $\sigma$ is assumed geodesically
complete. The topology on $\text{Geod}_{\sigma}(X)$ is metrizable. Indeed we
can construct a family of metrics on $\text{Geod}_{\sigma}(X)$ with the
following method.
Let $\mathcal{F}$ be the class 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)

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

For every $f\in\mathcal{F}$ we define the distance on
$\text{Geod}_{\sigma}(X)$:

$f(\gamma,\gamma^{\prime})=\int_{-\infty}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))f(s)ds.$
(2.3)

We remark that the choice of $f=\frac{1}{2e^{|s|}}$ gives exactly the distance
$d_{\text{Geod}}$. We are motivated to study the whole class $\mathcal{F}$
because of the applications in further works as [Cav21b].

######  Lemma 2.2.

The expression defined in (2.3) satisfies these properties:

  * (i)

it is a well defined distance on $\textup{Geod}_{\sigma}(X)$;

  * (ii)

for all $\gamma,\gamma^{\prime}\in\textup{Geod}_{\sigma}(X)$ it holds
$f(\gamma,\gamma^{\prime})\leq d(\gamma(0),\gamma(0))+C(f)$;

  * (iii)

for all $\gamma,\gamma^{\prime}\in\textup{Geod}_{\sigma}(X)$ it holds
$d(\gamma(0),\gamma^{\prime}(0))\leq f(\gamma,\gamma^{\prime})$;

  * (iv)

it induces the topology of Geod${}_{\sigma}(X)$.

###### Proof.

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

$\displaystyle d(\gamma(s),\gamma^{\prime}(s))$ $\displaystyle\leq
d(\gamma(s),\gamma(0))+d(\gamma(0),\gamma^{\prime}(0))+d(\gamma^{\prime}(0),\gamma^{\prime}(s))$
$\displaystyle\leq 2|s|+d(\gamma(0),\gamma^{\prime}(0)),$

so

$\int_{-\infty}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))f(s)ds\leq
d(\gamma(0),\gamma^{\prime}(0))+\int_{-\infty}^{+\infty}2|s|f(s)dt<+\infty.$

This shows (ii) and that the integral in (2.3) is finite. From the properties
of the integral and the positiveness of $f$ it is easy to prove that (2.3)
defines a distance. The proof of (iii) follows from the convexity of $\sigma$
and the symmetry of $f$. Indeed for all
$\gamma,\gamma^{\prime}\in\text{Geod}_{\sigma}(X)$ the function
$g(s)=d(\gamma(s),\gamma^{\prime}(s))$ is convex. This means that for all
$S,S^{\prime}\in\mathbb{R}$ and for all $\lambda\in[0,1]$ it holds

$g(\lambda S+(1-\lambda)S^{\prime})\leq\lambda g(S)+(1-\lambda)g(S^{\prime}).$

We take $s\geq 0$ and we use the inequality above with $S=s,S^{\prime}=-s$ and
$\lambda=\frac{1}{2}$, obtaining

$d(\gamma(0),\gamma^{\prime}(0))=g(0)\leq\frac{1}{2}g(-s)+\frac{1}{2}g(s)=\frac{d(\gamma(s),\gamma^{\prime}(s))+d(\gamma(-s),\gamma^{\prime}(-s))}{2}.$

We can now estimate the distance between $\gamma$ and $\gamma^{\prime}$ as

$\displaystyle f(\gamma,\gamma^{\prime})$
$\displaystyle=\int_{-\infty}^{0}d(\gamma(s),\gamma^{\prime}(s))f(s)ds+\int_{0}^{+\infty}d(\gamma(s),\gamma^{\prime}(s))f(s)ds$
$\displaystyle=\int_{0}^{+\infty}\big{(}d(\gamma(-s),\gamma^{\prime}(-s))+d(\gamma(s),\gamma^{\prime}(s))\big{)}f(s)ds\geq
d(\gamma(0),\gamma^{\prime}(0)),$

where we used the symmetry of $f$. This concludes the proof of (iii).
If a sequence $\gamma_{n}$ converges to $\gamma_{\infty}$ uniformly on compact
subsets then it is clear that for every $T\geq 0$ it holds

$\lim_{n\to+\infty}\int_{-T}^{+T}d(\gamma_{n}(s),\gamma_{\infty}(s))f(s)ds=0.$

For every $\varepsilon>0$ we pick $T_{\varepsilon}\geq 0$ such that
$\int_{T_{\varepsilon}}^{+\infty}2|s|f(s)<\varepsilon$. Then it is easy to
conclude, using the properties of $f$, that

$\lim_{n\to+\infty}\int_{-\infty}^{+\infty}d(\gamma_{n}(s),\gamma_{\infty}(s))f(s)ds\leq
2\varepsilon.$

By the arbitrariness of $\varepsilon$ we conclude that the sequence
$\gamma_{n}$ converges to $\gamma_{\infty}$ with respect to the metric $f$.
Now suppose the sequence $\gamma_{n}$ converges to $\gamma_{\infty}$ with
respect to $f$ and suppose it does not converge uniformly on compact subsets
to $\gamma_{\infty}$. Therefore there exists $T\geq 0$, $\varepsilon_{0}>0$
and a subsequence $\gamma_{n_{j}}$ such that
$d(\gamma_{n_{j}}(t_{j}),\gamma_{\infty}(t_{j}))>6\varepsilon_{0}$ for every
$j$, where $t_{j}\in[-T,T]$. We can suppose $t_{j}\to t_{\infty}$ and so
$d(\gamma_{n_{j}}(t_{\infty}),\gamma_{\infty}(t_{\infty}))>4\varepsilon_{0}$
for every $j$. For all
$t\in[t_{\infty}-\varepsilon_{0},t_{\infty}+\varepsilon_{0}]$ we get
$d(\gamma_{n_{j}}(t),\gamma_{\infty}(t))>2\varepsilon_{0}$. Therefore, if we
set
$m=\min_{t\in[t_{\infty}-\varepsilon_{0},t_{\infty}+\varepsilon_{0}]}f(s)>0$,
we obtain

$\int_{-\infty}^{+\infty}d(\gamma_{n_{j}}(s),\gamma_{\infty}(s))f(s)ds>4\varepsilon_{0}^{2}m$

for every $j$, which is a contradiction. ∎

A metric on $\text{Geod}_{\sigma}(X)$ inducing the topology of uniform
convergence on compact subsets is said geometric if the evaluation map $E$ is
Lipschitz with respect to this metric. Any metric induced by $f\in\mathcal{F}$
is geometric by Lemma 2.2.(iii).

##  3 Covering and volume entropy

In this section we will introduce the first two types of entropy: the covering
entropy, defined in terms of the covering functions, and the volume entropy of
a measure.

###  3.1 Covering entropy

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. It is
natural to define the upper covering entropy of $X$ as the number

$\overline{h_{\text{Cov}}}(X)=\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}(\overline{B}(x,T),r_{0}),$

where $x$ is any point of $X$. The lower covering entropy is defined taking
the limit inferior instead of the limit superior and it is denoted by
$\underline{h_{\text{Cov}}}(X)$. By triangular inequality it is easy to show
that the definitions of upper and lower covering entropy do not depend on the
point $x\in X$. In the next proposition, which is essentially Proposition 3.2
of [Cav21a], we can see that they do not depend on $r_{0}$ too and moreover we
can replace the covering function with the packing function.

######  Proposition 3.1.

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

$\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T),r)\underset{P_{0},r_{0},r,r^{\prime}}{\asymp}\frac{1}{T}\log\textup{Pack}(\overline{B}(x,T),r^{\prime})$

for all $r,r^{\prime}>0$. In particular any of these functions can be used in
the definition of the upper and lower covering entropy.

###### Proof.

For all $0<r\leq r^{\prime}$ and $x\in X$ clearly
$\text{Cov}(\overline{B}(x,T),r)\geq\text{Cov}(\overline{B}(x,T),r^{\prime})$
and
$\text{Cov}(\overline{B}(x,T),r)\leq\text{Cov}(\overline{B}(x,T),r^{\prime})\cdot\sup_{y\in
X}\text{Cov}(\overline{B}(y,r^{\prime}),r).$ By Proposition 2.1 we have
$\sup_{y\in
X}\text{Cov}(\overline{B}(y,r^{\prime}),r)=\text{Cov}(r^{\prime},r)$ which is
a finite number depending only on $P_{0},r_{0},r,r^{\prime}$. Therefore we
obtain

$\frac{1}{T}\log\text{Cov}(\overline{B}(x,T),r)\underset{P_{0},r_{0},r,r^{\prime}}{\asymp}\frac{1}{T}\log\text{Cov}(\overline{B}(x,T),r^{\prime}).$

Now the thesis follows from (2.1). ∎

The upper and lower covering entropies can be computed also using the covering
function of the metric spheres.

######  Proposition 3.2.

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$ and
$x\in X$. Then for all $r>0$

$\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T),r)\underset{P_{0},r_{0},r}{\asymp}\frac{1}{T}\log\textup{Cov}(S(x,T),r)$

###### Proof.

Clearly it holds $\text{Cov}(S(x,T),r)\leq\text{Cov}(\overline{B}(x,T),r).$
The other estimate is more involved. We divide the ball $\overline{B}(x,T)$ in
annulii $A(x,kr,(k+1)r)$ with $k=0,\ldots,\frac{T}{r}-1$. We easily obtain

$\text{Cov}(\overline{B}(x,T),2r)\leq\sum_{k=0}^{\frac{T}{r}-1}\text{Cov}(A(x,kr,(k+1)r),2r).$

Now we claim that for any $k$ it holds

$\text{Cov}(A(x,kr,(k+1)r),2r)\leq\text{Cov}(S(x,T),r).$

Indeed let $\\{y_{1},\ldots,y_{N}\\}$ be a set of points realizing
$\text{Cov}(S(x,T),r)$. For all $i=1,\ldots,N$ we consider the
$\sigma$-geodesic segment $\gamma_{i}=[x,y_{i}]$ and we call $x_{i}$ the point
along this geodesic segment at distance $kr$ from $x$. Then $x_{i}\in
A(x,kr,(k+1)r)$ for every $i=1,\ldots,N$. We claim that
$\\{x_{1},\dots,x_{N}\\}$ is a $2r$-dense subset of $A(x,kr,(k+1)r)$. We take
any $y\in A(x,kr,(k+1)r)$ and we consider the $\sigma$-geodesic segment
$[x,y]$. We extend this geodesic to a $\sigma$-geodesic segment
$\gamma=[x,y^{\prime}]$, where $y^{\prime}$ is at distance $T$ from $x$. Then
there exists $i$ such that $d(y^{\prime},y_{i})=d(\gamma(T),\gamma_{i}(T))\leq
r$. By convexity of $\sigma$ we have $d(\gamma(t),\gamma_{i}(t))\leq r$, where
$t=d(x,y)$. Therefore we conclude that $d(y,x_{i})\leq
d(y,\gamma_{i}(t))+d(\gamma_{i}(t),x_{i})\leq 2r.$ This ends the proof of the
claim, so
$\text{Cov}(\overline{B}(x,T),2r)\leq\frac{T}{r}\text{Cov}(S(x,T),r).$ The
thesis follows from these estimates and Proposition 3.1. ∎

Combining Proposition 2.1 and Proposition 3.1 we can find an uniform upper
bound to the covering entropy, see also Proposition 3.2 of [Cav21a].

######  Lemma 3.3.

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

$\overline{h_{\textup{Cov}}}(X)\leq\frac{\log(1+P_{0})}{r_{0}}.$

###### Proof.

For every $x\in X$ it holds $\text{Pack}(\overline{B}(x,R),r_{0})\leq
P_{0}(1+P_{0})^{\frac{R}{r_{0}}-1}.$ The thesis follows immediately. ∎

###  3.2 Volume entropy of homogeneous measures

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. The
upper volume entropy of a measure $\mu$ on $X$ is defined as

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

while the lower volume entropy $\underline{h_{\mu}}(X)$ is defined taking the
limit inferior. These definitions do not depend on the choice of the point
$x\in X$.
A measure $\mu$ on $X$ is called $H$-homogeneous at scale $r>0$ if

$\frac{1}{H}\leq\mu(\overline{B}(x,r))\leq H$

for all $x\in X$. We remark that the condition must hold only at scale $r$.

######  Proposition 3.4.

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$ and
let $\mu$ be a measure on $X$ which is $H$-homogeneous at scale $r$. Then

$\frac{1}{T}\log\mu(\overline{B}(x,T))\underset{P_{0},r_{0},H,r}{\asymp}\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T),r).$

In particular the upper (resp. lower) volume entropy of $\mu$ coincides with
the upper (resp. lower) covering entropy of $X$.

###### Proof.

For all $x\in X$ it holds $\mu(\overline{B}(x,T))\leq
H\cdot\text{Cov}(\overline{B}(x,T),r)$ and
$\mu(\overline{B}(x,T))\geq\frac{1}{H}\cdot\text{Pack}(\overline{B}(x,T-r),r).$
By Proposition 3.1 and since $\frac{T-r}{T}\underset{r}{\asymp}1$ we have the
thesis. ∎

######  Remark 3.5.

The proof of the proposition shows another fact: if a measure is
$H$-homogeneous at scale $r$ then it is $H(r^{\prime})$-homogeneous at scale
$r^{\prime}$ for all $r^{\prime}\geq r$ and $H(r^{\prime})$ depends only on
$H,P_{0},r_{0},r$ and $r^{\prime}$.

We provide here an example of homogeneous measure. If $X$ is a complete,
geodesically complete, CAT$(0)$ metric space that is $P_{0}$-packed at scale
$r_{0}$ then the natural measure on $X$ satisfies

$c\leq\mu_{X}(\overline{B}(x,r_{0}))\leq C$

for all $x\in X$, where $c$ and $C$ are constants depending only on $P_{0}$
and $r_{0}$ (Theorem 4.9 of [CS20b]). It follows immediately the following
result.

######  Corollary 3.6.

Let $X$ be a complete, geodesically complete, CAT$(0)$ metric space. If it is
$P_{0}$-packed at scale $r_{0}$ for some $P_{0}$ and $r_{0}$ then
$\overline{h_{\textup{Cov}}}(X)=\overline{h_{\mu_{X}}}(X).$ The same holds for
the lower entropies.

##  4 Lipschitz-topological entropy

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$. The
space $\text{Geod}_{\sigma}(X)$ is locally compact but not compact. The
topological entropy of the geodesic flow can be defined (see [Bow73], [HKR95])
as

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

where the infimum is taken among all metrics inducing the topology of
$\text{Geod}_{\sigma}(X)$, the supremum is taken among all compact subsets of
$\text{Geod}_{\sigma}(X)$ and $\text{Cov}_{\text{\texthtd}^{T}}(K,r)$ is the
covering function of the compact subset $K$ at scale $r$ with respect to the
metric T defined by

$\text{\texthtd}^{T}(\gamma,\gamma^{\prime})=\max_{t\in[0,T]}\text{\texthtd}(\Phi_{t}\gamma,\Phi_{t}\gamma^{\prime}).$

By the variational principle this quantity equals the measure-theoretic
entropy defined as the supremum of the entropies of the flow-invariant
probability measures on $\text{Geod}_{\sigma}(X)$ (cp. [HKR95], Lemma 1.5). An
easy computation shows that the topological entropy is always zero.

######  Lemma 4.1.

There are no flow-invariant probability measures on
$\textup{Geod}_{\sigma}(X)$. In particular the topological entropy of the
geodesic flow is $0$.

###### Proof.

Suppose there is a flow-invariant probability measure $\mu$ on
$\text{Geod}_{\sigma}(X)$. For $x\in X$ and $R\geq 0$ we define
$A_{R}=\\{\gamma\in\text{Geod}_{\sigma}(X)\text{ s.t.
}\gamma(0)\in\overline{B}(x,R)\\}.$ Clearly there exists $R\geq 0$ such that
$\mu(A_{R})>\frac{1}{2}$. By flow-invariance of $\mu$ we have that the set

$\Phi_{2R+1}^{-1}(A_{R})=\\{\gamma\in\text{Geod}_{\sigma}(X)\text{ s.t.
}\gamma(2R+1)\in\overline{B}(x,R)\\}$

has measure $>\frac{1}{2}$. This implies that
$\mu(A_{R}\cap\Phi_{2R+1}^{-1}(A_{R}))>0$, but this intersection is empty. ∎

Looking at the proof of the variational principle given in [HKR95] we can
observe that the sequence of metrics on $\text{Geod}_{\sigma}(X)$ that
approach the infimum in the definition of the topological entropy are the
restriction to $\text{Geod}_{\sigma}(X)$ of metrics defined on its one-point
compactification. These metrics are not the natural ones on
$\text{Geod}_{\sigma}(X)$, since they are not geometric. We propose a more
appropriate definition of topological entropy for proper GCB-spaces.
We define the upper Lipschitz-topological entropy of $\text{Geod}_{\sigma}(X)$
as

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

where the infimum is now taken only among all geometric metrics on
$\text{Geod}_{\sigma}(X)$. The lower Lipschitz-topological entropy is defined
by taking the limit inferior instead of the limit superior and it is denoted
by $\underline{h_{\text{Lip-top}}}(\text{Geod}_{\sigma}(X))$. The main result
of this section is the following.

######  Theorem 4.2.

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

$\overline{h_{\textup{Lip-
top}}}(\textup{Geod}_{\sigma}(X))=\overline{h_{\textup{Cov}}}(X).$

The same holds for the lower entropies.

One of the two inequalities is easy. In order to prove the other one we will
show that for the distances induced by the functions $f\in\mathcal{F}$ the
definition of topological entropy can be heavily simplified.

###  4.1 Topological entropy for the distances induced by $f\in\mathcal{F}$

For a metric $f\in\mathcal{F}$ we denote by $\overline{h_{f}}$ the upper
metric entropy of the $\sigma$-geodesic flow with respect to $f$, that is

$\overline{h_{f}}(\text{Geod}_{\sigma}(X))=\sup_{K}\lim_{r\to
0}\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(K,r).$

In the usual way it is defined the lower metric entropy with respect to $f$,
$\underline{h_{f}}(\text{Geod}(X))$. For a subset $Y$ of $X$ we denote by
$\text{Geod}_{\sigma}(Y)$ the set of $\sigma$-geodesic lines of $X$ passing
through $Y$ at time $0$.

######  Proposition 4.3.

Let $(X,\sigma)$ be a GCB-space that is $P_{0}$-packed at scale $r_{0}$ and
let $f\in\mathcal{F}$. Then

  * (i)

for all $x,y\in X$ it holds
$\overline{h_{f}}(\textup{Geod}_{\sigma}(x))=\overline{h_{f}}(\textup{Geod}_{\sigma}(y));$

  * (ii)

for all $x\in X$ and $R\geq 0$ it holds
$\overline{h_{f}}(\textup{Geod}_{\sigma}(\overline{B}(x,R)))=\overline{h_{f}}(\textup{Geod}_{\sigma}(x));$

  * (iii)

for all $x\in X$ it holds
$\overline{h_{f}}(\textup{Geod}_{\sigma}(X))=\overline{h_{f}}(\textup{Geod}_{\sigma}(x))\leq\overline{h_{\textup{Cov}}}(X)$;

  * (iv)

for all $x\in X$ the function
$r\mapsto\limsup_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(x),r)$
is constant.

The same conclusions hold for the lower Lipschitz-topological entropy.

The proposition is consequence of the following fundamental key lemma.

######  Lemma 4.4 (Key Lemma).

Let $f\in\mathcal{F}$, $\gamma\in\textup{Geod}_{\sigma}(X)$ and $0<r\leq
r^{\prime}$. Then

$\frac{1}{T}\log\textup{Cov}_{f^{T}}(\overline{B}_{f^{T}}(\gamma,r^{\prime}),r)\underset{P_{0},r_{0},r,r^{\prime},f}{\asymp}0,$

where $\overline{B}_{f^{T}}(\gamma,r^{\prime})$ is the closed ball of center
$\gamma$ and radius $r^{\prime}$ with respect to the metric $f^{T}$. We remark
that the convergence is uniform in $\gamma$.

###### Proof.

Let $P>0$ depending only on $f$ and $r^{\prime}$ such that

$\int_{-\infty}^{-P}2|u|f(u)du+\int_{P}^{+\infty}2|u|f(u)du<\frac{r}{4}.$

We fix $\varepsilon>0$ and $T\geq\frac{P}{\varepsilon}$. Let
$E_{T}=\\{x_{1},\ldots,x_{N}\\}$ be a maximal $\frac{r}{16}$-separated subset
of $B(\gamma(T),r^{\prime}+\varepsilon T)$, so it is also
$\frac{r}{16}$-dense, and $\\{y_{1},\ldots,y_{M}\\}$ be a $\frac{r}{16}$-dense
subset of $B(\gamma(-P),r^{\prime}+2P)$. For every $i=1,\ldots,M$ and
$j=1,\ldots,N$ we take a $\sigma$-geodesic line $\gamma_{ij}$ extending the
$\sigma$-geodesic segment $[y_{i},x_{j}]$. We parametrize $\gamma_{ij}$ in
such a way that $\gamma_{ij}(-P)=y_{i}$. The claim is that
$\\{\gamma_{ij}\\}_{i,j}$ is a $r$-dense subset of
$\overline{B}_{f^{T}}(\gamma,r^{\prime})$ with respect to the metric $f^{T}$.
We fix $\gamma^{\prime}\in\overline{B}_{f^{T}}(\gamma,r^{\prime})$. This means

$\max_{t\in[0,T]}f^{t}(\gamma^{\prime},\gamma)=\max_{t\in[0,T]}f(\Phi_{t}(\gamma^{\prime}),\Phi_{t}(\gamma))\leq
r^{\prime}.$

In particular for all $t\in[0,T]$ we get $d(\gamma^{\prime}(t),\gamma(t))\leq
r^{\prime}$, since

$d(\gamma^{\prime}(t),\gamma(t))=d(\Phi_{t}(\gamma^{\prime}),\Phi_{t}(\gamma))\leq
f(\Phi_{t}(\gamma^{\prime}),\Phi_{t}(\gamma))\leq r^{\prime}.$

Therefore $d(\gamma^{\prime}(-P),\gamma(-P))\leq r^{\prime}+2P.$ Moreover

$d(\gamma^{\prime}(T+\varepsilon T),\gamma(T))\leq
d(\gamma^{\prime}(T+\varepsilon
T),\gamma^{\prime}(T))+d(\gamma^{\prime}(T),\gamma(T))\leq\varepsilon
T+r^{\prime}.$

Thus there exists $x_{j}$ such that $d(x_{j},\gamma^{\prime}(T+\varepsilon
T))\leq\frac{r}{16}$ and $y_{i}$ such that
$d(y_{i},\gamma^{\prime}(-P))\leq\frac{r}{16}$. We have
$d(\gamma_{ij}(-P),\gamma^{\prime}(-P))\leq\frac{r}{16}$, so if we denote with
$t_{j}$ the time such that $\gamma_{ij}(t_{j})=x_{j}$ it holds
$|t_{j}-(T+\varepsilon T)|\leq\frac{r}{8}$. Then

$\displaystyle d(\gamma_{ij}(T+\varepsilon T),\gamma^{\prime}(T+\varepsilon
T))$ $\displaystyle\leq d(\gamma_{ij}(T+\varepsilon
T),\gamma_{ij}(t_{j}))+d(\gamma_{ij}(t_{j}),\gamma^{\prime}(T+\varepsilon T))$
$\displaystyle\leq\frac{r}{8}+\frac{r}{16}<\frac{r}{4}.$

From the convexity of $\sigma$ we have
$d(\gamma^{\prime}(u),\gamma_{ij}(u))<\frac{r}{4}$ for all
$u\in[-P,(1+\varepsilon)T]$. For $t\in[0,T]$ we have

$\displaystyle f^{t}(\gamma^{\prime},\gamma_{ij})$
$\displaystyle=\int_{-\infty}^{+\infty}d(\gamma^{\prime}(u),\gamma_{ij}(u))f(u-t)du$
$\displaystyle\leq\int_{-\infty}^{-P}\bigg{(}\frac{r}{4}+2|u+P|\bigg{)}f(u-t)du+$
$\displaystyle+\int_{-P}^{(1+\varepsilon)T}\frac{r}{4}f(u-t)du+$
$\displaystyle+\int_{(1+\varepsilon)T}^{+\infty}\bigg{(}\frac{r}{4}+2|u-(1+\varepsilon)T|\bigg{)}f(u-t)du.$

The first term can be estimated as follows

$\displaystyle\int_{-\infty}^{-P}\bigg{(}\frac{r}{4}+2|u+P|\bigg{)}f(u-t)du$
$\displaystyle\leq\frac{r}{4}+\int_{-\infty}^{-P-t}2|v+t+P|f(v)dv$
$\displaystyle\leq\frac{r}{4}+\int_{-\infty}^{-P}2|v|f(v)dv.$

The second term is less than or equal to $\frac{r}{4}$. The third term can be
controlled in this way:

$\displaystyle\int_{(1+\varepsilon)T}^{+\infty}\bigg{(}\frac{r}{4}+2|u-(1+\varepsilon)T|\bigg{)}f(u-t)du$
$\displaystyle\leq\frac{r}{4}+\int_{(1+\varepsilon)T-t}^{+\infty}2|v-(1+\varepsilon)T+t|f(v)dv$
$\displaystyle\leq\frac{r}{4}+\int_{(1+\varepsilon)T-t}^{+\infty}2|v|f(v)dv$
$\displaystyle\leq\frac{r}{4}+\int_{P}^{+\infty}2|v|f(v)dv.$

The last inequality follows from $T\geq\frac{P}{\varepsilon}$. Therefore

$f^{t}(\gamma^{\prime},\gamma_{ij})\leq\frac{r}{4}+\frac{r}{4}+\frac{r}{4}+\int_{-\infty}^{-P}2|v|f(v)dv+\int_{P}^{+\infty}2|v|f(v)dv\leq
r.$

We conclude that

$\text{Cov}_{f^{T}}(\overline{B}_{f^{T}}(\gamma,r^{\prime}),r)\leq\text{Cov}\bigg{(}r^{\prime}+2P,\frac{r}{16}\bigg{)}\cdot\\#E_{T}.$

From Proposition 2.1, if $\rho=\min\left\\{r_{0},\frac{r}{16}\right\\}$, we
get $\\#E_{T}\leq P_{0}(1+P_{0})^{\frac{r^{\prime}+\varepsilon T}{\rho}-1}.$
Thus

$\displaystyle\frac{1}{T}\log\text{Cov}_{f^{T}}(\overline{B}_{f^{T}}(\gamma,r^{\prime}),r)$
$\displaystyle\leq\frac{1}{T}K(P_{0},r_{0},r,r^{\prime},f)\cdot\frac{\varepsilon
T}{\rho}\log(1+P_{0})$ $\displaystyle=\varepsilon\cdot
K^{\prime}(P_{0},r_{0},r,r^{\prime},f).$

Here $K,K^{\prime}$ are constants depending only on
$P_{0},r_{0},r,r^{\prime},f$ and not on $\varepsilon$ or $\gamma$. So from the
arbitrariness of $\varepsilon$ we achieve the proof. ∎

The computation of $\overline{h_{f}}$ requires to consider the supremum among
all compact subsets of $\text{Geod}_{\sigma}(X)$. We notice that given a
compact subset $K\subseteq\text{Geod}_{\sigma}(X)$, the set $E(K)$ is compact
since $E$ is continuous. In particular it is bounded, hence contained in a
ball $\overline{B}(x,R)$ centered at a reference point $x\in X$. We observe
also that the set $\text{Geod}_{\sigma}(\overline{B}(x,R))$ is compact since
the evaluation map $E$ is proper. We conclude that any compact subset of
$\text{Geod}_{\sigma}(X)$ is contained in a compact subset of the form
$\text{Geod}_{\sigma}(\overline{B}(x,R))$ and therefore in order to compute
$\overline{h_{f}}$ it is enough to take the supremum among these sets. The
main consequence of Lemma 4.4 is the following result, which is the key
ingredient in the proof of Proposition 4.3.

######  Corollary 4.5.

Let $f\in\mathcal{F}$, $x\in X$, $R\geq 0$ and $0<r\leq r^{\prime}$. Then

$\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,R)),r)\underset{P_{0},r_{0},r,r^{\prime},f}{\asymp}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,R)),r^{\prime}).$

###### Proof.

The quantity
$\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,R)),r)$
is

$\displaystyle\leq\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,R)),r^{\prime})\cdot\sup_{\gamma\in\text{Geod}_{\sigma}(X)}\text{Cov}_{f^{T}}(\overline{B}_{f^{T}}(\gamma,r^{\prime}),r)$
$\displaystyle=\frac{1}{T}\big{(}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,R)),r^{\prime})+\log\sup_{\gamma\in
X}\text{Cov}_{f^{T}}(\overline{B}_{f^{T}}(\gamma,r^{\prime}),r)\big{)}$

The conclusion follows by Lemma 4.4. ∎

###### Proof of Proposition 4.3.(ii).

Let $\varepsilon>0$ and $T>\frac{R}{\varepsilon}$. Let
$\gamma_{1},\ldots,\gamma_{N}$ be a $r$-dense subset of Geod${}_{\sigma}(x)$
with respect to the metric $f^{(2+\varepsilon)T}$. The claim is that
$\\{\gamma_{i}\\}$ is a $K$-dense subset of
$\text{Geod}_{\sigma}(\overline{B}(x,R))$ with respect to $f^{T}$, where $K$
depends only on $r,R$ and $f$. We consider a $\sigma$-geodesic line
$\gamma\in\text{Geod}_{\sigma}(\overline{B}(x,R))$. Then there exists a
$\sigma$-geodesic line $\gamma^{\prime}\in\text{Geod}_{\sigma}(x)$ extending
the $\sigma$-geodesic segment $[x,\gamma((1+\varepsilon)T)]$. We call
$t_{\gamma^{\prime}}$ the time such that
$\gamma^{\prime}(t_{\gamma^{\prime}})=\gamma((1+\varepsilon)T)$. Then

$\displaystyle t_{\gamma^{\prime}}=d(x,\gamma((1+\varepsilon)T))$
$\displaystyle\leq d(x,\gamma(0))+d(\gamma(0),\gamma((1+\varepsilon)T))$
$\displaystyle\leq R+(1+\varepsilon)T\leq(1+2\varepsilon)T$

since $T\geq\frac{R}{\varepsilon}$. Moreover
$|t_{\gamma^{\prime}}-(1+\varepsilon)T|\leq R.$ We know there exists
$\gamma_{i}$ such that
$\max_{t\in[0,(1+2\varepsilon)T]}f(\Phi_{t}\gamma^{\prime},\Phi_{t}\gamma_{i})\leq
r.$ In particular
$d(\gamma^{\prime}(t_{\gamma^{\prime}}),\gamma_{i}(t_{\gamma^{\prime}}))\leq
r$. Then $d(\gamma((1+\varepsilon)T),\gamma_{i}(t_{\gamma^{\prime}}))\leq r$
and in conclusion

$d(\gamma((1+\varepsilon)T),\gamma_{i}((1+\varepsilon)T))\leq
d(\gamma((1+\varepsilon)T),\gamma_{i}(t_{\gamma^{\prime}}))+d(\gamma_{i}(t_{\gamma^{\prime}}),\gamma_{i}((1+\varepsilon)T))\leq
r+R.$

From the convexity of $\sigma$ we have $d(\gamma(t),\gamma_{i}(t))\leq R+r$
for all $t\in[0,(1+\varepsilon)T]$. We have to estimate
$f^{t}(\gamma,\gamma_{i})=\int_{-\infty}^{+\infty}d(\gamma(u),\gamma_{i}(u))f(u-t)du$
for every $t\in[0,T]$. Since $d(\gamma(0),\gamma_{i}(0))\leq R$ and
$d(\gamma((1+\varepsilon)T),\gamma_{i}((1+\varepsilon)T))\leq r+R$ then

$\displaystyle\int_{-\infty}^{+\infty}d(\gamma(u),\gamma_{i}(u))f(u-t)du\leq$
$\displaystyle\int_{-\infty}^{0}(R+2|u|)f(u-t)du+$ $\displaystyle+$
$\displaystyle\int_{0}^{(1+\varepsilon)T}(R+r)f(u-t)du+$ $\displaystyle+$
$\displaystyle\int_{(1+\varepsilon)T}^{+\infty}(R+r+2|u-(1+\varepsilon)T|)f(u-t)du$
$\leq
R+\int_{-\infty}^{-t}2|v+t|f(v)dv+(R+r)+\int_{(1+\varepsilon)T-t}^{+\infty}(R+r+2|v-(1+\varepsilon)T+t|)f(v)dv.$

We conclude that the above quantity is less than or equal to

$3R+2r+\int_{-\infty}^{0}2|v|f(v)dv+\int_{0}^{+\infty}2|v|f(v)dv\leq
3R+2r+C(f)=K(R,r,f).$

By the previous corollary
$\overline{h_{f}}(\text{Geod}_{\sigma}(\overline{B}(x,R)))$can be computed as

$\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,R)),K)$

which is

$\displaystyle\leq\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{(1+2\varepsilon)T}}(\text{Geod}_{\sigma}(x),r)$
$\displaystyle=(1+2\varepsilon)\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(x),r).$

Since this is true for all $\varepsilon>0$ then we obtain the thesis. ∎

###### Proof of Proposition 4.3.(i).

We have $y\in\overline{B}(x,R)$, where $R=d(x,y)$, so
$\text{Geod}_{\sigma}(y)\subseteq\text{Geod}_{\sigma}(\overline{B}(x,R))$.
Therefore

$\overline{h}_{f}(\text{Geod}_{\sigma}(y))\leq\overline{h}_{f}(\text{Geod}_{\sigma}(\overline{B}(x,R)))=\overline{h}_{f}(\text{Geod}_{\sigma}(x)).$

The other inequality can be proved in the same way. ∎

Finally we achieve the proof of the remaining parts of Proposition 4.3.

###### Proof of Proposition 4.3.(iii) & (iv).

The equality in (iii) follows directly from (ii), so

$\overline{h_{f}}(\text{Geod}_{\sigma}(X))=\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(x),r_{0}),$

where $x$ is a point of $X$. We fix $T>0$ and we consider a $r_{0}$-separated
subset $E_{T}$ of $S(x_{0},T)$ of maximal cardinality, which is also
$r_{0}$-dense. For all $y\in E_{T}$ we consider a $\sigma$-geodesic line
$\gamma_{y}$ extending the $\sigma$-geodesic segment $[x_{0},y]$ such that
$\gamma_{y}(0)=x_{0}$ and $\gamma_{y}(T)=y$. We claim that
$\\{\gamma_{y}\\}_{y\in E_{T}}$ is a $(r_{0}+C(f))$-dense subset of
$\text{Geod}_{\sigma}(x)$ with respect to $f^{T}$. We take a $\sigma$-geodesic
line $\gamma\in\text{Geod}_{\sigma}(x)$. Then there exists $y\in E_{T}$ such
that $d(\gamma(T),y)=d(\gamma(T),\gamma_{y}(T))\leq r_{0}.$ From the convexity
of $\sigma$ it holds $d(\gamma(u),\gamma_{y}(u))\leq r_{0}$ for all
$u\in[0,T]$. Moreover $d(\gamma(u),\gamma_{y}(u))\leq r_{0}+2|u-T|$ for all
$u\in[T,+\infty)$ and $d(\gamma(u),\gamma_{y}(u))\leq 2|u|$ for all
$u\in(-\infty,0]$. Then for all $t\in[0,T]$ we get

$\displaystyle f^{t}(\gamma,\gamma_{y})$
$\displaystyle=\int_{-\infty}^{+\infty}d(\gamma(u),\gamma_{y}(u))f(u-t)du$
$\displaystyle\leq\int_{-\infty}^{0}2|u|f(u-t)du+\int_{0}^{T}r_{0}f(u-t)du+$
$\displaystyle+\int_{T}^{+\infty}(r_{0}+2|u-T|)f(u-t)du\leq r_{0}+C(f).$

The last inequality follows from similar estimates given in the proofs of
Lemma 4.4. Therefore applying Corollary 4.5 we have

$\displaystyle\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(x),r_{0})\leq\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}(S(x,T),r_{0}).$

This, together with Proposition 3.2, proves (iii). We observe that (iv) is
exactly Corollary 4.5 with $R=0$. ∎

###  4.2 Proof of Theorem 4.2

We are ready to give the

###### Proof of Theorem 4.2.

Proposition 4.3.(iii) shows that $\overline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(X))$ is less than or equal to
$\overline{h_{\text{Cov}}}(X)$.
In order to prove the other inequality we fix a geometric metric on
$\text{Geod}_{\sigma}(X)$ and we denote by $M$ the Lipschitz constant with
respect to of the evaluation map $E$. Then we have

$\sup_{K}\lim_{r\to
0}\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(K,r)\geq\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(x),r_{0}),$

where $x\in X.$ We fix $T\geq 0$ and we consider a set
$\gamma_{1},\ldots,\gamma_{N}$ realizing
$\text{Cov}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(x),r_{0})$. The claim
is that $\gamma_{i}(T)$ is a $Mr_{0}$-dense subset of $S(x,T)$. Indeed we take
a point $y\in S(x,T)$ and we extend the $\sigma$-geodesic segment $[x,y]$ to a
$\sigma$-geodesic line $\gamma\in\text{Geod}_{\sigma}(x)$. Then there exists
$\gamma_{i}$ such that ${}^{T}(\gamma,\gamma_{i})\leq r_{0}$. Since the
evaluation map is $M$-Lipschitz we have

$d(y,\gamma_{i}(T))=d(\gamma(T),\gamma_{i}(T))=d(\Phi_{T}\gamma(0),\Phi_{T}\gamma_{i}(0))\leq
L\text{\texthtd}(\Phi_{T}\gamma,\Phi_{T}\gamma_{i})\leq Mr_{0}.$

Therefore

$\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(x),r_{0})\geq\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}(S(x,T),Mr_{0})$

and the conclusion follows by Proposition 3.2. ∎

######  Remark 4.6.

By Proposition 4.3 and Theorem 4.2 the upper Lipschitz-topological entropy of
$X$ can be computed as

$\overline{h_{\textup{Lip-
top}}}(X)=\limsup_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(x),r)$

independently of $f\in\mathcal{F}$, $x\in X$ and $r>0$. Moreover

$\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(x),r_{0})\underset{P_{0},r_{0},f}{\asymp}\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T),r_{0})$

by the proofs of Theorem 4.2 and Proposition 4.3 and by Proposition 3.2.

##  5 Gromov-hyperbolic metric spaces

In the second part of the paper we will study the versions of the entropies
introduced in the first part relative to subsets of the boundary at infinity.
In order to have meaningful definitions we will consider Gromov-hyperbolic
metric spaces.

Let $X$ be a geodesic 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:

$(x,z)_{w}\geq\min\\{(x,y)_{w},(y,z)_{w}\\}-\delta.$ (5.1)

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

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 $\approx$
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 $X$. The Gromov product can be extended to points
$z,z^{\prime}\in\partial X$ by

$(z,z^{\prime})_{x}=\sup_{(z_{n}),(z_{n}^{\prime})}\liminf_{n,m\to+\infty}(z_{n},z_{m}^{\prime})_{x}$

where the supremum is taken among all sequences such that $(z_{n})\approx z$
and $(z_{n}^{\prime})\approx z^{\prime}$. For every
$z,z^{\prime},z^{\prime\prime}\in\partial X$ it continues to hold

$(z,z^{\prime})_{x}\geq\min\\{(z,z^{\prime\prime})_{x},(z^{\prime},z^{\prime\prime})_{x}\\}-\delta.$
(5.2)

Moreover for all sequences $(z_{n}),(z_{n}^{\prime})$ converging to
$z,z^{\prime}$ respectively it holds

$(z,z^{\prime})_{x}-\delta\leq\liminf_{n,m\to+\infty}(z_{n},z_{m}^{\prime})_{x}\leq(z,z^{\prime})_{x}.$
(5.3)

The Gromov product between a point $y\in X$ and a point $z\in\partial X$ is
defined in a similar way and it satisfies a condition analogue of (5.3).

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
$\xi^{+}=z$, and we denote it by $[y,z]$. 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]). We denote any of these geodesic rays as
$\xi_{xz}=[x,z]$ even if it is possibly not unique.
Given different points $z=[(z_{n})],z^{\prime}=[(z^{\prime}_{n})]\in\partial
X$ there always exists a geodesic line $\gamma$ joining $z$ to $z^{\prime}$,
i.e. such that $\gamma|_{[0,+\infty)}$ and $\gamma|_{(-\infty,0]}$ join
$\gamma(0)$ to $z,z^{\prime}$ respectively. We call $z$ and $z^{\prime}$ the
positive and negative endpoints of $\gamma$, respectively, denoted
$\gamma^{+}$ and $\gamma^{-}$.

Here we recall some basic properties of Gromov-hyperbolic metric spaces.

######  Lemma 5.1 (Projection Lemma, cp. Lemma 3.2.7 of [CDP90]).

Let $X$ be a $\delta$-hyperbolic metric space and let $x,y,z\in X$. For every
geodesic segment $[y,z]$ we have $(y,z)_{x}\geq d(x,[y,z])-4\delta.$

We recall that a $(1,\nu)$-quasigeodesic is a curve $\alpha\colon I\to X$ such
that

$|t-t^{\prime}|-\nu\leq d(\alpha(t),\alpha(t^{\prime}))\leq|t-t^{\prime}|+\nu$

for all $t,t^{\prime}$ belonging to the interval $I$. As an immediate
consequence of the previous lemma and Proposition 2.7 of [CS20a] we get:

######  Lemma 5.2.

Let $X$ be a $\delta$-hyperbolic metric space, $x\in X$ and $\xi$ be a
geodesic ray such that $\xi(0)$ is a projection of $x$ on $\xi$. Then

  * (i)

for all $T\geq 0$ any curve $\alpha=[x,\xi(0)]\cup\xi|_{[0,T]}$ is a
$(1,4\delta)$-quasigeodesic. Moreover, if $\gamma$ is a geodesic segment
$[x,\xi(T)]$, then $d(\alpha(t),\gamma(t))\leq 72\delta$ for all possibles
$t$;

  * (ii)

any curve $\alpha=[x,\xi(0)]\cup\xi$ is a $(1,4\delta)$-quasigeodesic.
Moreover, if $\xi^{\prime}$ is a geodesic ray $[x,\xi^{+}]$, then
$d(\alpha(t),\xi^{\prime}(t))\leq 72\delta$ for all $t\geq 0$.

Furthermore:

######  Lemma 5.3.

Let $X$ be a proper, $\delta$-hyperbolic metric space. Let $\gamma$ be a
geodesic line and $x\in X$ with $S:=d(\gamma(0),x)$. Let $x^{\prime}$ be a
projection of $x$ on $\gamma$. Then

  * (i)

there exists an orientation of $\gamma$ such that
$[x,x^{\prime}]\cup[x^{\prime},\gamma^{+}]$ is a $(1,4\delta)$-quasigeodesic,
where the second segment is the subsegment of $\gamma$;

  * (ii)

with respect to the orientation of (i) then every geodesic ray
$\xi=[x,\gamma^{+}]$ satisfies $d(\xi(S+t),\gamma(t))\leq 76\delta$ for all
$t\geq 0$;

  * (iii)

for all orientations of $\gamma$ every geodesic ray $\xi=[x,\gamma^{+}]$
satisfies $d(\xi(S+t),\gamma(t))\leq 2S+76\delta$ for all $t\geq 0$.

###### Proof.

We choose the orientation of $\gamma$ such that $x^{\prime}$ belongs to the
negative ray $\gamma|_{(-\infty,0]}$ and we take a geodesic ray
$\xi=[x,\gamma^{+}]$. By Lemma 5.2 the path
$\alpha=[x,x^{\prime}]\cup[x^{\prime},\gamma^{+}]$ is a
$(1,4\delta)$-quasigeodesic and moreover it satisfies
$d(\xi(S+t),\alpha(S+t))\leq 72\delta$ for every $t\geq 0$. Furthermore the
time $t_{0}$ such that $\alpha(t_{0})=\gamma(0)$ is between $S$ and
$S+4\delta$ implying $d(\xi(S+t),\gamma(t))\leq 76\delta.$
For the second part of the proof we suppose to be in the situation above and
we consider a geodesic ray $\xi=[x,\gamma^{-}]$. By Lemma 5.2 the path
$\alpha=[x,x^{\prime}]\cup[x^{\prime},\gamma^{-}]$, where the second segment
is a subsegment of $\gamma$, satisfies $d(\xi(S+t),\alpha(S+t))\leq 72\delta$
for every $t\geq 0$. Furthermore for every $t\geq 0$ the point $\alpha(S+t)$
belongs to $\gamma$ and $d(\alpha(S+t),\gamma(0))\leq
d(\alpha(S+t),x)+d(x,\gamma(0))\leq 2S+t+4\delta$. So
$d(\xi(S+t),\gamma(t))\leq 76\delta+2S.$ ∎

We remark that if $\gamma(0)$ is a projection of $x$ on $\gamma$ then the
first part of the lemma holds for both the positive and negative ray of
$\gamma$.
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 essentially Lemma 2.5 of [Cav21a].

######  Lemma 5.4.

Let $X$ be a proper, $\delta$-hyperbolic metric space. Let $x\in X$ and
$C\subseteq\partial X$ be a subset with at least two points. Then for every
$z\in C$ it exists a geodesic line $\gamma$ with endpoints in $C$ such that
$d(\xi_{xz}(t),\gamma(t))\leq 22\delta+d(x,\textup{QC-Hull}(C))=:L$ for every
$t\geq 0$.

###### Proof.

Let $x^{\prime}\in\text{QC-Hull}(C)$ realizing $d(x,\text{QC-Hull}(C))$. By
Lemma 2.5 of [Cav21a] we know there exists $\gamma$ as desired such that
$d(\xi_{x^{\prime}z}(t),\gamma(t))\leq 14\delta$ for every $t\geq 0$. Now the
thesis follows by the fact that two geodesic rays with same endpoint are
$8\delta$-close, see for instance Proposition 8.10 of [BCGS17]. ∎

######  Remark 5.5.

Let $z,z^{\prime}\in C\subseteq\partial X$. It is clear that the conclusion of
Lemma 5.4 is true with $L=22\delta+d(x,[z,z^{\prime}])$.

######  Lemma 5.6.

Let $(X,\sigma)$ be a proper, $\delta$-hyperbolic GCB-space. Let QC-
Hull${}_{\sigma}(C)$ be the union of all $\sigma$-geodesic lines joining two
points of $C$. Then for every $x\in\textup{QC-Hull}(C)$ there exists
$x^{\prime}\in\textup{QC-Hull}_{\sigma}(C)$ with $d(x,x^{\prime})\leq
8\delta$.

###### Proof.

For all $z,z^{\prime}\in\partial X$ there exists a $\sigma$-geodesic line
joining them, see [CS20a]. Moreover two parallel geodesics are at most at
distance $8\delta$ ([BCGS17], Proposition 8.10), hence the conclusion. ∎

###  5.1 Minkowski dimension

When $X$ is a proper, $\delta$-hyperbolic metric space we define the
generalized visual ball of center $z\in\partial X$ and radius $\rho\geq 0$ to
be

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

It is comparable to the metric balls of the visual metrics on $\partial X$,
see Lemma 2.6 of [Cav21a]. Generalized visual balls are related to shadows.
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:

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

######  Lemma 5.7 (Lemma 2.7 of [Cav21a]).

Let $X$ be a proper, $\delta$-hyperbolic metric space. Let $z\in\partial X$,
$x\in X$ and $T\geq 0$. Then

  * (i)

$B(z,e^{-T})\subseteq\textup{Shad}_{x}\left(\xi_{xz}\left(T\right),7\delta\right)$;

  * (ii)

$\textup{Shad}_{x}\left(\xi_{xz}\left(T\right),r\right)\subseteq
B(z,e^{-T+r})$ for all $r>0$.

The upper and lower visual Minkowski dimension of a subset $C$ of $\partial X$
was defined in [Cav21a] as

$\overline{\text{MD}}(C)=\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}(C,e^{-T}),\qquad\underline{\text{MD}}(C)=\liminf_{T\to+\infty}\frac{1}{T}\log\text{Cov}(C,e^{-T})$

respectively, where $\text{Cov}(C,\rho)$ denotes the minimal number of
generalized visual balls of radius $\rho$ needed to cover $C$. Taking
$C=\partial X$ we get

######  Proposition 5.8.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$ and let $x\in X$. Then

$\frac{1}{T}\log\textup{Cov}(\partial
X,e^{-T})\underset{P_{0},r_{0},\delta}{\asymp}\frac{1}{T}\log\textup{Cov}(S(x,T),r_{0}).$

In particular the upper (resp. lower) visual Minkowski dimension of $\partial
X$ equals the upper (resp. lower) covering entropy of $X$.

We need:

######  Lemma 5.9 ([Cav21a], Lemma 2.2).

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
$(z,z^{\prime})_{x}>T-b$.

###### Proof of Proposition 5.8.

Let $z_{1},\ldots,z_{N}$ be points of $\partial X$ realizing
$\textup{Cov}(\partial X,e^{-T})$, and let $y_{i}$ be the point at distance
$T$ from $x$ along one geodesic ray $\xi_{xz_{i}}$. We claim that
$\\{y_{i}\\}$ covers $S(x,T)$ at scale $6\delta$. Indeed let $y\in S(x,T)$ and
let $z\in\partial X$ be the point at infinity of a $\sigma$-geodesic ray $\xi$
that extends the $\sigma$-geodesic $[x,y]$. We know it exists $i$ such that
$(z,z_{i})_{x}>T$, then by Lemma 5.9 we get $d(y,y_{i})\leq 6\delta$. This
shows $\textup{Cov}(\partial X,e^{-T})\geq\text{Cov}(S(x,T),6\delta)$.
Now let $\\{y_{i}\\}$ be points realizing $\text{Cov}(S(x,T+\delta),\delta)$.
For every $i$ let $z_{i}\in\partial X$ be the point at infinity of a
$\sigma$-geodesic ray $\xi_{i}$ that extends the $\sigma$-geodesic
$[x,y_{i}]$. For every $z\in\partial X$ we take a geodesic ray $\xi_{xz}$. We
know it exists $i$ such that $d(\xi_{xz}(T+\delta),y_{i})\leq\delta<2\delta$,
therefore $(z,z_{i})_{x}>T$ by Lemma 5.9. This shows $\textup{Cov}(\partial
X,e^{-T})\leq\text{Cov}(S(x,T+\delta),\delta)$. The conclusion follows by
Proposition 3.1. ∎

Putting together Proposition 3.1, Proposition 3.4, Proposition 5.8, Theorem
4.2 and Proposition 4.3 we get the proof of Theorem B.

##  6 Entropies of subsets of the boundary

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$. In this section we will consider a subset $C$ of $\partial X$
and we define the relative version, with respect to $C$, of all the different
definitions of entropies introduced in the previous sections. We observe that
when $C=\partial X$ then we are in the case yet studied.

###  6.1 Covering and volume entropy

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$ and let $C$ be a subset of $\partial X$. The upper covering
entropy of $C$ is defined as

$\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}(\overline{B}(x,T)\cap\overline{B}(\text{QC-
Hull}(C),\tau),r),$

where $r>0$, $\tau\geq 0$ and $x\in X$ and it is denoted by
$\overline{h_{\text{Cov}}}(C)$. The lower covering entropy of $C$, denoted by
$\underline{h_{\text{Cov}}}(C)$, is defined taking the limit inferior instead
of the limit superior. These quantities do not depend on $x\in X$ as usual.
The analogous of Proposition 3.1 holds.

######  Proposition 6.1.

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

$\displaystyle\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T)\cap\overline{B}(\textup{QC-
Hull}(C),\tau),r)\underset{P_{0},r_{0},r,r^{\prime},\tau,\tau^{\prime}}{\asymp}$
$\displaystyle\frac{1}{T}\log\textup{Pack}(\overline{B}(x,T)\cap\overline{B}(\textup{QC-
Hull}(C),\tau^{\prime}),r^{\prime})$

for all $r,r^{\prime}>0$ and $\tau,\tau^{\prime}\geq 0$. In particular any of
these functions can be used in the definition of the upper and lower covering
entropies of $C$.

###### Proof.

Once $\tau$ is fixed the asymptotic estimate can be proved exactly as in
Proposition 3.1. Moreover for all $\tau\geq 0$ it is easy to prove that

$\displaystyle\text{Cov}(\overline{B}(x,T)\cap\overline{B}(\text{QC-
Hull}(C),\tau),r)\leq$ $\displaystyle\text{Cov}(\overline{B}(x,T)\cap\text{QC-
Hull}(C),r_{0})\cdot\text{Cov}(r_{0}+\tau,r_{0}).$

and $\text{Cov}(r_{0}+\tau,r_{0})$ is uniformly bounded in terms of
$P_{0},r_{0}$ and $\tau$ by Proposition 2.1. This concludes the proof. ∎

Clearly when $C=\partial X$ we have $\overline{h_{\textup{Cov}}}(\partial
X)=\overline{h_{\textup{Cov}}}(X).$ Moreover if $C$ is a closed subset of
$\partial X$ then
$\overline{h_{\textup{Cov}}}(C)\leq\overline{h_{\textup{Cov}}}(\partial X),$
so $\overline{h_{\textup{Cov}}}(C)\leq\frac{\log(1+P_{0})}{r_{0}}$ by Lemma
3.3.
The analogous of Proposition 3.2 holds. We remark that in this case a
dependence on $\delta$ appears.

######  Proposition 6.2.

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

$\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T)\cap\textup{QC-
Hull}(C),r)\underset{P_{0},r_{0},r,\delta}{\asymp}\frac{1}{T}\log\textup{Cov}(S(x,T)\cap\textup{QC-
Hull}(C),r)$

In particular any of these functions can be used in the definition of the
upper and lower covering entropies of $C$.

###### Proof.

As in the proof of Proposition 3.2 one inequality is obvious, so we are going
to prove the other. We divide the ball $\overline{B}(x,T)$ in the annulii
$A(x,kr,(k+1)r)$ with $k=0,\ldots,\frac{T}{r}-1$. Therefore we can estimate
the quantity $\text{Cov}(\overline{B}(x,T)\cap\text{QC-Hull}(C),72\delta+2r)$
from above by

$\sum_{k=0}^{\frac{T}{r}-1}\text{Cov}(A(x,kr,(k+1)r)\cap\text{QC-
Hull}(C),72\delta+2r).$

We claim that every element of the sum is $\leq\text{Cov}(S(x,T)\cap\text{QC-
Hull}(C),r).$ Indeed let $y_{1},\ldots,y_{N}$ be a set of points realizing
$\text{Cov}(S(x,T)\cap\text{QC-Hull}(C),r)$. For all $i=1,\ldots,N$ we
consider the $\sigma$-geodesic segment $\gamma_{i}=[x,y_{i}]$ and we call
$x_{i}$ the point along this geodesic at distance $kr$ from $x$. We want to
show that $x_{1},\dots,x_{N}$ is a $(72\delta+2r)$-dense subset of
$A(x,kr,(k+1)r)\cap\text{QC-Hull}(C)$. Given a point $y\in
A(x,kr,(k+1)r)\cap\text{QC-Hull}(C)$ there exists a $\sigma$-geodesic line
$\gamma$ with endpoints in $C$ containing $y$. We parametrize $\gamma$ so that
$\gamma(0)$ is a projection of $x$ on $\gamma$ and
$y\in\gamma|_{[0,+\infty)}$. We take a point $y_{T}\in\gamma|_{[0,+\infty)}$
at distance $T$ from $x$, so that $y_{T}\in S(x,T)\cap\text{QC-Hull}(C)$ and
therefore there exists $i$ such that $d(y_{T},y_{i})\leq r$. By Lemma 5.2 the
path $\alpha=[x,\gamma(0)]\cup[\gamma(0),y_{T}]$, where the second geodesic is
a subsegment of $\gamma$, is a $(1,4\delta)$-quasigeodesic and, if $t_{y}$
denotes the real number such that $\alpha(t_{y})=y$, it holds
$t_{y}\in[kr,(k+1)r].$ By Lemma 5.2 we get
$d(y,\gamma_{i}^{\prime}(t_{y}))\leq 72\delta$, where $\gamma_{i}^{\prime}$ is
the $\sigma$-geodesic $[x,y_{T}]$. We conclude the proof of the claim since

$d(y,x_{i})\leq
d(y,\gamma_{i}^{\prime}(t_{y}))+d(\gamma_{i}^{\prime}(t_{y}),\gamma_{i}(t_{y}))+d(\gamma_{i}(t_{y}),x_{i})\leq
72\delta+2r,$

from the convexity of $\sigma$. The thesis follows by Proposition 6.1. ∎

The upper volume entropy of $C$ with respect to a measure $\mu$ is

$\overline{h_{\mu}}(C)=\sup_{\tau\geq
0}\limsup_{T\to+\infty}\frac{1}{T}\log\mu(\overline{B}(x,T)\cap\overline{B}(\text{QC-
Hull}(C),\tau)),$

where $x\in X$. Taking the limit inferior instead of the limit superior is
defined the lower volume entropy of $C$, $\underline{h_{\mu}}(C)$.

######  Proposition 6.3.

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 let $\mu$ be a measure
on $X$ which is $H$-homogeneous at scale $r$. Then for all $\tau\geq r$ it
holds

$\displaystyle\frac{1}{T}\log\mu(\overline{B}(x,T)\cap\overline{B}(\textup{QC-
Hull}(C),\tau))\underset{H,P_{0},r_{0},r,\tau}{\asymp}$
$\displaystyle\frac{1}{T}\log\textup{Cov}(\overline{B}(x,T)\cap\textup{QC-
Hull}(C),r_{0}).$

In particular the upper (resp. lower) volume entropy of $C$ with respect to
$\mu$ coincides with the upper (resp. lower) covering entropy of $C$ and it
can be computed using $\tau=r$ in place of the supremum.

###### Proof.

By Remark 3.5 we know that $\mu$ is $H(\tau)$-homogeneous at scale $\tau$ for
all $\tau\geq r$, where $H(\tau)$ depends on $P_{0},r_{0},\tau,r,H$. Therefore
the proof of Proposition 3.4 works in this case. ∎

###  6.2 Lipschitz topological entropy

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

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

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 reparametrization flow is well defined on
it. The upper Lipschitz-topological entropy of $\text{Geod}_{\sigma}(C)$ is
defined as

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

where the infimum is taken among all geometric metrics on
$\text{Geod}_{\sigma}(C)$. The lower Lipschitz-topological entropy is defined
taking the limit inferior instead of the limit superior and it is denoted by
$\underline{h_{\text{Lip-top}}}(\text{Geod}_{\sigma}(C))$. In the following
result we observe the difference between closed and non-closed subsets of
$\partial X$.

######  Theorem 6.4.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$ and $C$ be a subset of $\partial X$. Then

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

where the supremum is among closed subsets $C^{\prime}$ of $C$. The same holds
for the lower entropies.

We remark that the supremum of the covering entropies among the closed subsets
of $C$ can be strictly smaller than the covering entropy of $C$ (see
[Cav21b]), marking the distance between the equivalences of the different
notions of entropies in case of non-closed subsets of the boundary. We start
with an easy lemma.

######  Lemma 6.5.

Let $(X,\sigma)$ and $C$ be as in Theorem 6.4 and let $x\in X$. Then every
compact subset of $\textup{Geod}_{\sigma}(C)$ is contained in
$\textup{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime})$ for some $R\geq 0$ and
some $C^{\prime}\subseteq C$ closed. Moreover
$\textup{Geod}_{\sigma}(\overline{B}(x_{0},R),C^{\prime})$ is compact for all
$R\geq 0$ and all closed $C^{\prime}\subseteq C$.

###### Proof.

We fix a compact subset $K$ of $\textup{Geod}_{\sigma}(C)$. The continuity of
the evaluation map gives that $E(K)$ is contained in some ball
$\overline{B}(x,R)$. Moreover the maps
$+,-\colon\text{Geod}_{\sigma}(X)\to\partial X$, defined by
$\gamma\mapsto\gamma^{+},\gamma^{-}$ respectively, are continuous ([BL12],
Lemma 1.6). This means that $C^{\prime}=+(K)\cup-(K)$ is a closed subset of
$\partial X$ and clearly
$K\subseteq\text{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime})$. By a similar
argument, and since the evaluation map is proper, it follows that the set
$\text{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime})$ is compact for all $R\geq
0$ and all $C^{\prime}\subseteq C$ closed. ∎

For a metric $f\in\mathcal{F}$ and $C\subseteq\partial X$ we denote by
$\overline{h_{f}}$ the upper metric entropy of Geod${}_{\sigma}(C)$ with
respect to $f$, that is

$\overline{h_{f}}(\text{Geod}_{\sigma}(C))=\sup_{K}\lim_{r\to
0}\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{f^{T}}(K,r).$

Taking the limit inferior instead of the limit superior we define the lower
metric entropy of Geod${}_{\sigma}(C)$ with respect to $f$, denoted by
$\underline{h_{f}}(\text{Geod}_{\sigma}(C))$. The analogous of Proposition 4.3
is the following.

######  Proposition 6.6.

Let $(X,\sigma)$ be as in Theorem 6.4, $C^{\prime}$ be a closed subset of
$\partial X$, $f\in\mathcal{F}$, $x\in X$ and $L$ be the constant given by
Lemma 5.4. Then

  * (i)

$\overline{h_{f}}(\textup{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime}))=\overline{h_{f}}(\textup{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}))$
for all $R\geq L$;

  * (ii)

$\overline{h_{f}}(\textup{Geod}_{\sigma}(C^{\prime}))=\overline{h_{f}}(\textup{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}))\leq\overline{h_{\textup{Cov}}}(C^{\prime});$

  * (iii)

The function
$r\mapsto\limsup_{T\to+\infty}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}),r)$
is constant.

The same conclusions hold for the lower entropies.

We observe that applying the Key Lemma 4.4 we have directly the relative
version of Corollary 4.5.

######  Corollary 6.7.

Let $f\in\mathcal{F}$, $x\in X$, $R\geq 0$ and $0<r\leq r^{\prime}$. Then

$\displaystyle\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime}),r^{\prime})\underset{P_{0},r_{0},r,r^{\prime},f}{\asymp}$
$\displaystyle\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime}),r).$

###### Proof of of Proposition 6.6..

We fix $R\geq L$ and $T\geq 0$. We take a set $\gamma_{1},\ldots,\gamma_{N}$
of $\sigma$-geodesic lines realizing
$\text{Cov}_{f^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}),r_{0}).$
Our aim is to show that $\gamma_{1},\ldots,\gamma_{N}$ is a
$(4R+2L+C(f)+76\delta+r_{0})$-dense subset of
$\text{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime}).$ This, together with
Corollary 6.7, will prove (i).We consider a $\sigma$-geodesic line
$\gamma\in\text{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime})$, so
$d(\gamma(0),x)=:S\leq R$. By Lemma 5.3 there exists a $\sigma$-geodesic ray
$\xi$ starting at $x$ such that $d(\xi(S+t),\gamma(t))\leq 2S+76\delta$ for
all $t\geq 0$ and in particular $\xi^{+}$ belongs to $C$. Now we apply Lemma
5.4 to find a $\sigma$-geodesic line
$\gamma^{\prime}\in\text{Geod}(C^{\prime})$ such that
$d(\xi(t),\gamma^{\prime}(t))\leq L$ for all $t\geq 0$. Clearly we have
$\gamma^{\prime}\in\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$ and
$d(\gamma^{\prime}(S+t),\gamma(t))\leq 2S+L+76\delta$ for all $t\geq 0$.
Therefore $d(\gamma^{\prime}(t),\gamma(t))\leq 3S+L+76\delta$ for all $t\geq
0$. This implies that for all $t\in[0,T]$ we have

$f^{t}(\gamma,\gamma^{\prime})\leq\int_{-\infty}^{-t}\big{(}d(\gamma(0),\gamma^{\prime}(0))+2|s|\big{)}f(s)ds+\int_{-t}^{+\infty}\big{(}3S+L+76\delta\big{)}f(s)ds.$

Since $d(\gamma(0),\gamma^{\prime}(0))\leq L+S$ we get
$f^{t}(\gamma,\gamma^{\prime})\leq 4S+2L+C(f)+76\delta$ using the properties
of $f$, and so $f^{T}(\gamma,\gamma^{\prime})\leq 4R+2L+C(f)+76\delta.$
Moreover, since
$\gamma^{\prime}\in\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$, there
exists $\gamma_{i}$ such that $f^{T}(\gamma^{\prime},\gamma_{i})\leq r_{0}$.
This implies $f^{T}(\gamma,\gamma_{i})\leq 4R+2L+C(f)+76\delta+r_{0}.$
We observe that (iii) follows directly from the previous corollary.
The first equality in (ii) follows by (i). In order to prove the inequality we
fix $y_{1},\ldots,y_{N}$ realizing Cov$(S(x,T)\cap\text{QC-
Hull}(C^{\prime}),r_{0})$. Up to change $y_{i}$ with a point at distance at
most $8\delta$ from it we can suppose there are
$\gamma_{i}\in\text{Geod}_{\sigma}(C^{\prime})$ such that $y_{i}\in\gamma_{i}$
and $y_{1},\ldots,y_{N}$ is a $(8\delta+r_{0})$-dense subset of
$S(x,T)\cap\text{QC-Hull}(C^{\prime})$, as follows by Lemma 5.6. By Lemma 5.3
there exists an orientation of $\gamma_{i}$ such that, called
$S_{i}=d(x,\gamma_{i}(0))$ and $T_{i}\geq 0$ such that
$\gamma_{i}(T_{i})=y_{i}$, we have $T\leq S_{i}+T_{i}\leq T+4\delta$ and the
$\sigma$-geodesic ray $\xi_{i}=[x,\gamma_{i}^{+}]$ satisfies
$d(\xi_{i}(S_{i}+t),\gamma_{i}(t))\leq 76\delta$ for all $t\geq 0$. By Lemma
5.4 there exists
$\gamma_{i}^{\prime}\in\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$
such that $d(\gamma_{i}^{\prime}(t),\xi_{i}(t))\leq L$ for all $t\geq 0$. We
claim that the set $\\{\gamma_{i}^{\prime}\\}$ is
$(6L+176\delta+2r_{0}+2C(f))$-dense in
$\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$. By (i) and (iii) this
would imply the thesis. We fix
$\gamma\in\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$, so there exists
$y\in S(x,T)$ and $T_{y}\in[T-L,T+L]$ such that $\gamma(T_{y})=y$ and
therefore $d(y,y_{i})\leq 8\delta+r_{0}$ for some $i$. We observe that we have
$d(\gamma_{i}^{\prime}(S_{i}+T_{i}),y_{i})\leq L+76\delta$ and so
$d(\gamma_{i}^{\prime}(T),y_{i})\leq L+80\delta$. Moreover
$d(\gamma(T),y_{i})\leq L+8\delta+r_{0}$ implying
$d(\gamma(T),\gamma_{i}^{\prime}(T))\leq 2L+88\delta+r_{0}$. Furthermore by
definition $d(\gamma(0),\gamma_{i}^{\prime}(0))\leq 2L$, so by convexity of
$\sigma$ we get $d(\gamma(t),\gamma_{i}^{\prime}(t))\leq 2L+88\delta+r_{0}$
for all $t\in[0,T]$. The thesis follows by the classical subdivision of the
integral defining $f$ into three parts, each estimated by the constants above.
∎

###### Proof of Theorem 6.4.

We fix a geometric metric on $\text{Geod}_{\sigma}(C)$ and we denote by $M$
the Lipschitz constant with respect to of the evaluation map $E$. By Remark
5.5 the constant $L$ given by Lemma 5.4 can be chosen independently of
$C^{\prime}\subseteq C$, once $x$ is fixed. Clearly we have

$\sup_{R\geq 0,C^{\prime}\subseteq C}\lim_{r\to
0}\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,R),C^{\prime}),r)\geq$
$\sup_{C^{\prime}\subseteq
C}\limsup_{T\to+\infty}\frac{1}{T}\log\text{Cov}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}),r_{0}).$

We fix $\sigma$-geodesic lines $\gamma_{1},\ldots,\gamma_{N}$ realizing
Cov${}_{\text{\texthtd}^{T}}(\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime}),r_{0})$.
Since $d(\gamma_{i}(0),x)\leq L$ for all $i=1,\ldots,N$ then there exists
$t_{i}\in[T-L,T+L]$ such that $d(\gamma_{i}(t_{i}),x)=T$. We claim that the
points $y_{i}=\gamma_{i}(t_{i})\in S(x,T)\cap\text{QC-Hull}(C^{\prime})$ are
$(2L+80\delta+Mr_{0})$-dense. By Proposition 6.2 this would imply

$\overline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(C))\geq\sup_{C^{\prime}\subseteq
C}\overline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(C^{\prime}))\geq\sup_{C^{\prime}\subseteq
C}\overline{h_{\text{Cov}}}(C^{\prime}).$

We fix $y\in S(x,T)\cap\text{QC-Hull}(C^{\prime})$ and we select a geodesic
line $\gamma\in\text{Geod}(C^{\prime})$ containing $y$. Up to change $y$ with
a point at distance at most $8\delta$ we can suppose
$\gamma\in\text{Geod}_{\sigma}(C^{\prime})$, as follows by Lemma 5.6. By Lemma
5.3, with an appropriate choice of the orientation of $\gamma$, the
$\sigma$-geodesic ray $\xi=[x,\gamma^{+}]$ satisfies
$d(\xi(S+t),\gamma(t))\leq 76\delta$ for all $t\geq 0$, where
$S=d(x,\gamma(0))$. By Lemma 5.4 there exists
$\gamma^{\prime}\in\text{Geod}_{\sigma}(\overline{B}(x,L),C^{\prime})$ such
that $d(\xi(t),\gamma^{\prime}(t))\leq L$ for all $t\geq 0$, implying
$d(\gamma^{\prime}(S+t),\gamma(t))\leq L+76\delta$ for all $t\geq 0$. Denoting
by $T_{y}$ the real number such that $\gamma(T_{y})=y$ we have by Lemma 5.3
that $T\leq S+T_{y}\leq T+4\delta$. Therefore we apply the previous estimate
with $t=T_{y}$ obtaining $d(\gamma^{\prime}(T),y)\leq
d(\gamma^{\prime}(T),\gamma^{\prime}(S+T_{y}))+d(\gamma^{\prime}(S+T_{y}),y)\leq
L+80\delta$. Moreover there exists $i\in\\{1,\ldots,N\\}$ such that
$\text{\texthtd}^{T}(\gamma^{\prime},\gamma_{i})\leq r_{0}$ and in particular
$d(\gamma^{\prime}(T),\gamma_{i}(T))\leq Mr_{0}$. Therefore we get
$d(y_{i},y)\leq d(\gamma_{i}(t_{i}),\gamma_{i}(T))+d(\gamma_{i}(T),y)\leq
2L+80\delta+Mr_{0}$. Now, up to add $8\delta$, we obtain the inequality. The
other inequality follows by Proposition 6.6. Indeed we have

$\overline{h_{\text{Lip-
top}}}(\text{Geod}_{\sigma}(C))\leq\sup_{C^{\prime}\subseteq
C}\overline{h_{f}}(\text{Geod}_{\sigma}(C^{\prime}))\leq\sup_{C^{\prime}\subseteq
C}\overline{h_{\text{Cov}}}(C^{\prime}).$

∎

######  Remark 6.8.

Let $(X,\sigma)$ be as in Theorem 6.4, $C\subseteq\partial X$ closed and
$x\in\textup{QC-Hull}(C)$. By the proof of Theorem 6.4, Lemma 5.4 and Remark
5.5 we obtain

$\frac{1}{T}\log\textup{Cov}(S(x,T)\cap\textup{QC-
Hull}(C),r_{0})\underset{P_{0},r_{0},\delta,f}{\asymp}\frac{1}{T}\log\textup{Cov}_{f^{T}}(\textup{Geod}_{\sigma}(\overline{B}(x,L),C),r_{0})$

for all $f\in\mathcal{F}$, where $L=14\delta$.

###  6.3 Minkowski dimension

The relative version of Proposition 5.8 is:

######  Proposition 6.9.

Let $(X,\sigma)$ be a $\delta$-hyperbolic GCB-space that is $P_{0}$-packed at
scale $r_{0}$, let $C$ be a subset of $\partial X$, $x\in X$ and $L$ be the
constant given by Lemma 5.4. Then

$\frac{1}{T}\log\textup{Cov}(C,e^{-T})\underset{P_{0},r_{0},\delta,L}{\asymp}\frac{1}{T}\log\textup{Cov}(S(x,T)\cap\textup{QC-
Hull}(C),r_{0}).$

In particular the upper (resp. lower) Minkowski dimension of $C$ equals the
upper (resp. lower) covering entropy of $C$.

###### Proof.

We can suppose $T\geq L$. Let $z_{1},\ldots,z_{N}$ be points realizing
$\textup{Cov}(C,e^{-T})$. For every $i$ we take a geodesic ray $[x,z_{i}]$. By
Lemma 5.4 there exists a geodesic line $\gamma_{i}$ with both endpoints in $C$
such that $d(\gamma_{i}(t),\xi(t))\leq L$ for every $t\geq 0$. We take a point
$y_{i}=\gamma_{i}(t_{i})$ with $t_{i}\geq 0$ such that $d(y_{i},x)=T$. We know
$|t_{i}-T|\leq 2L$. We claim the set $\\{y_{i}\\}$ cover $S(x,T)\cap\text{QC-
Hull}(C)$ at scale $82\delta+3L$. Indeed let $y\in S(x,T)\cap\text{QC-
Hull}(C)$, i.e. it exists a geodesic line $\gamma$ with both endpoints in $C$
such that $y\in\gamma$. We parametrize $\gamma$ so that $\gamma(0)$ is a
projection of $x$ on $\gamma$ and $y\in\gamma|_{[0,+\infty)}$. We consider a
geodesic ray $\xi=[x,\gamma^{+}]$. By Lemma 5.3 we know that $d(\xi(T),y)\leq
76\delta+3L$. Moreover $\gamma^{+}\in C$, so there is $i$ such that
$(z_{i},\gamma^{+})>T$. By Lemma 5.9 we get $d(y_{i},\xi(T))\leq 6\delta$, so
$d(y,y_{i})\leq 82\delta+3L$. This shows

$\text{Cov}(S(x,T)\cap\text{QC-Hull}(C),82\delta+3L)\leq\text{Cov}(C,e^{-T}).$

Let $\\{y_{1},\ldots,y_{N}\\}$ be points realizing
$\text{Cov}(S(x,T+2L+38\delta)\cap\text{QC-Hull}(C),\frac{L}{2})$. Therefore
for every $i$ it exists a geodesic line $\gamma_{i}$ with both endpoints in
$C$ containing $y_{i}$. We parametrize each $\gamma_{i}$ so that
$\gamma_{i}(0)$ is a projection of $x$ on $\gamma_{i}$ and
$y_{i}\in\gamma_{i}|_{[0,+\infty)}$. We claim that the set
$\\{\gamma_{i}^{+}\\}$ cover $C$ at scale $e^{-T}$. Indeed for every $z\in C$
we take a geodesic ray $\xi=[x,z]$ and we set $y=\xi(T+2L+38\delta)$. By Lemma
5.4 we know it exists a geodesic line $\gamma$ with both endpoints in $C$ such
that $d(y,\gamma(T+2L+38\delta))\leq L$. Moreover there is a point
$y^{\prime}$ along $\gamma$ which is at distance exactly $T+2L+38\delta$ from
$x$ and that satisfies $d(y,y^{\prime})\leq 2L$. Now we know it exists $i$
such that $d(y^{\prime},y_{i})\leq\frac{L}{2}$, moreover for a fixed geodesic
ray $\xi_{i}=[x,z_{i}]$ it holds $d(\xi_{i}(T+2L+38\delta),y_{i})\leq
L+76\delta$ by Lemma 5.3. So in conclusion we get

$d(\xi(T+2L+38\delta),\xi_{i}(T+2L+38\delta))<4L+76\delta.$

By Lemma 5.9 we conclude that $(z,z_{i})_{x}>T$, i.e.

$\text{Cov}(C,e^{-T})\leq\text{Cov}\left(S(x,T+2L+38\delta)\cap\text{QC-
Hull}(C),\frac{L}{2}\right).$

Now the conclusion follows by Proposition 6.1. ∎

The proof of Theorems D and E follow by Proposition 6.1, Proposition 6.2,
Proposition 6.3, Theorem 6.4, Proposition 6.6, Remark 6.8 and Proposition 6.9.

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