ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR GEODESIC FLOWSgiommi/cusp_f.pdf · 2016-01-27 ·...

ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR

GEODESIC FLOWS

GODOFREDO IOMMI, FELIPE RIQUELME, AND ANIBAL VELOZO

Abstract. In this paper we study the geodesic flow for a particular class ofRiemannian non-compact manifolds with variable pinched negative sectionalcurvature. For a sequence of invariant measures we are able to prove resultsrelating the loss of mass and bounds on the measure entropies. We computethe entropy contribution of the cusps. We develop and study the correspond-ing thermodynamic formalism. We obtain certain regularity results for thepressure of a class of potentials. We prove that the pressure is real analyticuntil it undergoes a phase transition, after which it becomes constant. Ourtechniques are based on the one side on symbolic methods and Markov parti-tions and on the other on geometric techniques and approximation propertiesat level of groups.

1. Introduction

This paper is devoted to study thermodynamic formalism for a class of geodesicflows defined over non-compact manifolds of variable pinched negative curvature.These flows can be coded with suspension flows defined over Markov shifts, albeiton a countable alphabet. This paper addresses two di!erent but related prob-lems where the non-compactness of the ambient manifold plays a fundamental role.Inspired in some recent results proved in the context of homogeneous dynamics([ELMV, EKP]), we establish properties that relate the escape of mass of a se-quence of invariant probability measures for the geodesic flow with its measuretheoretic entropies (see section 5). The second goal of the paper is to construct aclass of potentials for which the pressure exhibits a phase transition (see section 6).To obtain these results we combine both geometric and symbolic methods.

The class of manifolds that we will be working on along the paper were introducedin [DP]. These manifolds are obtained as the quotient of a Hadamard manifoldwith an extended Schottky group (see sub-section 4.2 for precise definitions). Thisclass of groups have parabolic elements of rank 1, therefore the manifolds are non-compact. It was shown in [DP] that the geodesic flow over the unit tangent bundleof those manifolds can be coded as suspension flows over countable Markov shifts.The existence of a Markov coding for the geodesic flow is essential for ours results.

The idea of coding a flow in order to describe its dynamical and ergodic propertieshas long history and a great deal of interesting and important results have beenobtained with these methods. Probably, some of the most relevant results using this

Date: January 27, 2016.2010 Mathematics Subject Classification. 15A15; 26E60, 37A30, 60B20, 60F15.G.I. was partially supported by the Center of Dynamical Systems and Related Fields codigo

ACT1103 and by Proyecto Fondecyt 1150058.F.R. was supported by Programa de Cooperacion Cientıfica Internacional CONICYT-CNRS

codigo PCCI 14009.1

2 G. IOMMI, F. RIQUELME, AND A. VELOZO

technique are related to counting closed geodesics and also estimating the rate atwhich they grow [PP]. A landmark result is the construction of Markov partitionsfor Axiom A flows defined over compact manifolds done by Bowen [Bo1] and Ratner[Ra]. They actually showed that Axiom A flows can be coded with suspension flowsdefined over sub-shifts of finite type on finite alphabets with a regular (Holder) rooffunctions. The study in the non-compact setting is far less developed. However,some interesting results have been obtained. Recently, Hamenstadt [H] and alsoBufetov and Gurevich [BG] have coded Teichmuller flows with suspension flowsover countable alphabets and using this representation have proved, for example,the uniqueness of the measure of maximal entropy. Another important examplefor which codings on countable alphabets have been constructed is a type of Sinaibilliards [BS1, BS2].

As mentioned before, one goal of the paper is to investigate the loss of mass ofsequences of invariant measures for the geodesic flow. Recently, the loss of masshas been studied for the modular surface in [ELMV]. Despite being a particularcase, the method displayed in [ELMV] is quite flexible and has the advantage itcan be understood purely geometrically. A more general situation is studied in[EKP], the results apply to any geodesic flow on a finite volume hyperbolic surface(with respect to the constant sectional curvature metric). We begin introducingthe notion of entropy at infinity of a dynamical system defined over a non-compacttopological space.

Definition 1.1. Let Y be a non-compact topological space and S “ pStqtPR : Y Ñ Ya continuous flow. We define the “entropy at infinity” of the dynamical system asthe number

h8pY, tStuq “ supt!nuá0

lim supnÑ8

h!npSq,

where the supremum is taken over all the sequences of invariant probability mea-sures for the flow converging in the vague topology to the zero measure. If no suchsequence exists we set h8pY, tStuq “ 0. Here h!pSq denotes the measure-theoreticentropy of a probability S-invariant measure !.

Recall that the total mass of probability measures is not necessarily preservedunder vague convergence (as opposite to weak convergence). Note that Definition1.1 can be extended to more general group actions whenever an entropy theory hasbeen developed for the group in consideration. Amenable groups are a classicalexample of such.

In this paper we are able to compute h8pT 1X{", pgtqq, where X is a Hadamardmanifold with pinched negative sectional curvature, " is an extended Schottkygroup generated by N1 hyperbolic isometries and N2 parabolic ones, and pgtq isthe geodesic flow on the unit tangent bundle T 1X{" (see sub-section 4.2 for precisedefinitions). Define

"p,max :“ maxt"pi , 1 ! i ! N2u,where "pi is the critical exponent of the Poincare series of the group " pi # and piis a parabolic element of ". We prove that

h8pT 1X{", pgtqq “ "p,max.

It worth mentioning "p,max is strictly less than the topological entropy of the ge-odesic flow. In our context, the non-compact pieces of our space are modeled bycusps. That is why we refer to this quantity as entropy in the cusps. More concretely

ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR GEODESIC FLOWS 3

we prove that if a sequence of measures are dissipating through the cusps, then theentropy contribution of the sequence is at most "p,max. In [EKP] it is proven thath8p"zG,Aq “ htop{2 where G is a connected semisimple Lie group of real rank 1with finite center, " a lattice inG and A a one-parameter subgroup of diagonalizableelements over R acting by right multiplication. In particular h8pT 1S, pgtqq “ 1{2,where S is a hyperbolic surface with finite volume. We also obtain results in thecase where the sequence of measures keep some mass at the limit. Our bounds areless concrete than the analog result in the homogeneous dynamical case though.The following is one of our main results and gives the calculation of the entropyin the cusps mentioned before. The manifold under consideration is assumed tosatisfy some mild technical conditions, denoted by p‹q. This condition is explainedin detail in subsection 4.2.

Theorem 1.2. Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group of isometries of X with propertyp‹q. Assume that the derivatives of the sectional curvature are uniformly bounded.Then, for every c # "p,max there exists a constant m “ mpcq # 0, with the followingproperty: If p!nq is a sequence of pgtq-invariant probability measures on T 1X{"satisfying h!npgq $ c, then for every vague limit !n á !, we have

!pT 1X{"q $ m.

In particular if !n á 0 then lim suph!npgq ! "p,max. Moreover, the value "p,max

is optimal in the following sense: there exists a sequence p!nq of pgtq-invariantprobability measures on T 1X{" such that h!npgq Ñ "p,max and !n á 0.

We believe similar results can be obtained for the geodesic flow in the geometri-cally finite case (maybe under some strong conditions on the curvature).

Our second goal is to study regularity properties of the pressure function. Inorder to do so, we make strong use of the symbolic coding that the geodesic flowhas in the manifolds we are considering. The idea of using symbolic dynamics tostudy thermodynamic formalism of flows of geometric nature can be traced backto the work of Bowen and Ruelle [BR]. They studied in great detail ergodic theoryand thermodynamic formalism for Axiom A flows defined on compact manifolds.The techniques they used were symbolic in nature and were based on the symboliccodings obtained by Bowen [Bo1] and Ratner [Ra]. In this work we follow thisstrategy. We stress, however, that our symbolic models are non-compact. Thereare several di#culties related to the lack of compactness that have to be addressed,but also new phenomena are observed.

To begin with, in sub-section 6 we propose a definition of topological pressure,P p¨q, that satisfies not only the variational principle, but also an approximation bycompact invariant sets property. These provide symbolic proofs to results obtainedby di!erent (non-symbolic) methods in far more general settings by Paulin, Pollicottand Schapira [PPS]. The strength of our approach is perhaps better appreciated inour regularity results for the pressure (sub-section 6.2). Note that the techniquesin [PPS] do not provide these type of results. We say that the pressure functiont %Ñ P ptfq has a phase transition at t “ t0 if it is not analytic at that point. Itreadily follows from work by Bowen and Ruelle [BR] that the pressure for AxiomA flows and regular potentials is real analytic and hence has no phase transitions.Regularity properties of the pressure of geodesic flows defined on non-compact


manifolds, as far as we know, have not been studied, with the exception of thegeodesic flow defined on the modular surface (see [IJ, Section 6]).

There is a general strategy used to study regularity properties of the pressureof maps and flows with strong hyperbolic or expanding properties in most of thephase space but not in all of it. Indeed, if there exists a subset of the phase spaceB & X for which the restricted dynamics is not expansive and its entropy equalto A, then it is possible to construct potentials f : X Ñ R for which the pressurefunction has the form

(1) P ptfq :“#real analytic, strictly decreasing and convex if t " t1;A if t # t1.

Well known examples of this phenomena include the Manneville-Pomeau map (seefor example [Sa2]) in which the set B consist of a parabolic fixed point and thereforeA “ 0. The potential considered is the geometrical one: ´ log |T 1|. Similar resultsfor multimodal maps have been obtained, for example, in [DT, IT1, PR]. In thiscase the set B corresponds to the post-critical set and A “ 0. Examples of maps inwhich A # 0 have been studied in [DGR, IT2]. For suspension flows over countableMarkov shifts, similar examples were obtained in [IJ]. However, in that case thenumber A, which we denote by s8, remained unexplained for. In this paper weshow that the entropy at infinity of a suspension flow over a countable Markov shiftcorresponds to s8 (see Corollary 3.8).

In the case of geodesic flows, roughly speaking we are considering the set B asthe union of the cusps of the manifold. More interestingly, as we mentioned beforewe are able to compute the entropy contributions of the cusps in the geodesic flow.In sub-section 6.2 we construct a class of potentials, that we denote by F , forwhich the pressure exhibits similar behaviour as in equation (1). In those examplesA “ "p,max. Note that it is possible for t1 to be infinity and in that case the pressureis real analytic. The following is the precise statement,

Theorem 1.3. Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group of isometries of X with propertyp‹q. Assume that the derivatives of the sectional curvature are uniformly bounded.If f P F , then

(1) For every t P R we have that Pgptfq $ "p,max.(2) We have that limtÑ´8 Pgptfq “ "p,max.(3) Let t1 :“ suptt P R : Pgptfq “ "p,maxu, then

Pgptfq “#"p,max if t " t1;real analytic, strictly convex, strictly increasing if t # t1.

(4) If t # t1, the potential tf has a unique equilibrium measure. If t " t1 it hasno equilibrium measure.

In order to prove this result we need to relate symbolic quantities with geomet-rical ones. This is achieved in Theorem 4.12 in which a symbolic parameter of thesuspension flow, the number s8, is proven to be equal to the geometric parameterof the group "p,max. We stress that when coding a flow a great deal of geometricinformation is lost. With this result we are able to recover part of it.


Acknowledgements. We thank B. Schapira for a careful reading of the paperand for several interesting and useful comments. We also thank F. Dal’bo andF. Ledrappier for several enlighten and interesting discussions around the subjecttreated in this article. This work started when the second and third authors werevisiting Pontificia Universidad Catolica de Chile, they are very grateful for the greatconditions and hospitality received in PUC Mathematics Department.

2. Preliminaries on thermodynamic formalism and suspension flows

This section is devoted to provide the necessary background on thermodynamicformalism and on suspension flows required on the rest of the article.

2.1. Thermodynamic formalism for countable Markov shifts. Let M be anincidence matrix defined on the alphabet of natural numbers. The associated onesided countable Markov shift p$`,#q is the set

$` :“ tpxnqnPN : Mpxn, xn`1q “ 1 for every n P Nu ,together with the shift map # : $` Ñ $` defined by #px1, x2, . . . q “ px2, x3, . . . q.A standing assumption we will make throughout the article is that p$`,#q is topo-logically mixing. We equip $` with the topology generated by the cylinder sets

Ca1¨¨¨an “ tx P $` : xi “ ai for i “ 1, . . . , nu.We stress that, in general, $` is a non-compact space. Given a function $ : $` Ñ Rwe define the n´th variations of $ by

Vnp$q :“ supt|$pxq ´ $pyq| : x, y P $`, xi “ yi for i “ 1, . . . , nu,where x “ px1x2 ¨ ¨ ¨ q and y “ py1y2 ¨ ¨ ¨ q. We say that $ has summable variationif

!8n“1 Vnp$q "8 . We say that $ is locally Holder if there exists % P p0, 1q such

that for all n $ 1, we have Vnp$q ! Op%nq.This section is devoted to recall some of the notions and results of thermodynamic

formalism in this setting. The following definition was introduced by Sarig [Sa1]based on work by Gurevich [Gu].

Definition 2.1. Let $ : $` Ñ R be a function of summable variation. The Gure-vich pressure of $ is defined by

P p$q “ limnÑ8

1

nlog

ÿ

x:"nx“x

exp

˜n´1ÿ

i“0

$p#ixq¸&Ci1

pxq,

where &Ci1pxq is the characteristic function of the cylinder Ci1 & $`.

It is possible to show (see [Sa1, Theorem 1]) that the limit always exists andthat it does not depend on i1. The following two properties of the pressure will berelevant for our purposes (see [Sa1, Theorems 2 and 3] and [IJT, Theorem 2.10]).If $ : $` Ñ R is a function of summable variations, then

(1) (Approximation property)

P p$q “ suptPKp$q : K P Ku,where PKp$q is the classical topological pressure on K (see [Wa, Chapter9]) and

K :“ tK & $` : K ‰ H compact and #-invariantu.


(2) (Variational Principle) Denote by M" is the space of #´invariant probabil-ity measures and by hµp#q the entropy of the measure µ (see [Wa, Chapter4]). If $ : $` Ñ R is a function of summable variation then,

P"p$q “ sup

"hµp#q `

"$dµ : µ P M" and ´

"$dµ " 8

*.

A measure µ P M" attaining the supremum, that is, P"p$q “ hµp#q ` #$dµ is

called equilibrium measure for $. A potential of summable variations has at mostone equilibrium measure (see [BuS, Theorem 1.1]).

It turns out that under a combinatorial assumption on the incidence matrix M ,which roughly means to be similar to a full-shift, the thermodynamic formalism iswell behaved.

Definition 2.2. We say that a countable Markov shift p$`,#q, defined by thetransition matrix Mpi, jq with pi, jq P N ˆ N, satisfies the BIP (Big Images andPreimages) condition if and only if there exists tb1, . . . , bnu & N such that for everya P N there exists i, j P N with Mpbi, aqMpa, bjq “ 1.

The following theorem summarises results proven by Sarig in [Sa1, Sa2] and byMauldin and Urbanski, [MU], where they show that thermodynamic formalism inthis setting is similar to that observed for sub-shifts of finite type on finite alphabets.

Theorem 2.3. Let p$`,#q be a countable Markov shift satisfying the BIP conditionand $ : $` Ñ R a non-positive locally Holder potential. Then, there exists s8 # 0such that pressure function t Ñ P"pt$q has the following properties

P"pt$q “#

8 if t " s8;

real analytic if t # s8.

Moreover, if t # s8, there exists a unique equilibrium measure for t$.

2.2. Suspension flows. Let p$`,#q be a topologically mixing countable Markovshift and ' : $` Ñ R` a function of summable variations bounded away from zero.Consider the space

(2) Y “ tpx, tq P $` ˆ R : 0 ! t ! 'pxqu,with the points px, 'pxqq and p#pxq, 0q identified for each x P $`. The suspensionsemi-flow over # with roof function ' is the semi-flow % “ p$tqt!0 on Y defined by

$tpx, sq “ px, s ` tq whenever s ` t P r0, 'pxqs.In particular,

$#pxqpx, 0q “ p#pxq, 0q.2.3. Invariant measures. Let pY,%q be a suspension semi-flow defined over acountable Markov shift p$`,#q with roof function ' : $` Ñ R` bounded awayfrom zero. Denote by M! the space of invariant probability measures for the flow.It follows form a classical result by Ambrose and Kakutani [AK] that every measure! P M! can be written as

(3) ! “ pµ ˆ mq|Ypµ ˆ mqpY q ,

where µ P M" and m denotes the one dimensional Lebesgue measure. Whenp$`,#q is a sub-shift of finite type defined on a finite alphabet the relation in


equation (3) is actually a bijection between M" and M!. If p$`,#q is a countableMarkov shift with roof function bounded away from zero the map defined by

! %Ñ pµ ˆ mq|Ypµ ˆ mqpY q ,

is surjective. However, it can happen that pµˆmqpY q “ 8. In this case the imagecan be understood as an infinite invariant measure.

The case which is more subtle is when the roof function is only assumed to bepositive. We will not be interested in that case here, but we refer to [IJT] for adiscussion on the pathologies that might occur.

2.4. Of flows and semi-flows. In 1972 Sinai [Si, Section 3] observed that inorder to study thermodynamic formalism for suspension flows it su#ces to studythermodynamic formalism for semi-flows. Denote by p$,#q a two-sided countableMarkov shift. Recall that two continuous functions $, ( P Cp$q are said to becohomologous if there exists a bounded continuous function ) P Cp$q such that$ “ ( ` ) ˝ # ´ ). The relevant remark is that thermodynamic formalism for twocohomologous functions is exactly the same. Thus, if every continuous function$ P Cp$q is cohomologous to a continuous function ( P Cp$q which only dependsin future coordinates then thermodynamic formalism for the flow can be studied inthe corresponding semi-flow. The next result formalises this discussion.

Proposition 2.4. If $ P Cp$q has summable variation, then there exists ( P Cp$qof summable variation cohomologous to $ such that (pxq “ (pyq whenever xi “ yifor all i $ 0 (that is, ( depends only on the future coordinates).

Proposition 2.4 has been proved with di!erent regularity assumptions in thecompact setting and in the non-compact case in [Da, Theorem 7.1].

2.5. Abramov and Kac. The entropy of a flow with respect to an invariant mea-sure can be defined as the entropy of the corresponding time one map. The followingresult was proved by Abramov [Ab].

Proposition 2.5 (Abramov). Let ! P M! be such that ! “ pµˆmq|Y {pµˆmqpY q,where µ P M" then the entropy of ! with respect to the flow, that we denote h!p%q,satisfies

(4) h!p%q “ hµp#q#'dµ

.

In Proposition 2.5 a relation between the entropy of a measure for the flow anda corresponding measure for the base dynamics was established. We now provea relation between the integral of a function on the flow with the integral of arelated function on the base. Let f : Y Ñ R be a continuous function. Define&f : $` Ñ R by

&f pxq :“" #pxq

0fpx, tq dt.

Proposition 2.6 (Kac’s Lemma). Let f : Y Ñ R be a continuous function and! P M! an invariant measure that can be written as

! “ µ ˆ m

pµ ˆ mqpY q ,


where µ P M", then "

Yf d! “

#" &f dµ#" ' dµ

.

Propositions 2.5 and 2.6 together with the relation between the spaces of invari-ant measures for the flow and for the shift established by Ambrose and Kakutani(see subsection 2.3) allow us to study thermodynamic formalism for the flow bymeans of the corresponding one on the base.

2.6. Thermodynamic formalism for suspension flows. Let p$`,#q be a topo-logically mixing countable Markov shift and ' : $` Ñ R a positive functionbounded away from zero of summable variations. Denote by pY,%q the suspen-sion semi-flow over p$`,#q with roof function ' . Thermodynamic formalism hasbeen studied in this context by several people with di!erent degrees of gener-ality: Savchenko [Sav], Barreira and Iommi [BI], Kempton [Ke] and Jaerisch,Kessebohmer and Lamei [JKL]. Thermodynamic formalism for suspension flowswhere the base p$`,#q is a sub-shift of finite type defined on a finite alphabethas been studied, for example, in [BR, PP]. The next result provides equivalentdefinitions for the pressure, P!p¨q, on the flow.

Theorem 2.7. Let f : Y Ñ R be a function such that &f : $` Ñ R is of summablevariations. Then the following equalities hold

P!pfq :“ limtÑ8

1

tlog

¨

˝ÿ

$spx,0q“px,0q,0"s#t

exp

ˆ" s

0fp$kpx, 0qq dk

˙&Ci0

pxq˛

‚

“ inftt P R : P"p&f ´ t'q ! 0u “ suptt P R : P"p&f ´ t'q $ 0u“ suptP!|Kpfq : K P Kp%qu,

where Kp%q denotes the space of compact %´invariant sets.

In particular, the topological entropy of the flow is the unique number htopp%qsatisfying

(5) htopp%q “ inftt P R : P p´t'q ! 0u.Note that in this setting the Variational Principle also holds (see [BI, JKL, Ke,Sav]).

Theorem 2.8 (Variational Principle). Let f : Y Ñ R be a function such that&f : $` Ñ R is of summable variations. Then

P!pfq “ sup

"h!p%q `

"

Yf d! : ! P M! and ´

"

Yf d! " 8

*.

A measure ! P M! is called an equilibrium measure for f if

P!pfq “ h!p%q `"f d!.

It was proved in [IJT, Theorem 3.5] that potentials f for which &f is locallyHolder have at most one equilibrium measure. Moreover, the following result (see[BI, Theorem 4]) characterises functions having equilibrium measures.


Theorem 2.9. Let f : Y Ñ R be a continuous function such that &f is of summablevariations. Then there is an equilibrium measure !f P M! for f if and only if wehave that P"p&f ´ P!pfq'q “ 0 and there exists an equilibrium measure µf P M"

for &f ´ P!pfq' such that#'dµf " 8.

Remark 2.10. We stress that the situation is more complicated when ' is notassumed to be bounded away from zero. For results in that setting see [IJT].

3. Entropy and escape of mass

Over the last few years there has been interest, partially motivated for its connec-tions with number theory, in studying the relation between entropy and the escapeof mass of sequences of invariant measures for diagonal flows on homogenous spaces(see [EKP, ELMV, KLKM]). Some remarkable results have been obtained bound-ing the amount of mass that an invariant measure can give to an unbounded part ofthe domain (a cusp) in terms of the entropy of the measure (see for example [EKP,Theorem A] or [KLKM, Theorem 1.3]). The purpose of this section is to provesimilar results in the context of suspension flows defined over countable Markovshifts. As we will see, the proofs in this setting suggest a geometrical interpretationthat we pursue in section 5.

Let p$`,#q be a topologically mixing countable Markov shift of infinite topolog-ical entropy and ' : $` Ñ R` a potential of summable variations bounded awayfrom zero. Denote by pY,%q the associated suspension flow, which we assume tohave finite topological entropy. Note that since p$`,#q has infinite entropy and' is non-negative, the entropy htopp%q of the flow satisfies P"p´hp%q'q ! 0 (seeequation (5)). Therefore, there exists a real number s8 P p0, htopp%qs such that

P"p´t'q “#infinite if t " s8;

finite if t # s8.

As it turns out the number s8 will play a crucial role in our work.For geodesic flows defined in non-compact manifolds there are vectors that escape

through the cusps, they do not exhibit any recurrence property. That phenomenais impossible in the symbolic setting, every point will return to the base after sometime. The following definition describes the set of points that escape on average(compare with an analogous definition given in [KLKM]).

Definition 3.1. We say that a point px, tq P Y escapes on average if

limnÑ8

1

n

n´1ÿ

i“0

'p#ixq “ 8.

We denote the set of all points which escape on average by EAp'q.Remark 3.2. Note that if ! P M! is ergodic and ! “ pµ ˆ mq{pµ ˆ mqpY q withµ P M" then Birkho! ’s theorem implies that

limnÑ8

1

n

n´1ÿ

i“0

'p#ixq “"'dµ.

Thus, no measure in M! is supported on EAp'q. We can, however, describe thedynamics of the set EAp'q studying sequences of measures !n P M! such that the


associated measures µn P M" satisfy

limnÑ8

"'dµn “ 8.

In our first result we show that a measure of su#ciently large entropy can notgive too much weight to the set of points for which the return time to the base isvery high. More precisely,

Theorem 3.3. Let c P ps8, htopp%qq. There exists a constant C # 0 such that forevery ! P M! with h!p%q $ c, we have that

"'dµ ! C.

Proof. Let ! P M! with h!p%q “ c and let µ P M" be the invariant measuresatisfying ! “ pµ ˆ mq{ppµ ˆ mqpY qq. By the Abramov formula we have

hµp#q ´ c

"'dµ “ 0.

We will consider the straight line Lptq :“ hµp#q ´ t#'dµ. Note that Lpcq “ 0 and

Lp0q “ hµp#q. Let s P ps8, cq. Note that P"p´s'q "8 and by the variationalprinciple Lpsq ! P"p´s'q. This remark readily implies a bound on the slope ofLptq. Indeed,

"'dµ ! P"p´s'q

c ´ s.

Thus the constant C “ P"p´s'q{pc ´ sq satisfies the theorem. In order to obtainthe best possible constant we have to compute the infimum of the function definedfor s P ps8, cq by

s %Ñ P"p´s'qc ´ s

.

!

Remark 3.4. We stress that the constant C in Theorem 3.3 depends only on theentropy bound c and not on the measure !.

Remark 3.5. An implicit assumption in Theorem 3.3 is that s8 " htopp%q. InSection 4 we will see that in the geometrical context of geodesic flows this assump-tion has a very natural interpretation. Indeed, it will be shown to be equivalentto the parabolic gap property (see [DP, Section III] or Definition 4.13 for precisedefinitions).

Corollary 3.6. If p$`,#q is a Markov shift defined countable alphabet satisfyingthe BIP condition then the best possible constant C P R in Theorem 3.3 is given by

C “ P"p´sm'qc ´ sm

,

where sm P R is such that the equilibrium measure µsm for ´sm' satisfies

c “ hµsmp#q#

'dµsm

.


Proof. Since the system satisfies BIP condition the function P"p´s'q, when finite,is di!erentiable (see Theorem 2.3). Moreover, its derivative is given by (see [Sa3,Theorem 6.5]),

d

dsP"p´s'q

ˇˇs“sm

“ ´"'dµsm ,

where µsm is the (unique) equilibrium measure for ´sm' . The critical points of the

function s %Ñ P!p´s#qc´s are those which satisfy

(6) pc ´ sqP 1"p´s'q ` P"p´s'q “ 0.

Equivalently,

´pc ´ sq"'dµs ` hµsp#q ´ s

"'dµs “ 0.

Therefore, equation (6) is equivalent to

c “ hµsp#q#'dµs

.

!

In the next Theorem we prove that the entropy of the flow on EAp'q is boundedabove by s8 and that, under some additional assumptions, it is actually equal toit. This result could be thought of as a symbolic estimation for the entropy of aflow in a cusp. Theorem 3.7 is a refined version of a result first observed in [FJLR,Lemma 2.5] and used in the context of suspension flows in [IJ].

Theorem 3.7. Let p!nqn P M! be an sequence of invariant probability measurefor the flow of the form

!n “ µn ˆ m

pµn ˆ mqpY q ,

where µn P M". If limnÑ8#'dµn “ 8 then

lim supnÑ8

h!np%q ! s8.

Moreover, if s8 " htopp%q, then there exists a sequence p!nqn P M! such thatlimnÑ8

#'dµn “ 8 and

limnÑ8

h!np%q “ s8.

Proof. Observe that the first claim is a direct consequence of Theorem 3.3. Letus construct now a sequence p!nqn P M! with limnÑ8

#'dµn “ 8 such that

limnÑ8 h!np%q “ s8. First note that it is a consequence of the approximationproperty of the pressure, that there exists a sequence of compact invariant setspKN qN & $ such that limNÑ8 PKN p´t'q “ P"p´t'q. In particular, for everyn P N we have that

(7) limNÑ8

PKN p´ ps8 ´ 1{nq 'q “ 8.

For the same reason, for any n P N and N P N we have that

(8) PKN p´ ps8 ` 1{nq 'q ! P" p´ ps8 ` 1{nq 'q "8 .

Thus, given n P N there exists N P N such that

n2 " PKN p´ ps8 ´ 1{nq 'q ´ PKN p´ ps8 ` 1{nq 'q2{n .


Since the function t %Ñ PKN p´t'q is real analytic, by the mean value theorem thereexists tn P rs8 ´ 1{n, s8 ` 1{ns, such that P 1

KNp´tn'q # n2. Denote by µn the

equilibrium measure for ´tn' in KN . We have that

n2 " P 1KN

p´tn'q “"'dµn.

In particular the sequence pµnqn satisfies

limnÑ8

"'dµn “ 8.

Since s8 " htopp%q we have that for n P N large enough

hµnp#q ´ tn

"'dµn # 0.

In particular

tn " hµnp#q#'dµn

.

Since tn P ps8 ´ 1{n, s8 ` 1{nq we have that

(9) s8 “ limnÑ8

tn ! limnÑ8

hµnp#q#'dµn

“ limnÑ8

h!np%q.

But we already proved that the limit can not be larger than s8, thus the resultfollows. !

Corollary 3.8. If s8 " htopp%q, then the entropy at infinity h8pY,%q of thesuspension flow satisfies

h8pY,%q “ s8.

Proof. Let p!nq be a sequence of %-invariant probability measures such that !n á 0.Since

"fd!n “

#&fdµn#'dµn

,

the only way to have#fd!n Ñ 0 for every continuous function f P C0pY q with

compact support, is that

limnÑ8

"'dµn “ 8.

Hence, Theorem 3.7 implies that lim supnÑ8 h!np%q ! s8. On the other hand,Theorem 3.7 also says that there exists a sequence p!nq of %-invariant probabilitymeasures such that limnÑ8

#'dµn “ 8 and

limnÑ8

h!np%q “ s8.

Again, since#fd!n “

##fdµn##dµn

, we have that !n á 0, and the conclusion follows. !


4. The geodesic flow on Extended Schottky Groups

4.1. Some preliminaries in negative curvature. Let X be a Hadamard mani-fold with pinched negative sectional curvature, that is a complete simply connectedRiemannian manifold whose sectional curvature K satisfies ´b2 ! K ! ´1 (forsome fixed b $ 1). Denote by BX the boundary at infinity of X. Finally, denote byd the Riemannian distance on X. A crucial object on the study of the dynamics ofthe geodesic flow is the Busemann function. Let * P BX and x, y P X. For everygeodesic ray t %Ñ *t pointing to *, the limit

B%px, yq :“ limtÑ8

rdpx, *tq ´ dpy, *tqs,always exists, and is independent of the geodesic ray *t since X has negative sec-tional curvature. The Busemann function B : BX ˆ X2 Ñ R is the continuousfunction defined as Bp*, x, yq %Ñ B%px, yq. An (open) horoball based in * and pass-ing through x is the set of y P X such that B%px, yq # 0. In the hyperbolic case,when X “ D, an open horoball based in * P S1 and passing trough x P D is theinterior of an euclidean circle containing x and tangent to S1 at *.

Recall that every isometry ofX can be extended to a homeomorphism ofXYBX.A very important property of the Busemann function is the following. If $ : X Ñ Xis an isometry of X, then for every x, y P X, we have

(10) B$%p$x,$yq “ B%px, yq.Let o P X be a reference point, which is often called the origin of X. The

unit tangent bundle T 1X of X can be identified with B2X ˆ R, where B2X “pBX ˆ BXqzdiagonal, via Hopf’s coordinates. A vector v P T 1X is identified withpv´, v`, Bv` po,+pvqqq, where v´ (resp. v`) is the negative (resp. positive) end-point of the geodesic defined by v. Here + : T 1X Ñ X is the natural projection ofa vector to its base point. Observe that the geodesic flow pgtq : T 1X Ñ T 1X actsby translation in the third coordinate of this identification.

Another crucial object in this setting is the Poincare series. It is intimatelyrelated to the topological entropy of the geodesic flow.

Definition 4.1. Let " be a discrete subgroup of isometries of X and let x P X.The Poincare series P$ps, xq associated with " is defined by

P$ps, xq :“ÿ

gP$e´sdpx,gxq.

The critical exponent "$ of " is

"$ :“ inf ts P R : P$ps, xq " 8u .The group " is said to be of divergence type (resp. convergence type) if P$p"$, xq “8 (resp. P$p"$, xq "8 ).

Remark 4.2. As the sectional curvature of X is bounded from below, the criticalexponent is finite. Moreover, by the triangle inequality, it is independent of x P X.

The isometries of X are categorized in three types. Those fixing an unique pointin X called elliptic isometries. Those fixing an unique point in BX called parabolicisometries. And finally, those fixing uniquely two points in BX called hyperbolicisometries. For g a non-elliptic isometry of X, denote by "g the critical exponent


of the group " g #. If g is hyperbolic it is fairly straightforward to see that "g “ 0and that the group " g # is of divergence type. If g is parabolic, it was shown in[DP, Theorem III.1], that "g $ 1

2 .Let " be a discrete subgroup of isometries of X. Denote by ' the limit set of

", that is, the set ' “ " ¨ oz" ¨ o. The group " is non-elementary if ' containsinfinitely many elements. We recall the following fact proved in [DOP, Proposition2].

Theorem 4.3. Let " be a non-elementary discrete subgroup of isometries of aHadamard manifold X. If G is a divergence type subgroup of " and its limit set isstrictly contained in the limit set of ", then "$ # "G.

In particular, if " is a non-elementary discrete group of isometries and there is anelement g P " such that " g # is of divergence type, then "$ # ""g$ (see also [DP,Theorem III.1]). Note that a non-elementary group always contains a hyperbolicisometry (in fact, infinitely many non-conjugate of them), hence a non-elementarygroup " always satisfies "$ # 0.

We end this subsection giving an important relation between the topologicalentropy of the geodesic flow and the critical exponent of a group. Let X be aHadamard manifold with pinched negative sectional curvature and let " be a non-elementary free-torsion discrete subgroup of isometries of X. Denote by pgtq :T 1X{" Ñ T 1X{" the geodesic flow on the unit tangent bundle of the quotientmanifold X{". Otal and Peigne [OP, Theorem 1] proved that, if the derivatives ofthe sectional curvature are uniformly bounded, then the topological entropy htoppgqof the geodesic flow equals the critical exponent of the Poincare series of the group", that is

(11) htoppgq “ "$.

We stress the fact that the assumption on the derivatives of the sectional curvatureis crucial in order to compute the topological entropy of the geodesic flow. Thisassumption implies the Holder regularity of the strong unstable and stable foliations(see for instance [PPS, Theorem 7.3]), which is used in the proof of [OP, Theorem1].

4.2. The symbolic model for extended Schottky groups. In this section werecall the definition of an extended Schottky group. To the best of our knowledgethis definition has been introduced in 1998 by Dal’bo and Peigne [DP]. The basicidea is to extend the classical notion of Schottky groups to the context of manifoldswhere the non-wandering set of the geodesic flow is non-compact.

Let X be a Hadamard manifold as in subsection 4.1. Let N1, N2 be non-negativeintegers such that N1 ` N2 $ 2 and N2 $ 1. Consider N1 hyperbolic isometriesh1, ..., hN1 and N2 parabolic ones p1, ..., pN2 satisfying the following conditions:

(C1) For 1 ! i ! N1 there exist in BX a compact neighbourhood Chi of the at-tracting point *hi of hi and a compact neighbourhood Ch´1

iof the repelling

point *h`iof hi, such that

hipBXzCh´1i

q & Chi .


(C2) For 1 ! i ! N2 there exists in BX a compact neighbourhood Cpi of theunique fixed point *pi of pi, such that

@n P Z˚ pni pBXzCpiq & Cpi .

(C3) The 2N1 ` N2 neighbourhoods introduced in p1q and p2q are pairwise dis-joint.

(C4) The elementary parabolic groups " pi #, for 1 ! i ! N2, are of divergencetype.

The group " “" h1, ..., hN1 , p1, ..., pN2 # is a non-elementary free group whichacts properly discontinuously and freely on X (see [DP, Corollary II.2]). Such agroup " is called an extended Schottky group. Note that if N2 “ 0, that is the group" only contains hyperbolic elements, then " is a classical Schottky group and itsgeometric and dynamical properties are well understood. Indeed, in that case, thenon-wandering set ( & T 1X{" of the geodesic flow is compact, which implies thatpgtq|% is an Axiom A flow. If N2 $ 1, then X{" is a non-compact manifold and( is a non-compact subset of T 1X{". Figure 1 below is an example of a Schottkygroup acting on the hyperbolic disk D. It has two generators, one hyperbolic andthe other parabolic.

o

Cp‚

p

h

Figure 1. Schottky Group " “" h, p #.

‚

Ch´1

‚ ‚Ch

Let A˘ “ th˘11 , ..., h˘1

N1, p1, ..., pN2u. We now define an extra hypothesis that we

will use in large parts of this paper.

(C5) Let a1, a2 P A˘ be such that a1 ‰ a˘12 . Then, there exists a point o P X

such that for every * P Ca1 , we have

B%pa˘12 o, oq # 0.

In other words, we will assume that every horoball based in Ca1 and passing througha˘12 o contains the origin in his interior (see for instance Figure 2). This condition

is not very restrictive as we will see in Proposition 4.5.


‚o

‚z

‚*

‚

B%pz, oq

Figure 2.

For a P A˘ denote by Ua the convex hull in X Y BX of the set Ca.

Lemma 4.4. Let X be a Hadamard manifold with pinched negative sectional cur-vature and let " be an extended Schottky group. Fix o P X. Then, there exists anuniversal constant C # 0 (depending only on the generators of " and the fixed pointo) such that for every a1, a2 P A˘ satisfying a1 ‰ a˘1

2 , and for every x P Ua1 andy P Ua2 , we have

(12) dpx, yq $ dpx, oq ` dpy, oq ´ C.

Proof. Since Ca1 and Ca2 are disjoint, for every a1, a2 P A˘ satisfying a1 ‰ a˘12 ,

the same happens for the sets Ua1 and Ua2 . Let x P Ua1 and y P Ua2 . Thegeodesic segments ro, xs and ro, ys form an angle uniformly bounded below, hencedpx, yq $ dpx, oq ` dpy, oq ´ C for an universal constant C # 0. !Proposition 4.5. Let X be a Hadamard manifold with pinched negative sectionalcurvature and " “" h1, ..., hN1 , p1, ..., pN2 # an extended Schottky group. Then,for every o P X there exists an integer N $ 1 such that the group defined by" hN

1 , ..., hNN1

, pN1 , ..., pNN2# satisfies the Condition (C5).

Proof. Let a1, a2 P A˘ and * P Ca1 . Denote by Uan2the convex hull in X Y BX of

the set Can2, for n $ 1. Let zn P Uan

2be such that B%pzn, oq attains its minimum.

Since B is a continuous function and Ca1 is a compact set, it is su#cient to provethat B%pzn, oq # 0 for all n large enough. Consider p*tq “ ro, *tq the geodesic raystarting in o and pointing to *. Observe that there exists T # 0 such that for everyt $ T , we have that *t belongs to Ua1 . Since Ca1 and Ca2 are disjoint, by Lemma 4.4there exists a universal constant C # 0 such that dpzn, *tq $ dp*t, oq ` dpzn, oq ´C,for every t $ T . Therefore,

B%pzn, oq “ limtÑ`8

dpzn, *tq ´ dp*t, oq$ lim

tÑ`8dp*t, oq ` dpzn, oq ´ C ´ dp*t, oq

“ dpzn, oq ´ C.

Since dpzn, oq Ñ 8 as n Ñ 8, there exists a N $ 1 such that dpzn, oq # C for everyn $ N . In particular, we have B%pzn, oq # 0 for every n $ N and the conclusionfollows. !


In [DP] the authors proved that there exists a pgtq-invariant subset (0 of T 1X{",contained in the non-wandering set of pgtq, such that pgtq|%0 is topologically con-jugated to a suspension flow over a countable Markov shift p$,#q. The Theorembelow summarizes their construction together with some dynamical properties.

Theorem 4.6. Let X be a Hadamard manifold with pinched negative sectional cur-vature and let " be an extended Schottky group. Suppose that " satisfies Condition(C5). Then, there exists a pgtq-invariant subset (0 of T 1X{", a countable Markovshift p$,#q and a function ' : $ Ñ R, such that

(1) the function ' is locally Holder and is bounded away from zero,(2) the geodesic flow pgtq|%0 over (0 is topologically conjugated to the suspen-

sion flow over $ with roof function ' ,(3) the Markov shift p$,#q satisfies the BIP condition,(4) if N1 ` N2 $ 3, then p$,#q is topologically mixing.

Proof. Let A “ th1, ..., hN1 , p1, ..., pN2u and consider the symbolic space $ definedby

$ “ tpamii qiPZ : ai P A,mi P Z and ai`1 ‰ ai@i P Zu.

Note that the space $ is a sequence space defined on the countable alphabet tami :ai P A,m P Zu. Let '0 be the limit set ' minus the "-orbit of the fixed pointsof the elements of A. We denote by (0 the set of vectors in T 1X identified withp'0 ˆ '0zdiagonalq ˆ R via Hopf’s coordinates. Finally, define (0 :“ (0{", wherethe action of " is given by

( ¨ p*´, *`, sq “ p(p*´q, (p*`q, s ´ B%` po, (´10qq.Observe that (0 is invariant by the geodesic flow.

Fix now *0 P BXz $aPA Ca˘ , where Ca˘ “ Ca Y Ca´1 . Dal’bo and Peigne [DP,

Property II.5] established the following coding property: for every * P '0 thereexists an unique sequence ,p*q “ pami

i qi!1 with ai P A, mi P Z˚ and ai`1 ‰ aisuch that

limkÑ8

am11 ...amk

k *0 “ *.

For each a P A define '0a˘ “ '0 X Ca˘ and set B2'0 “ $

",#PA&‰'

'0&˘ ˆ '0

'˘ . For

any pair p*´, *`q P B2'0, if am is the first term of the sequence ,p*`q, define'p*`q “ B%` po, amoq and T p*´, *`q “ pa´m*´, a´m*`q. Define T # by the formula

T # p*´, *`, sq “ pT p*´, *`q, s ´ 'p*`qq.Observe that T # maps B2'0 ˆ R to itself. The set (0 can be identified with thequotient B2'0ˆR{ " T # #. In order to prove this claim, we first proceed to explainhow T # gives a suspension flow.

Let p*´, *`q P B2'0. Suppose that ,p*`q “ pamii q and let (n be the element

of " defined as (n “ am11 ...amn

n for n $ 1 and (0 “ Id. The geodesic determinedby p*´, *`q in X intersects the horosphere based in *` and passing through (noin only one point xn

o,%´,%` . Denote by vno,%´,%` the vector in T 1X based in xno,%´,%`

and pointing to *` (see Figure 3 below). Finally, denote by vno,%´,%` the projection

of vno,%´,%` over T 1X{". Set

S “ tv0o,%´,%` : p*´, *`q P B2'0u & T 1X{".


Condition (C5) implies that the first return time of every v0o,%´,%` P S is given

by 'p*`q (which is the distance between the horospheres based in *` and passingthrough o and (1o). Moreover, g#p%`qpv0o,%´,%` q “ v1o,%´,%` (see Figure 3).

*´

‚v1o,%´,%`

‚v0o,%´,%`

*`

'p*`q

‚o

Figure 3. Cross section for " “" h, p #

More generally, for every n $ 0 the first return time of vno,%´,%` P S is given by

'p(´1n *`q and

g#p(´1n %`qpvno,%´,%` q “ vn`1

o,%´,%` .

Note that for every vector v P (0 there exists a time t P R such that gtv belongs toS (otherwise, a lift of v to T 1M would have its positive endpoint v` in the "-orbitof a fixed point of an element in A). This fact give us the identification.

The coding property implies that the set B2'0 is identified with $ by consid-ering p*´, *`q as a bilateral sequence p,˚p*´q,,p*`qq. If ,p*´q “ pbni

i qi!1, wedefine ,˚p*´q as the sequence p..., b´n2

2 , b´n11 q, then p,˚p*´q,,p*`qq represent the

concatenated sequence. Let $` be the one sided symbolic space obtained from $by forgetting the negative time coordinates. We define the function ' : $` Ñ R as

'pxq “ 'p,´1pxqq “ B)´1pxqpo, amoq,where w : '0 Ñ $ is the coding function and am the first symbol in w´1pxq. Weextend ' to $ by making it independent of the negative time coordinates. Wetherefore have, that the geodesic flow pgtq on (0 can be coded as the suspensionflow on Y “ tpx, tq P $ ˆ R : 0 ! t ! 'pxqu{ „. This implies (2) in the conclusionof Theorem 4.6. Property (1) follows from Lemma 4.7 below.

Lemma 4.7. Under the hypothesis of Theorem 4.6, the function ' : $ Ñ R dependsonly on future coordinates, it is locally Holder and it is bounded away from zero.

Proof. The fact that it depends on the future is by definition and the regularityproperty was established in [DP, Lemma VII.]. Let x P $` and * “ ,´1pxq thepoint in '0 associated to x by the coding property. The function ' satisfies

'pxq “ B%po, amoq “ Ba´m%pa´mo, oq.The last term above is greater than 0 by Condition (C5). Indeed, note that a´m* “,´1p#xq R Ca˘ and a´mo is contained in the convex-hull of Ca˘ in X ˆ BX,


hence Condition (C5) applies directly. Since the domains Ca, for a P A˘, arecompact and B is continuous, there exists an uniform strictly positive lower boundfor Ba´m%pa´mo, oq. Thus ' is bounded away from zero. !Lemma 4.8. Under the hypothesis of Theorem 4.6 the countable Markov shiftp$,#q satisfies the BIP condition. Moreover, if N1 ` N2 $ 3, then the countableMarkov shift p$,#q is topologically mixing.

Proof. It is not hard to see that the set A satisfies the required conditions in orderfor p$,#q to be BIP (see definition 2.2). Suppose now that N1`N2 $ 3. Recall thatthe Markov shift p$,#q is topologically mixing if for every a, b P tami : ai P A,m P Zuthere exists Npa, bq P N such that for every n # Npa, bq there exists an admissibleword of length n of the form ai1i2 . . . in´1b. The set of allowable sequences is givenby

tpamii qiPZ : ai P A,mi P Z and ai`1 ‰ ai@i P Nu.

Since N1 ` N2 $ 3 then given any pair of symbols in tami : ai P A,m P Zu, sayam11 , am2

2 we can consider the symbol a3 R ta1, a2u. Hence the following words areadmissible:

am11 a3a1a3 . . . a1a

m22 , am1

1 a3a1a3a1 . . . a3am22 .

Thus, the system is topologically mixing. !Since Lemma 4.8 above shows the points (3) and (4), we have concluded the

proof of Theorem 4.6.!

Remark 4.9. Under the conditions pC5q and N1 ` N2 $ 3 we have proved thatp$,#q is a topologically mixing countable Markov shift satisfying the BIP condition(Lemma 4.8) and that the roof function ' is locally Holder and bounded away fromzero (Lemma 4.7). Therefore, the associated suspension semi-flow pY,%q can bestudied with the techniques presented in Section 2.

So far we have proved that the geodesic flow restricted to the set (0 can becoded by a suspension flow over a countable Markov shift. We now describe, fromthe ergodic point of view, the geodesic flow in the complement

`T 1X{"

˘z(0 of (0.

We denote by M%0 the space of pgtq-invariant probability measures supported inthe set where we have coding, in other words in B2'0ˆR{ " T # #. We describe thedi!erence between the spaceM%0 and the spaceMg of all pgtq-invariant probabilitymeasures. Recall that in " there are hyperbolic isometries h1, ..., hN1 , each of whichfixes a pair of points in BX. The geodesic connecting the fixed points of hi willdescend to a closed geodesic in the quotient by ". We denote !hi the probabilitymeasure equidistributed along such a geodesic.

Proposition 4.10. The set of ergodic measures in MgzM%0 is finite, those areexactly the set t!hi : 1 ! i ! N1u. Moreover, for every ! P MgzM%0 we haveh!pgq “ 0.

Proof. Let ! P MgzM%0 be an ergodic measure. Take v a generic vector for !.Since a generic vector is recurrent, the orbit gtv does not goes to infinity, thereforev` is not parabolic. Now consider the case when v points toward a hyperbolic fixedpoint z. Let ( : R Ñ X be a geodesic flowing at positive time to z with initialcondition (1p0q “ v and let (i be the geodesic connecting z with the associatedhyperbolic fixed point. By reparametrization we can assume (ip`8q “ z and that


(ip0q lies in the same horosphere centered at z than v. By estimates in [HI] we havedp(iptq, (ptqq Ñ 0 exponentially fast (here d stands for hyperbolic distance, actuallyin [HI] the stronger exponential decay in the horospherical distance is obtained).Since the vectors along the geodesics are perpendicular to the horospheres centeredat z we have the desired geometric convergence in TX. Observe (i descends to aperiodic orbit in TX{". This gives the convergence of ( to the periodic orbit andBirkho! ergodic theorem gives that the measure generated by such a geodesic isexactly one of !hi .

The fact that h!pgq “ 0 for every ! P MgzM%0 is a classical result for measuressupported on periodic orbits. !

We end up this section with a definition that includes the standing assumptionson the Schottky groups considered in most of our statements. The important pointis that we will be in position to use Lemma 4.7, Lemma 4.8 and Theorem 4.12below.

Definition 4.11. We say a extended Schottky group " satisfies property p‹q ifconditions (C5) and N1 ` N2 $ 3 hold.

4.3. Geometric meaning of s8. One of our main technical results is the follow-ing. Let ' be the roof function constructed in sub-section 4.2. In the next Theoremwe give a geometrical characterisation of the value s8 defined as the unique realnumber satisfying

P"p´t'q “#infinite if t " s8;

finite if t # s8.

One of the key ingredients in this paper, and the important result of this section, isthe relation between s8 and the largest parabolic critical exponent. This relationwill allow us to translate several results at the symbolic level into the geometricalone.

Theorem 4.12. Let " be an extended Schottky group with property p‹q. Let p$,#qand ' : $ Ñ R be the base space and the roof function of the symbolic representationof the geodesic flow pgtq on (0. Then s8 “ maxt"pi , 1 ! i ! N2u.Proof. We first show that s8 ! maxt"pi , 1 ! i ! N2u.

P"p´t'q “ limnÑ8

1

n ` 1log

ÿ

x:"n`1x“x

exp

ñÿ

i“0

´t'p#ixq¸&Ch1

pxq

“ limnÑ8

1

n ` 1log

ÿ

%“h1x2...xnxn`1%0

exp

ñÿ

i“0

´tB)´1p"ixqpo, xi`1oq¸

$ limnÑ8

1

n ` 1log

ÿ

%“h1x2...xn`1%0

exp

ñÿ

i“0

´tdpo, xi`1oq¸

The last inequality follows from dpx, yq $ B%px, yq. By removing words havinghm1 (some m) in more places than just the first coordinate, we conclude that the

argument of the function log in the limit above is greater than

e´tdpo,h1oq ÿ

pc1,...,cnqPpAzh1qn˚

ÿ

pm1,...,mnqPZn

exp

ñÿ

i“1

´tdpo, cmii oq

¸,


where pAzh1qn˚ represent the set of admissible words of length n for the code, i.e.ci ‰ c˘1

i`1. Let k $ 1. For all 0 ! j ! k ´ 1 and 1 ! i ! N1 ` N2 ´ 1, define

bi`jpN1`N2´1q “#hi`1, if 1 ! i ! N1 ´ 1

pi`1Ń1 , if N1 ! i ! N1 ` N2 ´ 1.

Consider n ` 1 “ kpN1 ` N2 ´ 1q. By restricting the above sum to words withci “ bi for every i “ 1, ..., n, we can continue the sequence of inequalities above toget

P"p´t'q $ÿ

m1,...,mnPZexp

ñÿ

i“1

´tdpo, bmii oq

¸,

where the right-hand side is equal ton%

i“1

ÿ

mPZexpp´tdpo, bmi oqq.

By definition of the bi’s, the last term is equal to˜

N1%

i“2

ÿ

mPZexpp´tdpo, hm

i oqq¸k˜

N2%

i“1

ÿ

mPZexpp´tdpo, pmi oqq

¸k

.

Hence, it follows that

P"p´t'q $ 1

N1 ` N2log

Ñ1%

i“2

ÿ

mPZexpp´tdpo, hm

i oqq¸˜

N2%

i“1

ÿ

mPZexpp´tdpo, pmi oqq

¸

“ 1

N1 ` N2log

%

aPAzh1

P"a$pt, oq.

In particular, if t " maxt"pi , 1 ! i ! N2u then P"p´t'q “ `8. This shows thats8 $ maxt"pi , 1 ! i ! N2u.

We prove now the other inequality. Let p*itq be the geodesic ray ro,,´1p#i`1xqq.Using (12), we have

'p#ixq “ B)´1p"ixqpo, xioq“ B)´1p"i`1xqpx´1

i o, oq“ lim

tÑ8dp*it, xioq ´ dp*it, oq

$ rdp*it, oq ` dpo, xioq ´ Cs ´ dp*it, oq“ dpo, xioq ´ C.

Thus,expp´t'p#ixqq ! expptCq expp´tdpo, xioqq.

Therefore,

P"p´t'q ! limnÑ8

1

nlog

ÿ

a1,...,an

ÿ

m1,...,mn

n%

i“1

expptCq expp´tdpo, amii oqq

“ log

˜Ct

%

aPAP"a$pt, oq

¸.


In particular, the pressure P"p´t'q is finite for every t # maxt"pi , 1 ! i ! N2u. !Denote by "p,max :“ maxt"pi , 1 ! i ! N2u. The simplest example to consider

is a real hyperbolic space X. In this case ""pi$ “ 1{2 for any i P t1, ..., N2u.In particular "p,max “ 1{2. More generally, if we replace hyperbolic space by amanifold of constant negative curvature equal to ´b2 then ""p$ “ b{2. In thecase of non-constant curvature some bounds are known, indeed if the curvature isbounded above by ´a2 then ""p$ $ a{2 (see [DOP]).

Recall that at a symbolic level we have htopp%q “ inf tt : P"p´t'q ! 0u. Inparticular, if the derivatives of the sectional curvature are uniformly bounded, thenTheorem 4.6, Proposition 4.10 and equality (11) imply

(13) htoppgq “ "$ “ htopp%q.The existence of a measure of maximal entropy for the flow pgtq is related to con-vergence properties of the Poincare series at the critical exponent. Indeed, usingthe construction of Patterson and Sullivan ([Pa],[Su]) of a "-invariant measure onB2X, it is possible to construct a measure on T 1X which is invariant under theaction of " and the geodesic flow. This measure induces a pgtq-invariant measureon T 1X{" called the Bowen-Margulis measure. It turns out that, if the group " isof convergence type then the Bowen-Margulis measure is infinite and dissipative.Hence the geodesic flow does not have a measure of maximal entropy. On the otherhand, if the group " is of divergence type then the Bowen-Margulis measure isergodic and conservative. If finite, it is the measure of maximal entropy.

It is, therefore, of interest to determine conditions that will ensure that thegroup is of divergence type and that the Bowen-Margulis measure is finite. It isalong these lines that Dal’bo, Otal and Peigne [DOP] introduced the following:

Definition 4.13. A geometrically finite group " satisfies the parabolic gap condi-tion (PGC) if its critical exponent "$ is strictly greater than the one of each of itsparabolic subgroups.

It was shown in [DOP, Theorem A] that if a group satisfies the PGC-conditionthen the group is divergent and the measure of Bowen-Margulis is finite [DOP, The-orem B]. In particular it has a measure of maximal entropy. Note that a divergentgroup in the case of constant negative curvature satisfies the PGC-property.

In our context, an extended Schottky group is a geometrically finite group suchthat all the parabolic subgroups have rank 1. Moreover, by Condition (C4) andTheorem 4.3, it satisfies the PGC-condition. Thus, the following property is adirect consequence of Theorem 4.12, Theorem 4.3 and (13).

Proposition 4.14. Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group. Assume that the derivatives ofthe sectional curvature are uniformly bounded. If pY,%q is the symbolic representa-tion of the geodesic flow on T 1X{", then s8 " htopp%q.

5. Escape of Mass for geodesic flows

This section contains our main results relating the escape of mass of a sequenceof invariant probability measures for a class of geodesic flows defined over non-compact manifolds. We prove that there is a uniform bound, depending only inthe entropy of a measure, for the amount of mass a measure can give to the cusps.We also characterise the amount of entropy that the cusp can have. These results


are similar in spirit to those obtained in [EKP, ELMV, KLKM] for other types offlows.

Theorem 5.1. Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group of isometries of X with propertyp‹q. Assume that the derivatives of the sectional curvature are uniformly bounded.Then, for every c # "p,max there exists a constant M “ Mpcq # 0 such that forevery ! P M%0 with h!pgq $ c, we have

"'dµ ! M,

where ! has a symbolic representation as pµ ˆ mq{ppµ ˆ mqpY qq. Moreover, thevalue "p,max is optimal in the following sense: there exists a sequence !n P M%0 ofg-invariant probability measures such that limnÑ8

#'dµn “ 8 and

limnÑ8

h!npgq “ "p,max.

Proof. This is a direct consequence of Theorems 3.3 and 3.7 using the symbolicmodel for the geodesic flow on T 1X{". !

The following corollary is just an equivalence of the first conclusion in Theorem5.1 (see also Theorem 3.7).

Corollary 5.2. Assume X and " as in Theorem 5.1. If !n P M%0 is a sequenceof pgtq-invariant probability measures such that limnÑ8

#'dµn “ 8, then

lim supnÑ8

h!npgq ! "p,max.

We are now in position to prove the main result about escape of mass.

Theorem (1.2). Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group of isometries of X with propertyp‹q. Assume that the derivatives of the sectional curvature are uniformly bounded.Then, for every c # "p,max there exists a constant m “ mpcq # 0, with the followingproperty: If p!nq is a sequence of ergodic pgtq-invariant probability measures onT 1X{" satisfying h!npgq $ c, then for every vague limit !n á !, we have

!pT 1X{"q $ m.

In particular if !n á 0 then lim suph!npgq ! "p,max. Moreover, the value "p,max

is optimal in the following sense: there exists a sequence p!nq of pgtq-invariantprobability measures on T 1X{" such that h!npgq Ñ "p,max and !n á 0.

Proof. Since every ergodic measure in MgzM%0 has zero entropy, we can supposethat !n belongs to M%0 for every n P N. Observe now that the cross-sectionS & T 1X{" defined in the proof of Theorem 4.6 is bounded. Hence, using theidentification ) : (0 Ñ Y given by Theorem 4.6 and fixing 0 " r ! infxP" 'pxq,there exists a compact set Kr & T 1X{" such that

$ ˆ r0, rs{ „ & )pKrq.Let µn be the probability measure on $ associated to the symbolic representationof !n. By Theorem 5.1, we have

"'dµn ! M.


Hence,

!npKrq “ )˚!np)pKrqq $ )˚!np$ ˆ r0, rs{ „q

“#"

#r0 dtdµn#'dµn

$ r

M.

In other words, every vague limit ! of the sequence of ergodic probability measuresp!nqn satisfies !pKrq $ r{M . In particular, we obtain !pT 1X{"q $ r{M . By set-ting m “ infxP" 'pxq{M , the conclusion follows.

Before giving the proof of the optimality of "p,max, we need the following result.

Proposition 5.3. Let " be an extended Schottky group of isometries of X. Letp P A be a parabolic isometry. We can choose a hyperbolic isometry h P " for whichthe groups "n “" p, hn # satisfy the following conditions:

(1) The group "n is of divergence type for every n $ 1,(2) The sequence p"$nqn of critical exponents satisfy "$n Ñ "P as n goes to 8,(3) The following limit holds

limnÑ8

n!(PP e´*!ndpx,(xq “ 0.

Proof. The proof is based on that of [DOP, Theorem C]. Let G be a group, wewill use the notation G˚ for Gztidu. Define P “" p # and take UP & X Y BXa connected compact neighbourhood of the fixed point *p of p such that for everym P Z˚ we have pmpBXzUPq & UP . We could take UP so that UP X BX is afundamental domain for the action of P in BXzt*pu. Because " is non-elementaryand '$ is not contained in UP , it is possible to choose h P " a hyperbolic isometryof X such that its two fixed points *h´ , *h do not lie in UP . We have used the factthat pair of points fixed by a hyperbolic isometry is dense in 'ˆ'. Fix x P X overthe axis of h. Since " is an extended Schottky group, for every k P N the elementsp and hk are in Schottky position. In particular, for Hk “" hk # we can find acompact subset UHk & X Y BX satisfying the following three conditions

(1) H˚pBXzUHkq & UHk .(2) UHk X UP “ H.(3) x R UHk Y UP .

Since P and UHk are in Schottky position it is a consequence of the Ping PongLemma that the group generated by them is a free product. By the same argumentto that one of Lemma 4.4, there is a positive constant C P R such that for everyy P UHk and z P UP , we have

(14) dpy, zq $ dpx, yq ` dpx, zq ´ C.

Applying inequality (14) and the inclusion properties described above we obtain

(15) dpx, pm1hkn1 ..pmjhknjxq $ÿ

i

dpx, pmixq `ÿ

i

dpx, hknixq ´ 2kC,

where mi P Z˚. As remarked in [DOP] the sum

P psq “ÿ

j!1

ÿ

ni,miPZ˚expp´sdpx, pm1hkn1 ...pmjhknjxqq,


is comparable with the Poincare series of "k. Indeed, since h is hyperbolic bothhave the same critical exponent. Using the inequality (15) we obtain

P psq !ÿ

j!1

˜e2sC

ÿ

nPZ˚e´sdpx,hknxq ÿ

mPZ˚e´sdpx,pmxq

¸j

.

Because of our choice of x, if l :“ dpx, hxq then dpx, hNxq “ |N |l for all N P Z.Thus,

ÿ

nPZ˚e´sdpx,hknxq ! 2

e´slk

1 ´ e´slk.

Let s+ :“ "P ` - # "P and denoting Ps “ !mPZ e

´p*P`sqdpx,pmxq, then the sum P+

is finite. Assuming - small, we get a constant D such that

e2s$C2e´s$kl

1 ´ e´s$kl

ÿ

mPZ˚e´s$dpx,pmxq " De´s$klP+.

Hence, if logpDP+q{s+l " k, then De´s$klP+ " 1 and therefore "$k ! s+. Observethat the function t %Ñ logpDPtq{st is continuous, decreasing and unbounded in theinterval p0, .q, for any 0 " . ! 1. We can then solve the equation logpDPtq{stl “k ´ 1, where t P p"p, "p ` .q and k is large enough. We call this solution -k. Byconstruction "$k ! s+k . It follows from the definition of -k that limnÑ8 P+k “ 8.Observe that

k!

(PP e´*!kdpx,(xq ! k

P+k

“ logpDP+kq{ps+k lq ` 1

P+k

,

but the RHS goes to 0 as k Ñ 8. Since p is of divergence type, it follows from[DOP, Theorem A] that "n is of divergence type. !

We proceed to show an explicit family of measures satisfying the property claimedin the second part of Theorem 1.2. We remark that the measures constructed inTheorem 5.1 can not be used at this point, since a compact set in T 1X{" is notnecessarily a compact set in the topology of Y . Hence, the fact that

#'dµn Ñ 8

does not imply that !n á 0. Despite this di#culty, we can use the geometry toconstruct the desired family.

Denote by p a parabolic isometry in the generator set A with maximal criticalexponent, that is "p,max “ "p. Take "n “" p, hn # as in Proposition 5.3. Let mBM

n

be the Bowen-Margulis measure on T 1X{"n. Since an extended Schottky groupis a geometrically finite group, the measure mBM

n is finite [DOP, Theorem B].Moreover, it maximises the entropy of the geodesic flow on T 1X{Hn [OP, Theorem2]. In other words hmBM

npgq equals "$n . Recall that the critical exponent "$n

converges to "p,max as n goes to infinity, therefore

(16) hmBMn

ppgtqq Ñ "p,max.

Using the coding property, we know that T 1X{"n (except vectors defining geo-desics pointing to the "n-orbit of the fixed points of h and p) is identified withYn “ tpx, tq P $n ˆ R : 0 ! t ! 'pxqu{ „, where

$n “ tpamii qiPZ : ai P tp, hnu,mi P Zu,

and the geodesic flow is conjugated to the suspension flow on pYn, 'q (same ' asbefore, but for this coding). It is convenient to think p$n,#q as a sub-shift of p$,#q.


Since the Bowen-Margulis measure mBMn is ergodic and has positive entropy it need

to be supported in Yn under the corresponding identification, i.e. in the space ofgeodesics modeled by the suspension flow. In particular we can consider mBM

n assupported in some invariant subset of Y . Let us call !BM

n the image measure ofmBM

n induced by the inclusion Yn ãÑ Y and normalized so that !BMn is a probability

measure. Observe that (16) implies that

limnÑ8

h!npgq “ "p,max.

We just need to prove that !BMn á 0 to end the proof of Theorem 1.2. This

sequence actually dissipates through the cusp associated to the parabolic elementp. Recall that *p denotes the fixed point of p at infinity. Define N%ppsq :“ tx P X :B%ppo, xq # su, where o P X is a reference point. Since " is geometrically finite, for slarge enough N%ppsq{ " p # embeds isometrically into T 1X{", i.e. it is a standardmodel for the cusp at *p. By definition, the group " p # acts co-compactly on'$zt*pu. In other words, if we consider a fundamental domain for the action ofP on '$zt*pu, say D, then '$

&D is relatively compact in D. Clearly the other

fundamental domains are given by (D where ( P P.In [DOP] it is proven that for any geometrically finite group " the Bowen-

Margulis measure in the cusp C satisfies a bound of the type

(17)1

A$,C

ÿ

pPPdpx, pxqe´*!dpx,pxq ! m$

BM pT 1Cq ! A$,C

ÿ

pPPdpx, pxqe´*!dpx,pxq.

Here the point x is chosen inside C and the constant A$,C basically depend on thesize of C and the minimal distance between '$

&D and BD.

Define Qi “ N%ppsiq{ " p #, where the sequence tsiui!1 is increasing withlimi si “ 8. We assume Q1 provides a standard cusp neighborhood. Denote by pnthe projection

pn : T 1X{"n Ñ T 1X{",induced by the inclusion at the level of groups. By definition

!BMn “ 1

mBMn pT 1X{"nq ppnq˚mBM

n .

We will prove that limnÑ8 !BMn pT 1ppX{"qzQiqq “ 0 for any i. For this it is enough

to prove the limit

limnÑ8

mBMn pp´1

n T 1ppX{"qzQiqqmBM

n pp´1n T 1Qiq

“ 0.

Observe that, if +n : X{"n Ñ X{" is the natural projection, then the sets +´1n Qi

are represented by the same one in the universal covering. We denote Si this cuspneighborhood.

Lemma 5.4. The measure mBMn pp´1

n T 1ppX{"qzQiqqq growth at most linearly inkn, that is for certain positive constant Ci, we have

(18) mBMn pp´1

n T 1ppX{"qzQiqq ! Cin.

Proof. Let D0 (resp. Dn) be the fundamental domain of " (resp. "n) containingo P X. By definition of fundamental domain, there exists a set Tn & " such that

(1) for any (1, (2 P Tn and (1 ‰ (2, we have (1intpD0q X (2intpD0q “ H, and(2)

$(PTn

(D0 “ Dn.


Denote by Ki the compact Ki “ pX{"qzQi and let rKi the lift of Ki into X in-tersecting D0. By definition, any lift of Ki into X intersecting Dn is a translationof rKi by an element in Tn. Since mBM

n is supported in '$n , the Bowen-Margulismeasure mBM

n on X satisfies

mBMn pp´1

n T 1pKiqq !ÿ

%PTn

('KiXCp$nq‰H

mBMn pT 1p( rKiqq,

where Cp"nq is the convex hull of Lp"nq ˆ Lp"nq in X Y BX. By construction andconvexity of the domains C( , there exists a positive integer N $ 1 such that

#t( P Tn : ( rKi X Cp"nq ‰ Hu ! Nn.

In particular, we have

mBMn pp´1

n T 1pKiqq ! NnmBMn pT 1p rKiqq.

But again, by estimates given in [DOP], the measure mBMn pT 1p rKiqq satisfies

mBMn pT 1p rKiqq ! Li,

where Li is a constant depending on the diameter of rKi. By setting Ci “ NLi, theconclusion follows. !

Using the comments just below equation (17) we know that the constants AHn,Qi

can be all considered equal to AH1,Qi . We have then

(19) mBMn pT 1Siq —AH1,Qi

ÿ

pPPdpx, pxqe´*!ndpx,pxq.

Hence, from (18) and (19), we get

mBMn pp´1

n T 1pXzQiqqmBM

n pp´1n T 1Qiq

! AH1,QiCin!pPP dpx, pxqe´*!ndpx,pxq

! C 1in!

pPP e´*!ndpx,pxq .

Finally, property (3) in Proposition 5.3 implies that the last term above convergesto 0. Therefore

limnÑ8

mBMn pp´1

n T 1pXzQiqqmBM

n pp´1n T 1Qiq

“ 0,

which concludes the proof of Theorem 1.2. !

Corollary 5.5. Let X and " as in Theorem 1.2. Then, the entropy at infinity ofthe geodesic flow satisfies

h8pT 1X{", pgtqq “ "p,max.


6. Thermodynamic Formalism

In this section we always consider X a Hadamard manifold with pinched neg-ative sectional curvature and " an extended Schottky group of isometries of Xwith property p‹q. We also assume that the derivatives of the sectional curvatureare uniformly bounded. Our goal is to obtain several results on thermodynamicformalism for the geodesic flow over X{". Some of these results were already ob-tained, without symbolic methods, by Coudene (see [Cou]) and Paulin, Pollicottand Schapira (see [PPS]). However, the strength of our symbolic approach will beclear in the study of regularity properties of the pressure (sub-section 6.2).

Here we keep the notation of sub-section 4.2. Thus, the geodesic flow pgtq inthe set (0 can be coded by a suspension semi-flow pY,%q with base p$,#q and rooffunction ' : $ Ñ R.

6.1. Equilibrium measures. We will consider the following class of potentials.

Definition 6.1. A continuous function f : T 1X{" Ñ R belongs to the class ofregular functions, that we denote by R, if the symbolic representation &f : $ Ñ Rof f |%0 has summable variations.

We begin studying thermodynamic formalism for the geodesic flow restrictedto the set (0. The following results can be deduced from the general theory ofsuspension flows over countable Markov shifts and from the symbolic model for thegeodesic flow. With a slight abuse of notation, using the indentification explicitedbefore, we still denote by f : Y Ñ R the given map f : (0 Ñ R.

Definition 6.2. Let f P R, then the pressure of f with respect to the geodesic flowg :“ pgtq restricted to the set (0 is defined by

P%0pfq :“ limtÑ8

1

tlog

¨

˝ÿ

$spx,0q“px,0q,0"s#t

exp

ˆ" s

0fp$kpx, 0qq dk

˙&Ci0

pxq˛

‚.

This pressure satisfies the following properties:

Proposition 6.3 (Variational Principle). Let f P R, then

P%0pfq “ sup

"h!pgq `

"

%0

f d! : ! P M%0 and ´"

%0

f d! " 8*,

where M%0 denotes the set of pgtq-invariant probability measures supported in (0.

Proposition 6.4. Let f P R, then

P%0pfq “ suptPg|Kpfq : K P K%0pgqu,where K%0pgq denotes the space of compact g´invariant sets in (0.

Remark 6.5 (Convexity). It is well known that for any K P K%0pgq the pressurefunction Pg|Kp¨q is convex. Since the supremum of convex functions is a convexfunction, it readily follows that P%0p¨q is convex.

Proposition 6.6. Let f P R. Then there is an equilibrium measure !f P M%0 ,that is,

P%0pfq “ h!f pgq `"

%0

f d!f ,


for f if and only if we have that P"p&f´P!pfq'q “ 0 and there exists an equilibriummeasure µf P M" for &f ´ P!pfq' such that

#'dµf " 8. Moreover, if such an

equilibrium measure exists then it is unique.

In order to extend these results to the geodesic flow in T 1X{" we use the secondconclusion of Proposition 4.10.

Definition 6.7. Let f P R, then the pressure of f with respect to the geodesic flowg :“ pgtq in T 1X{" is defined by

Pgpfq :“ max

"P%0pfq,

"f d!h1 , . . . ,

"f d!hN1

*.

Proposition 6.8 (Variational Principle). Let f P R, then

Pgpfq “ sup

"h!pgq `

"f d! : ! P Mg and ´

"f d! " 8

*,

Proposition 6.9. Let f P R, then

Pgpfq “ suptPg|Kpfq : K P Kpgqu,where Kpgq denotes the space of compact g´invariant sets.

Proposition 6.10. Let f P R be such that sup f " Pgpfq then there is an equilib-rium measure !f P Mg, for f if and only if we have that P%0p&f ´P!pfq'q “ 0 andthere exists an equilibrium measure µf P M" for &f´P!pfq' such that

#'dµf " 8.

Moreover, if such an equilibrium measure exists then it is unique.

Proof. Note that if sup f " Pgpfq then an equilibrium measure for f , if it exists,must have positive entropy. Since the measures !hi , with i P t1, . . . , N1u have zeroentropy (see Proposition 4.10). The result follows from Proposition 6.6. !

The next result shows that potentials with small oscillation do have equilibriummeasures, this result can also be deduced from [Cou, PPS]. Our proof is short anduses the symbolic structure.

Theorem 6.11. Let f P R. If

sup f ´ inf f " htoppgq ´ "p,max

then f has an equilibrium measure.

Proof. Assume that the measures !hi are not equilibrium measures for f , otherwisethe theorem is proved. Therefore, we have that Pgpfq “ P%0pfq. We first showthat P"p&f ´ Pgpfq'q “ 0. Note that for every x P $,

'pxq inf f ! &f pxq ! 'pxq sup f.By monotonicity of the pressure we obtain

P"ppinf f ´ tq'q ! P"p&f ´ t'q ! P"ppsup f ´ tq'q.Let t P ps8 ` sup f, htoppgq ` inf fq and recall that s8 “ "p,max. Then

0 " P"ppinf f ´ tq'q ! P"p&f ´ t'q ! P"ppsup f ´ tq'q " 8.

Since Pgpfq "8 and the function t Ñ P"p&f´t'q is continuous with limtÑ8 P"p&f´t'q “ ´8, we obtain that P"p&f ´ Pgpfq'q “ 0. Since the system $ has the BIP


condition and the potential &f ´ Pgpfq' is of summable variations, it has an equi-librium measure µ. It remains to prove the integrability condition. Recall that

BBtP"p&f ´ t'q

ˇˇt“Pgpfq

“ ´"'dµ.

But we have proved that the function t Ñ P"p&f ´ t'q is finite (at least) in aninterval of the form rPgpfq ´ -, Pgpfq ` -s. The result now follows, because whenfinite the function is real analytic. !6.2. Phase transitions. This sub-section is devoted to study the regularity prop-erties of pressure functions t %Ñ Pgptfq for a certain class of functions f . We saythat the pressure function t %Ñ Pgptfq has a phase transition at t “ t0 if the pres-sure function is not real analytic at t “ t0. The set of points at which the pressurefunction exhibits phase transitions might be a very large set. However, since thepressure is a convex function it can only have a countable set of points where it isnot di!erentiable. Regularity properties of the pressure are related to importantdynamical properties, for example exponential decay of correlations of equilibriummeasures. In the Axiom A case the pressure is real analytic. Indeed, this can beproved noting that, in that setting, the function t %Ñ P"p&f ´ t'q is real analyticand that P"p&f ´ P!'q “ 0. The result then follows from the implicit functiontheorem noticing that the non-degeneracy condition is fulfilled:

BBtP"p&f ´ t'q “ ´

"'dµ " 0,

where µ is the equilibrium measure corresponding to &f ´ t' . The inequalityabove, together with the coding properties established in [Bo1, BR, Ra], allow usto establish that the pressure is real analytic for regular potentials in the Axiom Asetting. In the non-compact case the situation can be di!erent. However, the onlyresults involving the regularity properties of the pressure function for geodesic flowsdefined on non-compact manifolds, that we are aware of, are those concerning themodular surface (see [IJ, Section 6]). In this section we establish regularity resultsfor pressure functions of geodesic flows defined on extended Schottky groups. Webegin by defining conditions (F1) and (F2) on the potentials.

Definition 6.12. Consider a non-negative continuous function f : T 1X{" Ñ R.We will say f satisfies Condition (F1) or (F2) if the corresponding property belowholds.

(F1) The symbolic representation &f : $` Ñ R is locally Holder and boundedaway from zero in every cylinder Cam & $`, where a P A, m P Z.

(F2) Consider any indexation pCnqnPN of the cylinders of the form Cam . Then,

limnÑ8

supt&f pxq : x P Cnuinft'pxq : x P Cnu “ 0.

We say f belong to the class F if it satisfies (F1) and (F2).

In the following Lemma we establish two properties of potentials in F that willbe used in the sequel.

Lemma 6.13. Let f be a potential satisfying (F1) and p!nq a sequence of measures

in M%0 such that !n “ µnˆm|YpµnˆmqpY q . Then,

(1) if limnÑ8#%0

fd!n “ 0, then limnÑ8#'dµn “ 8.


(2) if f satisfies Condition (F2) and limnÑ8#'dµn “ 8, then limnÑ8

#%0

fd!n “0.

Proof. To prove (1) we will argue by contradiction. Assume, passing to a sub-sequence if necessary, that

limnÑ8

"'dµn “ C.

Let - # 0, there exists N P N such that for every n # N we have thatˇˇ"'dµn ´ C

ˇˇ " -.

Lemma 6.14. Let r # 1 then for every n # N we have that

µnptx : 'pxq ! ruq # 1 ´ C ` -

r.

Proof of Lemma 6.14. Since the function ' is positive we have"'dµn $ rµnptx : 'pxq $ ruq `

"

tx:#pxq#ru'dµn.

Thus,

C ` - $ rµnptx : 'pxq $ ruq.

C ` -

r$ µnptx : 'pxq $ ruq.

Finally

µnptx : 'pxq ! ruq # 1 ´ C ` -

r.

!

Note that the set tx : 'pxq ! ru is contained in a finite union of cylinders on $.This follows from the inequality dpo, amoq ´ C ! 'pxq, which is a consequence ofLemma 4.4, and the fact that A is finite. Since &f is bounded away from zero inevery one of them, there exist a constant Gprq # 0 such that

&f pxq # Gprq,

on tx : 'pxq ! ru. Thus"

%0

fd!n “#" &f pxqdµn#

" 'dµn$

#tx:#pxq#ru &f pxqdµn#

" 'dµn$ Gprq

`1 ´ C`+

r

˘

C ´ -.

If we choose r large enough so that 1´ C`+r # 0 we obtain the desired contradiction.

To prove (2) observe that for every / # 0 there exists N $ 1 such that for everyk $ N , we have

supt&f pxq : x P Ckuinft'pxq : x P Cku " /.


Hence,

limnÑ8

"fd!n “ lim

nÑ81#

'dµn

ÿ

k!1

"

Ck

&fdµn

“ limnÑ8

1#'dµn

ÿ

k!N

"

Ck

&fdµn

! limnÑ8

1#'dµn

ÿ

k!N

"

Ck

supt&f pxq : x P Ckuinft'pxq : x P Cku inft'pxq : x P Ckudµn

! limnÑ8

1#'dµn

ÿ

k!N

"

Ck

/ inft'pxq : x P Ckudµn

! /.

Since / # 0 is arbitrary, it follows the conclusion of the second claim. !

Combining Theorem 3.7 and Lemma 6.13, we obtain the following

Lemma 6.15. Let " be an extended Schottky group with property p‹q and let fa function satisfying property (F1). If p!nqn & Mg is a sequence of invariantprobability measure for the geodesic flow such that

limnÑ8

"

T 1X{$fd!n “ 0.

Then

lim supnÑ8

h!npgq ! "p,max.

Proof. If !n P M%0 the Lemma follows directly from (1) in Lemma 6.13 and Theo-

rem 3.7. If !n P MgzM%0 then we can consider the measure !n :“ !n ´ !N1

i“1 cni !

hi

where the constants cni $ 0 are chosen so that !n(periodic orbit associated to hy-perbolic generator hiq “ 0. Let Cn “ !npX{"q. If Cn “ 0 then hp!nq “ 0 and thereis not contribution to the desired lim suphp!nq. Otherwise define un “ C´1

n !n.By definition un is a probability measure in M%0 , we claim limnÑ8

#fdun “ 0.

Observe#fd!n “ #

fd!n ` !N1

i“1 cni !

hipfq has non-negative summands and it isconverging to zero, it follows that

#fd!n Ñ 0, cni Ñ 0 as n Ñ 8. By definition

Cn “ 1´ !N1

i“1 cni , therefore Cn Ñ 1. Recalling un “ C´1

n !n we get lim#fdun “ 0.

Because !n “ Cnun ` p1 ´ Cnqp1 ´ Cnq´1!N1

i“1 cni !

hi , we have h!npgq “ Cnhunpgq.Finally since Cn Ñ 1 and lim supnÑ8 hunpgq ! "p,max (because un P M%0), we getlim supnÑ8 h!npgq ! "p,max. !

The next Theorem is the main result of this sub-section and it is an adaptation ofresults obtained at a symbolic level in [IJ]. It is possible to translate those symbolicresults into this geometric setting thanks to Theorem 4.12.

Theorem (1.3). Let X be a Hadamard manifold with pinched negative sectionalcurvature and let " be an extended Schottky group of isometries of X with propertyp‹q. Assume that the derivatives of the sectional curvature are uniformly bounded.If f P F , then

(1) For every t P R we have that Pgptfq $ "p,max.(2) We have that limtÑ´8 Pgptfq “ "p,max.


(3) Let t1 :“ suptt P R : Pgptfq “ "p,maxu, then

Pgptfq “#"p,max if t " t1;real analytic, strictly convex, strictly increasing if t # t1.

(4) If t # t1, the potential tf has a unique equilibrium measure. If t " t1 it hasno equilibrium measure.

Note that Theorem 1.3 shows that when t1 is finite then the pressure functionexhibits a phase transition at t “ t1 whereas when t1 “ ´8 the pressure functionis real analytic where defined (see Figure below). Recall that "p,max “ s8.

No phase transitions Phase transition at t “ t1t t

Pgptfq Pgptfq

‚!! ‚!!

‚!p,max ‚

!p,max

‚t1

Proof of (1). The first claim follows from the variational principle. By Theorem5.1 there exists a sequence p!nq & Mg such that limnÑ8 h!npgq “ s8 and theircorresponding probability #-invariant measures pµnq in $ satisfy limnÑ8

#'dµn “

8. Therefore, by (2) in Lemma 6.13, we also have that limnÑ8#fd!n “ 0. Hence,

for every t P R, we have

s8 “ "p,max “ limnÑ8

ˆh!npgq ` t

"

%0

fd!n

˙

! sup

"h!pgq ` t

"fd! : ! P Mg

*“ Pgptfq.

!

Proof of (2). Since t %Ñ Pgptfq is non-decreasing and bounded below, the followinglimit limtÑ´8 Pgptfq exists. Define A P R as the limit limtÑ´8 Pgptfq :“ A. Usingthe Variational Principle, we can choose a sequence of measures p!nqn in Mg forwhich

limnÑ8

h!npgq ´ n

"fd!n “ A.

Since A is finite it follows that limnÑ8#fd!n “ 0. Hence, from Lemma 6.15, we

obtain lim supnÑ8 h!npgq ! s8. In particular,

s8 ! limtÑ´8

Pgptfq

“ limnÑ8

h!npgq ´ n

"fd!n

! limnÑ8

h!npgq ! s8.


Therefore, we have that A “ "p,max. !

Proof of (3): Real analyticity. We first prove Pgptfq “ P%0ptfq. After this is donewe can proceed with standard regularity arguments in the symbolic picture. Ob-serve that for t " 0 the pressure P%0ptfq is always positive while the contribution ofthe pressure on pT 1X{"qz(0 is negative, so Pgptfq “ P%0ptfq for every t ! 0. Con-sider now t # 0. Pick !hi as in Proposition 4.10 (see also Definition 6.7). Denote byx´ (resp. x`) the repulsor (attractor) of hi and (hi the geodesic defined by thosepoints. Consider p a parabolic element in A and let (n be the geodesic connectingthe points *´ and *` where ,p*´q “ p´1h´n and ,p*`q “ hnp. Denote (8 thegeodesic connecting p´1x´ and x`. Observe (n descends to a closed geodesic inT 1X{". By comparing (n and (8 we see that for any - # 0, the amount of time(n leaves a --neighborhood of (hi is uniformly bounded for big enough n. Let !nbe the invariant probability measure defined by the closed geodesic (n, then we getthe weak convergence !n Ñ !hi . Then

t

"fd!hi “ lim

nÑ8t

"fd!n ! lim

nÑ8ph!npgq ` t

"fd!nq ! P%0ptfq.

This give us Pgptfq “ P%0ptfq.The pressure function t %Ñ Pgptfq is convex, non-decreasing and bounded from

below by s8. We now prove that for t # t1 it is real analytic. Note that since t1 " twe have that

P"pt&f ´ s8'q # 0,

possibly infinity and that there exists p # s8 such that 0 " P"pt&f ´ p'q "8(see [IJ, Lemma 4.2]). Moreover, Condition pF2q implies that Pgptfq " 8 for everyt # t1, hence

P"p&tf ´ Pgptfq'q ! 0.

Since ' is a positive, the function s %Ñ P"pt&f ´ s'q is decreasing and

limsÑ`8

P"pt&f ´ s'q “ ´8.

Moreover, since the base map of the symbolic model satisfies the BIP condition,the function ps, tq %Ñ P"pt&f ´ s'q is real analytic in both variables. Hence, thereexists a unique real number sf # s8 such that P"pt&f ´ sf'q “ 0 and

BBsP"pt&f ´ s'q

ˇˇs“sf

" 0.

Therefore, Pgptfq “ sf and by Implicit Function Theorem, the function t %Ñ Pgptfqis real analytic in pt1, t‹q. !

Proof of (4). First note that the previous claims imply that no zero entropy mea-sure can be an equilibrium measure. Moreover, in the proof of (3) we obtained thatfor t P pt1,8q we have that P"pt&f ´ Pgptfq'q “ 0. Since the system satisfies theBIP condition there exists an equilibrium measure µf P M" for t&f ´Pgptfq' suchthat

#'dµf " 8 (see Theorem 2.9). Therefore it follows from Proposition 6.6 that

tf has an equilibrium measure.In order to prove the last claim, assume by contradiction that for some t1 " t1

the potential t1f has an equilibrium measure !t1 . Then s8 “ Pgpt1fq “ h!t1pgq `

t1#%0

fd!t1 . Since f # 0 on (0, we have that#%0

fd!t1 :“ B # 0. Thus the straight


line r Ñ h!t1pgq ` r

#%0

fd!t1 is increasing with r, therefore for t P pt1, t1q we havethat

h!t1pgq ` t

"

%0

fd!t1 # s8 “ Pgptfq.

This contradiction proves the statement. !

6.3. Examples. We will use the following criterion, first introduced in [IJ], toconstruct phase transitions.

Proposition 6.16. Let f P F . Then

(1) If there exist t0 P R such that P"pt0&f ´s8'q "8 , then there exists t1 " t0such that for every t " t1 we have Pgptfq “ s8.

(2) Suppose that there exists an interval I such that P"pt&f ´ s8'q “ 8 forevery t P I. Then t %Ñ Pgptfq is real analytic on I. In particular, if forevery t P R we have P"pt&f ´ s8'q “ 8, then t %Ñ Pgptfq is real analyticin R.

The proof of this Lemma follows as in [IJ, Lemma 4.5, Theorem 4,1]. We nowpresent an example of a phase transition (Example 6.18) and another one withpressure real analytic everywhere (Example 6.19). A useful lemma in order toconstruct an example of a phase transition, is the following

Lemma 6.17. Let panqn be a sequence of positive real numbers such that!8

n“1 atn

converges for every t # t˚ and diverges at t “ t˚. Then there exists a sequencep/nqn of positive numbers such that limnÑ8 /n “ 0 and

8ÿ

n“1

at˚`,nn " 8.

Proof of Lemma 6.17. Let p0mqm any sequence of real numbers in p0, 1s convergingto zero. Note that, for every m $ 1 we have

8ÿ

n“1

at˚`&mn " 8.

Then, there exists an integer Nm $ 1 such that

8ÿ

n“Nm

at˚`&mn " 1{m2.


We can suppose without loss of generality that Nm " Nm`1. Define /n for everyNm ! n " Nm`1 as /n “ 0m, and /n “ 1 for 1 ! n " N1. Thus

8ÿ

n“1

at˚`,nn “

N1´1ÿ

n“1

at˚`,nn `

8ÿ

m“1

Nm`1´1ÿ

n“Nm

at˚`,nn

“N1´1ÿ

n“1

at˚`1n `

8ÿ

m“1

Nm`1´1ÿ

n“Nm

at˚`&mn

!N1´1ÿ

n“1

at˚`1n `

8ÿ

m“1

8ÿ

n“Nm

at˚`&mn

!N1´1ÿ

n“1

at˚`1n `

8ÿ

m“1

1{m2

" 8.

!

Example 6.18. (Phase transition) Let " be a Schottky group with property p‹qand assume that there are at least 2 di!erent cusps, i.e. N2 $ 2. Moreover assumethere exists a unique parabolic generator p with "p “ "p,max. Recall that theseries

!mPZ e

´*pdpo,pmoq diverges since p is a parabolic isometry of divergence type.Take a decreasing sequence of real numbers /m # 0, as in Lemma 6.17, such thatlimmÑ8 /m “ 0 and

!mPZ e

´p*p`,mqdpo,pmoq " 8. Define a function f0 : $ Ñ R`

by

(1) f0pxq “ /m'pxq if the first symbol of x is pm for some m P Z.(2) f0pxq “ 1 otherwise.

Observe that since ' is locally Holder, the function f0 is also locally Holder. Wefirst see that f0 P F , for this it is enough to check that Condition (F2) holds.There exist constant C independent of m such that dpo, pmoq ´C ! 'pxq wheneverx P Cpm . Then, if x, y P Cpm we have 'pxq{'pyq ! dpo, pmoq{pdpo, pmoq ´ Cq, i.e.supxPCpm

'pxq{ infxPCpm'pxq is uniformly bounded in m, this implies Condition

(F2). As shown in [BRW, Section 2], we can construct a continuous functionf : Y Ñ R with &f “ f0. We define t : $ Ñ R by tpxq “ ps8 ` /mq if the firstsymbol of x is pm, and s8 otherwise. By simplicity we will denote spamq “ tpxq ifthe first symbol of x is am. Following notations and ideas of the second part of theproof of Theorem 4.12, we obtain,


P"p´&f ´ s8'q “ limnÑ8

1

nlog

ÿ

x:"nx“x

exp

˜n´1ÿ

i“0

´p&f p#ixq ` s8'p#ixqq¸&Ch1

pxq

! limnÑ8

1

nlog

ÿ

x:"nx“x

exp

˜n´1ÿ

i“0

´'p#ixqtp#ixq¸&Ch1

pxq

! limnÑ8

1

nlog

ÿ

a1,...,an

ÿ

m1,...,mn

n%

i“1

Cspamii qe´spami

i qdpo,amii oq

! limnÑ8

1

nlogCnps8`1q ÿ

a1,...,an

ÿ

m1,...,mn

n%

i“1

e´spamii qdpo,ami

i oq

“ limnÑ8

1

nlogCnps8`1q

˜ÿ

aPA

ÿ

mPZe´spamqdpo,amoq

¸n

“ logCs8`1

˜ÿ

aPA

ÿ

mPZe´spamqdpo,amoq

¸.

Observe that!

m e´spamqdpo,amoq converges for every s # "a and every a ‰ p. On theother hand, the series

!mPZ e

´p*p`,mqdpo,pmoq is finite by construction. In particu-lar P"p´&f ´ s8'q is finite. Observe that f is a potential belonging to the familyF , then from Proposition 6.16 it follows that t %Ñ Pgptfq exhibits a phase transition.

Example 6.19. (No phase transition) Let " be a Schottky group with propertyp‹q. Define f0 : $ Ñ R` to be constant of value 1 and construct a continuousfunction f : Y Ñ R with &f “ f0. Observe

P"pt ´ s8'q “ t ` P"p´s8'q “ 8.

Recall that P"p´s8'q “ 8, because the maximal parabolic generator is of diver-gence type (see the first part of Theorem 4.12). Since ' is unbounded and f0 isconstant, we can apply Proposition 6.16 to show that t %Ñ Pgptfq is real analytic inR. In particular, it never attains the lower bound s8.

Remark 6.20. In Example 6.18 and Example 6.19, the potential f is defined (apriori) only on the set (0. To extend it continuously to the entire manifold T 1X{",it is enough to define it to be equal to 0 on the complement pT 1X{"qz(0.

References

[Ab] L. M. Abramov, On the entropy of a flow. Dokl. Akad. Nauk SSSR 128 (1959) 873–875.[AK] W. Ambrose and S. Kakutani, Structure and continuity of measurable flows. Duke Math.

J. 9 (1942) 25–42.[BI] L. Barreira and G. Iommi, Suspension flows over countable Markov shifts. Journal of

Statistical Physics 124 no.1, 207-230 (2006).[Bo1] R. Bowen, Symbolic dynamics for hyperbolic flows, Amer. J. Math. 95 (1973), 429–460.[Bo2] R. Bowen, Equilibrium states and the ergodic theory of Anosov di!eomorphisms. volume

470 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, revised edition, 2008. Witha preface by David Ruelle, Edited by Jean-Rene Chazottes.

[BR] R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows. Invent. Math. 29 (1975),181–202.


[BRW] L. Barreira, L. Radu and C. Wolf, Dimension of measures for suspension flows, Dyn.Syst. 19 (2004), 89107.

[BG] A.I. Bufetov and B.M. Gurevich, Existence and uniqueness of the measure of maximalentropy for the Teichmuller flow on the moduli space of Abelian di!erentials. SbornikMathematics 202 (2011), 935–970.

[BS1] L.A. Bunimovich and Y.G. Sinai, Markov partitions for dispersed billiards. Comm. Math.Phys. 78 (1980/81), 247–280.

[BS2] L.A. Bunimovich and Y.G. Sinai, Statistical properties of Lorentz gas with periodic con-figuration of scatterers. Comm. Math. Phys. 78 (1980/81), 479–497.

[BuS] J. Buzzi and O. Sarig, Uniqueness of equilibrium measures for countable Markov shifts andmultidimensional piecewise expanding maps. Ergodic Theory Dynam. Systems 23 (2003),1383–1400.

[Cou] Y. Coudene, Gibbs measures on negatively curved manifolds. J. Dynam. Control Syst. 9(2003), 89–101.

[DP] F. Dal’bo and M. Peigne, Some negatively curved manifolds with cusps, mixing and count-ing. Journal fur die Reine und Angewandte Mathematik 497 (1998), 141–169.

[DOP] F. Dal’bo, J-P. Otal and M. Peigne, Series de Poincare des groupes geometriquement finis.Israel Journal of Mathematics 118 (2000), 109–124.

[Da] Y. Daon, Bernoullicity of equilibrium measures on countable Markov shifts. Discrete Con-tin. Dyn. Syst. 33 (2013), 4003–4015.

[DGR] L. Dıaz, K. Gelfert, and M. Rams, Rich phase transitions in step skew products. Nonlin-earity 24 (2011), no. 12, 3391–3412.

[DT] N. Dobbs and M. Todd, Free energy jumps up. Preprint available at arXiv:1512.09245.[EKP] M. Einsiedler, S. Kadyrov, A. D. Pohl, Escape of mass and entropy for diagonal flows in

real rank one situations, arXiv:1110.0910 .[ELMV] M. Einsiedler, E. Linderstrauss, P. Michel, A. Venkatesh, The distribution of closed

geodesics on the modular surface, and Duke’s theorem, L’Enseignement Mathematique,58, Issue 3/4 (2012), 249-313.

[FJLR] A. Fan, T. Jordan, L. Liao and M. Rams, Multifractal analysis for expanding intervalmaps with infinitely many branches. Trans. Amer. Math. Soc. 367 (2015), no. 3, 1847–1870.

[Gu] B.M. Gurevic, Topological entropy for denumerable Markov chains. Dokl. Akad. NaukSSSR 10 (1969), 911–915.

[Ha] J. Hadamard, Les surfaces a courbures oppose et leurs lignes geodesiques. J.Math.PuresAppl. (5) 4 (1898), 27–73.

[H] U. Hamenstadt, Symbolic dynamics for the Teichmuller flow. Preprint available atarXiv:1112.6107.

[HI] E. Heintze and H. Im Hof, Geometry of horospheres. J. Di!erential Geom. vol. 12, Number4 (1977), 481–491.

[IJ] G. Iommi and T. Jordan, Phase transitions for suspension flows. Comm. Math. Phys. 320(2013), np. 2, 475–498.

[IJT] G. Iommi, T. Jordan and M. Todd, Recurrence and transience for suspension flows. IsraelJournal of Mathematics 209 ,(2015) Issue 2, 547–592.

[IT1] G. Iommi and M. Todd, Natural equilibrium states for multimodal maps. Comm. Math.Phys. 300 (2010), no. 1, 65–94.

[IT2] G. Iommi and M. Todd, Transience in dynamical systems. Ergodic Theory Dynam. Sys-tems 33 (2013), no. 5, 1450–1476.

[JKL] J. Jaerisch, M. Kessebohmer and S. Lamei, Induced topological pressure for countable stateMarkov shifts. Stoch. Dyn. 14 (2014), no. 2, 1350016, 31 pp.

[KLKM] S. Kadyrov, E. Lindestrauss, D. Y. Kleinbock, and G. A. Margulis. Entropy in the cuspand singular systems of linear forms. Preprint available at arXiv:1407.5310.

[Ke] T. Kempton, Thermodynamic formalism for suspension flows over countable Markovshifts. Nonlinearity 24 (2011), 2763–2775.

[Led] F. Ledrappier, Entropie et principe variationnel pour le flot geodesique en courburenegative pincee. Monogr. Enseign. Math. 43 (2013), 117–144.

[MU] R. Mauldin and M. Urbanski, Graph directed Markov systems: geometry and dynamics oflimit sets. Cambridge tracts in mathematics 148, Cambridge University Press, Cambridge2003.


[OP] J-P. Otal and M. Peigne, Principe variationnel et groupes kleiniens. Duke Math. J. 125(2004), no. 1, 15–44.

[PP] W. Parry and M. Pollicott, Zeta Functions and the Periodic Orbit Structure of HyperbolicsDynamics, Asterisque 187–188, 1990.

[Pa] S.J. Patterson, The limit set of a Fuchsian group. Acta Math. 136 (1976), no. 3-4, 241–273.[PPS] F. Paulin, M. Pollicott and B. Schapira Equilibrium states in negative curvature. to appear

in Asterisque.[PR] F. Przytycki and J. Rivera-Letelier Geometric pressure for multimodal maps of the inter-

val. Preprint available at arXiv:1405.2443[Ra] M. Ratner, Markov partitions for Anosov flows on n-dimensional manifolds. Israel J.

Math. 15 (1973), 92–114.[Sa1] O. Sarig, Thermodynamic formalism for countable Markov shifts. Ergodic Theory Dynam.

Systems 19 (1999), 1565–1593.[Sa2] O. Sarig, Phase transitions for countable Markov shifts. Comm. Math. Phys. 217 (2001),

555–577.[Sa3] O. Sarig, Thermodynamic formalism for countable Markov shifts. Proc. of Symposia in

Pure Math. 89 (2015), 81–117.[Sav] S. Savchenko, Special flows constructed from countable topological Markov chains. Funct.

Anal. Appl. 32 (1998), 32–41.[Si] Y.G. Sinai. Gibbs measures in ergodic theory (English translation). Russian Math. Surveys

27 (1972) 21–64.[Su] D. Sullivan, Entropy, Hausdor! measures old and new, and limit sets of geometrically

finite Kleinian groups. Acta Math. 153 (1984), no. 3-4, 259–277.[Wa] P. Walters, An Introduction to Ergodic Theory. Graduate Texts in Mathematics 79,

Springer, 1981.

Facultad de Matematicas, Pontificia Universidad Catolica de Chile (PUC), AvenidaVicuna Mackenna 4860, Santiago, Chile

E-mail address: [email protected]: http://http://www.mat.uc.cl/~giommi/

IRMAR-UMR 6625 CNRS, Universite de Rennes 1, Rennes 35042, FranceE-mail address: [email protected]

Princeton University, Princeton NJ 08544-1000, USA.E-mail address: [email protected]

ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR GEODESIC FLOWSgiommi/cusp_f.pdf · 2016-01-27 ·...

Documents

Transcript of ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR GEODESIC FLOWSgiommi/cusp_f.pdf · 2016-01-27 ·...