aa r X i v : . [ m a t h . D S ] F e b Entropies of non-positively curved metric spaces
Nicola Cavallucci
Abstract.
We show the equivalences of several notions of entropy, such as a version ofthe 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. A uniform Ahlfors regularity ofthe limit set of quasiconvex-cocompact actions on Gromov-hyperbolic packed metric spaceswith convex geodesic bicombing will be shown, implying a uniform rate of convergenceto the entropy. As a consequence we prove the continuity of the critical exponent forquasiconvex-cocompact groups with bounded codiameter.
Contents f ∈ F . . . . 284.2 Proof of Theorem 4.2 . . . . . . . . . . . . . . . . . . . . . . . 334.3 Lipschitz-topological entropy of the geodesic semi-flow . . . . 341 Dimension of the boundary 34
Introduction
This paper is devoted to the investigation of different asymptotic quantitiesassociated to a metric space, some of them classical and widely studied. Theattention will be held on (sometimes Gromov-hyperbolic) metric spaces sup-porting a geodesically complete, convex, geodesic bicombing and satisfyinga uniform covering (or packing) condition. A metric space is C ( · , · ) -covered,where C is a function C : R > × R > → N , if every ball of radius R can becovered by at most C ( R, r ) balls of radius r , for every < r ≤ R . Thisconditions mymics a macroscopic version of a lower bound of the sectionalcurvature for Riemannian manifolds and it plays an important role in themetric space setting as highlighted by Gromov’s precompatness Theorem([Gro81]). For metric spaces supporting a geodesically complete, convex,geodesic bicombing it is enough to have a control of the covering function atsome fixed scale to guarantee a uniform control of the covering function atevery scale (see Proposition 2.1, Section 2 and [CS20a]). This is the mainmotivation for the restriction to GCB-spaces, namely complete metric spacessupporting a geodesically complete, convex, consistent, reversible bicombing(see Section 2.1 for precise definitions). It is preferable to work in this settingbecause this class of metric spaces is closed under limit operations ([CS20a]),while it is not the case for Busemann convex spaces, so it is the natural set-ting for compactness and continuity results we will present later. Howevergeodesically complete Busemann convex spaces, and so CAT (0) -spaces, sup-port a unique geodesically complete, convex, consistent, reversible geodesicbicombing, so they fit perfectly in this setting. Another reason behind thechoose of GCB-space is that every word-hyperbolic group acts geometricallyon a proper metric space supporting a convex geodesic bicombing ([Des15]).Lastly it is also a natural setting for the study of the geodesic flow. Weare going to present the different notions of entropies we are interested in,starting from the Lipschitz-topological entropy. The topological entropy of the geodesic flow has been intensively studiedin case of Riemannian manifolds, especially in the negatively curved set-ting. If such a manifold is denoted by ¯ M = M/ Γ , where M is its universalcover and Γ is its fundamental group, then the set of parametrized geodesiclines is identified with the unit tangent bundle S ¯ M and the non-wanderingset of the geodesic flow is the set of unit tangent vectors whose lift to M generate a geodesic with endpoints in the limit set of Γ : we denote it by S ¯ M nw . Two cornerstones of the theory of the geodesic flow are the works ofEberlein ([Ebe72]), who proved that the geodesic flow restricted to S ¯ M nw is topologically transitive, and Sullivan [Sul84], who proved the ergodicityof the geodesic flow when M is the -dimensional hyperbolic space and Γ is3eometrically finite.Probably the most important invariant associated to the geodesic flow isthe topological entropy of its restriction to the non-wandering set, denoted h nwtop ( ¯ M ) . It equals the Hausdorff dimension of the limit set of Γ and thecritical exponent of Γ (see [Sul84], [OP04]). Moreover if ¯ M is compact thenit coincides also with the volume entropy of M ([Man79]), while this is nomore true in general, even when ¯ M has finite volume (cp. [DPPS09]): wewill come back to these examples at the end of the introduction. The topo-logical entropy of the non-wandering set of the geodesic flow characterizesthe hyperbolic metrics among Riemannian manifolds with pinched, negativecurvature and with finite volume ([PS19]).A GCB-space is denoted by ( X, σ ) , where σ is the bicombing on X .For such spaces we restrict the attention to σ -geodesic lines and the topo-logical entropy of the σ -geodesic flow is defined as the topological entropy(in the sense of Bowen, cp. [Bow73], [HKR95]) of the dynamical system ( Geod σ ( X ) , Φ t ) , where Geod σ ( X ) is the space of parametrized σ -geodesiclines, endowed with the topology of uniform convergence on compact sub-sets, and Φ t is the reparametrization flow. It is: h top ( Geod σ ( X )) = inf â sup K ⊆ Geod σ ( X ) lim r → lim T → + ∞ T log Cov â T ( K, r ) , where the infimum is taken among all metrics on Geod σ ( X ) inducing itstopology, the supremum is taken among all compact subsets of Geod σ ( X ) , â T is the distance â T ( γ, γ ′ ) = max t ∈ [0 ,T ] â (Φ t ( γ ) , Φ t ( γ ′ )) and Cov â T ( K, r ) is the minimal number of balls (with respect to the metric â T ) of radius r needed to cover K . We remark that in case of Busemann convex (or CAT (0) )metric spaces the space of σ -geodesic lines coincides with the set of geodesiclines.For this flow the non-wandering set is empty and applying the variationalprinciple (cp. [HKR95]) it is straightforward to conclude that its topologicalentropy is zero since there are no flow-invariant probability measures (Lemma4.1, Section 4). Looking carefully at the proof of the variational principle itturns out that the metrics on Geod σ ( X ) almost realizing the infimum in thedefinition of the topological entropy are restriction to Geod σ ( X ) of metricson its one-point compactification. In particular they are no the natural onesto consider: indeed the general idea behind the topological entropy is tocompute the number of geodesic lines needed to stay at small distance r from any other geodesic line for a long time T . But a general metric â on Geod σ ( X ) does not take into account this information. That is why, inSection 4, we will restrict the attention to the class of geometric metrics â :those with the property that the evaluation map E : ( Geod σ ( X ) , â ) → ( X, d ) defined as E ( γ ) = γ (0) is Lipschitz. Notice that for a geometric metric twogeodesic lines are not close if they are distant at time . Accordingly the4 ipschitz-topological entropy of the geodesic flow is defined as h Lip-top ( Geod σ ( X )) = inf â sup K ⊆ Geod σ ( X ) lim r → lim T → + ∞ T log Cov â T ( K, r ) , where now the infimum is taken only among the geometric metrics of Geod σ ( X ) .Although the definition of the Lipschitz-topological entropy is quite compli-cated, its computation can be remarkably simplified. Indeed one of the mostused metric on Geod σ ( X ) (see for instance [BL12]) is: d Geod ( γ, γ ′ ) = Z + ∞−∞ d ( γ ( s ) , γ ′ ( s )) 12 e | s | ds that induces the topology of Geod σ ( X ) and is geometric, and it turns outthat it realizes the infimum in the definition of the Lipschitz-topologicalentropy. Theorem A (Extract from Theorem 4.2 & Proposition 4.3) . Let ( X, σ ) bea GCB -space that is P -packed at scale r . Then h Lip-top ( Geod σ ( X )) = lim T → + ∞ T log Cov d T Geod ( Geod σ ( x ) , r ) , where Geod σ ( x ) is the set of σ -geodesic lines passing through x at time . Therefore the infimum in the definition of the Lipschitz topological entropy isactually realized by the metric d Geod and the supremum among the compactsets can be replaced by a fixed (relative small) compact set. Moreover alsothe scale r can be fixed to be r (or any other positive real number). The second definition of entropy we consider (see Section 3.2) is the volumeentropy. If X is a metric space equipped with a measure µ it is classical toconsider the exponential growth rate of the volume of balls, namely: h µ ( X ) := lim T → + ∞ T log µ ( B ( x, T )) . It is called the volume entropy of X with respect to the measure µ and it doesnot depend on the choice of the basepoint x ∈ X by triangular inequality.This invariant has been studied intensively in case of complete Riemannianmanifolds with non positive sectional curvature, where µ is the Riemannianvolume on the universal cover. It is related to other interesting invariants asthe simplicial volume of the manifold (see [Gro82]), [BS20]), a macroscopicalcondition on the scalar curvature (cp. [Sab17]) and the systole in case ofcompact, non-geometric -manifolds (cp. [CS19]). Moreover the infimum ofthe volume entropy among all the possible Riemannian metrics of volume5 on a fixed closed manifold is a subtle homotopic invariant (see [Bab93],[Bru08] for general considerations and [BCG95], [Pie19] for the computationof the minimal volume entropy in case of, respectively, closed n -dimensionalmanifolds supporting a locally symmetric metric of negative curvature and -manifolds).Another example, besides the Riemannian setting, is the counting mea-sure of the orbit of a discrete, cocompact group of isometries of a Gromov-hyperbolic, GCB-space. Both this case and the Riemannian one (at leastwhen the curvature of the Riemannian manifold considered is pinched) sharethe following property: the measure µ under consideration satisfies H ≤ µ ( B ( x, r )) ≤ H for every x ∈ X , where r, H are positive real numbers. A measure with thisproperty is called H -homogeneous at scale r . Among homogeneous mea-sure there is a remarkable example: the volume measure µ X of a complete,geodesically complete, CAT (0) metric space X that is P -packed at scale r (see [CS20b] and [LN19] for a description of the measure). In case X is aRiemannian manifold of non-positive sectional curvature then µ X coincideswith the Riemannian volume, up to a universal multiplicative constant.A more combinatoric and intrinsic version of the volume entropy of ageneric metric space is the covering entropy , defined as: h Cov ( X ) := lim T → + ∞ T log Cov ( B ( x, T ) , r ) , where x is a point of X and Cov ( B ( x, T ) , r ) is the minimal number of ballsof radius r needed to cover B ( x, T ) . It does not depend on x but it candepend on the choice of r . This is not the case when X is a GCB-space thatis P -packed at scale r , as follows by Proposition 3.1, Section 3. Moreoverit is always finite (cp.Lemma 3.3, Section 3). The expression of the Lipschitz-topological entropy given by Theorem Asuggests the possibility to relate that invariant to some property of the σ -boundary at infinity of X . Once fixed a basepoint x ∈ X we define the shadow dimension of the σ -boundary ∂ σ X of X asShad-D ( ∂ σ X ) = lim T → + ∞ T log Shad-Cov r ( ∂X, e − T ) , where Shad-Cov r ( ∂X, e − T ) is the minimal number of points y , . . . , y N at dis-tance T from x such that every σ -geodesic ray issuing from x passes throughone of the balls of radius r and center y i . The limit above does not dependneither on x nor on r . It describes the asymptotic behaviour of the number6f shadows casted by points at distance T from x needed to cover ∂ σ X , when T goes to + ∞ . The shadows, and especially their relations with other prop-erties of the boundary at infinity, have been intensively studied during theyears (starting from [Sul79]). In particular if X is Gromov-hyperbolic theboundary at infinity can be equipped with a metric and it turns out that themetric balls are approximately equivalent to the shadows. This equivalenceremains true when we consider the generalized visual balls. If we denote by ( · , · ) x the Gromov product based on x then the generalized visual ball ofcenter z ∈ ∂X and radius ρ is B ( z, ρ ) = { z ′ ∈ ∂X s.t. ( z, z ′ ) x > log ρ } . The visual Minkowski dimension of ∂X (that coincides in this case with ∂ σ X andwith the Gromov boundary ∂ G X ) is:MD ( ∂X ) = lim T → + ∞ T log Cov ( ∂X, e − T ) , where Cov ( ∂X, e − T ) is the minimal number of generalized visual balls ofradius e − T needed to cover ∂X . If the generalized visual balls are metric ballsfor some visual metric D x,a then we refind the usual definition of Minkowskidimension of the metric space ( ∂X, D x,a ) , once the obvious change of variable ρ = e − T is made. These invariants are presented in Section 5. One of the main results of the paper is:
Theorem B.
Let ( X, σ ) be a GCB -space that is P -packed at scale r . Then h Lip-top ( Geod σ ( X )) = h µ ( X ) = h Cov ( X ) = Shad-D ( ∂ σ X ) , where µ is every homogeneous measure on X . Moreover if X is also δ -hyperbolic then they coincide also with MD ( ∂X ) . Actually something more is true but in order to state it we need tointroduce the notion of equivalent asymptotic behaviour of two functions.Given f, g : [0 , + ∞ ) → R we say that f and g have the same asymptoticbehaviour , and we write f ≍ g , if for all ε > there exists T ε ≥ such thatif T ≥ T ε then | f ( T ) − g ( T ) | ≤ ε . The function T ε is called the thresholdfunction . Usually we will write f ≍ P ,r ,δ,... g meaning that the thresholdfunction can be expressed only in terms of ε and P , r , δ, . . . In particular if g is constantly equal to g and f ≍ P ,r ,δ,... g then the function f tends to g when T goes to + ∞ and moreover the rate of convergence of the limit canbe expressed only in terms of P , r , δ, . . . Theorem C.
Let ( X, σ ) be a GCB -space that is P -packed at scale r . Thenthe functions defining the quantities of Theorem B have the same asymptoticbehaviour and the threshold functions depend only on P , r , δ and the homo-geneous constants of µ . Theorem D.
Let ( X n , x n , σ n ) be a sequence of pointed GCB -spaces that are P -packed at scale r converging in the pointed Gromov-Hausdorff sense to ( X ∞ , x ∞ ) . If for every n it holds T log Cov ( B ( x n , T ) , r ) ≍ h n and the threshold functions do not depend on n , then h Cov ( X ∞ ) = lim n → + ∞ h n . Under the assumptions of Theorem D we have that X ∞ is naturally a properGCB-space and moreover for every n the covering entropy of X n is exactly h n ,so it gives a continuity result for the covering entropy. Actually, by TheoremC, the assumption on the asymptotic behaviour of the covering entropy canbe replaced by an equivalent assumption on the asymptotic behaviour of anyother notion of entropy. This continuity result can be compared with [Rev05],where a continuity of the volume entropy is established in case of specialclasses of compact metric spaces. Clearly Theorem D is interesting if we canfind a family of metric spaces whose asymptotic behaviour of the coveringentropy (or any other notion of entropy) is uniformly controlled. We willsee in a minute how it is possible to bound the asymptotic behaviour of theMinkowski dimension in case of quasiconvex-cocompact groups of isometries.Before doing this we need to introduce the relative versions of the severalnotions of entropy that will be discussed in Section 6. In case X is also δ -hyperbolic it is possible to define the version of all thedifferent notions of entropies relative to subsets of the boundary ∂X . Forevery subset C ⊆ ∂X we denote by Geod σ ( C ) the set of parametrized σ -geodesic lines with endpoints belonging to C and with QC-Hull ( C ) the unionof the points belonging to the geodesics joining any two points of C . Actuallythe hyperbolicity assumption (or at least a visibility assumption on ∂X )is necessary since otherwise the sets Geod σ ( C ) and QC-Hull ( C ) could be8mpty. The numbers h Cov ( C ) = lim T → + ∞ T log Cov ( B ( x, T ) ∩ QC-Hull ( C ) , r ) h Lip-top ( Geod σ ( C )) = inf â sup K ⊆ Geod σ ( C ) lim r → + ∞ lim T → + ∞ T log Cov â T ( K, r ) Shad-D ( C ) = lim T → + ∞ T log Shad-Cov r ( C, e − T ) MD ( C ) = lim T → + ∞ T log Cov ( C, e − T ) are called, respectively, covering entropy of C , Lipschitz-topological entropyof Geod σ ( C ) , shadow dimension of C and visual Minkowski dimension of C .The volume entropy of C with respect to a measure µ is h µ ( C ) = sup τ ≥ lim T → + ∞ T log µ ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ )) , where B ( Y, τ ) means the closed τ -neighbourhood of Y ⊆ X . If µ is H -homogeneous at scale r then the volume entropy can be computed putting τ = r instead of the supremum over τ ≥ (Proposition 6.7, Section 6). Forinstance when X is a Riemannian manifold with pinched negative curvatureand µ X is the Riemannian volume then it is H ( r ) -homogeneous at everyscale r > , so the definition does not depend on τ at all. Most of therelations of Theorem B remain true for subsets of the boundary, but theasymptotic behaviour of the different functions involved in the definitions ofthe entropies depend also on the choice of the basepoint x ∈ X . The bestpossible choice, x ∈ QC-Hull ( C ) , allows us to give again uniform asymptoticestimates. Theorem E.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and let C ⊆ ∂X . Then h Cov ( C ) = Shad-D ( C ) = MD ( C ) = h µ ( C ) for every homogeneous measure µ on X . All the functions defining the quan-tities above have the same asymptotic behaviour and the threshold functionscan be expressed only in terms of P , r , δ and the homogeneous constants of µ , if the basepoint x belongs to QC-Hull ( C ) . The proof of this result does not follow by the same arguments of The-orem B, indeed it will be based heavily on the Gromov-hyperbolicity of X .The relation between the Lipschitz-topological entropy of Geod σ ( C ) and theother definitions of entropy is more complicated. We have9 heorem F. Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and let C ⊆ ∂X . Then(i) if C is closed then h Lip-top ( Geod σ ( C )) = h Cov ( C ) and the functionsdefining these two quantities have the same asymptotic behaviour withthresholds function depending only on P , r , δ .Moreover if x ∈ QC-Hull ( C ) then h Lip-top ( Geod σ ( C )) = lim T → + ∞ T log Cov d T Geod ( Geod σ ( B ( x, L ) , C ) , r ) , where Geod σ ( B ( x, L ) , C ) is the set of geodesic lines with endpoints in C and passing through B ( x, L ) at time and L is a constant dependingonly on δ .(ii) if C is not closed then h Lip-top ( Geod σ ( C )) = sup C ′ ⊆ C h Lip-top ( Geod σ ( C ′ )) ≤ h Cov ( C ) , where the supremum is taken among the closed subsets of C . When C is the limit set Λ(Γ) of a discrete group of isometries Γ of X thereis another largely studied invariant: the Γ -entropy of X defined as h Γ ( X ) = lim T → + ∞ T log Γ x ∩ B ( x, T ) , which is precisely, when x ∈ QC-Hull (Λ(Γ)) , the volume entropy of the set
Λ(Γ) with respect to the counting measure µ Γ x of the orbit Γ x . It equals thecritical exponent of Γ , whose definition will be recalled in Section 7. Thecritical exponent of Γ has a dynamical and a measure-theoretical counterpart.Indeed when ¯ M = M/ Γ is a complete Riemannian manifold with pinchednegative sectional curvature then it equals the topological entropy of thenon-wandering set of the geodesic flow, h nwtop ( ¯ M ) , as shown in [OP04]. Thesame result will be generalized in case of discrete and torsion-free group of σ -invariant isometries of complete, Gromov-hyperbolic, packed, GCB-spacesin [Cav21]. On the other hand, by Bishop-Jones’ Theorem, it equals theHausdorff dimension of the radial limit set of Γ ([BJ97], [DSU17]).We recall that a group Γ is said quasiconvex-cocompact if the actionof Γ on QC-Hull (Λ(Γ)) is compact. Quasiconvex-cocompact groups havebeen extensively studied, due to their regularity. For instance in [Coo93]is shown that the limit set of a quasiconvex cocompact group of isometriesof a proper Gromov-hyperbolic metric space is Ahlfors-regular. Our mainresult of Section 7 is a refinement of this result, in terms of quantification ofthe Ahlfors-regularity constants. In the following the Γ -entropy of X will bedenoted simply by h Γ . 10 heorem G. Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and let Γ be a discrete, quasiconvex-cocompact group of σ -isometriesof X with codiameter ≤ D . Then the Patterson-Sullivan measure µ PS on Λ(Γ) is ( A, h Γ ) -Ahlfors regular, i.e. for every z ∈ Λ(Γ) and every < ρ ≤ it holds A ρ h Γ ≤ µ PS ( B ( z, ρ )) ≤ Aρ h Γ , where A is a constant depending only on P , r , δ and D . Moreover T log Cov (Λ(Γ) , e − T ) ≍ P ,r ,δ,D h Γ . We recall that an isometry of X is σ -invariant if it preserves the bicombing.If Γ is elementary the proof is trivial, while in the non-elementary case itdepends heavily on a uniform estimate of the critical exponent. Indeed byCorollary 1.3 of [CS20a] in this case we have < h − ≤ h Γ ≤ h + , where h − and h + depend only on P , r , δ and D (see Remark 7.3, Section 7).Theorem G, together with its proof, has two main consequences: the firstone is a quantified equidistribution of the orbit (see again [Coo93] for a nonquantified version). Theorem H.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and let Γ be a discrete, quasiconvex-cocompact group of σ -isometriesof X with codiameter ≤ D and let x ∈ QC-Hull (Λ(Γ)) . Then there exists
K > depending only on P , r , δ and D such that for all T ≥ it holds K · e T · h Γ ≤ Γ x ∩ B ( x, T ) ≤ K · e T · h Γ . The second one is the continuity of the critical exponent, which is not surpris-ing at this point since under the assumptions of Theorem G we have a univer-sal bounded asymptotic behaviour of the function defining the Minkowski di-mension of
Λ(Γ) . The lower semicontinuity of the critical exponent is knownin some cases (see [BJ97] and [Pau97]) but several restrictions on the classof groups are made.In order to state this last result we denote by GCB qc ( P , r , δ ; D ) the spaceof -uples ( X, x, σ, Γ) where ( X, σ ) is a δ -hyperbolic GCB-space that is P -packed at scale r , Γ is a discrete, non-elementary, quasiconvex-cocompactgroup of σ -isometries of X with codiameter ≤ D and x is a point belongingto QC-Hull (Λ(Γ)) . Theorem I.
The class
GCB qc ( P , r , δ ; D ) is compact with respect to thepointed Gromov-Hausdorff convergence and with respect to this convergencethe critical exponent is continuous, i.e. if ( X n , x n , σ n , Γ n ) is a sequence ofspaces of GCB qc ( P , r , δ ; D ) converging to ( X ∞ , x ∞ , σ ∞ , Γ ∞ ) then h Γ ∞ =lim n → + ∞ h Γ n .
11n the proof of the compactness part we will show the interesting fact thatunder the assumptions of the theorem the boundary at infinity of the limitspace is homeomorphic (and actually isometric for a suitable choice of ametric) to the limit space of the boundaries. Moreover in the quasiconvex-cocompact case with bounded codiameter we will see that the limit sets
Λ(Γ n ) converge to the limit set Λ(Γ ∞ ) . Many tools necessary to the proofhave been developed in [CS20b] and [CS20a], in terms of ultralimits.We remark that since the CAT (0) condition is stable under limits then The-orem I is true even under this stronger assumption. In this last part of the introduction we restrict the attention to the case ofRiemannian manifolds ¯ M = M/ Γ with pinched negative sectional curvature.If Γ is geometrically finite then the limit set of Γ is the union of the radiallimit set Λ r (Γ) and the bounded parabolic points. The latter is a countableset, therefore the Hausdorff dimension of the limit set coincides with theHausdorff dimension of the radial limit set, so by Bishop-Jones’ Theorem itholds: HD (Λ(Γ)) = HD (Λ r (Γ)) = h Γ . (1.1)We remark that this is not true if Γ is not geometrically finite, even when M is the hyperbolic space. Example 1.1.
In general it can happen HD (Λ r (Γ))) < HD (Λ(Γ)) . Indeedlet Γ be a cocompact group of H and let Γ ′ be a normal subgroup of Γ such that Γ / Γ ′ is non amenable. Let F ⊆ Λ(Γ ′ ) be the subsets of points z that are fixed by some g ∈ Γ ′ . For every z ∈ F and every h ∈ Γ we have hz = hgz = g ′ hz for some g ′ ∈ Γ ′ since Γ ′ is normal. Then hz is fixed by g ′ and so it belongs to F , i.e. F is Γ -invariant. By minimality of Λ(Γ) we get
Λ(Γ ′ ) = Λ(Γ) , so HD (Λ(Γ ′ )) = HD (Λ(Γ)) . But by the growth tightness of Γ (cp. [Sam02]) we haveHD (Λ r (Γ ′ )) = h Γ ′ < h Γ = HD (Λ(Γ)) = HD (Λ(Γ ′ )) . However if M is the hyperbolic space and Γ is geometrically finite then evensomething more is true, indeed by [SU96]: h Γ = HD (Λ(Γ)) = MD (Λ(Γ)) . (1.2)This equality fails to be true for geometrically finite (actually of finite co-volume) 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 Rieman-nian manifold M with pinched negative sectional curvature admitting a (non-uniform) lattice (i.e. a group of isometries Γ with Vol ( M/ Γ) < + ∞ ) such12hat h Γ < h µ M ( X ) . We observe that since Γ is a lattice then Λ(Γ) = ∂M , so h µ M ( M ) = MD (Λ(Γ)) by Theorem B, while h Γ = HD (Λ r (Γ)) = HD (Λ(Γ)) by (1.1).The example above is due to a relevant variation of the curvature of M . In-deed in [DPPS19] is shown that for non-uniform lattices Γ of asymptotically / -pinched manifolds with negative curvature M it holds h µ M ( M ) = h Γ .The general situation in the geometrically finite case is:HD (Λ(Γ)) = h Γ = h nwtop ( M/ Γ) h Lip-top ( Geod (Λ(Γ))) = h Cov (Λ(Γ)) = h µ M (Λ(Γ)) = MD (Λ(Γ)) , (1.3)where the equalities follow by Theorem E, Theorem F and (1.1). Moreover itis clear that the first line is always less than or equal to the second one, sincethe Hausdorff dimension is always smaller than the Minkowski dimension.We remark that in case M is of constant curvature − then the coveringentropy of the limit set of a group Γ coincides with the convex-core entropyof Γ and in this setting some of the equalities above are known (see [FM15]).The relations in (1.3) allow us to give new interpretations of the phenomenaoccurring in Example 1.2, i.e. the possible difference between the criticalexponent of the group and the volume entropy of Λ(Γ) :• measure-theoretic interpretation: it can be seen as the difference be-tween the Hausdorff and the Minkowski dimension of the limit set Λ(Γ) ,so it is related to the fractal structure of the limit set;• dynamical interpretation: it can be seen as the difference between thetopological entropy of the non-wandering set of the geodesic flow andthe Lipschitz-topological entropy of Geod (Λ(Γ)) .• combinatoric interpretation: it can be seen as the difference between h Γ and h Cov (Λ(Γ)) , where the former counts the exponential growthrate of an orbit while the latter counts the exponential growth rate ofthe cardinality of r -nets, for some (any) r > . Here the differencearises in terms of sparsity of the orbit. Throughout the paper X will denote a metric space and d will denote themetric on X . The open (resp.closed) ball of radius r and center x is denotedby B ( x, r ) (resp. 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 ′ ) to denote the closedannulus of center x and radii < r < r ′ , i.e. the set of points y ∈ X suchthat r ≤ d ( x, y ) ≤ r ′ . A geodesic segment is an isometry γ : I → X where13 = [ a, b ] is a a bounded interval of R . The points γ ( a ) , γ ( b ) are called theendpoints of γ . A metric space X is said geodesic if for all couple of points x, y ∈ X there exists a geodesic segment whose endpoints are x and y . Wewill denote any geodesic segment between two points x and y , with an abuseof notation, as [ x, y ] . A geodesic ray is an isometry γ : [0 , + ∞ ) → X while ageodesic line is an isometry γ : R → X .Let Y be any subset of a metric space X :– a subset S of Y is called r -dense if ∀ y ∈ Y ∃ z ∈ S such that d ( y, z ) ≤ r ;– a subset S of Y is called r -separated if ∀ y, z ∈ S it holds d ( y, z ) > r .The packing number of Y at scale r is the maximal cardinality of a r -separated subset of Y and it is denoted by Pack ( Y, r ) . The covering numberof Y is the minimal cardinality of a r -dense subset of Y and it is denoted byCov ( Y, r ) . The following inequalities are classical:Pack ( Y, r ) ≤ Cov ( Y, r ) ≤ Pack ( Y, r ) . (2.1)The packing and the covering functions of X are respectivelyPack ( R, r ) = sup x ∈ X Pack ( B ( x, R ) , r ) , Cov ( R, r ) = sup x ∈ X Cov ( B ( x, R ) , r ) . They take values on [0 , + ∞ ] . By (2.1) it holdsPack ( R, r ) ≤ Cov ( R, r ) ≤ Pack ( R, r ) . (2.2)Let C , P , r > . We say that a metric space X is P -packed at scale r if Pack (3 r , r ) ≤ P , that is every ball of radius r contains no more than P points that are r -separated. The space X is C -covered at scale r ifCov (3 r , r ) ≤ C , that is every ball of radius r can be covered by at most C balls of radius r . Let X be a metric space. A geodesic bicombing is a map σ : X × X × [0 , → X with the property that for all ( x, y ) ∈ X × X the map σ xy : t σ ( x, y, t ) isa (constant speed) geodesic from x to y , i.e. d ( σ xy ( t ) , σ xy ( t ′ )) = | t − t ′ | d ( x, y ) for all t, t ′ ∈ [0 , and σ xy (0) = x, σ xy (1) = y . When X is equipped with a geodesic bicombing then for all x, y ∈ X we willdenote by [ x, y ] the geodesic σ xy parametrized by arc-length. A geodesic bicombing is:• convex if the map t d ( σ xy ( t ) , σ x ′ y ′ ( t )) is convex on [0 , for all x, y, x ′ , y ′ ∈ X ;• consistent if for all x, y ∈ X , for all ≤ s ≤ t ≤ and for all λ ∈ [0 , it holds σ pq ( λ ) = σ xy ((1 − λ ) s + λt ) , where p := σ xy ( s ) and q := σ xy ( t ) ;14 reversible if σ xy ( t ) = σ yx (1 − t ) for all t ∈ [0 , .For instance every convex metric space in the sense of Busemann (so alsoany CAT (0) metric space) admits a unique convex, consistent, reversiblegeodesic bicombing.Given a geodesic bicombing σ we say that a geodesic (segment, ray, line) γ is a σ -geodesic (segment, ray, line) if for all x, y ∈ γ we have that [ x, y ] coincides with the subsegment of γ between x and y .A geodesic bicombing is geodesically complete if every σ -geodesic segmentis contained in a σ -geodesic line. A couple ( X, σ ) is said a GCB-space if σ is a convex, consistent, reversible, geodesically complete geodesic bicombingon the complete metric space X . The packing condition has a controlledbehaviour in GCB-spaces. Proposition 2.1 (Proposition 3.2 of [CS20a]) . Let ( X, σ ) be a GCB -spacethat is P -packed at scale r . Then:(i) for all r ≤ r , the space X is P -packed at scale r and is proper;(ii) for every < r ≤ R and every x ∈ X it holds: Pack ( R, r ) ≤ P (1 + P ) Rr − , if r ≤ r ; Pack ( R, r ) ≤ P (1 + P ) Rr − , if r > r . The same result can be expressed in terms of the covering function using(2.2).
Corollary 2.2.
Let ( X, σ ) be a GCB -space. Suppose X is C -covered atscale r . Then for every < r ≤ R it holds: Cov ( R, r ) ≤ C (1 + C ) Rr − , if r ≤ r ; Cov ( R, r ) ≤ C (1 + C ) Rr − , if r > r . Basic examples of GCB-spaces that are P -packed at scale r are:i) complete and simply connected Riemannian manifolds with sectionalcurvature pinched between two nonpositive constants κ ′ ≤ κ < ;ii) simply connected M κ -complexes, with κ ≤ , without free faces and bounded geometry (i.e., with valency at most V , size at most S andpositive injectivity radius);iii) complete, geodesically complete, CAT (0) metric spaces X with dimen-sion at most n and volume of balls of radius R bounded above by V . 15or further details on the second and the third class of examples we refer to[CS20b].When ( X, σ ) is a proper GCB-space we can consider the space of parametrizedgeodesic lines of X , Geod ( X ) = { γ : R → X isometry } , endowed with the topology of uniform convergence on compact subsets of R ,and its subset Geod σ ( X ) made of elements whose image is a σ -geodesic line.By the continuity of σ we have that Geod σ ( X ) is closed in Geod ( X ) . Thereis a natural action of R on Geod ( X ) defined by reparametrization: Φ t γ ( · ) = γ ( · + t ) for every t ∈ R . It is easy to see it is a continuous action, i.e. Φ t ◦ Φ s = Φ t + s for all t, s ∈ R and for every t ∈ R the map Φ t is a homeomorphism ofGeod ( X ) . Moreover the action restricts as an action on Geod σ ( X ) . Thisaction on Geod σ ( X ) is called the σ -geodesic flow on X . The evaluationmap E : Geod ( X ) → X , which is defined as E ( γ ) = γ (0) , is continuous andproper ([BL12], Lemma 1.10), so its restriction to Geod σ ( X ) has the sameproperties. Moreover it is surjective since σ is assumed geodesically complete.The topology on Geod σ ( X ) is metrizable. Indeed we can construct a familyof metrics on Geod σ ( X ) with the following method.Let F be the class of continuous functions f : R → R satisfying(a) f ( s ) > for all s ∈ R ;(b) f ( s ) = f ( − s ) for all s ∈ R ;(c) R + ∞−∞ f ( s ) ds = 1 ;(d) R + ∞−∞ | s | f ( s ) ds = C ( f ) < + ∞ .For every f ∈ F we define the distance on Geod σ ( X ) : f ( γ, γ ′ ) = Z + ∞−∞ d ( γ ( s ) , γ ′ ( s )) f ( s ) ds. (2.3)We remark that the choice of f = e | s | gives exactly the distance d Geod . Lemma 2.3.
The expression defined in (2.3) satisfies these properties:(i) it is a well defined distance on
Geod σ ( X ) ;(ii) for all γ, γ ′ ∈ Geod σ ( X ) it holds f ( γ, γ ′ ) ≤ d ( γ (0) , γ (0)) + C ( f ) ;(iii) for all γ, γ ′ ∈ Geod σ ( X ) it holds d ( γ (0) , γ ′ (0)) ≤ f ( γ, γ ′ ) ;(iv) it induces the topology of Geod σ ( X ) . roof. For all γ, γ ′ ∈ Geod σ ( X ) we have d ( γ ( s ) , γ ′ ( s )) ≤ d ( γ ( s ) , γ (0)) + d ( γ (0) , γ ′ (0)) + d ( γ ′ (0) , γ ′ ( s )) ≤ | s | + d ( γ (0) , γ ′ (0)) , so Z + ∞−∞ d ( γ ( s ) , γ ′ ( s )) f ( s ) ds ≤ d ( γ (0) , γ ′ (0)) + Z + ∞−∞ | s | f ( s ) dt < + ∞ . This shows (ii) and that the integral in (2.3) is finite. From the properties ofthe integral and the positiveness of f it is easy to prove that f is actually adistance. The proof of (iii) follows from the convexity of σ and the symmetryof f . Indeed for all γ, γ ′ ∈ Geod σ ( X ) the function g ( s ) = d ( γ ( s ) , γ ′ ( s )) isconvex. This means that for all S, S ′ ∈ R and for all λ ∈ [0 , it holds g ( λS + (1 − λ ) S ′ ) ≤ λg ( S ) + (1 − λ ) g ( S ′ ) . We take s ≥ and we use the inequality above with S = s, S ′ = − s and λ = , obtaining d ( γ (0) , γ ′ (0)) = g (0) ≤ g ( − s ) + 12 g ( s ) = d ( γ ( s ) , γ ′ ( s )) + d ( γ ( − s ) , γ ′ ( − s ))2 . We can now estimate the distance between γ and γ ′ as f ( γ, γ ′ ) = Z −∞ d ( γ ( s ) , γ ′ ( s )) f ( s ) ds + Z + ∞ d ( γ ( s ) , γ ′ ( s )) f ( s ) ds = Z + ∞ (cid:0) d ( γ ( − s ) , γ ′ ( − s )) + d ( γ ( s ) , γ ′ ( s )) (cid:1) f ( s ) ds ≥ d ( γ (0) , γ ′ (0)) , where we used the symmetry of f . This concludes the proof of (iii).If a sequence γ n converges to γ ∞ uniformly on compact subsets then it isclear that for every T ≥ it holds lim n → + ∞ Z + T − T d ( γ n ( s ) , γ ∞ ( s )) f ( s ) ds = 0 . For every ε > we pick T ε ≥ such that R + ∞ T ε | s | f ( s ) < ε . Then it is easyto conclude, using the properties of f , that lim n → + ∞ Z + ∞−∞ d ( γ n ( s ) , γ ∞ ( s )) f ( s ) ds ≤ ε. By the arbitrariness of ε we conclude that the sequence γ n converges to γ ∞ with respect to the metric f .Now suppose the sequence γ n converges to γ ∞ with respect to f and supposeit does not converge uniformly on compact subsets to γ ∞ . Therefore there17xists T ≥ , ε > and a subsequence γ n j such that d ( γ n j ( t j ) , γ ∞ ( t j )) > ε for every j , where t j ∈ [ − T, T ] . We can suppose t j → t ∞ and so d ( γ n j ( t ∞ ) , γ ∞ ( t ∞ )) > ε for every j . For all t ∈ [ t ∞ − ε , t ∞ + ε ] we get d ( γ n j ( t ) , γ ∞ ( t )) > ε . Therefore, if we set m = min t ∈ [ t ∞ − ε ,t ∞ + ε ] f ( s ) > ,we obtain Z + ∞−∞ d ( γ n j ( s ) , γ ∞ ( s )) f ( s ) ds > ε m for every j , which is a contradiction.A metric â on Geod σ ( X ) inducing the topology of uniform convergence oncompact subsets is said geometric if the evaluation map E is Lipschitz withrespect to this metric. Any metric induced by f ∈ F is geometric by Lemma2.3.(iii).Similar definitions can be given for the space of σ -geodesic rays, Ray σ ( X ) ,which isRay σ ( X ) = { ξ : [0 , + ∞ ) → X isometry with image a σ -geodesic ray } , endowed with the topology of uniform convergence on compact subsets of [0 , + ∞ ) . Any f ∈ F defines a distance on Ray σ ( X ) by f ( ξ, ξ ′ ) = Z + ∞ d ( ξ ( s ) , ξ ′ ( s )) f ( s ) ds that induces the topology of Ray σ ( X ) . The evaluation map E : Ray σ ( X ) → X that sends ξ to ξ (0) is again continuous, surjective and proper. A metric â on Ray σ ( X ) inducing its topology is geometric if the evaluation map isLipschitz with respect to â . The reparametrization flow on Ray σ ( X ) isdefined only for positive times and therefore it is a semi-flow called the geodesic semi-flow .The σ -boundary at infinity of X is defined as the set Ray σ ( X ) modulothe equivalence relation ξ ∼ ξ ′ if and only if sup t ∈ [0 , + ∞ ) d ( ξ ( t ) , ξ ′ ( t )) < + ∞ and it is denoted by ∂ σ X . Since X is proper then for every σ -geodesic ray ξ and every point x ∈ X there exists a unique σ -geodesic ray ξ ′ which isequivalent to ξ and with ξ ′ (0) = x . If z ∈ ∂ σ X then the unique σ -geodesicray ξ in the class of z with ξ (0) = x ∈ X is denoted by ξ z = [ x, z ] . Thereexists a topology on X ∪ ∂ σ X that induces the original metric topology on X and the quotient topology on ∂ σ X (as quotient of Ray σ ( X ) ). With thistopology the space X ∪ ∂ σ X is compact. Let X be a geodesic space. Given three points x, y, z ∈ X , the Gromovproduct of y and z with respect to x is defined as ( y, z ) x = 12 (cid:0) d ( x, y ) + d ( x, z ) − d ( y, z ) (cid:1) . X is said δ -hyperbolic if for every four points x, y, z, w ∈ X thefollowing hold: ( x, z ) w ≥ min { ( x, y ) w , ( y, z ) w } − δ (2.4)or, equivalently, d ( x, y ) + d ( z, w ) ≤ max { d ( x, z ) + d ( y, w ) , d ( x, w ) + d ( y, z ) } + 2 δ. (2.5)The space X is Gromov hyperbolic if it is δ -hyperbolic for some δ ≥ .This formulation of δ -hyperbolicity is convenient when interested in takinglimits (since they are preserved under ultralimits). We will also make use ofanother classical characterization of δ -hyperbolicity.A geodesic triangle in X is the union of three geodesic segments [ x, y ] , [ y, z ] , [ z, x ] and is denoted by ∆( x, y, z ) . For every geodesic triangle there exists aunique tripod ∆ with vertices ¯ x, ¯ y, ¯ z such that the lengths of [¯ x, ¯ y ] , [¯ y, ¯ z ] , [¯ z, ¯ x ] equal the lengths of [ x, y ] , [ y, z ] , [ z, x ] respectively. There exists a uniquemap f ∆ from ∆( x, y, z ) to the tripod ∆ that identifies isometrically thecorresponding edges, and there are exactly three points c x ∈ [ y, z ] , c y ∈ [ x, z ] , c z ∈ [ x, y ] such that f ∆ ( c x ) = f ∆ ( c y ) = f ∆ ( c z ) = c , where c is thecenter of the tripod ∆ . By definition of f ∆ it holds: d ( x, c z ) = d ( x, c y ) , d ( y, c x ) = d ( y, c z ) , d ( z, c x ) = d ( z, c y ) . The triangle ∆( x, y, z ) is called δ -thin if for every u, v ∈ ∆( x, y, z ) suchthat f ∆ ( u ) = f ∆ ( v ) it holds d ( u, v ) ≤ δ ; in particular the mutual distancesbetween c x , c y and c z are at most δ . It is well-known that every geodesictriangle in a geodesic δ -hyperbolic metric space (as defined above) is δ -thin.Moreover, the last condition is equivalent to the above definition of hy-perbolicity, up to slightly increasing the hyperbolicity constant δ in (2.4). Let X be a proper, δ -hyperbolic metric space x be a point of X .The Gromov boundary of X is defined as the quotient ∂ G X = { ( z n ) n ∈ N ⊆ X | lim n,m → + ∞ ( z n , z m ) x = + ∞} / ≈ , where ( z n ) n ∈ N is a sequence of points in X and ≈ is the equivalence relationdefined by ( z n ) n ∈ N ≈ ( z ′ n ) n ∈ N if and only if lim n,m → + ∞ ( z n , z ′ m ) x = + ∞ .We will write z = [( z n )] ∈ ∂ G 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 ∪ ∂ G X that extends the metric topologyof X . The Gromov product can be extended to points z, z ′ ∈ ∂ G X by ( z, z ′ ) x = sup ( z n ) , ( z ′ n ) lim inf n,m → + ∞ ( z n , z ′ m ) x ( z n ) ≈ z and ( z ′ n ) ≈ z ′ . For every z, z ′ , z ′′ ∈ ∂ G X it continues to hold ( z, z ′ ) x ≥ min { ( z, z ′′ ) x , ( z ′ , z ′′ ) x } − δ. (2.6)Moreover for all sequences ( z n ) , ( z ′ n ) converging to z, z ′ respectively it holds ( z, z ′ ) x − δ ≤ lim inf n,m → + ∞ ( z n , z ′ m ) x ≤ ( z, z ′ ) x . (2.7)The Gromov product between a point y ∈ X and a point z ∈ ∂ G X is definedin a similar way and it satisfies a condition analogue of (2.7).Every geodesic ray ξ defines a point ξ + = [( ξ ( n )) n ∈ N ] of the Gromov bound-ary ∂ G X : we say that ξ joins ξ (0) = y to ξ + = z , and we denote it by [ y, z ] .Moreover for every z ∈ ∂ G X and every x ∈ X it is possible to find a geodesicray ξ such that ξ (0) = x and ξ + = z . Indeed if ( z n ) is a sequence of pointsconverging to z then, by properness of X , the sequence of geodesics [ x, z n ] converges to a geodesic ray ξ which has the properties above (cp. LemmaIII.3.13 of [BH13]). We denote this geodesic ray as ξ z = [ x, z ] even if it ispossibly not unique.If X admits also a geodesic bicombing σ such that ( X, σ ) is a GCB-spacethen the construction above gives a unique σ -geodesic ray ξ z for all z ∈ ∂ G X .This fact defines a natural identification between ∂ σ X and ∂ G X : indeed, abase point x ∈ X being fixed, for every σ -geodesic ray ξ starting at x onedefines a point in the Gromov boundary as ξ + = [( ξ ( n ) n ∈ N ] . This formulaprovides a well defined homeomorphism between ∂ σ X and ∂ G X .The Busemann function associated to z ∈ ∂ G X with basepoint x is the map B xz : X → R , y lim T → + ∞ ( d ( ξ z ( T ) , y ) − T ) , where ξ z denotes a geodesic ray [ x, z ] . It depends on the choice of thegeodesic ray [ x, z ] but two maps obtained taking two different geodesic raysare at bounded distance and the bound depends only on δ . Every Busemannfunction is -Lipschitz.Here we recall two basic properties of Gromov-hyperbolic metric spaces. Lemma 2.4 (Projection Lemma, cp. Lemma 3.2.7 of [CDP90]) . Let X bea δ -hyperbolic metric space and let x, y, z ∈ X . For every geodesic segment [ x, y ] we have ( y, z ) x ≥ d ( x, [ y, z ]) − δ. We recall that a (1 , ν ) -quasigeodesic is a curve α : I → X such that | t − t ′ | − ν ≤ d ( α ( t ) , α ( t ′ )) ≤ | t − t ′ | + ν for all t, t ′ belonging to the interval I . As an immediate consequence of theprevious lemma and Proposition 2.7 of [CS20a] we get:20 emma 2.5. Let X be a δ -hyperbolic metric space, x ∈ X and ξ be a geodesicray such that ξ (0) is a projection of x on ξ . Then(i) for all T ≥ the curve α = [ x, ξ (0)] ∪ [ ξ (0) , ξ ( T )] is a (1 , δ ) -quasigeodesicand d ( α ( t ) , γ ( t )) ≤ δ for all possibles t , where γ = [ x, ξ ( T )] ;(ii) the curve α = [ x, ξ (0)] ∪ [ ξ (0) , ξ + ] is a (1 , δ ) -quasigeodesic and d ( α ( t ) , ξ ′ ( t )) ≤ δ for all t ≥ , where ξ ′ = [ x, ξ + ] ; The quasiconvex hull of a subset C of ∂ G X is the union of all the geodesiclines joining two points of C and it is denoted by QC-Hull ( C ) . If ( X, σ ) isa proper, δ -hyperbolic GCB-space then for any two points z, z ′ ∈ ∂ G X = ∂ σ X = ∂X there exists a σ -geodesic line joining z and z ′ . Given C ⊆ ∂X wedenote by QC-Hull σ ( C ) the union of all σ -geodesic lines joining two pointsof C . Lemma 2.6.
Let ( X, σ ) be a proper, δ -hyperbolic GCB -space. Then(i) every two geodesic lines γ, γ ′ with same endpoints at infinity are atdistance at most δ ;(ii) for every C ⊆ ∂X and for every x ∈ QC-Hull ( C ) there exists x ′ ∈ QC-Hull σ ( C ) with d ( x, x ′ ) ≤ δ .Proof. The first statement is classic, see for instance Lemma 3.1 of [CS20a].The second one follows from the fact that when X is proper then for all z, z ′ ∈ ∂X there exists a σ -geodesic line joining them. When X is a proper, δ -hyperbolic metric space it is known that the bound-ary ∂ G X is metrizable. A metric D x,a on ∂ G X is called a visual metric ofparameter a ∈ (cid:16) , δ · log e (cid:17) and center x ∈ X if there exists V > such thatfor all z, z ′ ∈ ∂ G X it holds V e − a ( z,z ′ ) x ≤ D x,a ( z, z ′ ) ≤ V e − a ( z,z ′ ) x . (2.8)A visual metric is said standard if for all z, z ′ ∈ ∂ G X it holds (3 − e aδ ) e − a ( z,z ′ ) x ≤ D x,a ( z, z ′ ) ≤ e − a ( z,z ′ ) x . For all a as before and x ∈ X there exists always a standard visual metricof parameter a and center x , see [Pau96]. We remark that the constantsinvolved in the definition of a standard visual metric depend only on δ . Asin [Pau96] we define the generalized visual ball of center z ∈ ∂ G X and radius ρ ≥ as B ( z, ρ ) = (cid:26) z ′ ∈ ∂ G X s.t. ( z, z ′ ) x > log 1 ρ (cid:27) . It is comparable to the metric balls of the visual metrics on ∂ G X .21 emma 2.7. Let D x,a be a visual distance of center x and parameter a on ∂ G X . Then for all z ∈ ∂ G X and for all ρ > it holds B D x,a (cid:18) z, V ρ a (cid:19) ⊆ B ( z, ρ ) ⊆ B D x,a ( z, V ρ a ) . Proof.
For all z ′ ∈ B ( z, ρ ) by definition it holds ( z, z ′ ) x > log ρ and therefore D x,a ( z, z ′ ) ≤ V e − a ( z,z ′ ) x < V ρ a . If z ′ ∈ B D x,a ( z, V ρ a ) then V e − a ( z,z ′ ) x ≤ D x ,a ( z, z ′ ) < V ρ a . This easilyimplies z ′ ∈ B ( z, ρ ) .A compact metric space Z is ( A, s ) -Ahlfors regular if there exists a prob-ability measure µ on Z such that A ρ s ≤ µ ( B ( z, ρ )) ≤ Aρ s for all z ∈ Z and all ≤ ρ ≤ Diam ( Z ) , where Diam ( Z ) is the diameter of Z . In case Z = ∂ G X we say that Z is visual ( A, s ) -Ahlfors regular if thereexists a probability measure µ on ∂ G X such that A ρ s ≤ µ ( B ( z, ρ )) ≤ Aρ s for all z ∈ Z and all ≤ ρ ≤ , where B ( z, ρ ) is the generalized visual ball ofcenter z and radius ρ . From Lemma 2.7 it follows immediately the following. Lemma 2.8. If ∂ G X is ( A, s ) -Ahlfors regular with respect to a visual metricof center x and parameter a , then it is visual ( AV s , as ) -Ahlfors regular, where V is the constant of (2.8) . If X is a proper metric space we denote by Isom ( X ) its group of isometries,endowed with the uniform convergence on compact subsets of X . A subgroup Γ of Isom ( X ) is called discrete if the following equivalent conditions (see[BCGS17]) hold:(a) Γ is discrete as a subspace of Isom ( X ) ;(b) ∀ x ∈ X and R ≥ the set Σ R ( x ) = { g ∈ Γ s.t. gx ∈ B ( x, R ) } is finite.In particular taking R = 0 we observe that the stabilizer of any point isfinite. Moreover the pointwise systole is strictly positive: for all x ∈ X thereexists ε x > such that d ( x, gx ) < ε x implies gx = x . The supremum of such ε x ’s is denoted by sys ⋄ (Γ , x ) and it is called the free-systole of Γ at x .When X is a proper and δ -hyperbolic metric space then every isometryof X acts naturally on ∂ G X and the resulting map on X ∪ ∂ G X is a home-omorphism. The limit set Λ(Γ) of a discrete group of isometries Γ is the set22f accumulation points of the orbit Γ x on ∂ G X , where x is any point of X ; itis the smallest Γ -invariant closed set of the Gromov boundary (cp. [Coo93],Theorem 5.1). The group Γ is called elementary if ≤ .The set Λ(Γ) is Γ -invariant so it is its quasiconvex hull. A discrete groupof isometries Γ is called quasiconvex-cocompact if its action on QC-Hull (Λ(Γ)) is cocompact, i.e. if there exists D ≥ such that for all x, y ∈ QC-Hull (Λ(Γ)) it holds d ( gx, y ) ≤ D for some g ∈ Γ . The smallest D satisfying this propertyis called the codiameter of Γ . A quasiconvex-cocompact group Γ is said cocompact if Λ(Γ) = ∂X .The free-systole of the group Γ is the quantitysys ⋄ (Γ , X ) = inf x ∈ X sys ⋄ (Γ , x ) . If ( X, σ ) is a GCB-space then we say that an isometry g of X is a σ -isometry iffor all x, y ∈ X it holds σ g ( x ) g ( y ) = g ( σ xy ) . We say that a group of isometriesof X is a group of σ -isometries if every element of the group is a σ -isometry.Example 5.8 of [CS20a] shows that if ( X, σ ) is a δ -hyperbolic GCB-spacewhich is P -packed at scale r and if Γ is a discrete, non-elementary, quasiconvex-cocompact group of σ -isometries of X with codiameter ≤ D thensys ⋄ (Γ , X ) ≥ s ( P , r , δ, D ) , (2.9)where s ( P , r , δ, D ) depends only on P , r , δ and D .The radial limit set of a discrete group of isometries Γ is defined as follows.Once fixed x ∈ X a point z ∈ ∂ G X is said τ -radial if there exists a sequence { g n } of elements of Γ such that { g n x } is unbounded and d ( g n x, [ x, z ]) ≤ τ for some geodesic ray [ x, z ] and for all n ∈ N . We denote the set of τ -radialpoints by Λ r ,τ (Γ) and the set of radial points by Λ r (Γ) = [ τ ≥ Λ r ,τ (Γ) . The set Λ r (Γ) is Γ -invariant, so its closure is Λ(Γ) . In this section we will introduce the first two types of entropy: the coveringentropy, defined in terms of the covering functions, and the volume entropyof a measure.
Let ( X, σ ) be a GCB-space that is P -packed at scale r . It is natural todefine the upper covering entropy of X as the number h Cov ( X ) = lim sup T → + ∞ T log Cov ( B ( x, T ) , r ) , x is any point of X . The lower covering entropy is defined takingthe limit inferior instead of the limit superior and it is denoted by h Cov ( X ) .By triangular inequality it is easy to show that the definitions of upper andlower covering entropy do not depend on the point x ∈ X . In the nextproposition we can see that they do not depend on r too and moreover wecan replace the covering function with the packing function. Proposition 3.1.
Let ( X, σ ) be a GCB -space that is P -packed at scale r and let x ∈ X . Then T log Cov ( B ( x, T ) , r ) ≍ P ,r ,r,r ′ T log Pack ( B ( x, T ) , r ′ ) for all r, r ′ > . In particular any of these functions can be used in thedefinition of the upper and lower covering entropy.Proof. For all < r ≤ r ′ and x ∈ X clearly Cov ( B ( x, T ) , r ) ≥ Cov ( B ( x, T ) , r ′ ) and Cov ( B ( x, T ) , r ) ≤ Cov ( B ( x, T ) , r ′ ) · sup y ∈ X Cov ( B ( y, r ′ ) , r ) . By Corol-lary 2.2 we have sup y ∈ X Cov ( B ( y, r ′ ) , r ) = Cov ( r ′ , r ) which is a finite numberdepending only on P , r , r, r ′ . Therefore we obtain T log Cov ( B ( x, T ) , r ) ≍ P ,r ,r,r ′ T log Cov ( B ( x, T ) , r ′ ) . The thesis follows from (2.1).The upper and lower covering entropies can be computed also using thecovering function of the metric spheres.
Proposition 3.2.
Let ( X, σ ) be a GCB -space that is P -packed at scale r and x ∈ X . Then for all r > T log Cov ( B ( x, T ) , r ) ≍ P ,r ,r T log Cov ( S ( x, T ) , r ) Proof.
Clearly it holds Cov ( S ( x, T ) , r ) ≤ Cov ( B ( x, T ) , r ) . The other esti-mate is more involved. We divide the ball B ( x, T ) in annulii A ( x, kr, ( k +1) r ) with k = 0 , . . . , Tr − . We easily obtainCov ( B ( x, T ) , r ) ≤ Tr − X k =0 Cov ( A ( x, kr, ( k + 1) r ) , r ) . Now we claim that for any k it holdsCov ( A ( x, kr, ( k + 1) r ) , r ) ≤ Cov ( S ( x, T ) , r ) . Indeed let { y , . . . , y N } be a set of points realizing Cov ( S ( x, T ) , r ) . Forall i = 1 , . . . , N we consider the σ -geodesic segment γ i = [ x, y i ] and we24all x i the point along this geodesic segment at distance kr from x . Then x i ∈ A ( x, kr, ( k + 1) r ) for every i = 1 , . . . , N . We claim that { x , . . . , x N } isa r -dense subset of A ( x, kr, ( k + 1) r ) . We take any y ∈ A ( x, kr, ( k + 1) r ) and we consider the σ -geodesic segment [ x, y ] . We extend this geodesic toa σ -geodesic segment γ = [ x, y ′ ] , where y ′ is at distance T from x . Thenthere exists i such that d ( y ′ , y i ) = d ( γ ( T ) , γ i ( T )) ≤ r . By convexity of σ we have d ( γ ( t ) , γ i ( t )) ≤ r , where t = d ( x, y ) . Therefore we concludethat d ( y, x i ) ≤ d ( y, γ i ( t )) + d ( γ i ( t ) , x i ) ≤ r. This ends the proof of theclaim, so Cov ( B ( x, T ) , r ) ≤ Rr Cov ( S ( x, T ) , r ) . The thesis follows from theseestimates and Proposition 3.1.Combining Lemma 2.1 and Proposition 3.1 we can find an uniform upperbound to the covering entropy.
Lemma 3.3.
Let ( X, σ ) be a GCB -space that is P -packed at scale r . Then h Cov ( X ) ≤ log(1 + P ) r . Proof.
For every x ∈ X it holds Pack ( B ( x, R ) , r ) ≤ P (1 + P ) Rr − . Thethesis follows immediately.
Let ( X, σ ) be a GCB-space that is P -packed at scale r . The upper volumeentropy of a measure µ on X is defined as h µ ( X ) = lim sup T → + ∞ T log µ ( B ( x, T )) , while the lower volume entropy h µ ( X ) is defined taking the limit inferior.These definitions do not depend on the choice of the point x ∈ X .A measure µ on X is called H -homogeneous at scale r if H ≤ µ ( B ( x, r )) ≤ H for all x ∈ X . We remark that the condition must hold only at scale r . Proposition 3.4.
Let ( X, σ ) be a GCB -space that is P -packed at scale r and let µ be a measure on X which is H -homogeneous at scale r . Then T log µ ( B ( x, T )) ≍ P ,r ,H,r T log Cov ( B ( x, T ) , r ) . In particular the upper (resp. lower) volume entropy of µ coincides with theupper (resp. lower) covering entropy of X . roof. For all x ∈ X it holds µ ( B ( x, T )) ≤ H · Cov ( B ( x, T ) , r ) and µ ( B ( x, T )) ≥ H · Pack ( B ( x, T − r ) , r ) . By Proposition 3.1 and since T − rT ≍ r we have the thesis. Remark 3.5.
The proof of the proposition shows another fact: if a measureis H -homogeneous at scale r then it is H ( r ′ ) -homogeneous at scale r ′ for all r ′ ≥ r and H ( r ′ ) depends just on H, P , r , r and r ′ . We provide here two examples of homogeneous measures. If X is a com-plete, geodesically complete, CAT (0) metric space that is P -packed at scale r then the natural measure on X satisfies c ≤ µ X ( B ( x, r )) ≤ C for all x ∈ X , where c and C are constants depending only on P and r (Theorem 4.9 of [CS20b]). It follows immediately the following result. Corollary 3.6.
Let X be a complete, geodesically complete, CAT (0) metricspace. If it is P -packed at scale r for some P and r then h Cov ( X ) = h µ X ( X ) . The same holds for the lower entropies.
The second example is the counting measure of a cocompact group ofisometries on a δ -hyperbolic metric space. Corollary 3.7.
Let ( X, σ ) be a δ -hyperbolic GCB -space and Γ be a discrete,non-elementary, cocompact group of σ -isometries of X . Then for all x ∈ X it holds h Cov ( X ) = h µ Γ x ( X ) , where µ Γ x is the counting measure of the orbit Γ · x . The same holds for thelower entropies. We will see in Section 7 that in this case the upper and lower entropiescoincide.
Proof.
By the cocompactness assumption we know that X is complete and P -packed at scale r for some P , r . Then by (2.9) we have sys ⋄ (Γ , X ) ≥ s ( P , r , δ, D ) , where D is an upper bound on the codiameter of the actionand δ is the Gromov-hyperbolicity constant. Then for all y ∈ X it holds ≤ µ Γ x ( B ( y, D )) ≤ Pack (cid:16)
D, s (cid:17) . This shows, by Proposition 2.1, that µ Γ x is H ( P , r , δ, D ) -homogeneous atscale D . The thesis follows from Proposition 3.4.26 Lipschitz-topological entropy
Let ( X, σ ) be a GCB-space that is P -packed at scale r . The space Geod σ ( X ) is locally compact but not compact. The upper topological entropy of thegeodesic flow is defined (see [Bow73], [HKR95]) as h top ( Geod σ ( X )) = inf â sup K lim r → lim sup T → + ∞ T log Cov â T ( K, r ) , where the infimum is taken among all metrics â inducing the topology ofGeod σ ( X ) , the supremum is taken among all compact subsets of Geod σ ( X ) and Cov â T ( K, r ) is the covering function of the compact subset K at scale r with respect to the metric â T defined by â T ( γ, γ ′ ) = max t ∈ [0 ,T ] â (Φ t γ, Φ t γ ′ ) . By the variational principle this quantity equals the measure-theoretic en-tropy defined as the supremum of the entropies of the flow-invariant probabil-ity measures on Geod σ ( X ) (cp. [HKR95], Lemma 1.5). An easy computationshows that the upper topological entropy is always zero. Lemma 4.1.
There are no flow-invariant probability measures on
Geod σ ( X ) .In particular the upper topological entropy of the geodesic flow is .Proof. Suppose there is a flow-invariant probability measure µ on Geod σ ( X ) .For x ∈ X we define A R = { γ ∈ Geod σ ( X ) s.t. γ (0) ∈ B ( x, R ) } , for every R ≥ . Clearly there exists R ≥ such that µ ( A R ) > . By flow-invarianceof µ we have that the set Φ − R +1 ( A R ) = { γ ∈ Geod σ ( X ) s.t. γ (2 R + 1) ∈ B ( x, R ) } has measure > . This implies that µ ( A R ∩ Φ − R +1 ( A R )) > , but thisintersection is empty.Looking at the proof of the variational principle given in [HKR95] wecan observe that the sequence of metrics on Geod σ ( X ) that approach theinfimum in the definition of the upper topological entropy are the restrictionto Geod σ ( X ) of metrics defined on its one-point compactification. Thesemetrics are not the natural ones on Geod σ ( X ) , since they are not geometric.We propose a more appropriate definition of topological entropy for properGCB-spaces.We define the upper Lipschitz-topological entropy of Geod σ ( X ) as h Lip-top ( Geod σ ( X )) = inf â sup K lim r → lim sup T → + ∞ T log Cov â T ( K, r ) , where the infimum is now taken only among all geometric metrics on Geod σ ( X ) .The lower Lipschitz-topological entropy is defined by taking the limit inferiorinstead of the limit superior and it is denoted by h Lip-top ( Geod σ ( X )) . Themain result of this section is the following.27 heorem 4.2. Let ( X, σ ) be a GCB -space that is P -packed at scale r .Then h Lip-top ( Geod σ ( X )) = h Cov ( X ) . The same holds for the lower entropies.
One of the two inequalities is easy. In order to prove the other one we willshow that for the distances induced by the functions f ∈ F the definition oftopological entropy can be heavily simplified. f ∈ F For a metric f ∈ F we denote by h f the upper metric entropy of the σ -geodesic flow with respect to f , that is h f ( Geod σ ( X )) = sup K lim r → lim sup T → + ∞ T log Cov f T ( K, r ) . In the same way is defined the lower metric entropy with respect to f , h f ( Geod ( X )) . For a subset Y of X we denote by Geod σ ( Y ) the set of σ -geodesic lines of X passing through Y at time . Proposition 4.3.
Let ( X, σ ) be a GCB -space that is P -packed at scale r and let f ∈ F . Then(i) for all x, y ∈ X it holds h f ( Geod σ ( x )) = h f ( Geod σ ( y )); (ii) for all x ∈ X and R ≥ it holds h f ( Geod σ ( B ( x, R ))) = h f ( Geod σ ( x )); (iii) for all x ∈ X it holds h f ( Geod σ ( X )) = h f ( Geod σ ( x )) ≤ h Cov ( X ) ;(iv) for all x ∈ X the function r lim sup T → + ∞ T log Cov f T ( Geod σ ( x ) , r ) is constant.The same conclusions hold for the lower Lipschitz-topological entropy. The proposition is a consequence of the following key lemma.
Lemma 4.4 (Key Lemma) . Let f ∈ F , γ ∈ Geod σ ( X ) and < r ≤ r ′ .Then T log Cov f T ( B f T ( γ, r ′ ) , r ) ≍ P ,r ,r,r ′ ,f , where B f T ( γ, r ′ ) is the closed ball of center γ and radius r ′ with respect tothe metric f T . As a consequence the convergence is uniform in γ .Proof. Let
P > depending only on f and r ′ such that Z − P −∞ | u | f ( u ) du + Z + ∞ P | u | f ( u ) du < r .
28e fix ε > and T ≥ Pε . Let E T = { x , . . . , x N } be a maximal r -separatedsubset of B ( γ ( T ) , r ′ + εT ) , so it is also r -dense, and { y , . . . , y M } be a r -dense subset of B ( γ ( − P ) , r ′ + 2 P ) . For every i = 1 , . . . , M and j = 1 , . . . , N we take a σ -geodesic line γ ij extending the σ -geodesic segment [ y i , x j ] . Weparametrize γ ij in such a way that γ ij ( − P ) = y i . The claim is that { γ ij } i,j is a r -dense subset of B f T ( γ, r ′ ) with respect to the metric f T . We fix γ ′ ∈ B f T ( γ, r ′ ) . This means max t ∈ [0 ,T ] f t ( γ ′ , γ ) = max t ∈ [0 ,T ] f (Φ t ( γ ′ ) , Φ t ( γ )) ≤ r ′ . In particular for all t ∈ [0 , T ] we get d ( γ ′ ( t ) , γ ( t )) ≤ r ′ , since d ( γ ′ ( t ) , γ ( t )) = d (Φ t ( γ ′ ) , Φ t ( γ )) ≤ f (Φ t ( γ ′ ) , Φ t ( γ )) ≤ r ′ . Therefore d ( γ ′ ( − P ) , γ ( − P )) ≤ r ′ + 2 P. Moreover d ( γ ′ ( T + εT ) , γ ( T )) ≤ d ( γ ′ ( T + εT ) , γ ′ ( T )) + d ( γ ′ ( T ) , γ ( T )) ≤ εT + r ′ . Thus there exists x j such that d ( x j , γ ′ ( T + εT )) ≤ r and y i such that d ( y i , γ ′ ( − P )) ≤ r . We have d ( γ ij ( − P ) , γ ′ ( − P )) ≤ r , so if we denote with t j the time such that γ ij ( t j ) = x j it holds | t j − ( T + εT ) | ≤ r . Then d ( γ ij ( T + εT ) , γ ′ ( T + εT )) ≤ d ( γ ij ( T + εT ) , γ ij ( t j )) + d ( γ ij ( t j ) , γ ′ ( T + εT )) ≤ r r < r . From the convexity of σ we have d ( γ ′ ( u ) , γ ij ( u )) < r for all u ∈ [ − P, (1+ ε ) T ] .For t ∈ [0 , T ] we have f t ( γ ′ , γ ij ) = Z + ∞−∞ d ( γ ′ ( u ) , γ ij ( u )) f ( u − t ) du ≤ Z − P −∞ (cid:18) r | u + P | (cid:19) f ( u − t ) du ++ Z (1+ ε ) T − P r f ( u − t ) du ++ Z + ∞ (1+ ε ) T (cid:18) r | u − (1 + ε ) T | (cid:19) f ( u − t ) du. The first term can be estimated as follows Z − P −∞ (cid:18) r | u + P | (cid:19) f ( u − t ) du ≤ r Z − P − t −∞ | v + t + P | f ( v ) dv ≤ r Z − P −∞ | v | f ( v ) dv. r . The third term can be controlledin this way: Z + ∞ (1+ ε ) T (cid:18) r | u − (1 + ε ) T | (cid:19) f ( u − t ) du ≤ r Z + ∞ (1+ ε ) T − t | v − (1 + ε ) T + t | f ( v ) dv ≤ r Z + ∞ (1+ ε ) T − t | v | f ( v ) dv ≤ r Z + ∞ P | v | f ( v ) dv. The last inequality follows from T ≥ Pε . Therefore f t ( γ ′ , γ ij ) ≤ r r r Z − P −∞ | v | f ( v ) dv + Z + ∞ P | v | f ( v ) dv ≤ r. We conclude thatCov f T ( B f T ( γ, r ′ ) , r ) ≤ Cov (cid:18) r ′ + 2 P, r (cid:19) · E T . From Proposition 2.1, if ρ = min (cid:8) r , r (cid:9) , we get E T ≤ P (1 + P ) r ′ + εTρ − . Thus T log Cov f T ( B f T ( γ, r ′ ) , r ) ≤ T K ( P , r , r, r ′ , f ) · εTρ log(1 + P )= ε · K ′ ( P , r , r, r ′ , f ) . Here
K, K ′ are constants depending only on P , r , r, r ′ , f and not on ε or γ .So from the arbitrariness of ε we achieve the proof.The computation of h f requires to consider the supremum among allcompact subsets of Geod σ ( X ) . We notice that given a compact subset K ⊆ Geod σ ( X ) then the set E ( K ) is compact since E is continuous. In particularit is bounded, hence contained in a ball B ( x, R ) centered at a referencepoint x ∈ X . We observe also that the set Geod σ ( B ( x, R )) is compact sincethe evaluation map E is proper. We conclude that any compact subset ofGeod σ ( X ) is contained in a compact subset of the form Geod σ ( B ( x, R )) andtherefore in order to compute h f it is enough to take the supremum amongthese sets. The main consequence of Lemma 4.4 is the following result, whichis the key ingredient in the proof of Proposition 4.3. Corollary 4.5.
Let f ∈ F , x ∈ X , R ≥ and < r ≤ r ′ . Then T log Cov f T ( Geod σ ( B ( x, R )) , r ) ≍ P ,r ,r,r ′ ,f T log Cov f T ( Geod σ ( B ( x, R )) , r ′ ) . roof. The quantity T log Cov f T ( Geod σ ( B ( x, R )) , r ) is ≤ T log Cov f T ( Geod σ ( B ( x, R )) , r ′ ) · sup γ ∈ Geod σ ( X ) Cov f T ( B f T ( γ, r ′ ) , r )= 1 T (cid:0) log Cov f T ( Geod σ ( B ( x, R )) , r ′ ) + log sup γ ∈ X Cov f T ( B f T ( γ, r ′ ) , r ) (cid:1) The conclusion follows by Lemma 4.4.
Proof of Proposition 4.3.(ii).
Let ε > and T > Rε . Let γ , . . . , γ N bea r -dense subset of Geod σ ( x ) with respect to the metric f (2+ ε ) T . Theclaim is that { γ i } is a K -dense subset of Geod σ ( B ( x, R )) with respect to f T , where K depends only on r, R and f . We consider a σ -geodesic line γ ∈ Geod σ ( B ( x, R )) . Then there exists a σ -geodesic line γ ′ ∈ Geod σ ( x ) extending the σ -geodesic segment [ x, γ ((1 + ε ) T )] . We call t γ ′ the time suchthat γ ′ ( t γ ′ ) = γ ((1 + ε ) T ) . Then t γ ′ = d ( x, γ ((1 + ε ) T )) ≤ d ( x, γ (0)) + d ( γ (0) , γ ((1 + ε ) T )) ≤ R + (1 + ε ) T ≤ (1 + 2 ε ) T since T ≥ Rε . Moreover | t γ ′ − (1 + ε ) T | ≤ R. We know there exists γ i suchthat max t ∈ [0 , (1+2 ε ) T ] f (Φ t γ ′ , Φ t γ i ) ≤ r. In particular d ( γ ′ ( t γ ′ ) , γ i ( t γ ′ )) ≤ r .Then d ( γ ((1 + ε ) T ) , γ i ( t γ ′ )) ≤ r and in conclusion d ( γ ((1+ ε ) T ) , γ i ((1+ ε ) T )) ≤ d ( γ ((1+ ε ) T ) , γ i ( t γ ′ ))+ d ( γ i ( t γ ′ ) , γ i ((1+ ε ) T )) ≤ r + R. From the convexity of σ we have d ( γ ( t ) , γ i ( t )) ≤ R + r for all t ∈ [0 , (1+ ε ) T ] .We have to estimate f t ( γ, γ i ) = R + ∞−∞ d ( γ ( u ) , γ i ( u )) f ( u − t ) du for every t ∈ [0 , T ] . Since d ( γ (0) , γ i (0)) ≤ R and d ( γ ((1 + ε ) T ) , γ i ((1 + ε ) T )) ≤ r + R then Z + ∞−∞ d ( γ ( u ) , γ i ( u )) f ( u − t ) du ≤ Z −∞ ( R + 2 | u | ) f ( u − t ) du ++ Z (1+ ε ) T ( R + r ) f ( u − t ) du ++ Z + ∞ (1+ ε ) T ( R + r + 2 | u − (1 + ε ) T | ) f ( u − t ) du ≤ R + Z − t −∞ | v + t | f ( v ) dv +( R + r )+ Z + ∞ (1+ ε ) T − t ( R + r +2 | v − (1+ ε ) T + t | ) f ( v ) dv. We conclude that the above quantity is less than or equal to R + 2 r + Z −∞ | v | f ( v ) dv + Z + ∞ | v | f ( v ) dv ≤ R + 2 r + C ( f ) = K ( R, r, f ) .
31y the previous corollary h f ( Geod σ ( B ( x, R ))) can be computed as lim sup T → + ∞ T log Cov f T ( Geod σ ( B ( x, R )) , K ) which is ≤ lim sup T → + ∞ T log Cov f (1+2 ε ) T ( Geod σ ( x ) , r )= (1 + 2 ε ) lim sup T → + ∞ T log Cov f T ( Geod σ ( x ) , r ) . Since this is true for all ε > then we obtain the thesis. Proof of Proposition 4.3.(i).
We have y ∈ B ( x, R ) , where R = d ( x, y ) , soGeod σ ( y ) ⊆ Geod σ ( B ( x, R )) . Therefore h f ( Geod σ ( y )) ≤ h f ( Geod σ ( B ( x, R ))) = h f ( Geod σ ( x )) . The other inequality can be proved in the same way.Finally we achieve the proof of the remaining parts of Proposition . . Proof of Proposition 4.3.(iii) & (iv).
The equality in (iii) follows directly from(ii), so h f ( Geod σ ( X )) = lim sup T → + ∞ T log Cov f T ( Geod σ ( x ) , r ) , where x is a point of X . We fix T > and we consider a r -separatedsubset E T of S ( x , T ) of maximal cardinality, which is also r -dense. For all y ∈ E T we consider a σ -geodesic line γ y extending the σ -geodesic segment [ x , y ] such that γ y (0) = x and γ y ( T ) = y . We claim that { γ y } y ∈ E T isa ( r + C ( f )) -dense subset of Geod σ ( x ) with respect to f T . We take a σ -geodesic line γ ∈ Geod σ ( x ) . Then there exists y ∈ E T such that d ( γ ( T ) , y ) = d ( γ ( T ) , γ y ( T )) ≤ r . From the convexity of σ it holds d ( γ ( u ) , γ y ( u )) ≤ r forall u ∈ [0 , T ] . Moreover d ( γ ( u ) , γ y ( u )) ≤ r + 2 | u − T | for all u ∈ [ T, + ∞ ) and d ( γ ( u ) , γ y ( u )) ≤ | u | for all u ∈ ( −∞ , . Then for all t ∈ [0 , T ] we get f t ( γ, γ y ) = Z + ∞−∞ d ( γ ( u ) , γ y ( u )) f ( u − t ) du ≤ Z −∞ | u | f ( u − t ) du + Z T r f ( u − t ) du ++ Z + ∞ T ( r + 2 | u − T | ) f ( u − t ) du ≤ r + C ( f ) . The last inequality follows from similar estimates given in the proofs ofLemma 4.4. Therefore applying Corollary 4.5 we have lim sup T → + ∞ T log Cov f T ( Geod σ ( x ) , r ) ≤ lim sup T → + ∞ T log Cov ( S ( x, T ) , r ) . R = 0 . We are ready to give the
Proof of Theorem 4.2.
Proposition 4.3.(iii) shows that h Lip-top ( Geod σ ( X )) isless than or equal to h Cov ( X ) .In order to prove the other inequality we fix a geometric metric â on Geod σ ( X ) and we denote by M the Lipschitz constant with respect to â of the evalu-ation map E . Then we have sup K lim r → lim sup T → + ∞ T log Cov â T ( K, r ) ≥ lim sup T → + ∞ T log Cov â T ( Geod σ ( x ) , r ) , where x ∈ X. We fix T ≥ and we consider a set γ , . . . , γ N realizingCov â T ( Geod σ ( x ) , r ) . The claim is that γ i ( T ) is a M r -dense subset of S ( x, T ) . Indeed we take a point y ∈ S ( x, T ) and we extend the σ -geodesicsegment [ x, y ] to a σ -geodesic line γ ∈ Geod σ ( x ) . Then there exists γ i suchthat â T ( γ, γ i ) ≤ r . Since the evaluation map is M -Lipschitz we have d ( y, γ i ( T )) = d ( γ ( T ) , γ i ( T )) = d (Φ T γ (0) , Φ T γ i (0)) ≤ L â (Φ T γ, Φ T γ i ) ≤ M r . Therefore lim sup T → + ∞ T log Cov â T ( Geod σ ( x ) , r ) ≥ lim sup T → + ∞ T log Cov ( S ( x, T ) , M r ) 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 h Lip-top ( X ) = lim sup T → + ∞ T log Cov f T ( Geod σ ( x ) , r ) independently of f ∈ F , x ∈ X and r > . Moreover T log Cov f T ( Geod σ ( x ) , r ) ≍ P ,r ,f T log Cov ( B ( x, T ) , r ) by the proofs of Theorem 4.2 and Proposition 4.3 and by Proposition 3.2. .3 Lipschitz-topological entropy of the geodesic semi-flow We consider now the space of σ -geodesic rays Ray σ ( X ) and the correspondinggeodesic semi-flow. The definition of upper and lower Lipschitz-topologicalentropy of the geodesic semi-flow can be given in an analogous way to thecase of the geodesic flow and they are denoted by h Lip-top ( Ray σ ( X )) and h Lip-top ( Ray σ ( X )) respectively. We denote the space of σ -geodesic rays withstarting point belonging to Y ⊆ X as Ray σ ( Y ) . Proposition 4.7.
Let ( X, σ ) be a GCB -space that is P -packed at scale r .Then:(i) h Lip-top ( Ray σ ( X )) equals lim sup T →∞ T log Cov f T ( Ray σ ( x ) , r ) indepen-dently of f ∈ F , the point x ∈ X and r > .(ii) h Lip-top ( Ray σ ( X )) = h Lip-top ( Geod σ ( X )) = h Cov ( X ) .The same conclusions hold for the lower topological entropy.Proof. The proof of h Lip-top ( Ray σ ( X )) ≥ h Cov ( X ) is the same given in theproof of Theorem 4.2. On the other hand it is clear thatCov f T ( Ray σ ( B ( x, R )) , r ) ≤ Cov f T ( Geod σ ( B ( x, R )) , r ) , therefore, using Theorem 4.2 and Proposition 4.3, h Cov ( X ) = h Lip-top ( Geod σ ( X )) = lim sup T → + ∞ T log Cov f T ( Geod σ ( B ( x, R )) , r ) ≥ lim sup T → + ∞ T log Cov f T ( Ray σ ( B ( x, R )) , r ) . Since this is true for all R ≥ and for all r > we obtain h f ( Ray σ ( X )) = h Cov ( X ) , which shows (ii).Moreover the number lim sup T → + ∞ T log Cov f T ( Ray σ ( B ( x, R )) , r ) does notdepend on f ∈ F , x ∈ X , r > and R ≥ , proving (i).We remark that we have T log Cov f T ( Geod σ ( x ) , r ) ≍ P ,r ,r,f T log Cov f T ( Ray σ ( x ) , r ) . Let ( X, σ ) be a GCB-space that is P -packed at scale r . In this section wewill show how the covering entropy equals the σ -shadow dimension of theboundary, which is a sort of Minkowski dimension relative to appropriateopen sets of the boundary: the σ -shadows. In the second part we will seethat if X is also Gromov-hyperbolic then there is a precise link between theshadow dimension and the Minkowski dimension of the visual metrics.34 .1 Shadow dimension We fix a point x ∈ X . For all y ∈ X and r ≥ we define the σ -shadow ofradius r casted by y with center x asShad x ( y, r ) = { z ∈ ∂ σ X s.t. [ x, z ] ∩ B ( y, r ) = ∅} . Recall that for z ∈ ∂ σ X and x ∈ X we denote by [ x, z ] the unique σ -geodesicray in the class of z starting from x .We define the r -shadow covering number of ∂ σ X at scale ρ > as theminimum number of σ -shadows of radius r casted by points at distanceat least log ρ from x with center x needed to cover ∂ σ X. It is denoted byShad-Cov r ( ∂ σ X, ρ ) . The upper shadow dimension of ∂ σ X is defined asShad-D ( ∂ σ X ) = lim sup ρ → log Shad-Cov r ( ∂ σ X, ρ )log ρ . Taking the limit inferior instead of the limit superior we define the lowershadow dimension , denoted by Shad-D ( ∂ σ X ) . We can see that if we do thechange of variable ρ = e − T we can writeShad-D ( ∂ σ X ) = lim sup T → + ∞ T log Shad-Cov r ( ∂ σ X, e − T ) . A priori the upper and the lower shadow dimension may depend on r , butwe will see it is not the case. Lemma 5.1.
Let ( X, σ ) be a GCB -space that is P -packed at scale r . Then Cov ( S ( x, T ) , r ) ≤ Shad-Cov r ( ∂ σ X, e − T ) ≤ Cov ( S ( x, T ) , r ) for all x ∈ X , T ≥ and r > .Proof. Let y , . . . , y N be a subset of S ( x, T ) realizing Cov ( S ( x, T ) , r ) . Thenany σ -geodesic ray starting at x passes through a closed ball of radius r andcenter some of the y i and in particular it passes through the open ball ofradius r and center y i . In other words the boundary ∂ σ X is covered by theshadows Shad x ( y i , r ) , showing the right inequality.Now let y , . . . , y N be a set realizing Shad-Cov r ( ∂ σ X, e − T ) . This meansthat d ( x, y i ) ≥ T for all i and that any σ -geodesic ray [ x, z ] passes throughsome open ball of radius r and center y i . First of all it is possible to suppose y i ∈ S ( x, T ) . Indeed we consider the σ -geodesic [ x, y i ] and we take the point y ′ i at distance T from x . By convexity of σ it follows that also the shadowscasted by y ′ i of radius r and center x cover the σ -boundary of X . So wesuppose y i ∈ S ( x, T ) and we want to show that { y i } covers S ( x, T ) at scale r . We take y ∈ S ( x, T ) and we extend the σ -geodesic [ x, y ] to a σ -geodesicray [ x, z ] . This ray passes through B ( y i , r ) for some i . Let y ′ ∈ [ x, y ] be apoint such that d ( y ′ , y i ) < r . Then d ( y, y ′ ) < r and so d ( y, y i ) < r . Thisconcludes the proof. 35s a consequence the covering entropy of X equals the shadow dimension ofthe σ -boundary of X . Proposition 5.2.
Let ( X, σ ) be a GCB -space that is P -packed at scale r .Then T log Shad-Cov r ( ∂ σ X, e − T ) ≍ P ,r ,r T log Cov ( S ( x, T ) , r ) . In particular the upper (resp. lower) shadow dimension of ∂ σ X does notdepend on r and equals the upper (resp. lower) covering entropy of X .Proof. It follows directly from the previous lemma and Proposition 3.2.
In case X is also δ -hyperbolic the shadow dimension of ∂ σ X = ∂X = ∂ G X is equivalent to a modified version of the Minkowski dimension. We recallthat upper and lower Minkowski dimension of ∂X with respect to a visualmetric D x,a are respectively classically defined asMD D x,a ( ∂X ) = lim sup ρ → log Cov D x,a ( ∂X, ρ )log ρ , MD D x,a ( ∂X ) = lim inf ρ → log Cov D x,a ( ∂X, ρ )log ρ , where the covering is considered with respect to the metric D x,a . If wecover ∂X with generalized visual balls we define the upper and lower visualMinkowski dimension asMD ( ∂X ) = lim sup ρ → log Cov ( ∂X, ρ )log ρ , MD ( ∂X ) = lim inf ρ → log Cov ( ∂X, ρ )log ρ respectively, where Cov ( ∂X, ρ ) denotes the minimal number of generalizedvisual balls of radius ρ needed to cover ∂X . Also in this case if we put ρ = e − T we have MD ( ∂X ) = lim sup T → + ∞ T log Cov ( ∂X, e − T ) , and the analogous formula for the lower Minkowski dimension. The followingresult follows directly from Lemma 2.7. Lemma 5.3.
Let D x,a be a visual metric of center x and parameter a . Then MD ( ∂X ) = a · MD D x,a ( ∂X ) , MD ( ∂X ) = a · MD D x,a ( ∂X ) . z ∈ ∂ G X and x ∈ X then ξ z denotes any geodesic ray such that ξ z (0) = x and ξ + z = z . Lemma 5.4.
Let X be a proper, δ -hyperbolic metric space, z, z ′ ∈ ∂ G X and x ∈ X . Then(i) if ( z, z ′ ) x ≥ T then d ( ξ z ( T − δ ) , ξ z ′ ( T − δ )) ≤ δ ;(ii) for all b > , if d ( ξ z ( T ) , ξ z ′ ( T )) < b then ( z, z ′ ) x > T − b .Proof. Assume ( z, z ′ ) x ≥ T and suppose d ( ξ z ( T − δ ) , ξ z ′ ( T − δ )) > δ .We fix S ≥ T − δ and we consider the triangle ∆( x, ξ z ( S ) , ξ z ′ ( S )) . Weknow there exist a ∈ [ x, ξ z ( S )] , b ∈ [ x, ξ z ′ ( S )] , c ∈ [ ξ z ( S ) , ξ z ′ ( S )] such that d ( a, b ) < δ, d ( b, c ) < δ, d ( a, c ) < δ and T δ := d ( x, a ) = d ( x, b ) ,d ( ξ z ( S ) , a ) = d ( ξ z ( S ) , c ) , d ( ξ z ′ ( S ) , b ) = d ( ξ z ′ ( S ) , c ) . Since this triangle is δ -thin we conclude that T − δ > T δ .Moreover d ( ξ z ( S ) , ξ z ′ ( S )) = d ( ξ z ( S ) , c ) + d ( c, ξ z ′ ( S )) = 2( S − T δ ) . Hence ( z, z ′ ) x ≤ lim inf S → + ∞ (cid:0) S − d ( ξ z ( S ) , ξ z ′ ( S )) (cid:1) + δ = T δ + δ < T where we have used (2.7). This contradiction concludes the first part.Now we assume d ( ξ z ( T ) , ξ z ′ ( T )) < b . Using d ( ξ z ( S ) , ξ z ′ ( S )) < S − T )+ 2 b for all S ≥ T we obtain, again by (2.7), ( z, z ′ ) x ≥ lim inf S → + ∞ (cid:0) S − d ( ξ z ( S ) , ξ z ′ ( S )) (cid:1) > T + b. The shadows in a proper, δ -hyperbolic metric space can be defined in twoways: Shad + x ( y, r ) = { z ∈ ∂ G X s.t. [ x, z ] ∩ B ( y, r ) = ∅ for some [ x, z ] } ; Shad − x ( y, r ) = { z ∈ ∂ G X s.t. [ x, z ] ∩ B ( y, r ) = ∅ for all [ x, z ] } . . Clearly if ( X, σ ) is also a GCB-space then we haveShad − x ( y, r ) ⊆ Shad x ( y, r ) ⊆ Shad + x ( y, r ) . Lemma 5.5 (Shadow’s Lemma, [Sul79]) . Let X be a proper, δ -hyperbolicmetric space. Let z ∈ ∂ G X , x ∈ X and T ≥ . Then B ( z, e − T ) ⊆ Shad − x ( ξ z ( T ) , δ ) ⊆ Shad + x ( ξ z ( T ) , δ ) ⊆ B ( z, e − T +7 δ ) . roof. Let z ′ ∈ B ( z, e − T ) , i.e. ( z, z ′ ) x > T . By the previous lemma we get d ( ξ z ( T − δ ) , ξ z ′ ( T − δ )) ≤ δ. So d ( ξ z ′ ( T, ξ z ( T )) ≤ δ < δ . This implies z ′ ∈ Shad − x ( ξ z ( T ) , δ ) , showing the first containment.Now we fix z ′ ∈ Shad + x ( ξ z ( T ) , δ ) , which means that there exists a geodesicray ξ z ′ = [ x, z ′ ] that passes through B ( ξ z ( T ) , δ ) . Let T ′ ≥ such that d ( ξ z ′ ( T ′ ) , ξ z ( T )) < δ . Then it holds | T ′ − T | < δ and so d ( ξ z ′ ( T ) , ξ z ( T )) < δ . By the previous lemma we get ( z, z ′ ) x > T − δ implying the secondcontainment.As a corollary we get another characterization of the covering entropy of X in case it is also δ -hyperbolic. Proposition 5.6.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r . Then T log Cov ( ∂X, e − T ) ≍ P ,r ,δ T log Cov ( S ( x , T ) , r ) . In particular the upper (resp. lower) visual Minkowski dimension of ∂X equals the upper (resp. lower) covering entropy of X .Proof. It follows directly from the previous lemma and Proposition 5.2.Putting together Proposition 3.1, Proposition 3.4, Proposition 5.2, Propo-sition 5.6, Theorem 4.2 and Proposition 4.3 we get the proof of Theorem B.
Let ( X, σ ) be a δ -hyperbolic GCB-space that is P -packed at scale r . In thissection we will consider a subset C of ∂X and we define the relative version,with respect to C , of all the different definitions of entropies introduced inthe previous sections. We observe that when C = ∂X then we are in thecase yet studied.We start with a couple of basic, although fundamental, lemmas relatinggeodesic rays and lines with endpoints in C . Lemma 6.1.
Let X be a proper, δ -hyperbolic metric space. Let γ be ageodesic line and x ∈ X with S := d ( γ (0) , x ) . Let x ′ be a projection of x on γ . Then(i) there exists an orientation of γ such that [ x, x ′ ] ∪ [ x ′ , γ + ] is a (1 , δ ) -quasigeodesic;(ii) with respect to the orientation of (i) then every geodesic ray ξ = [ x, γ + ] satisfies d ( ξ ( S + t ) , γ ( t )) ≤ δ for all t ≥ ;(iii) for all orientations of γ every geodesic ray ξ = [ x, γ + ] satisfies d ( ξ ( S + t ) , γ ( t )) ≤ S + 76 δ for all t ≥ . roof. We choose the orientation of γ such that x ′ belongs to the negativeray γ | ( −∞ , and we take a geodesic ray ξ = [ x, γ + ] . By Lemma 2.5 thepath α = [ x, x ′ ] ∪ [ x ′ , γ + ] is a (1 , δ ) -quasigeodesic and moreover it satisfies d ( ξ ( S + t ) , α ( S + t )) ≤ δ for every t ≥ . Furthermore the time t suchthat α ( t ) = γ (0) is between S and S + 4 δ implying d ( ξ ( S + t ) , γ ( t )) ≤ δ. For the second part of the proof we suppose to be in the situation above andwe consider the geodesic ray ξ = [ x, γ − ] . By Lemma 2.5 the path α = [ x, x ′ ] ∪ [ x ′ , γ − ] satisfies d ( ξ ( S + t ) , α ( S + t )) ≤ δ for every t ≥ . Furthermorefor every t ≥ the point α ( S + t ) belongs to γ and d ( α ( S + t ) , γ (0)) ≤ d ( α ( S + t ) , x ) + d ( x, γ (0)) ≤ S + t + 4 δ . So d ( ξ ( S + t ) , γ ( t )) ≤ δ + 2 S. We remark that if γ (0) is a projection of x on γ then the first part of thelemma holds for both the positive and negative rays of γ . Lemma 6.2.
Let X be a proper, δ -hyperbolic metric space. Let x ∈ X and C ⊆ ∂X be a subset with at least two points. Then there exists L > such that for every geodesic ray ξ with ξ + ∈ C there exists a geodesic line γ with γ ± ∈ C such that d ( ξ ( t ) , γ ( t )) ≤ L for all t ∈ [0 , + ∞ ) . Moreover if x ∈ QC-Hull ( C ) then L depends only on δ .Proof. Let z, z ′ be two distinct points of C . Let D x,a be a standard visualdistance of parameter a centered at x and let m = D x,a ( z,z ′ )2 . We have thateither D x,a ( ξ + , z ) ≥ m or D x,a ( ξ + , z ′ ) ≥ m . We suppose it holds the firstcase and we choose a geodesic line γ joining z and ξ + . We parametrize γ in such a way that γ (0) is a projection of x on γ . Then m ≤ D x,a ( z, ξ + ) ≤ e − a ( z,ξ + ) x . If S denotes d ( x, γ (0)) = d ( ξ (0) , γ (0)) and ξ z = [ x, z ] then bythe previous lemma and the remark below we have d ( ξ ( S + t ) , ξ z ( S + t )) ≤ d ( γ ( t ) , γ ( − t )) + 152 δ = 2 t + 152 δ . Therefore ( z, ξ + ) x ≥
12 lim inf t → + ∞ [2( S + t ) − d ( ξ + ( S + t ) , ξ z ( S + t ))] ≥ S − δ implying e − a ( S − δ ) ≥ m . This means d ( γ (0) , ξ (0)) ≤ a log m + 76 δ =: L ′ .The thesis follows with L := L ′ + 76 δ by the previous lemma.We observe that L depends on δ, a and m , and the choice of a depends onlyon δ . Moreover when x ∈ QC-Hull ( C ) we can take z, z ′ ∈ C such that x ∈ [ z, z ′ ] . With this choice m equals showing that L depends only on δ . Remark 6.3. If C ′ ⊆ C ⊆ ∂X and x ∈ X is fixed then the constant L givenby the previous lemma relative to C works also for C ′ , provided C ′ has atleast two points, as follows by the proof. Remark 6.4.
In the proof of Lemma 6.2 we can take any geodesic lineconnecting z and ξ + . So if ( X, σ ) is also a GCB -space we can take γ in thethesis of Lemma 6.2 to be a σ -geodesic line. .1 Covering and volume entropy Let ( X, σ ) be a δ -hyperbolic GCB-space that is P -packed at scale r andlet C be a subset of ∂X . The upper covering entropy of C is defined as lim sup T → + ∞ T log Cov ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ ) , r ) , where r > , τ ≥ and x ∈ X and it is denoted by h Cov ( C ) . The lowercovering entropy of C , denoted by h Cov ( C ) , is defined taking the limit inferiorinstead of the limit superior. These quantities do not depend on x ∈ X asusual. The analogous of Proposition 3.1 holds. Proposition 6.5.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r , C be a subset of ∂X and x ∈ X . Then T log Cov ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ ) , r ) ≍ P ,r ,r,r ′ ,τ,τ ′ T log Pack ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ ′ ) , r ′ ) for all r, r ′ > and τ, τ ′ ≥ . In particular any of these functions can beused in the definition of the upper and lower covering entropies of C .Proof. Once τ is fixed the asymptotic estimate can be proved exactly as inProposition 3.1. Moreover for all τ ≥ it is easy to prove thatCov ( B ( x , T ) ∩ B ( QC-Hull ( C ) , τ ) , r ) ≤ Cov ( B ( x , T ) ∩ QC-Hull ( C ) , r ) · Cov ( r + τ, r ) . and Cov ( r + τ, r ) is uniformly bounded in terms of P , r and τ by Propo-sition 2.1. This concludes the proof.Clearly when C = ∂X we have h Cov ( ∂X ) = h Cov ( X ) . Moreover if C is aclosed subset of ∂X then h Cov ( C ) ≤ h Cov ( ∂X ) , so h Cov ( C ) ≤ log(1+ P ) r byLemma 3.3.The analogous of Proposition 3.2 holds. We remark that in this case adependence on δ appears. Proposition 6.6.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r , C be a subset of ∂X and x ∈ X . Then T log Cov ( B ( x, T ) ∩ QC-Hull ( C ) , r ) ≍ P ,r ,r,δ T log Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) In particular any of these functions can be used in the definition of the upperand lower covering entropies of C . roof. As in the proof of Proposition 3.2 one inequality is obvious, so weare going to prove the other. We divide the ball B ( x, T ) in the annulii A ( x, kr, ( k + 1) r ) with k = 0 , . . . , Tr − . Therefore we can estimate thequantity Cov ( B ( x, T ) ∩ QC-Hull ( C ) , δ + 2 r ) from above by Tr − X k =0 Cov ( A ( x, kr, ( k + 1) r ) ∩ QC-Hull ( C ) , δ + 2 r ) . We claim that every element of the sum is ≤ Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) . Indeed let y , . . . , y N be a set of points realizing Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) .For all i = 1 , . . . , N we consider the σ -geodesic segment γ i = [ x, y i ] and wecall x i the point along this geodesic at distance kr from x . We want toshow that x , . . . , x N is a (72 δ + 2 r ) -dense subset of A ( x, kr, ( k + 1) r ) ∩ QC-Hull ( C ) . Given a point y ∈ A ( x, kr, ( k + 1) r ) ∩ QC-Hull ( C ) there existsa σ -geodesic line γ with endpoints in C containing y . We parametrize γ so that γ (0) is a projection of x on γ and y ∈ γ | [0 , + ∞ ) . We take a point y T ∈ γ | [0 , + ∞ ) at distance T from x , so that y T ∈ S ( x, T ) ∩ QC-Hull ( C ) andtherefore there exists i such that d ( y T , y i ) ≤ r . By Lemma 2.5 the path α = [ x, γ (0)] ∪ [ γ (0) , y T ] is a (1 , δ ) -quasigeodesic and, if t y denotes the realnumber such that α ( t y ) = y , it holds t y ∈ [ kr, ( k +1) r ] . By Lemma 2.5 we get d ( y, γ ′ i ( t y )) ≤ δ , where γ ′ i is the σ -geodesic [ x, y T ] . We conclude the proofof the claim since d ( y, x i ) ≤ d ( y, γ ′ i ( t y )) + d ( γ ′ i ( t y ) , γ i ( t y )) + d ( γ i ( t y ) , x i ) ≤ δ + 2 r , from the convexity of σ . We remark that using the ideas of Lemma5.4 it is possible to obtain a similar estimate without using the convexity.The thesis follows by Proposition 6.5.The upper volume entropy of C with respect to a measure µ is h µ ( C ) = sup τ ≥ lim sup T → + ∞ T log µ ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ )) , where x ∈ X . Taking the limit inferior instead of the limit superior is definedthe lower volume entropy of C , h µ ( C ) . Proposition 6.7.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r , let C be a subset of ∂X and let µ be a measure on X which is H -homogeneous at scale r . Then for all τ ≥ r it holds T log µ ( B ( x, T ) ∩ B ( QC-Hull ( C ) , τ )) ≍ H,P ,r ,r,σ T log Cov ( B ( x, T ) ∩ QC-Hull ( C ) , r ) . In particular the upper (resp. lower) volume entropy of C with respect to µ coincides with the upper (resp. lower) covering entropy of C and it can becomputed using τ = r in place of the supremum. roof. By Remark 3.5 we know that µ is H ( τ ) -homogeneous at scale τ forall τ ≥ r , where H ( τ ) depends on P , r , τ, r, H . Therefore the proof ofProposition 3.4 works in this case. Let ( X, σ ) be a δ -hyperbolic GCB-space that is P -packed at scale r . For asubset C of ∂X and Y ⊆ X we setGeod σ ( Y, C ) = { γ ∈ Geod σ ( X ) s.t. γ ± ⊆ C and γ (0) ∈ Y } . If Y = X we simply write Geod σ ( C ) . Clearly Geod σ ( C ) is a Φ -invariantsubset of Geod σ ( X ) , so the geodesic flow is well defined on it. The upperLipschitz-topological entropy of Geod σ ( C ) is defined as h Lip-top ( Geod σ ( C )) = inf â sup K lim r → lim sup T → + ∞ T log Cov â T ( K, r ) , where the infimum is taken among all geometric metrics on Geod σ ( C ) . The lower Lipschitz-topological entropy is defined taking the limit inferior insteadof the limit superior and it is denoted by h Lip-top ( Geod σ ( C )) . In the followingcrucial result we observe the difference between closed and non-closed subsetsof ∂X which will be at the basis of the difference between the Hausdorff andthe Minkowski dimension of the limit set of a discrete group of isometries. Theorem 6.8.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and C be a subset of ∂X . Then h Lip-top ( Geod σ ( C )) = sup C ′ ⊆ C h Cov ( C ′ ) , where the supremum is among closed subsets C ′ of C . The same holds forthe lower entropies. We remark that the supremum of the covering entropies among the closedsubsets of C can be strictly smaller than the covering entropy of C , markingthe distance between the equivalences of the different notions of entropies incase of non-closed subsets of the boundary. We start with an easy lemma. Lemma 6.9.
Let ( X, σ ) and C be as in Theorem 6.8 and let x ∈ X . Thenevery compact subset of Geod σ ( C ) is contained in Geod σ ( B ( x, R ) , C ′ ) forsome R ≥ and some C ′ ⊆ C closed. Moreover Geod σ ( B ( x , R ) , C ′ ) iscompact for all R ≥ and all closed C ′ ⊆ C .Proof. We fix a compact subset K of Geod σ ( C ) . The continuity of theevaluation map gives that E ( K ) is contained in some ball B ( x, R ) . Moreoverthe maps + , − : Geod σ ( X ) → ∂X , defined by γ γ + , γ − respectively, arecontinuous ([BL12], Lemma 1.6). This means that C ′ = +( K ) ∪ − ( K ) is42 closed subset of ∂X and clearly K ⊆ Geod σ ( B ( x, R ) , C ′ ) . By a similarargument, and since the evaluation map is proper, it follows that the setGeod σ ( B ( x, R ) , C ′ ) is compact for all R ≥ and all C ′ ⊆ C closed.For a metric f ∈ F and C ⊆ ∂X we denote by h f the upper metricentropy of Geod σ ( C ) with respect to f , that is h f ( Geod σ ( C )) = sup K lim r → lim sup T → + ∞ T log Cov f T ( K, r ) . Taking the limit inferior instead of the limit superior we define the lowermetric entropy of Geod σ ( C ) with respect to f , denoted by h f ( Geod σ ( C )) .The analogous of Proposition 4.3 is the following. Proposition 6.10.
Let ( X, σ ) be as in Theorem 6.8, C ′ be a closed subsetof ∂X , f ∈ F , x ∈ X and L be the constant given by Lemma 6.2. Then(i) h f ( Geod σ ( B ( x, R ) , C ′ )) = h f ( Geod σ ( B ( x, L ) , C ′ )) for all R ≥ L ;(ii) h f ( Geod σ ( C ′ )) = h f ( Geod σ ( B ( x, L ) , C ′ )) ≤ h Cov ( C ′ ); (iii) The function r lim sup T → + ∞ T log Cov f T ( Geod σ ( B ( x, L ) , C ′ ) , r ) isconstant.The same conclusions hold for the lower entropies. We observe that applying the Key Lemma 4.4 we have directly the rela-tive version of Corollary 4.5.
Corollary 6.11.
Let f ∈ F , x ∈ X , R ≥ and < r ≤ r ′ . Then T log Cov f T ( Geod σ ( B ( x, R ) , C ′ ) , r ′ ) ≍ P ,r ,r,r ′ ,f T log Cov f T ( Geod σ ( B ( x, R ) , C ′ ) , r ) . Proof of of Proposition 6.10.
We fix R ≥ L and T ≥ . We take a set γ , . . . , γ N of σ -geodesic lines realizing Cov f T ( Geod σ ( B ( x, L ) , C ′ ) , r ) . Ouraim is to show that γ , . . . , γ N is a (4 R + 2 L + C ( f ) + 76 δ + r ) -dense subsetof Geod σ ( B ( x, R ) , C ′ ) . This, together with Corollary 6.11, will prove (i).We consider a σ -geodesic line γ ∈ Geod σ ( B ( x, R ) , C ′ ) , so d ( γ (0) , x ) =: S ≤ R . By Lemma 6.1 there exists a σ -geodesic ray ξ starting at x such that d ( ξ ( S + t ) , γ ( t )) ≤ S + 76 δ for all t ≥ and in particular ξ + belongs to C .Now we apply Lemma 6.2 to find a σ -geodesic line γ ′ ∈ Geod ( C ′ ) such that d ( ξ ( t ) , γ ′ ( t )) ≤ L for all t ≥ . Clearly we have γ ′ ∈ Geod σ ( B ( x, L ) , C ′ ) and d ( γ ′ ( S + t ) , γ ( t )) ≤ S + L + 76 δ for all t ≥ . Therefore d ( γ ′ ( t ) , γ ( t )) ≤ S + L + 76 δ for all t ≥ . This implies that for all t ∈ [0 , T ] we have f t ( γ, γ ′ ) ≤ Z − t −∞ (cid:0) d ( γ (0) , γ ′ (0)) + 2 | s | (cid:1) f ( s ) ds + Z + ∞− t (cid:0) S + L + 76 δ (cid:1) f ( s ) ds. d ( γ (0) , γ ′ (0)) ≤ L + S we get f t ( γ, γ ′ ) ≤ S + 2 L + C ( f ) + 76 δ usingthe properties of f , and so f T ( γ, γ ′ ) ≤ R + 2 L + C ( f ) + 76 δ. Moreover,since γ ′ ∈ Geod σ ( B ( x, L ) , C ′ ) , there exists γ i such that f T ( γ ′ , γ i ) ≤ r . Thisimplies f T ( γ, γ i ) ≤ R + 2 L + C ( f ) + 76 δ + r . We observe that (iii) follows directly from the previous corollary.The first equality in (ii) follows by (i). In order to prove the inequalitywe fix y , . . . , y N realizing Cov ( S ( x, T ) ∩ QC-Hull ( C ′ ) , r ) . Up to change y i with a point at distance at most δ from it we can suppose there are γ i ∈ Geod σ ( C ′ ) such that y i ∈ γ i and y , . . . , y N is a (8 δ + r ) -dense subsetof S ( x, T ) ∩ QC-Hull ( C ′ ) , as follows by Lemma 2.6. By Lemma 6.1 thereexists an orientation of γ i such that, called S i = d ( x, γ i (0)) and T i ≥ suchthat γ i ( T i ) = y i , we have T ≤ S i + T i ≤ T + 4 δ and the σ -geodesic ray ξ i = [ x, γ + i ] satisfies d ( ξ i ( S i + t ) , γ i ( t )) ≤ δ for all t ≥ . By Lemma6.2 there exists γ ′ i ∈ Geod σ ( B ( x, L ) , C ′ ) such that d ( γ ′ i ( t ) , ξ i ( t )) ≤ L for all t ≥ . We claim that the set { γ ′ i } is (6 L + 176 δ + 2 r + 2 C ( f )) -dense inGeod σ ( B ( x, L ) , C ′ ) . By (i) and (iii) this would imply the thesis. We fix γ ∈ Geod σ ( B ( x, L ) , C ′ ) , so there exists y ∈ S ( x, T ) and T y ∈ [ T − L, T + L ] such that γ ( T y ) = y and therefore d ( y, y i ) ≤ δ + r for some i . We observethat we have d ( γ ′ i ( S i + T i ) , y i ) ≤ L + 76 δ and so d ( γ ′ i ( T ) , y i ) ≤ L + 80 δ .Moreover d ( γ ( T ) , y i ) ≤ L + 8 δ + r implying d ( γ ( T ) , γ ′ i ( T )) ≤ L + 88 δ + r .Furthermore by definition d ( γ (0) , γ ′ i (0)) ≤ L , so by convexity of σ we get d ( γ ( t ) , γ ′ i ( t )) ≤ L + 88 δ + r for all t ∈ [0 , T ] . The thesis follows by theclassical subdivision of the integral defining f into three parts, each estimatedby the constants above. Proof of Theorem 6.8.
We fix a geometric metric â on Geod σ ( C ) and wedenote by M the Lipschitz constant with respect to â of the evaluationmap E . By Remark 6.3 the constant L given by Lemma 6.2 can be chosenindependently of C ′ ⊆ C , once x is fixed. Clearly we have sup R ≥ ,C ′ ⊆ C lim r → lim sup T → + ∞ T log Cov â T ( Geod σ ( B ( x, R ) , C ′ ) , r ) ≥ sup C ′ ⊆ C lim sup T → + ∞ T log Cov â T ( Geod σ ( B ( x, L ) , C ′ ) , r ) . We fix σ -geodesic lines γ , . . . , γ N realizing Cov â T ( Geod σ ( B ( x, L ) , C ′ ) , r ) .Since d ( γ i (0) , x ) ≤ L for all i = 1 , . . . , N then there exists t i ∈ [ T − L, T + L ] such that d ( γ i ( t i ) , x ) = T . We claim that the points y i = γ i ( t i ) ∈ S ( x, T ) ∩ QC-Hull ( C ′ ) are (2 L + 80 δ + M r ) -dense. By Proposition 6.6 this wouldimply h Lip-top ( Geod σ ( C )) ≥ sup C ′ ⊆ C h Lip-top ( Geod σ ( C ′ )) ≥ sup C ′ ⊆ C h Cov ( C ′ ) . We fix y ∈ S ( x, T ) ∩ QC-Hull ( C ′ ) and we select a geodesic line γ ∈ Geod ( C ′ ) containing y . Up to change y with a point at distance at most δ we can44uppose γ ∈ Geod σ ( C ′ ) , as follows by Lemma 2.6. By Lemma 6.1, with anappropriate choice of the orientation of γ , the σ -geodesic ray ξ = [ x, γ + ] satisfies d ( ξ ( S + t ) , γ ( t )) ≤ δ for all t ≥ , where S = d ( x, γ (0)) . ByLemma 6.2 there exists γ ′ ∈ Geod σ ( B ( x, L ) , C ′ ) such that d ( ξ ( t ) , γ ′ ( t )) ≤ L for all t ≥ , implying d ( γ ′ ( S + t ) , γ ( t )) ≤ L + 76 δ for all t ≥ . Denotingby T y the real number such that γ ( T y ) = y we have by Lemma 6.1 that T ≤ S + T y ≤ T + 4 δ . Therefore we apply the previous estimate with t = T y obtaining d ( γ ′ ( T ) , y ) ≤ d ( γ ′ ( T ) , γ ′ ( S + T y )) + d ( γ ′ ( S + T y ) , y ) ≤ L + 80 δ .Moreover there exists i ∈ { , . . . , N } such that â T ( γ ′ , γ i ) ≤ r and in par-ticular d ( γ ′ ( T ) , γ i ( T )) ≤ M r . Therefore we get d ( y i , y ) ≤ d ( γ i ( t i ) , γ i ( T )) + d ( γ i ( T ) , y ) ≤ L + 80 δ + M r . Now, up to add δ , we obtain the inequality.The other inequality follows by Proposition 6.10. Indeed we have h Lip-top ( Geod σ ( C )) ≤ sup C ′ ⊆ C h f ( Geod σ ( C ′ )) ≤ sup C ′ ⊆ C h Cov ( C ′ ) . Remark 6.12.
Let ( X, σ ) be as in Theorem 6.8, C ⊆ ∂X closed and x ∈ QC-Hull ( C ) . By the proof of Theorem 6.8, Lemma 6.2 and Remark6.3 we obtain T log Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) ≍ P ,r ,δ,f T log Cov f T ( Geod σ ( B ( x, L ) , C ) , r ) for all f ∈ F , where L depends only on δ . For all Y ⊆ X and C ⊆ ∂X we denote by Ray σ ( Y, C ) the set of σ -geodesic rays ξ with ξ (0) ∈ Y and ξ + ∈ C . When Y = X we use the notationRay σ ( C ) and in this case this set is invariant by the geodesic semi-flow. So itis defined in the usual way its upper and lower Lipschitz-topological entropy,denoted respectively by h Lip-top ( Ray σ ( C )) and h Lip-top ( Ray σ ( C )) . Proposition 6.13.
Let ( X, σ ) and C be as in Theorem 6.8. Then(i) h Lip-top ( Ray σ ( C )) equals sup C ′ ⊆ C lim sup T →∞ T log Cov f T ( Ray σ ( x, C ′ ) , r ) independently of f ∈ F , the point x ∈ X and r > , where the supre-mum is taken among the closed subsets of C .(ii) h Lip-top ( Ray σ ( C )) = h Lip-top ( Geod σ ( C )) ;(iii) the equivalent asymptotic estimate of Remark 6.12 holds for the geodesicsemi-flow.The same conclusions hold for the lower entropies.Proof. The inequality h Lip-top ( Ray σ ( C ′ )) ≥ h Cov ( C ′ ) for all closed C ′ ⊆ C follows by the same proof of Theorem 6.8. The remaining part of thethesis can be proved in a similar way of Proposition 6.10 and we omit thedetails. 45 .3 Shadow and Minkowski dimension The notions of shadow covering, shadow dimension and visual Minkowskidimension can be directly generalized to the case of subsets C of ∂X . Theupper (resp. lower) shadow dimension of C will be denoted by Shad-D ( C ) (resp. Shad-D ( C ) ), while the upper (resp. lower) visual Minkowski dimen-sion of C will be denoted by MD ( C ) (resp. MD ( C ) ). Proposition 6.14.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r and C be a subset of ∂X . Let C be a subset of ∂X , x ∈ X and L be the constant of Lemma 6.2. Then T log Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) ≍ P ,r ,δ,r,L T log Shad-Cov r ( C, e − T ) . In particular the upper (resp. lower) shadow dimension of C equals the upper(resp. lower) covering entropy of C .Proof. Let y , . . . , y N be a set realizing Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) . Wefix z ∈ C and we consider the σ -geodesic ray ξ = [ x, z ] . By Lemma 6.2 thereexists γ ∈ Geod σ ( B ( x, L ) , C ) such that d ( ξ ( t ) , γ ( t )) ≤ L for all t ≥ . Let t y ∈ [ T − L, t + L ] such that d ( γ ( t y ) , x ) = T and call y = γ ( t y ) . Then thereexists i ∈ { i, . . . , N } such that d ( y, y i ) ≤ r and moreover d ( ξ ( T ) , y ) ≤ L ,implying [ x, z ] ∩ B ( y i , L + 2 r ) = ∅ . This shows thatShad-Cov L +2 r ( C, e − T ) ≤ Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) . Now let y i , . . . , y N be points realizing Shad-Cov r ( C, e − T ) . By the same ar-gument used in the proof of Lemma 5.1 we can suppose d ( y i , x ) = T . Let y ∈ S ( x, T ) ∩ QC-Hull σ ( C ) and let γ ∈ Geod σ ( C ) such that y ∈ γ orientedin such a way that, by Lemma 6.1, the σ -geodesic ray ξ = [ x, γ + ] satisfies d ( ξ ( S + t ) , γ ( t )) ≤ δ for all t ≥ , where S = d ( x, γ (0)) . By the samelemma we know, indicated by t y ≥ the real number such that γ ( t y ) = y ,that T ≤ S + t y ≤ T + 4 δ implying d ( ξ ( T ) , y ) ≤ δ . Moreover there exists i ∈ { , . . . , N } such that d ( ξ ( T ) , y i ) < r , therefore d ( y, y i ) < δ + 2 r . Thisshows that, adding the usual δ given by Lemma 2.6,Cov ( S ( x, T ) ∩ QC-Hull ( C ) , δ + 2 r ) ≤ Shad-Cov r ( C, e − T ) . The thesis follows by Proposition 6.6 together with Proposition 6.5.By Lemma 5.1 we get immediately the following.
Proposition 6.15.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packedat scale r and C be a subset of ∂X , let C be a subset of ∂X , x ∈ X and L be the constant given by Lemma 6.2. Then T log Cov ( C, e − T ) ≍ P ,r ,δ,L T log Cov ( S ( x, T ) ∩ QC-Hull ( C ) , r ) . n particular the upper (resp. lower) Minkowski dimension of C equals theupper (resp. lower) covering entropy of C . We remark that the upper visual Minkowski dimension of C equals theupper visual Minkowski dimension of its closure C while it can happen that sup C ′ ⊆ C MD ( C ′ ) < MD ( C ) , where the supremum is taken among the closedsubsets of C .The proof of Theorems E and F follow by Proposition 6.5, Proposition6.6, Proposition 6.7, Theorem 6.8, Proposition 6.10, Remark 6.12, Proposi-tion 6.14 and Proposition 6.15. In this section we will specialize the study of the entropies to special subsetsof the boundary at infinity of a δ -hyperbolic GCB-space that is P -packed atscale r . In the first subsection we will introduce the entropy of X relativeto a discrete group Γ of σ -isometries of X and the critical exponent of Γ .In the second one we will study the special case of quasiconvex cocompactgroups. Let X be a proper metric space and let Γ be a discrete group of isometriesof X . The critical exponent . of Γ is h Γ := inf (cid:26) s ≥ s.t. X g ∈ Γ e − sd ( x,gx ) < + ∞ (cid:27) . It does not depend on x ∈ X . We remark that for every s ≥ the series P g ∈ Γ e − sd ( x,gx ) , which is called the Poincaré series of Γ , is Γ -invariant. Inother words P g ∈ Γ e − sd ( x,gx ) = P g ∈ Γ e − sd ( x ′ ,gx ′ ) for all x ′ ∈ Γ x .The upper Γ -entropy of X is defined as h Γ ( X ) = lim sup T → + ∞ T log x ∩ B ( x, T ) = lim sup T → + ∞ T log T ( x ) , where the last equality follows from the finiteness of the stabilizers of adiscrete group. The lower Γ -entropy of X is defined taking the limit inferiorinstead of the limit superior and it is denoted by h Γ ( X ) . They do not dependon x ∈ X . The following proposition is proved in the δ -hyperbolic case in[Coo93], but it remains true for proper metric spaces. Lemma 7.1 (Proposition 5.3 of [Coo93]) . Let X be a proper metric spaceand let Γ be a discrete group of isometries of X . Then h Γ ( X ) = h Γ .
47e remark that for CAT ( − metric spaces X it holds h Γ ( X ) = h Γ ( X ) forevery discrete group of isometries of X , see [Rob02]. The Γ -entropy of X isalso related to the covering entropy of the limit set Λ(Γ) . Lemma 7.2.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r and let Γ be a discrete group of σ -isometries of X . Then(i) h Γ ( X ) ≤ h Cov (Λ(Γ)) . (ii) if Γ is non-elementary, quasiconvex-cocompact with codiameter ≤ D and if x ∈ QC-Hull (Λ(Γ)) then T log x ∩ B ( x, T ) ≍ P ,r ,δ,D T log Cov ( B ( x, T ) ∩ QC-Hull (Λ(Γ))) . In particular h Γ ( X ) = h Cov (Λ(Γ)) .The same conclusions hold for the lower entropies.Proof.
We fix x ∈ QC-Hull (Λ(Γ)) and ε = sys ⋄ (Γ , x ) > . By the Γ -invariance of QC-Hull (Λ(Γ)) and the definition of the systole we get x ∩ B ( x, T ) ≤ Pack (cid:18) B ( x, T ) ∩ QC-Hull (Λ(Γ)) , ε (cid:19) , showing (i) by Proposition 6.5. In order to prove (ii) we fix x ∈ QC-Hull (Λ(Γ)) and we call D the codiameter of the action. By (2.9) the free-systole of theaction is bounded from below by a constant depending only on P , r , δ and D , so the proof of (i) shows the first half of the asymptotic estimate. Fur-thermore we claim thatPack ( B ( x, T ) ∩ QC-Hull (Λ(Γ)) , D ) ≤ x ∩ B ( x, T + D ) . Indeed let y , . . . , y N be points realizing Pack ( B ( x, T ) ∩ QC-Hull (Λ(Γ)) , D ) ,so d ( y i , y j ) > D for every i = j . For every i let x i be a point of the orbitof x at distance at most D from y i . It follows that the points x i are alldistinct, concluding the proof of the claim. The conclusion follows applyingProposition 6.5. Remark 7.3.
Under the assumptions of Lemma 7.2 then by Lemma 7.1and the discussion after Proposition 6.5 we always have h Γ ≤ log(1+ P ) r =: h + . Moreover if Γ is as in (ii) then there exists h − > depending only on P , r , δ, D such that h Γ ≥ h − . This follows from Lemma 7.1 and Example5.8 of [CS20a]. Let X be a proper, δ -hyperbolic metric space, let x ∈ X and B be aBorelian subset of ∂ G X . Following [Pau96], for all α ≥ and all η > weset H αη ( B ) = inf (X i ∈ N ρ αi s.t. B ⊆ [ i ∈ N B ( z i , ρ i ) and ρ i ≤ η ) .
48s in the classical case the visual α -dimensional Hausdorff measure of B isdefined as lim η → H αη ( B ) =: H α ( B ) , while the visual Hausdorff dimension of B is defined as the unique α ≥ such that H α ′ ( B ) = 0 for all α ′ > α and H α ′ ( B ) = + ∞ for all α ′ < α . The visual Hausdorff dimension of the boreliansubset B is denoted by HD ( B ) . By Lemma 2.7, see also [Pau96], we haveHD ( B ) = a · HD D x,a ( B ) for all visual metrics D x,a of center x and parameter a , where HD D x,a ( B ) denotes the classical Hausdorff dimension with respectto the metric D x,a . Therefore we directly obtain the usual inequalitiesHD ( B ) ≤ MD ( B ) ≤ MD ( B ) for all Borelian subsets B of ∂ G X . In Section 1.7 we have seen how theseinequalities can be strict.There is a canonical way to construct a measure on ∂ G X starting fromthe Poincaré series. For every s > τ (Γ) the measure µ s = 1 P g ∈ Γ e − sd ( x,gx ) X g ∈ Γ e − sd ( x,gx ) ∆ gx , where ∆ gx is the Dirac measure at gx , is a probability measure on the com-pact space X ∪ ∂ G X . Then there exists a sequence s i converging to h Γ suchthat µ s i converges ∗ -weakly to a probability measure on X ∪ ∂ G X . Any ofthese limits is called a Patterson-Sullivan measure and it is denoted by µ PS . Proposition 7.4 (Theorem 5.4 of [Coo93].) . Let X be a proper, δ -hyperbolicmetric space and let Γ be a discrete group of isometries of X with h Γ < + ∞ .Then every Patterson-Sullivan measure is supported on Λ(Γ) . Moreover it isa Γ -quasi conformal density of dimension h Γ , i.e. it satisfies Q e h Γ ( B xz ( x ) − B xz ( g − x )) ≤ d ( g ∗ µ PS ) dµ PS ( z ) ≤ Qe h Γ ( B xz ( x ) − B xz ( g − x )) for every g ∈ Γ and every z ∈ Λ(Γ) , where Q is a constant depending onlyon δ and an upper bound on h Γ . The quantification of Q is not explicitated in the original paper, but it followsfrom the proof therein. Let Γ be a discrete, quasiconvex-cocompact group of isometries of a proper, δ -hyperbolic metric space X . Then it is proved in [Coo93] that the Patterson-Sullivan measure on Λ(Γ) is ( A, h Γ ) -Ahlfors regular for some A > . We willprecise this result quantifying the constant A in terms of universal constantsin case X is also a packed GCB-space.49 heorem 7.5. Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r . Let Γ be a discrete, quasiconvex-cocompact group of σ -isometries of X with codiameter ≤ D and x be a point of QC-Hull (Λ(Γ)) . Then:(i)
Λ(Γ) is visually ( A, h Γ ) -Ahlfors regular with respect to any Patterson-Sullivan measure, where A depends only on P , r , δ and D .(ii) it holds T log Cov (Λ(Γ) , e − T ) ≍ P ,r ,δ,D h Γ ; (iii) MD (Λ(Γ)) = MD (Λ(Γ)) = h Γ . We observe that (iii) follows immediately from (ii), while (ii) is essentiallystraightforward once proved (i). Indeed we have
Lemma 7.6.
Suppose C ⊆ ∂X is visually ( A, s ) -Ahlfors regular. Then T log Cov ( C, e − T ) ≍ δ,A,s s. We define the packing ∗ number at scale ρ of a subset C of ∂X as the maximalnumber of disjoint generalized visual balls of radius ρ with center in C andwe denote it by Pack ∗ ( C, ρ ) . Lemma 7.7.
For all T ≥ it holds Pack ∗ ( C, e − T + δ ) ≤ Cov ( C, e − T ) and Cov ( C, e − T + δ ) ≤ Pack ∗ ( C, e − T ) . Proof.
Let z , . . . , z N be points of C realizing Cov ( C, e − T ) . Suppose thereexist points w , . . . , w M of C such that B ( w i , e − T + δ ) are disjoint, in partic-ular ( w i , w j ) x ≤ T − δ for every i = j . If M > N then two different points w i , w j belong to the same ball B ( z k , ρ ) , i.e. ( z k , w i ) x > T and ( z k , w j ) x > T. By (2.6) we have ( w i , w j ) x > T − δ which is a contradiction. This shows thefirst inequality.Now let z , . . . , z N be a maximal collection of points of C such that B ( z i , ρ ) are disjoint. Then for every z ∈ C there exists i such that B ( z, ρ ) ∩ B ( z i , ρ ) = ∅ . Therefore there exists w ∈ ∂X such that ( z i , w ) x > T and ( z, w ) x > T .As before we get ( z i , z ) x > T − δ, proving the second inequality. Proof of Lemma 7.6.
Since the measure µ in the definition of Ahlfors regu-larity is assumed to be of total measure one, we have µ ( C ) ≤ Ae − sT · Cov ( C, e − T ) and µ ( C ) ≥ A e − sT · Pack ( C, e − T ) implying Cov ( C, e − T ) ≥ A e sT and Pack ( C, ρ ) ≤ Ae sT . Therefore T log Cov ( C, e − T ) ≥ s + 1 T log 1 A and T log Cov ( C, e − T ) ≤ T log Pack ∗ ( C, e − T − δ ) ≤ s + 1 T log A + sδT . roof of Theorem 7.5. As observed (iii) follows from (ii) and (ii) follows from(i) applying Lemma 7.6 and the fact that h Γ ≤ log(1+ P ) r . In order to prove(i) we consider two cases: if Γ is elementary then ∈ { , } and h Γ = 0 .If this cardinality is there is nothing to prove. If Λ(Γ) = { z − , z + } then itis straightfoward to see that µ PS ( z − ) = µ PS ( z + ) = .If Γ is non-elementary we denote by < h − ≤ h + < + ∞ the numbers intro-duced in Remark 7.3. They depend only on P , r , δ and D . We will prove (i’) Λ(Γ) is visually ( A, h Γ ) -Ahlfors regular with respect to the Patterson-Sullivan measure, where A depends only on δ, h − , h + and D . We denote by L the constant given by Lemma 6.2 relative to x and Λ(Γ) ,remarking that it depends only on δ . Step 1: ∀ z ∈ ∂X and ∀ ρ > it holds µ PS ( B ( z, ρ )) ≤ e h Γ (21 δ +3 D +3 L ) ρ h Γ . We suppose first z ∈ Λ(Γ) and we take the set ˜ B ( z, ρ ) = (cid:26) y ∈ X ∪ ∂X s.t. ( y, z ) x > log 1 ρ (cid:27) . It is open (cp. Observation 4.5.2 of [DSU17]) and ˜ B ( z, ρ ) ∩ ∂X = B ( z, ρ ) ,so µ PS ( ˜ B ( z, ρ )) = µ PS ( B ( z, ρ )) since µ PS is supported on Λ(Γ) ⊆ ∂X . Let T = log ρ , ξ z = [ x, z ] and z T be the point on ξ z at distance T from x . Forevery y ∈ Γ x ∩ ˜ B ( z, ρ ) we have d ( x, y ) > T − δ and d ( x, y ) > d ( x, z T ) + d ( z T , y ) − δ. (7.1)Indeed from d ( y, ξ z ( S )) ≥ S − d ( x, y ) for all S ≥ we get T < ( y, z ) x ≤ d ( x, y ) + δ . In order to prove the second inequality we extend the σ -geodesicsegment [ x, y ] to a σ -geodesic ray ξ w with ξ + w = w . By the analogue of (2.6)we have ( w, z ) x ≥ min { ( y, z ) x , ( y, w ) x } − δ > T − δ, where the last inequality followsfrom d ( x, y ) > T − δ . By Lemma 5.4 we have d ( ξ z ( T − δ ) , ξ w ( T − δ )) ≤ δ and applying the triangular inequality we get d ( y, z T ) ≤ δ and the secondestimate in (7.1).Moreover by Lemma 6.2, since x ∈ QC-Hull (Λ(Γ)) , there exists γ ∈ Geod (Λ(Γ)) such that d ( z T , γ ( T )) ≤ L . By the cocompactness of the action on QC-Hull (Λ(Γ)) we can find a point x ∈ Γ x such that d ( x , γ ( T )) ≤ D , so d ( z T , x ) ≤ L + D .This actually implies d ( x, y ) > d ( x, x ) + d ( x , y ) − δ − D − L for all y ∈ Γ x ∩ ˜ B ( z, ρ ) . Therefore X y ∈ Γ x ∩ ˜ B ( z,ρ ) e − sd ( x,y ) ≤ X y ∈ Γ x ∩ ˜ B ( z,ρ ) e − s ( d ( x,x )+ d ( x ,y ) − δ − D − L ) = e s (20 δ +2 D +2 L ) e − sd ( x,x ) · X y ∈ Γ x ∩ ˜ B ( z,ρ ) e − sd ( x ,y ) ≤ e s (20 δ +3 D +3 L ) e − sd ( x,z T ) · X g ∈ Γ e − sd ( x ,gx ) = e s (20 δ +3 D +3 L ) · ρ s · X g ∈ Γ e − sd ( x,gx ) .
51n other words we have µ s ( ˜ B ( z, ρ )) ≤ e s (20 δ +3 D +3 L ) ρ s , and by ∗ -weak con-vergence we conclude that µ PS ( B ( z, ρ )) = µ PS ( ˜ B ( z, ρ )) ≤ lim inf i → + ∞ µ s i ( ˜ B ( z, ρ )) ≤ e h Γ (20 δ +3 D +3 L ) ρ h Γ . In the general case of z ∈ ∂X we observe that if B ( z, ρ ) ∩ Λ(Γ) = ∅ thenthe thesis is obviously true since µ PS is supported on Λ(Γ) . Otherwise thereexists w ∈ Λ(Γ) such that ( z, w ) x > log ρ . It is straightforward to check that B ( w, ρ ) ⊆ B ( z, ρe δ ) by (2.6). Then µ PS ( B ( z, ρ )) ≤ e h Γ (21 δ +3 D +3 L ) ρ h Γ . Step 2: for every R ≥ R := log 2 h Γ + 21 δ + 3 D + 3 L + 5 δ and for every g ∈ Γ it holds µ PS ( Shad x ( gx, R )) ≥ Q e − h Γ d ( x,gx ) , where Q is the constantof Proposition 7.4 that depends only on δ and h + . From the first step we know that for every ρ ≤ ρ := 2 − h Γ e − (21 δ +3 D +3 L ) and for every z ∈ ∂X it holds µ PS ( B ( z, ρ )) ≤ . A direct computationshows that R = log ρ + 5 δ . We claim that for every R ≥ R and every g ∈ Γ the set ∂X \ g ( Shad x ( g − x, R )) is contained in a generalized visualball of radius at most ρ . Indeed if z, w ∈ ∂X \ g ( Shad x ( g − x, R )) then the σ -geodesic rays [ gx, z ] , [ gx, w ] do not intersect the ball B ( x, R ) . Therefore,setting ξ = [ gx, z ] , we get ( ξ ( T ) , gx ) x ≥ d ( x, [ gx, ξ ( T )]) − δ ≥ R − δ byLemma 2.4. This implies ( z, gx ) x ≥ lim inf T → + ∞ ( ξ ( T ) , gx ) x ≥ R − δ andthe same holds for w . Thus by (2.6) we get ( z, w ) x ≥ R − δ, proving theclaim. By Proposition 7.4 we get µ PS ( Shad x ( gx, R )) µ PS ( g − ( Shad x ( gx, R ))) ≥ Q e − h Γ ( B xz ( x ) − B xz ( gx )) . Since R ≥ R the measure of g − ( Shad x ( gx, R )) is at least and the Buse-mann function is -Lipschitz, so µ PS ( Shad x ( gx, R )) ≥ Q e − h Γ d ( x,gx ) . Step 3.
For every z ∈ QC-Hull (Λ(Γ)) and every ρ > the following is true: µ PS ( B ( z, ρ )) ≥ Q e − h Γ ( R + δ +2 D +2 L ) ρ h Γ . For every ρ > we set T = log ρ . We first want to show that if z ∈ ∂X and R ≥ then Shad x ( ξ z ( T + R ) , R ) ⊆ B ( z, e − T ) . Indeed if w ∈ ∂X is a pointsuch that the σ -geodesic ray ξ w = [ x, w ] passes through B ( ξ z ( T + R ) , R ) then d ( ξ z ( T + R ) , ξ w ( T + R )) < R and by Lemma 5.4 we get ( z, w ) x > T .We take R = R + L + D , where R is the constant of the second stepand we conclude that Shad x ( ξ z ( T + R ) , R ) is contained in B ( z, ρ ) . ByLemma 6.2 it is possible to find a geodesic line γ ∈ Geod (Λ(Γ)) such that d ( γ ( T + R ) , ξ z ( T + R )) ≤ L . Moreover there exists g ∈ Γ such that d ( gx, γ ( T + R )) ≤ D , implying Shad x ( gx, R ) ⊆ Shad x ( ξ z ( T + R ) , R ) ⊆ ( z, ρ ) . From the second step we obtain µ PS ( B ( z, ρ )) ≥ Q e − h Γ d ( x,gx ) . Fur-thermore d ( x, gx ) ≤ T + R + 2 L + 2 D, so finally µ PS ( B ( z, ρ )) ≥ Q e − h Γ ( R + δ +2 L +2 D ) ρ h Γ . The explicit description of the constants shows as they depend only on δ, h − , h + and D , proving (i’) and so the theorem.As a consequence we have a uniform asymptotic behaviour for the Γ -entropy of a discrete, non-elementary, quasiconvex-cocompact group of σ -isometries. Indeed by Lemma 7.2 and Proposition 6.15 we get T log x ∩ B ( x, T ) ≍ P ,r ,δ,D h Γ , where x ∈ QC-Hull (Λ(Γ)) . We remark that similar results can be obtainedfor the covering entropy, the Lipschitz-topological entropy, and the shadowdimension. This uniform convergence to the limit will be the key of thecontinuity results we will prove in the following sections.Actually for the Γ -entropy we can improve the rate of convergence fol-lowing again the ideas of [Coo93]. Theorem 7.8.
Let ( X, σ ) be a δ -hyperbolic GCB -space that is P -packed atscale r . Let Γ be a discrete, quasiconvex-cocompact group of σ -isometriesof X with codiameter ≤ D and x be a point of QC-Hull (Λ(Γ)) . Then thereexists
K > depending only on P , r , δ and D such that for all T ≥ itholds K · e T · h Γ ≤ Γ x ∩ B ( x, T ) ≤ K · e T · h Γ . Proof.
We denote by s = s ( P , r , δ, D ) the number given by (2.9), by R = R ( P , r , δ, D ) the number of Step 2 of Theorem 7.5, by Q the constantof Proposition 7.4 and by L the constant of Lemma 6.2 that depends onlyon δ . Moreover we set N = Pack (cid:0) R + 1 , s (cid:1) , which depends only on P , r , δ, D by Proposition 2.1. It is easy to check that if [ x, z ] ∩ B ( y, R ) = ∅ and [ x, z ] ∩ B ( y ′ , R ) = ∅ , where z ∈ ∂X and y, y ′ are points of X with | d ( x, y ) − d ( x, y ′ ) | ≤ , then d ( y, y ′ ) ≤ R + 1 . Therefore for every k ∈ N we have { y ∈ Γ x s.t. y ∈ A ( x, k, k + 1) and z ∈ Shad x ( y, R ) } ≤ N . Step 1.
For all k ∈ N it holds x ∩ B ( x, k ) ≤ QN e h Γ k . Let A j = Γ x ∩ A ( x, j, j + 1) . By the observation made before we concludethat among the set of shadows { Shad x ( y, R ) } y ∈ A j there are at least A j N disjoint sets. Thus ≥ µ PS [ y ∈ A j Shad x ( y, R ) ≥ A j N · Q e − h Γ ( j +1) , A j ≤ QN e h Γ ( j +1) for every j ∈ N . Finally we have x ∩ B ( x, k ) ≤ k − X j =0 A k ≤ QN e h Γ k . Step 2.
For all k ∈ N it holds x ∩ B ( x, k ) ≥ e − h Γ (21 δ +6 D +6 L ) e h Γ k . For every z ∈ Λ(Γ) we consider the σ -geodesic ray ξ z = [ x, z ] . Then byLemma 6.2 there exists γ ∈ Geod (Λ(Γ)) such that d ( ξ z ( t ) , γ ( t )) ≤ L for every t ≥ . Moreover for every t ≥ we can find y t ∈ Γ x such that d ( y t , γ ( t )) ≤ D , so d ( ξ z ( t ) , y t ) ≤ D + L . This implies that z ∈ Shad x ( y t , D + L ) and | d ( x, y t ) − t | ≤ D + L . Therefore for every t ≥ we can cover Λ(Γ) withshadows casted by points of Γ x at distance between t − D − L and t + D + L from x and with radius D + L . Choosing t = k − D − L we get Λ(Γ) ⊆ S y ∈ Γ x ∩ A ( x,k − D − L,k ) Shad x ( y, D + L ) . By the same argument of Lemma5.5 we have Shad x ( y, D + L ) ⊆ B ( z y , e − d ( x,y )+ D + L ) ⊆ B ( z y , e − k +3 D +3 L ) forevery y ∈ Γ x ∩ A ( x, k − D − L, k ) , where z y is the point at infinity of a σ -extension of the σ -geodesic segment [ x, y ] . So by Step 1 of Theorem 7.5we conclude µ PS (Λ(Γ)) ≤ x ∩ B ( x, k ) · e h Γ (21 δ +6 D +6 L ) e − h Γ k . The thesis follows by the bounded quantification of all the constantsinvolved in terms of P , r , δ and D . In this last section we will study stability properties under ultralimits andpointed Gromov-Hausdorff convergence. We denote by GCB ( P , r ) theclass of pointed GCB-spaces ( X, x, σ ) that are P -packed at scale r andby GCB ( P , r , δ ) its subclass made of δ -hyperbolic metric spaces. We recallthat both GCB ( P , r ) and GCB ( P , r , δ ) are closed under ultralimits, see[CS20a]. In particular for every sequence ( X n , x n , σ n ) ⊆ GCB ( P , r ) andany non-principal ultrafilter ω there exists a canonic way to define a bicomb-ing σ ω on the ultralimit space X ω in such a way that ( X ω , x ω , σ ω ) belongsto GCB ( P , r ) .In [CS20a] it is introduced the notion of ultralimit of groups: if ( X n , x n ) is a sequence of pointed metric spaces and if Γ n is a group of isometries of X n for every n , then a sequence { g n } , where g n ∈ Γ n , is said admissible if sup n d ( x n , g n x n ) < + ∞ . For every non-principal ultrafilter ω we have thatan admissible sequence defines an isometry g ω = ω - lim g n of the ultralimitspace X ω that acts as g ω y ω = ω - lim g n y n , where ( y n ) is any sequence suchthat ω - lim y n = y ω . The set of isometries of Γ ω defined by admissible se-quences is called the ultralimit group and it is denoted by Γ ω . For further54roperties of the ultralimit group we refer to [CS20a]. We just recall that if ( X n , x n , σ n ) ⊆ GCB ( P , r ) and if Γ n is σ n -invariant then Γ ω is σ ω -invariant. The boundary is stable under ultralimits.
Proposition 8.1.
Let ( X n , x n , σ n ) ⊆ GCB ( P , r , δ ) and let D x n ,a be astandard visual metric of parameter a and center x n on ∂X n . Let ω be anon-principal ultrafilter and let ( X ω , x ω , σ ω ) be the ultralimit of the sequence ( X n , x n , σ n ) . Then there exists a visual metric D x ω ,a of parameter a andcenter x ω on ∂X ω such that ω - lim( ∂X n , D x n ,a ) is isometric to ( ∂X ω , D x ω ,a ) . We observe that since the spaces ∂X n are compact with diameter at most then the ultralimit ω - lim ∂X n does not depend on the basepoints. Proof.
A point of ω - lim ∂X n is a class of a sequence of points ( z n ) ∈ ∂X n and each point z n is identified to the σ n -geodesic ray ξ n = [ x n , z n ] . Thesequence of σ n -geodesic rays ( ξ n ) defines a σ ω -geodesic ray ξ ω of X ω with ξ ω (0) = x ω which provides a point of ∂X ω . It is then defined the map Ψ : ω - lim ∂X n → ∂X ω that sends the sequence ( z n ) to the boundary pointidentified by the σ ω -geodesic ray ξ ω . Good definition.
We need to show that Ψ is well defined. Let ( z ′ n ) beanother sequence of points equivalent to ( z n ) , i.e. ω - lim D x n ,a ( z n , z ′ n ) = 0 .Since D x n ,a is a standard visual metric for every n this implies that for all ε > and for ω -a.e. ( n ) it holds ( z n , z ′ n ) x n > log ε =: T ε . By Lemma 5.4 wehave d ( ξ n ( T ε − δ ) , ξ ′ n ( T ε − δ )) ≤ δ and, by convexity of σ n , we have that d ( ξ n ( S η ) , ξ ′ n ( S η )) < η , where S η = η · T ε δ for all η > . This means thatfor every T ≥ and every η > we have d ( ξ n ( T ) , ξ ′ n ( T )) < η for ω -a.e. ( n ) .Since η is arbitrary we obtain that ξ ω and ξ ′ ω coincide up to time T for every T ≥ and therefore ξ ω = ξ ′ ω . Bijectivity.
The next step is to show that Ψ is bijective. It is clearlysurjective since ∂X ω = ∂X and every σ ω -geodesic ray of X ω is ultralimitof σ n -geodesic rays of X n by definition. Let us show it is injective: if twosequence of points ( z n ) , ( z ′ n ) have the same image under Ψ then for all T ≥ and for every η > we have that for ω -a.e. ( n ) the σ n -geodesic rays ξ z n and ξ z ′ n stay at distance less than η up to time T . By Lemma 5.4 weconclude that ( z n , z ′ n ) x n > T − η and therefore D x n ,a ( z n , z ′ n ) ≤ e − a ( T − η ) .Since this is true for ω -a.e. ( n ) we get ω - lim D x n ,a ( z n , z ′ n ) ≤ e − a ( T − η ) implying ω - lim D x n ,a ( z n , z ′ n ) = 0 , i.e. ( z n ) = ( z ′ n ) as elements of ω - lim ∂X n , by thearbitrariness of T and η . Homeomorphism
Let us show Ψ is continuous. Both ω - lim ∂X n and ∂X ω are metrizable, then it is enough to check the continuity on sequences ofpoints. We take a sequence ( z kn ) k ∈ N converging to ( z ∞ n ) in ω - lim ∂X n . Thismeans that for every ε > there exists k ε ≥ such that if k ≥ k ε then55 - lim D x n ,a ( z kn , z ∞ n ) < ε . Arguing as before we obtain that for every ε > there exists k ε ≥ such that for every fixed k ≥ k ε it holds ( z kn , z ∞ n ) x n ≥ log ε =: T ε for ω -a.e. ( n ) . Therefore by the same argument used before weconclude that for every T ≥ there exists k T ≥ such that for every fixed k ≥ k T then ξ z kn and ξ z ∞ n stay at distance at most up to time T for ω -a.e. ( n ) . So the same conclusion holds for ξ z kω and ξ z ∞ ω and by Lemma 5.4we have (Ψ( z kn ) , Ψ( z ∞ n )) x ω ≥ T − . This implies exactly that the sequence Ψ( z kn ) converges to Ψ( z ∞ n ) . To prove the continuity of the inverse map wesuppose Ψ( z kn ) converges to Ψ( z ∞ n ) . By similar arguments used before we getthat the σ ω -geodesic rays ξ kω and ξ ∞ ω stay at bounded distance up to time T ,provided k ≥ k T . So the same happens for ξ kn and ξ ∞ n for ω -a.e. ( n ) implyingonce again the convergence of ( z kn ) to ( z ∞ n ) . The metric on ∂X ω . Since Ψ is an homeomorphism we can endow ∂X ω with the metric induced by Ψ , i.e. D ( z ω , z ′ ω ) = ω - lim D x n ,a ( z n , z ′ n ) , where z n and z ′ n are sequences such that Ψ( z n ) = z ω and Ψ( z ′ n ) = z ′ ω . It remains toshow it is a visual metric. We show one of the two conditions since the otheris similar. We take z ω = Ψ( z n ) , z ′ ω = Ψ( z ′ n ) and we set D n := D x n ,a ( z n , z ′ n ) .By definition D ω = ω - lim D n = D ( z ω , z ′ ω ) . Since each D x n ,a is a standardvisual metric we get ( z n , z ′ n ) x n ≤ a log D n =: T n for every n and by Lemma5.4 we conclude that d ( ξ z n ( T n + 3 δ ) , ξ z n ( T n + 3 δ )) ≥ δ for every n . Thereare two possibilities: T ω := ω - lim T n is either + ∞ or a positive real number.In the first case we have D ω = 0 and so there is nothing to prove. In thesecond case we know that d ( ξ z ω ( T ω + 3 δ ) , ξ z ω ( T ω + 3 δ )) ≥ δ and so byLemma 5.4 we conclude that ( z ω , z ′ ω ) x ω < T ω + δ = a log D ω + δ , implying D ω < e δ e − a ( z ω ,z ′ ω ) xω .We denote by GCB qc ( P , r , δ ; D ) the class of -uples ( X, x, σ, Γ) suchthat ( X, x, σ ) ∈ GCB ( P , r , δ ) , Γ is a discrete, non-elementary, quasiconvex-cocompact group of σ -isometries of X with codiameter ≤ D and finally x ∈ QC-Hull (Λ(Γ)) . This class is closed under ultralimits.
Theorem 8.2.
Let ( X n , x n , σ n , Γ n ) ⊆ GCB qc ( P , r , δ ; D ) , ω be a non-principal ultrafilter and let ( X ω , x ω , σ ω , Γ ω ) be the ultralimit -uple.Then Ψ( ω - lim Λ(Γ n )) = Λ(Γ ω ) , where Ψ is the isometry of Proposition 8.1.Moreover Γ ω is a discrete, non-elementary, quasiconvex-cocompact group of σ ω -isometries of X ω with codiameter ≤ D and x ω ∈ QC-Hull (Λ(Γ ω )) .Proof. Let L be the constant of Lemma 6.2, depending only on δ . We fix asequence z n ∈ Λ(Γ n ) and we observe that by Lemma 6.2 and the cocompact-ness of the action of Γ n on QC-Hull (Λ(Γ n )) we can find a sequence ( g kn ) k ∈ N such that(a) g kn x n converges to z n when k tends to + ∞ ;(b) g n = id; 56c) d ( g kn x n , g k +1 n x n ) ≤ L + 2 D ;(d) d ( g kn x n , ξ z n ( k )) ≤ L + D .For every k ∈ N the sequence g kn is admissible by (b) and (c), so it definesa limit isometry g kω ∈ Γ ω . Moreover we have d ( g kω x ω , ξ Ψ( z n ) ( k )) ≤ L + D forevery k ∈ N , as follows by the definition of Ψ . This clearly implies that thesequence g kω x ω converges to Ψ( z n ) and so Ψ( z n ) ∈ Λ(Γ ω ) . In other words Ψ( ω - lim Λ(Γ n )) ⊆ Λ(Γ ω ) . It is easy to show that Γ ω acts on ω - lim Λ(Γ n ) by ( g n )( z n ) = ( g n z n ) and that the action commutes with Ψ . Moreoverthe set ω - lim Λ(Γ n ) is Γ ω -invariant and closed, so it is Ψ( ω - lim Λ(Γ n )) .The Γ ω -invariance is trivial, while if ( z kn ) k ∈ N ∈ ω - lim Λ(Γ n ) is a sequenceconverging to ( z ∞ n ) and z ∞ n / ∈ ω - lim Λ(Γ n ) then there exists ε > suchthat for ω -a.e. ( n ) we have D x n ,a ( z ∞ n , Λ(Γ n )) ≥ ε and this is a contradic-tion. Therefore the set Ψ( ω - lim Λ(Γ n )) is a closed Γ ω -invariant subset of ∂X ω , that implies it contains Λ(Γ ω ) and so the equality between these twosets. This also implies that ω - lim QC-Hull (Λ(Γ n )) = QC-Hull (Λ(Γ ω )) andso x ω ∈ QC-Hull (Λ(Γ ω )) .By Theorem 6.13, Proposition 6.15 and Example 5.8 of [CS20a] we knowthat Γ ω is a non-elementary and discrete group. Moreover for every twopoints y ω , y ′ ω ∈ QC-Hull (Λ(Γ ω )) there exist sequences of points y n , y ′ n ∈ QC-Hull (Λ(Γ n )) such that y ω = ω - lim y n and y ′ ω = ω - lim y ′ n and so there are g n ∈ Γ n such that d ( g n y n , y ′ n ) ≤ D . The sequence g n is clearly admissible soit defines an element g ω = ω - lim g n of Γ ω and d ( g ω y ω , y ′ ω ) ≤ D , implying thatthe action of Γ ω on QC-Hull (Λ(Γ)) is cocompact with codiameter ≤ D . In this section we will find sufficient conditions to ensure the continuity ofthe entropy under convergence of metric spaces. In general it is false that theupper (resp.lower) entropies of the ultralimit is the ultralimit of the upper(resp.lower) entropies of the spaces.
Example 8.3.
Let X be any complete, geodesically complete, CAT (0) met-ric space X that is P -packed at scale r and let X ′ be the metric spaceobtained by gluing a ray [0 , + ∞ ) to a point of X . The space X ′ is againcomplete, geodesically complete, CAT (0) and packed. We take the sequence ( X n , x n ) = ( X ′ , n ) , where n ∈ [0 , + ∞ ) . Clearly X ω is isometric to R withrespect to every non-principal ultrafilter ω , so h Cov ( X ω ) = h Cov ( X ω ) = 0 .On the other hand h Cov ( X n ) = h Cov ( X ) and h Cov ( X n ) = h Cov ( X ) for every n . The same holds for all the other definition of entropies.If we require an uniformity condition on the entropy function, as ex-plained in the following theorem, then we have continuity. Later we will sea relative version of this result. 57 heorem 8.4. Let ( X n , x n , σ n ) ⊆ GCB ( P , r ) and ω be a non-principalultrafilter. Suppose that for every n it holds T log Pack ( B ( x n , T ) , r ) ≍ B h n and that the threshold functions do not depend on n . Then the upper andlower covering entropies of X ω coincide and equals h ω = ω - lim h n . Remark 8.5.
We remark that:(i) under the assumptions of the theorem then for every n the upper andlower covering entropies coincide and h n is their common value. More-over, and that is the important hypothesis, the rate of convergence tothe limit is uniform in n .(ii) Furthermore by Proposition 3.1, Proposition 3.4, Theorem 4.2 and Re-mark 4.6, Proposition 5.2 and Theorem 5.6 the assumption of the heo-rem is equivalent, up to change B , to a control of the rate of convergenceto the limit of the functions definining the volume entropies, the Lip-schitz topological entropies, the shadow dimensions or the Minkowskidimensions. So, if one has a uniform control on the rate of convergenceof one of these functions then it has the continuity of all the entropies.Proof of Theorem 8.4. The first step is the following: we claim that for every T ≥ it holds ω - lim Pack ( B ( x n , T ) , r ) ≤ Pack ( B ( x ω , T ) , r ) ≤ ω - lim Pack ( B ( x n , T ) , r ) . Let y ω , . . . , y Nω be a maximal r -separated subset of B ( x ω , T ) . By LemmaA.8 of [CS20b] each y iω can be written as y iω = ω - lim y in with y in ∈ B ( x n , T ) .Since d ( y iω , y jω ) > r for every i = j and since they are a finite number thenfor ω -a.e. n it holds d ( y in , y jn ) > r for all i = j , so for ω -a.e. ( n ) there is a r -separated subset of B ( x n , T ) with at least N elements. This impliesPack ( B ( x ω , T ) , r ) ≤ ω - lim Pack ( B ( x n , T ) , r ) . Now let y n , . . . , y N n n be a maximal r -separated subset of B ( x n , T ) . Weconsider the set A ω = { ω - lim y i n n s.t. ≤ i n ≤ N n } . Clearly every element of A ω belongs to B ( x ω , T ) . Moreover for every twodistinct points y ω , z ω ∈ A ω it holds d ( y ω , z ω ) ≥ r . Indeed ω - lim y i n n = ω - lim y j n n if and only if ω ( { n ∈ N s.t. i n = j n } ) = 1 , otherwise for ω -a.e. n itholds d ( y i n n , y j n n ) > r . This implies that if y ω , z ω ∈ A ω are distinct pointsthen d ( y ω , z ω ) ≥ r > r , so A ω is a r -separated subset of B ( x ω , T ) .Since X ω is proper the set A ω is of finite cardinality N ω . We claim thatthe set I = { n ∈ N s.t. N n = N ω } satisfies ω ( I ) = 1 . In order to proveit we rename the elements of A ω as y ω , . . . , y N ω ω , where y kω = ω - lim y i kn n for58ome ≤ i kn ≤ N n . From what said before we know that for k = l we have ω ( { n ∈ N s.t. i kn = i ln } ) = 1 . So ω (cid:18) \ ≤ k
Let C be a class of pointed, proper metric spaces and h : C → R be a function. Suppose that C is closed under ultralimits and h is continuous under ultralimits, i.e. for every non-principal ultrafilter ω and every sequence ( X n , x n ) ∈ C it holds h ( X ω ) = ω - lim h ( X n ) . Suppose that ( X n , x n ) ⊆ C converges in the pointed Gromov-Hausdorff sense to ( X ∞ , x ∞ ) .Then X ∞ ∈ C and h ( X ∞ ) = lim n → + ∞ h ( X n ) .
59e need the following lemma.
Lemma 8.7.
Let a n be a bounded sequence of real numbers. Let a n j be asubsequence converging to ˜ a . Then there exists a non-principal ultrafilter ω such that ω - lim a n = ˜ a .Proof. The set { n j } j is infinite, then there exists a non-principal ultrafilter ω containing { n j } j (cp. [Jan17], Lemma 3.2). Moreover for every ε > there exists j ε such that for all j ≥ j ε it holds | a n j − ˜ a | < ε . The set ofindices where the inequality is true belongs to ω since the complementary isfinite. This implies exactly that ˜ a = ω - lim a n . Proof of Proposition 8.6.
We fix every non-principal ultrafilter ω . Since theclass C is made of proper metric spaces then X ω is isometric to X ∞ (cp.[CS20b],Proposition A.11). Therefore h ( X ∞ ) = h ( X ω ) = ω - lim h ( X n ) . This impliesthat ω - lim h ( X n ) does not depend on the ultrafilter ω . By the previouslemma we conclude that every converging subsequence of h ( X n ) has h ( X ∞ ) as a limit, i.e. lim inf n → + ∞ h ( X n ) = lim sup n → + ∞ h ( X n ) = h ( X ∞ ) .We now state the relative version of theorem 8.4. For every sequence ( X n , x n , σ m ) ∈ GCB ( P , r , δ ) , for every sequence of subsets C n ⊆ ∂X n andevery non-principal ultrafilter ω we denote by C ω the set Ψ( ω - lim C n ) , where Ψ is the map of Proposition 8.1. Theorem 8.8.
Let ( X n , x n , σ n ) ⊆ GCB ( P , r , δ ) , C n ⊆ ∂X n for every n and ω be a non-principal ultrafilter. Suppose that for every n it holds T log Pack ( B ( x n , T ) ∩ QC-Hull ( C n ) , r ) ≍ B h n and that the threshold functions do not depend on n . Then the upper andlower covering entropies of C ω coincide and equals h ω = ω - lim h n . Proof.
The proof is the same of Theorem 8.4. The only delicate point is thefirst estimate on the packing number. But by definition of C ω we observethat QC-Hull ( C ω ) = ω - lim QC-Hull ( C n ) , so that estimate can be proved inthe same way.The analogue of Remark 8.5 holds for Theorem 8.8. As a consequence weget the desired corollary on continuity of the critical exponent of quasiconvex-cocompact groups. Corollary 8.9.
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