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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 class of Riemanniannon-compact manifolds with variable pinched negative sectional curvature. Wedevelop and study the corresponding thermodynamic formalism. We proposea definition of pressure that satisfies the variational principle and establishconditions under which a potential has a (unique) equilibrium measure. Wecompute the entropy contribution of the cusps. This allow us to prove certainregularity results for the pressure of a class of potentials. We prove that thepressure is real analytic until it undergoes a phase transition, after which itbecomes constant. Our techniques are based on the one side on symbolicmethods and Markov partitions and on the other on 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. The idea of coding a flow in order to describe its dynam-ical and ergodic properties can be traced back to the work of Hadamard [Ha] in1898. During the 1920’s these ideas, and new methods of coding, were developed byMorse [Mo] and Artin [Ar] (among others). See the work of Katok and Ugarcovici[KU] for more details. During the next decades these methods were enhanced andrefined. But it was in 1973 that landmark results were obtained. Indeed, Bowen[Bo1] and Ratner [Ra] constructed Markov partitions for Axiom A flows definedover compact manifolds. They actually showed that Axiom A flows can be codedwith suspension flows defined over sub-shifts of finite type on finite alphabets witha regular (Holder) roof functions. In particular, this remarkable coding allowed forthe proof of many important results for geodesic flows defined on compact manifoldsof constant negative curvature (see also the work of Series [Se1]). This coding waslater used by Bowen and Ruelle [BR] to study, in great detail, ergodic theory andthermodynamic formalism for Axiom A flows. This was possible because the er-godic theory of compact sub-shifts of finite type is very well understood. Recently,Lima and Sarig [LS] constructed symbolic dynamics for smooth flows of positivetopological entropy defined on three-dimensional compact Riemannian manifolds.In this generality, they prove that these flows can be coded with suspension flows

Date: November 11, 2015.2010 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

over countable alphabets. It is worth stressing that the examples covered by thistheory include geodesic flows of positive entropy in positively curvatured Riemann-ian surfaces. These results were used in [LLS] to prove that some of those flowstogether with its measure of maximal entropy are Bernoulli.

The case of geodesic flows defined over non-compact manifolds is more subtleand the theory is far less developed that in its compact counterpart. Perhaps thecase which is best understood is that of the geodesic flow defined over the modularsurface. Codings for this flow have been constructed at least since the work ofArtin [Ar] and have been throughly studied by Series (see [Se2, Se3]). A relatedflow, the so called positive geodesic flow over the modular surface was studiedby Gurevich and Katok [GK]. They proved that this flow can be coded with asuspension flow defined over a countable alphabet with an (unbounded) regularroof function. Making use of this coding it is possible to study its correspondingthermodynamic formalism [IJ]. Other examples include certain Teichmuller flows.Hamenstadt [H] and also Bufetov and Gurevich [BG] have coded Teichmuller flowswith suspension flows over countable alphabets and using this representation haveproved, for example, the uniqueness of the measure of maximal entropy. Yet anotherimportant example for which codings on countable alphabets have been constructedis a type of Sinai billiards [BS1, BS2, BChS1].

In this paper we will study a class of geodesic flows defined over non-compactmanifolds introduced by Dal’bo and Peigne [DP]. These are geodesic flows definedon the unit tangent bundle of a manifold obtained as the quotient of a Hadamardmanifold with an extended Schottky group (see sub-section 4.2 for precise defini-tions). A relevant point is that the group has parabolic elements and thereforethe manifold have cusps, thus it is non-compact. It was shown in [DP] that thesegeodesic flows can be coded as suspension flows defined over countable Markovshifts. We prove several results on thermodynamic formalism for these flows. Insub-section 4.5 we propose a definition of topological pressure that satisfies notonly the variational principle, but also an approximation by compact invariant setsproperty. These results were recently obtained by di!erent (non-symbolic) meth-ods in far more general settings by Paulin, Pollicott and Schappira [PPS]. A newresult in this setting is the characterization of regular potentials having (unique)equilibrium measures (see Proposition 4.5.5 and Theorem 4.5.10).

The strength of our approach is perhaps better appreciated in our regularityresults for the pressure (sub-section 4.7), note that the techniques in [PPS] do notprovide these type of results. We say that the pressure function t P tf has aphase transition at t t0 if it is not analytic at that point. It readily follows fromwork by Bowen and Ruelle [BR] that the pressure for Axiom A flows and regularpotentials is real analytic and hence has no phase transitions. Regularity propertiesof the pressure of geodesic flows defined on non-compact manifolds, as far as weknow, have not been studied. With the exception of the geodesic flow defined onthe modular surface (see [IJ, Section 6]). There is a general strategy used to studyregularity properties of the pressure of maps and flows with strong hyperbolic orexpanding properties in most of the phase space but not in all of it. Indeed, ifthere exists a subset of the phase space B X for which the restricted dynamics isnot expansive and its entropy equal to A, then it is possible to construct potentials

ENTROPY IN THE CUSP AND PHASE TRANSITIONS FOR GEODESIC FLOWS 3

f : X R for which the pressure function has the form

(1) P tf :real analytic, strictly decreasing and convex if t t ;

A if t t .

Well known examples of this phenomena include the Manneville-Pomeau map (see[LSV, PM, Sa2] or [IT2, Section 4.2]) in which the set B consist of a parabolicfixed point and therefore A 0. The potential considered is the geometrical one:log T . Similar results for multimodal maps have been obtained, for example,

in [IT1, PR]. In this case the set B corresponds to the post-critical set and A 0.Examples of maps in which A 0 have been studied in [DGR, IT2]. For suspensionflows over countable Markov shifts similar examples were obtained in [IJ], howeverin that case the number A remained unexplained for. In this paper we show thatfor the geodesic flows we are considering the set B is the union of the cusps of themanifold. Moreover, and more interestingly, we are able to compute the entropycontributions of the cusps in the geodesic flow. We prove (see sub-section 4.6) thatthe entropy in the cusp is given by

!p,max : max !pi , 1 i N2 ,

where !pi is the critical exponent of the Poincare series of the group pi ,where pi is a parabolic element of the extended Schottky group " (see sub-section4.1 for precise definitions). In sub-section 4.7 we construct potentials for whichthe pressure exhibits similar (although it is an increasing, rather than decreasing,function) behaviour as in equation (1). In those examples A !p,max. Note thatit is possible for t to be infinity and in that case the pressure is real analytic.

Technically, an interesting result is Theorem 4.4.1 in which a symbolic parameterof the suspension flow, the number s , is proven to be equal to the geometricparameter of the group !p,max. We stress that when coding a flow a great dealof geometric information is lost. With this result we are able to recover part ofit. A result of independent interest, that is used in the proof of Theorem 4.4.1, isTheorem 4.1.5, where it is proven that when we consider parabolic isometries ofdivergence type, their critical exponents can be approximated by critical exponentsof non elementary subgroups of isometries.

Acknowledgements. We thank F. Dal’bo, F. Ledrappier and B. Schapira for sev-eral useful and interesting discussions around the subject treated in this article. Thiswork started when the second and third authors were visiting Pontificia Universi-dad Catolica de Chile, they are very grateful for the great conditions and hospitalityreceived in PUC Mathematics Department.

2. Preliminaries

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 bean incidence matrix defined on the alphabet of natural numbers. The associatedcountable Markov shift #," is the set

# : xn n N : M xn, xn 1 1 for every n N ,


together with the shift map " : # # defined by " x1, x2, . . . x2, x3, . . . . Astanding assumption we will make throughout the article is that #," is topologi-cally mixing. We equip # with the topology generated by the cylinder sets

Ca1 an x # : xi ai for i 1, . . . , n .

We stress that, in general, # is a non-compact space. Given a function # : # Rwe define the n th variations of # by

Vn # : sup # x # y : x, y #, xi yi for i 1, . . . , n ,

where x x1x2 and y y1y2 . We say that # has summable variation if

n 1 Vn # . We say that # locally Holder if there exists $ 0, 1 such thatfor all n 1 we have Vn # O $n .

This section is devoted to recall some of the notions and results of thermodynamicformalism in this setting. The following definition was introduced by Sarig [Sa1]based on work by Gurevich [Gu1, Gu2].

Definition 2.1.1. Let # : # R be a function of summable variation. The Gure-vich pressure of # is defined by

P # limn

1

nlog

x:!nx x

expn 1

i 0

# "ix %Ci1x ,

where %Ci1x 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 # : # R is a function of summable variations, then

(1) (Approximation property)

P # sup PK # : K K ,

where PK # is the classical topological pressure on K (see [Wa, Chapter9]) and

K : K # : K compact and "-invariant .

(2) (Variational Principle) Denote by M! is the space of " invariant probabil-ity measures and by h µ the entropy of the measure µ (see [Wa, Chapter4]). Then,

P # sup h µ #d& : µ M! and #dµ ,

A measure µ M! attaining the supremum, that is, P # h µ #dµ is calledequilibrium measure for #. A potential of summable variations has at most oneequilibrium 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 2.1.2. We say that a countable Markov shift #," , defined by thetransition matrix M i, j with i, j N N, satisfies the BIP condition if and onlyif there exists b1, . . . , bn N such that for every a N there exists i, j N withM bi, a M a, bj 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 2.1.3. Let #," be a countable Markov shift satisfying the BIP con-dition and # : # R a non-positive locally Holder potential. Then, there existss 0 such that pressure function t P t# has the following properties

P t#if t s ;

real analytic if t s .

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

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

(2) Y x, t # R : 0 t ' x ,

with the points x, ' x and " x , 0 identified for each x #. The suspensionsemi-flow over " with roof function ' is the semi-flow $ #t t 0 on Y defined by

#t x, s x, s t whenever s t 0, ' x .

In particular,#" x x, 0 " x , 0 .

2.3. Invariant measures. Let Y,$ be a suspension semi-flow defined over acountable Markov shift #," with roof function ' : # R bounded away fromzero. Denote by M! the space of invariant probability measures for the flow. Itfollows form a classical result by Ambrose and Kakutani [AK] that every measure& M! can be written as

(3) &µ m Y

µ m Y,

where µ M! and m denotes the one dimensional Lebesgue measure. When #,"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 #," is a countable Markov shiftwith roof function bounded away from zero the map defined by

&µ m Y

µ m Y,

is surjective. However, it can happen that µ m Y . 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 in orderto study thermodynamic formalism for suspension flows it su%ces to study ther-modynamic formalism for semi-flows. Denote by # ," a two-sided countableMarkov shift. Recall that two continuous functions #, ( C # are said to becohomologous if there exists a bounded continuous function ) C # such that# ( ) " ). The relevant remark is that thermodynamic formalism for twocohomologous functions is exactly the same. Thus, if every continuous function


# C # is cohomologous to a continuous function ( C # which onlydepends in future coordinates then thermodynamic formalism for the flow can bestudied in the corresponding semi-flow. The next result formalises this discussion.

Proposition 2.4.1. If # C # has summable variation, then there exists( C # of summable variation cohomologous to # such that ( x ( ywhenever xi yi for all i 0 (that is, ( depends only on the future coordinates).

Proposition 2.4.1 has been proved with di!erent regularity assumptions in thecompact setting by [Si, Section 3], [Bo2, Lemma 1.6], [CQ, Theorem 2] and in thenon-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]. Savchenko [Sav, Theorem 1] established theversion corresponding to this setting.

Proposition 2.5.1 (Abramov-Savchenko). Let & M! be such that & µm Y µ m Y , where µ M! then the entropy of & with respect to the flow,that we denote h! & , satisfies

(4) h! &h! µ

'dµ.

In Proposition 2.5.1 a relation between the entropy of a measure for the flowand a 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 R be a continuous function. Define&f : # R by

&f x" x

0f x, t dt.

Proposition 2.5.2 (Kac’s Lemma). Let f : Y R be a continuous function and& M! an invariant measure that can be written as

&µ m

µ m Y,

where µ M!, then

Yf d& " &f dµ

" ' dµ.

Propositions 2.5.1 and 2.5.2 together with the relation between the spaces ofinvariant measures for the flow and for the shift established by Ambrose and Kaku-tani (see subsection 2.3) allow us to study thermodynamic formalism for the flowby means of the corresponding one on the base.

2.6. Thermodynamic formalism for suspension flows. Let #," be a topo-logically mixing countable Markov shift and ' : # R a positive function bundedaway from zero of summable variations. Denote by Y,$ the suspension semi-flowover #," with roof function ' . Thermodynamic formalism has been studied inthis context by several people with di!erent degrees of generality: Savchenko [Sav],Barreira and Iommi [BI], Kempton [Ke] and Jaerisch, Kessebohmer and Lamei[JKL]. The pioneering work in which thermodynamic formalism for suspensionflows where the base #," is a sub-shift of finite type defined on a finite alphabet


has been studied, for example, in [BR, PP]. The next result provides equivalentdefinitions for the pressure, P! , on the flow.

Theorem 2.6.1. Let f : Y R be a function such that &f : # R is of summablevariations. Then the following equalities hold

P! f : limt

1

tlog

#s x,0 x,0 ,0 s t

exps

0f #k x, 0 dk %Ci0

x

inf t R : P &f t' 0 sup t R : P &f t' 0

sup P! K f : K K $ ,

where K $ denotes the space of compact $ invariant sets.

In particular the entropy of the flow is the unique number h $ satisfying

(5) h $ inf t R : P t' 0 .

Note that in this setting the Variational Principle also holds (see [BI, JKL, Ke,Sav]).

Theorem 2.6.2 (Variational Principle). Let f : Y R be a function such that&f : # R is of summable variations. Then

P! f sup h! &Yf d& : & M! and

Yf d& .

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

P! f h! & 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 2.6.3. Let f : Y R be a continuous function such that &f is of sum-mable variations. Then there is an equilibrium measure &f M! for f if andonly if we have that P &f P! f ' 0 and there exists an equilibrium measureµf M! for &f P! f ' such that 'dµf .

Remark 2.6.4. 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 in the cusp 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 sub-section 4.6.


Let #," be a topologically mixing countable Markov shift of infinite entropyand ' : # R a potential of summable variations bounded away from zero.Denote by Y,$ the associated suspension flow, which we assume to have finiteentropy. Note that since #," has infinite entropy and ' is non-negative theentropy of the flow h $ satisfies P h $ ' 0 (see equation (5)). Therefore,there exists a real number s 0, h $ such that

P t'infinite if t s ;

finite if t s .

As it turns out the number s 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 3.0.5. We say that a point x, t Y escapes on average if

limn

1

n

n 1

i 0

' "ix .

We denote the set of all points which escape on average by EA.

Remark 3.0.6. Note that if & M! is ergodic and & µ m µ m Y withµ M! then Birkho! ’s theorem implies that

limn

1

n

n 1

i 0

' "ix 'dµ.

Thus, no measure in M! is supported on EA. We can, however, describe thedynamics of the set EA studying sequences of measures &n M! such that theassociated measures µn M! satisfy

limn

'dµn .

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 3.0.7. Let c s , h $ . There exists a constant C 0 such that forevery & M! with h! & c we have that

'dµ C.

Proof. Let & M! with h! & c and let µ M! be the invariant measuresatisfying & µ m µ m Y . By Abramov-Savchenko formula we have

h µ c 'dµ 0.

We will consider the straight line L t : h µ t 'dµ. Note that L c 0 andL 0 h µ . Let s s , c . Note that P s' and by the variational


principle L s P s' . This remark readily implies a bound on the slope ofL t . Indeed,

'dµP s'

c s.

Thus the constant C P s' c s satisfies the theorem. In order to obtainthe best possible constant we have to compute the infimum of the function definedfor s s , c by

sP s'

c s.

!Remark 3.0.8. We stress that the constant C in Theorem 3.0.7 depends only onthe entropy bound c and not on the measure &.

Remark 3.0.9. An implicit assumption in Theorem 3.0.7 is that s h $ . 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 equivalent tothe parabolic gap property (see [DP, Section III] or Definition 4.5.11 for precisedefinitions).

Corollary 3.0.10. If #," is a Markov shift defined countable alphabet satisfyingthe BIP condition then the best possible constant C R in Theorem 3.0.7 is givenby

CP sm'

c sm,

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

ch µsm

'dµsm

.

Proof. Since the system is BIP the function P s' , when finite, is di!erentiable(see Theorem 2.1.3). Moreover, its derivative is given by (see [Sa3, Theorem 6.5]),

d

dsP s'

s sm'dµsm ,

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

function s P s"c s are those which satisfy

(6) c s P s' P s' 0.

Equivalently,

c s 'dµs h µs s 'dµs 0.

Therefore, equation (6) is equivalent to

ch µs

'dµs.

!In the next Theorem we prove that the entropy of the flow on EA is bounded

above by s and that, under some additional assumptions, it is actually equal to it.This result could be thought of as a symbolic estimation for the entropy of a flowin a cusp. Theorem 3.0.11 is a version of a result first observed in [FJLR, Lemma2.5] and used in the context of suspension flows in [IJ].


Theorem 3.0.11. Let &n n M! be an sequence of invariant probability measurefor the flow of the form

&nµn m

µn m Y,

where µn M!. If limn 'dµn then

lim supn

h! &n s .

Moreover, if s h $ then there exists a sequence &n n M! with limn 'dµn

such thatlimn

h! &n s .

Proof. In order to prove the first claim note that for every * 0 we have thatP s * ' . Let µn n be a sequence of " invariant measures such thatlimn 'dµn . By the variational principle we have that

h µn s * 'dµn P s * ' .

Thus

lim supn

h µn

'dµns * lim sup

n

P s * '

'dµns * .

Since * 0 was arbitrary, using Abramov-Savchenko formula we obtain

lim supn

h &n s .

Let us construct now a sequence &n n M! with limn 'dµn such thatlimn h! &n s . First note that it is a consequence of the approximationproperty of the pressure, that there exists a sequence of compact invariant setsKN N # such that limN PKN t' P t' . In particular, for every n Nwe have that

(7) limN

PKN s 1 n ' .

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

(8) PKN s 1 n ' P s 1 n ' .

Thus, given n N there exists N N such that

n2 PKN s 1 n ' PKN s 1 n '

2 n.

Since the function t PKN t' is real analytic, by the mean value theorem thereexists tn s 1 n, s 1 n , such that PKN

tn' n2. Denote by µn theequilibrium measure for tn' in KN . We have that

n2 PKNtn' 'dµn.

In particular the sequence µn n satisfies

limn

'dµn .

Since s h $ we have that for n N large enough

h µn tn 'dµn 0.


In particular

tnh µn

'dµn.

Since tn s 1 n, s 1 n we have that

(9) s limn

tn limn

h µn

'dµnlimn

h &n .

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

4. The geodesic flow on Extended Schottky Groups

This section contain our main results. We will develop thermodynamic formalismfor a class of geodesic flows defined over non-compact manifolds. Recent work ofPaulin, Pollicott and Schapira, [PPS] is related to ours. However, their work is moregeneral, since it does not require symbolic models and ours gives a more detaileddescription of the thermodynamic properties.

4.1. The Poincare series. Let X be a Hadamard manifold with pinched negativesectional curvature, that is a complete simply connected Riemannian manifold forwhich the sectional curvature K satisfies b2 K 1 (for some fixed b 1).Denote by X the corresponding visual boundary relatively to a reference pointo X. Finally, denote by d the Riemannian distance on X.

Definition 4.1.1. Let " be a discrete subgroup of isometries of X. The Poincareseries P# s, x associated with ", where x X, is defined by

P# s, x :g #

e sd x,gx .

The critical exponent of " is defined by

!# : inf s R : P# s, x .

The group " is said to be of divergence type (resp. convergence type) if P# !#, x(resp. P# !#, x ).

Remark 4.1.2. Since the sectional curvature of X is bounded below, the criticalexponent is finite. Moreover, by the triangle inequality, it is independent of x X.

For g a non elliptic isometry of X, denote by !g the critical exponent of thegroup g . If g is hyperbolic it is fairly straightforward to see that !g 0 andthat the group g is of divergence type. If g is parabolic, it was shown in [DP,Theorem III.1], that !g

12 .

If a group " is divergent several geometrical and dynamical consequences canbe established. A criteria for this to happen in the setting of geometrically finitegroups was obtained by Dal’bo, Otal and Peigne [DOP, Theorem A]. Indeed thegroup is divergent if its critical exponent is strictly larger than the one of each ofits parabolic subgroups. We will come back to this property later (see sub-section4.5).

Let " be a discrete subgroup of isometries of X. Denote by ' the limit set of", that is, the set ' " o " o. The group " is non elementary if ' containsinfinitely many elements.


Remark 4.1.3. If " is a non elementary discrete subgroup of isometries of X,then !# 0 (see for instance [Su1]).

We recall the following fact proved in [DOP, Proposition 2].

Theorem 4.1.4. Let " be a non elementary discrete group 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 group and we have an element g " suchthat g is of divergence type, then !# ! g (this fact was proven earlier in[DP, Theorem III.1]). This property will give us the divergence type of our extendedSchottky groups (compare with PGC-condition discussed in next sections).

We now state and prove Theorem 8, which says that when we consider para-bolic isometries of divergence type, their critical exponents can be approximatedby critical exponents of non elementary subgroups of isometries of X.

Theorem 4.1.5. Let " be a discrete non elementary subgroup of isometries ofX which contains a parabolic isometry p generating a subgroup of divergence type.Then there exists a decreasing sequence "n of non elementary subgroups of "containing P p as a subgroup satisfying the following conditions.

(1) !P !#n for every n 1;(2) the group "n is of divergence type for every n 1, and(3) !#n !P as n goes to .

Proof. The proof is based on that of [DOP, Theorem C]. Let G be a group, wewill use the notation G for G id . Let UP X X be a connected com-pact neighbourhood of the fixed point of p such that for every m Z we havepm X UP UP . We could take UP so that UP X is a fundamental domainfor the action of P in X fixed point of p . Consider h " a hyperbolic isometryof X such that its two fixed points +h , +h do not lie in UP . This is possible be-cause the pair of points fixed by a hyperbolic isometry is dense in ' '. Fix x Xover the axis of h. There exists k N such that for the group H hk we canfind a compact subset UH X X satisfying the following three conditions

(1) H X UH UH .(2) UH UP .(3) x UH UP .

Since P and H are in Schottky position it is a consequence of the Ping PongLemma that the group generated by them is a free product. We will denote by"k : hk, p the group generated by H and P.

Considering k N large enough, we can assume that for every y UH , z UPthe geodesic segments x, y and x, z form an angle uniformly bounded below. Inother words, there is a positive constant C R such that

(10) d y, z d x, y d x, z C.

Applying inequality (10) and the inclusion properties described above we obtain

(11) d x, pm1hkn1 ..pmkhknkxi

d x, pmixi

d x, hknix 2kC,

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

P sk 1 ni,mi Z

exp sd x, pm1hkn1 ...pmkhknkx ,


is comparable with the Poincare series of "k. Indeed, since h is hyperbolic bothhave the same critical exponent. Using the inequality (11) we obtain

P sk 1

e2sC

n Ze sd x,hknx

m Ze sd x,pmx

k

.

Because of our choice of x, if l : d x, hx then d x, hNx N l for all N Z.Thus,

n Ze sd x,hknx 2

e sl

1 e sl.

Let s$ : !P * !P , then the sum m Z e s!d x,pmx is finite. Choose k Nlarge enough so that

e2s!C2e sl

1 e slm Z

e sd x,pmx 1.

In particular, the Poincare series for "k converges at s$, that is P#k s$, x .Hence, !#k s$. Moreover, since P is of divergence type, it follows directly fromTheorem 4.1.4 that, !P !#k . Thus,

!P !#k s$.

Given a decreasing sequence s$i converging to !P , we can choose an appropriateincreasing sequence ki such that the sequence of groups "ki i satisfies the desiredproperties. !

4.2. Extended Schottky groups. In this section we recall the definition of a ex-tended Schottky group. To the best of our knowledge this definition was introducedin 1998 by Dal’bo and Peigne [DP]. The basic idea is to extend the classical notionof Schottky groups to the context of non-compact manifolds.

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 X a compact neighbourhood Chi of the at-tracting point +hi of hi and a compact neighbourhood Ch 1

iof the repelling

point +hiof hi, such that

hi X Ch 1i

Chi .

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

n Z pni X Cpi Cpi .

(C3) The 2N1 N2 neighbourhoods introduced in 1 and 2 are pairwise dis-joint.

(C4) The elementary groups pi , for 1 i N2, are of divergence type.

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 thegroup only consists on hyperbolic elements, then " is a classical Schottky groupand its geometric and dynamical properties are reasonably well understood. Figure


1 below is an example of a Schottky group for the hyperbolic model X D andtwo generators; one of them hyperbolic and the other parabolic.

o

Cp

ChCh 1

p

h

Figure 1. Schottky Group " h, p .

For our purposes we will suppose that " satisfies an additional geometrical con-dition. Let + X and x, y X. For every geodesic ray t +t pointing to +, thelimit

B% x, y : limt

d x, +t d y, +t ,

always exists, and is independent of the geodesic ray +t since X has negativesectional curvature. The Busemann function B : X X2 R is defined byB +, x, y B% x, y . We remark that B is a continuous function. An (open)horoball based in + and passing through x is the set of y X such that B% x, y 0.In the hyperbolic case, when X D, an horoball based in + S1 and passing troughx D is the interior of a circle containing x and tangent to S1 at +.

o

z

+

B% z, o

Figure 2.

Let A h 11 , ..., h 1

N1, p1, ..., pN2 . We now define an extra hypothesis that we

will be used in large parts of this paper.


(C5) Let a1, a2 A be such that a1 a 12 . Then, for every + Ca1 , we have

B% a 12 o, o 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). Note that this

condition is not very restrictive as we can see in Proposition 4.2.1 below.

Proposition 4.2.1. Let " h1, ..., hN1 , p1, ..., pN2 be a extended Schottkygroup. Then, there exists an integer N 1 such that the group defined byhN1 , ..., hN

N1, pN1 , ..., pNN2

satisfies the condition C5 .

Proof. Let a1, a2 A and + Ca1 . Denote by Uan2the convex hull in X X

of the set Can2, for n 1. Let z Uan

2. It is su%cient to prove that B% z, o 0

for all n large enough. Let +t be a geodesic ray pointing to +. Observe that thereexists T 0, such that for every t T we have that +t belongs to Ua1 . Since Ca1

and Ca2 are disjoint, there exists a universal constant C 0 (depending only onthe elements of A ) such that d z, +t d +t, o d z, o C, for every t T .Therefore,

B% z, o limt

d z, +t d +t, o

limt

d +t, o d z, o C d +t, o

d z, o C.

Since d z, o when n , there exists a N 1 such that d z, o C forevery n N . In particular, we have B% z, o 0.

!

4.3. The geodesic flow and its symbolic representation. Let " be an ex-tended Schottky group and T 1X " be the unit tangent bundle of the manifoldX ". Denote by gt the geodesic flow on T 1X ". Observe that the assumptionN2 1 implies that X " is a non-compact manifold.

In [DP] the authors proved that there exists a gt -invariant subset ( of T 1X ",contained in the nonwandering set of gt , such that gt $ is topologically conju-gated to a suspension flow over a countable Markov shift # ," . We now recallthat construction.

We first give a brief geometrical description of the set (. Let

A h1, ..., hN1 , p1, ..., pN2 ,

we associate to it the symbolic space # defined by

# amii i Z : ai A,mi Z and ai 1 ai i Z .

Note that the space # is a sequence space defined on the countable alphabetami : ai A,m Z . Denote by 2X the set X X diagonal. The unit tangentbundle T 1X is identified with 2X R via Hopf’s coordinates. A vector v T 1Xis identified with v , v , Bv o,, v , where v (resp. v ) is the negative (resp.positive) endpoint of the geodesic defined by v. Here , : T 1X X is the naturalprojection of a vector to its base point. Observe that the geodesic flow acts bytranslation in the third coordinate of this identification. Let '0 be the limit set 'minus the "-orbit of the fixed points of the elements of A. We denote by ( the


set of vectors in T 1X identified with '0 '0 diagonal R. Finally, we define( : ( ", where the action of " is given by

g + , + , t g + , g + , t B% o, g 10 .

Observe that ( is invariant by the geodesic flow.Fix now +0 X a A Ca , where Ca Ca Ca 1 . Dal’bo and Peigne [DP,

Property II.5] established the following coding property: for every + '0 thereexists an unique sequence - + ami

i i 1 with ai A, mi Z and ai 1 aisuch that

limk

am11 ...amk

k +0 +.

For each a A define '0a '0 Ca and set 2'0

",# A& '

'0& '0

' . For

any pair + , + 2'0 with am as the first term of the sequence - + , define' + B% o, amo and T + , + a m+ , a m+ . Define T " by the formula

T " + , + , s T + , + , s ' + .

Observe T " maps 2'0 R to itself. We claim that ( can be identified with thequotient 2'0 R T " . In order to prove the claim, we first proceed to explainhow T " gives a suspension flow.

Let + , + 2'0 and let D the fundamental domain of " containing o. Fromcondition (C5) it follows that the geodesic determined by + , + in X intersectsthe horosphere based in + and passing through o in only one point xo,% ,% be-longing to D (see Figure 3 below). Denote by vo,% ,% the vector in T 1X based inxo,% ,% and pointing to + . Finally, denote by vo,% ,% the projection of vo,% ,%

on T 1X ". SetS vo,% ,% : + , + 2'0 .

Then, for every vo,% ,% S, the first return time of vo,% ,% to S is given by ' + .Moreover, for every vector v ( there exists a time t R such that the gtv belongsto S (otherwise, a lift on T 1M is pointing to a "-orbit of a fixed point of an elementsin A). This give us the identification.

+

xo,% ,%

+

o

Figure 3. Cross section for " h, p

The coding property implies that the set 2'0 is identified with # by con-sidering + , + as a bilateral sequence - + ,- + . If - + bni

i i 1, wedefine - + as the sequence ..., b n2

2 , b n11 , then - + ,- + represent the


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

' x ' - 1 x B( 1 x o, amo ,

where w : '0 # is the coding function and am the first symbol in w 1 x . Weextend ' to # making it independent of the negative time coordinates.

Lemma 4.3.1. The function ' : # R depends only on future coordinates, itis 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 # and + - 1 x the pointin '0 associated to x by the coding property. The function ' satisfies

' x B% o, amo Ba m% a mo, o .

The last term above is greater than 0 by condition (C5). Since the domains Ca, fora A , are compact and B is continuous, there exists an uniform strictly positivelower bound for Ba m% a mo, o . Thus ' is bounded away from zero. !

We therefore have, that the geodesic flow gt on ( can be coded as the suspen-sion flow on Y x, t # R : 0 t ' x . Moreover, since the rooffunction ' depends only on future coordinates, we can consider the one-sided shift#," associated to # ," and in virtue of the discussion in sub-section 2.4 wehave:

Proposition 4.3.2. The thermodynamic formalism of the geodesic flow gt on (is equivalent to that of the suspension semi-flow with base #," and roof function' .

Lemma 4.3.3. If N1 N2 3, then the countable Markov shift #," is topolog-ically mixing and it satisfies the BIP condition.

Proof. Recall that the the Markov shift #," is topologically mixing if for everya, b ami : ai A,m Z there exists N a, b N such that for every n N a, bthere exists an admissible word of length n of the form ai1i2 . . . in 1b. The set ofallowable sequences is given by

amii i Z : ai A,mi Z and ai 1 ai i N .

Since N1 N2 3 then given any pair of symbols in ami : ai A,m Z , sayam11 , am2

2 we can consider the symbol a3 a1, a2 . Hence the following words areadmissible:

am11 a3a1a3 . . . a1a

m22 , am1

1 a3a1a3a1 . . . a3am22 .

Thus, the system is topologically mixing. Moreover, the set A satisfies the requiredconditions in order for #," to be BIP (see definition 2.1.2). !

Remark 4.3.4. Under the conditions N1 N2 3 and (C5) we have proved that#," is a topologically mixing countable Markov shift satisfying the BIP condition(Lemma 4.3.3) and that the roof function ' is locally Holder (Lemma 4.3.1) andbounded away from zero (condition (C5) and Lemma 4.3.1). Therefore, the asso-ciated suspension semi-flow Y,$ can be studied with the techniques presented inSection 2.


So far we have proved that the geodesic flow restricted to the set ( can be codedby a suspension flow over a countable Markov shift. We now describe, from theergodic point of view, the geodesic flow in T 1X " (. We denote by M$ thespace of gt -invariant probability measures supported in the set where we havecoding, in other words in 2'0 R. We describe the di!erence between the spaceM$ and the space Mg of all gt -invariant probability measures. Recall that in "there are hyperbolic isometries h1, ..., hN1 , each of which fixes a pair of points inX. The geodesic connecting the fixed points of hi will descend to a closed geodesic

in the quotient by ". We denote &hi the probability measure equidistributed alongsuch a geodesic.

Proposition 4.3.5. Let & Mg M$ to be an ergodic measure. Then & &hi forsome i 1, ..., N1 .

Proof. Take v a vector in the support of &. Consider first the case when v pointstoward a parabolic fixed point. If that was the case, by flowing toward the cusp wefind a disjoint countably family of balls with positive measure, the measure can notbe a probability measure. Now consider the case when v points toward a hyperbolicfixed point z. Let ( : R X be a geodesic flowing at positive time to z with initialcondition ( 0 v and let (i to be the geodesic connecting z with the associatedhyperbolic fixed point. By reparametrization we can assume (i z and that(i 0 lies in the same horosphere centered at z than v. By estimates in [HI] we haved (i t , ( t 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 . !

Remark 4.3.6. Note that it is a consequence of Proposition 4.3.5 that the setMg M$ contains only a finite number of ergodic measures. Moreover, for every& Mg M$ we have h & 0.

We end up this section with a definition that includes the standing assumptionson the Schottky groups considered in mostly all of our statements. The importantpoint is that we will be in position to use Lemma 4.3.1, Lemma 4.3.3 and Theorem4.4.1 below.

Definition 4.3.7. We say a extended Schottky group " satisfies property if thefollowing two conditions hold

(1) N1 N2 3.(2) Condition (C5) holds.

4.4. Parabolic critical exponent and s . One of our main technical results isthe following. Let ' be the roof function constructed in sub-section 4.3. In thenext Theorem we give a geometrical characterisation of the value s defined as theunique real number satisfying

P t'infinite if t s ;

finite if t s .


It turns out that this value equals the largest parabolic critical exponent. Thisrelation will allow us to translate several results at the symbolic level into thegeometrical one.

Theorem 4.4.1. Let " be an extended Schottky group with property . Let #,"and ' : # R be the base space and the roof function of the symbolic representationof the geodesic flow gt on (. Then s max !pi , 1 i N2 .

Proof. We first show that s max !pi , 1 i N2 .

P t' limn

1

n 1log

x:!n 1x x

expn

i 0

t' "ix %Ch1x

limn

1

n 1log

% h1x2...xnxn 1%0

expn

i 0

tB( 1 !ix o, xi 1o

limn

1

n 1log

% h1x2...xn 1%0

expn

i 0

td o, xi 1o

The last inequality follows from d x, y B% x, y . 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 td o,h1o

c1,...,cn A h1n m1,...,mn Zn

expn

i 1

td o, cmii o ,

where A h1n represent the set of vectors admissible 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 j N1 N2 1hi 1, if 1 i N1 1

pi 1 N1 , if N1 i N1 N2 1.

Consider n 1 k N1 N2 1 . By restricting ourselves to words with ci biwe can continue the sequence of inequalities above with

m1,...,mn Zexp

n

i 1

td o, bmii o ,

which is equal ton

i 1m Zexp td o, bmi o .

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

N1

i 2m Zexp td o, hm

i o

k N2

i 1m Zexp td o, pmi o

k

.

Hence, it follows that


P t'1

N1 N2log

N1

i 2m Zexp td o, hm

i oN2

i 1m Zexp td o, pmi o

1

N1 N2log

a A h1

P a t, 0 .

In particular, if t max !pi , 1 i N2 then P t' . This shows thats max !pi , 1 i N2 .

We prove now the other inequality. Recall that

' "ix B( 1 !ix 0, an0 B( 1 !ix an0, 0 B( 1 !i 1x 0, a n0 ,

thus,

exp t' "ix exp B( 1 !i 1x 0, a n0 t.

By [DP, Lemmas I.3 and I.4], there exists C 1 and sequences Kan n Z such thatfor any y '0 '0

a one has exp B!i 1x 0, a n0 CKan (see the proof of [DP,Proposition VII.3] where the authors use this fact). Therefore,

P t' limn

1

nlog

a1,...,an m1,...,mn

n

i 1

CtKtamii

limn

1

nlog

a1,...,an m1,...,mn

Ctnn

i 1

Ktamii

limn

1

nlogCtn

a Am ZKt

am

n

logCt

a Am ZKt

am .

We claim that m Z Ktam is finite for every t max !pi , 1 i N2 and every

a A. In that case P t' is finite and hence s max !pi , 1 i N2 . Notethat [DP, Lemma I.3] says that m Z K

tam is finite for every t 0 and a A

a hyperbolic isometry. Thus, we only have to prove our claim for the case whena A is a parabolic isometry. The proof follows from Lemma 4.4.2 below (see [DP,Lemma VII.5] for a stronger statement).

Lemma 4.4.2 (Dal’bo-Peigne). Let p be a parabolic isometry of X of divergencetype and let " be any extended Schottky group containing p as a generator. Then

m Z K)!pm is finite.

Consider p A be a parabolic isometry in our extended Schottky group ". FromTheorem 4.1.5 there exists a sequence "n of discrete non elementary subgroupsof isometries of X, of divergence type, containing p as a subgroup, such that!#n !p as n goes to . Moreover "n is an extended Schottky group. Using

Lemma 4.4.2 we obtain that for any n 1 the sum m Z K)!npm is finite. Since

!#n !p, it follows that m Z Ktpm is finite for every t max !pi , 1 i N2 .

This concludes the proof Theorem 4.4.1.!


Denote by !p,max : max !pi , 1 i N2 . The simplest example is to considerX as a real hyperbolic space, in this case ! pi 1 2 for any i 1, ..., N2 .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, Pe2]).

4.5. Thermodynamic formalism. In this section all Schottky groups consideredsatisfy property . Combining the results from sub-section 4.5 with those insub-section 4.3 we can obtain several results on thermodynamic formalism for thegeodesic flow defined by extended Schottky groups with property . Some ofthese results (like the variational principle) were already obtained, without symbolicmethods, by Paulin, Pollicott and Schapira in [PPS]. Others are new, like Theorem4.5.10, in which we show that potentials with small oscillation do have equilibriummeasures. However, the strength of our symbolic approach will be clear in the nextsections where regularity properties of the pressure (sub-section 4.7) and the escapeof mass of certain measures (sub-section 4.6) will be discussed.

Here we keep the notation of sub-section 4.3. Thus, the geodesic flow gt t inthe set ( can be coded by a suspension semi-flow Y,$ with base #," and rooffunction ' : # R. We will consider the following class of potentials.

Definition 4.5.1. A function f : T 1X " R belongs to the class of regularfunctions, that we denote by R, if f $ has a symbolic representation and &f : #R has summable variations.

We begin by studying thermodynamic formalism for the geodesic flow restrictedto the set (. 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 we denote by f : Y R the symbolicrepresentation of f : ( R.Definition 4.5.2. Let f R, then the pressure of f with respect to the geodesicflow g : gt restricted to the set ( is defined by

P$ f : limt

1

tlog

#s x,0 x,0 ,0 s t

exps

0f #k x, 0 dk %Ci0

x .

This pressure satisfies the following properties:

Proposition 4.5.3 (Variational Principle). Let f R, then

P$ f sup hg &$f d& : & M$ and

$f d& ,

where M$ denotes the set of gt -invariant probability measures supported in (.

Proposition 4.5.4. Let f R, then

P$ f sup Pg K f : K K$ g ,

where K$ g denotes the space of compact g invariant sets in (.

Proposition 4.5.5. Let f R. Then there is an equilibrium measure &f M$,that is,

P$ f hg &f$f d&f ,


for f if and only if we have that P &f P! f ' 0 and there exists an equilibriummeasure µf M! for &f P! f ' such that 'dµf . Moreover, if such anequilibrium measure exists then it is unique.

In order to extend these results to the geodesic flow in T 1X " we make use ofRemark 4.3.6.

Definition 4.5.6. Let f R, then the pressure of f with respect to the geodesicflow g : gt in T 1X " is defined by

Pg f : max P$ f , f d&h1 , . . . , f d&hN1 .

Proposition 4.5.7 (Variational Principle). Let f R, then

Pg f sup hg & f d& : & Mg and f d& ,

Proposition 4.5.8. Let f R, then

Pg f sup Pg K f : K K g ,

where K g denotes the space of compact g invariant sets.

Proposition 4.5.9. Let f R be such that sup f Pg f then there is an equilib-rium measure &f Mg, for f if and only if we have that P$ &f P! f ' 0 andthere exists an equilibrium measure µf M! for &f P! f ' such that 'dµf .Moreover, if such an equilibrium measure exists then it is unique.

Proof. Note that if sup f Pg f then an equilibrium measure for f , if it exists,must have positive entropy. Since the measures &hi , with i 1, . . . , N1 have zeroentropy (see Remark 4.3.6). The result follows from Proposition 4.5.5. !

The next result shows that potentials with small oscillation do have equilibriummeasures.

Theorem 4.5.10. Let f R. If

sup f inf f htop g !p,max

then f has an equilibrium measure.

Proof. Assume that the measures &hi are not equilibrium measures for f , otherwisethe theorem is proved. We therefore have that Pg f P$ f . We first show thatP &f Pg f ' 0. Note that for every x #,

' x inf f &f x ' x sup f.

By monotonicity of the pressure we obtain

P inf f t ' P &f Pg f ' P sup f t ' .

Let t s sup f, htop g inf f and recall that s !p,max. Then

0 P inf f t ' P &f t' P sup f t ' .

Since Pg f and the function t P &f Pg f ' is continuous we obtain thatP &f Pg f ' 0. Since the system # has the BIP condition and the potential


&f Pg f ' is of summable variations, it has an equilibrium measure µ. It remainsto prove the integrability condition. Recall that

tP &f t'

t Pg f&f Pg f 'dµ.

But we have proved that the function t P &f Pg f ' is finite (at least) in aninterval of the form Pg f *, Pg f * . The result now follows, because whenfinite the function is real analytic. !

It was shown by Otal and Peigne [OP, Theorem 1], in a more general setting,that the topological entropy of the geodesic flow, that we denote by htop g , equalsthe critical exponent of the Poincare series of the group ", that is

htop g !#.

Recall that at a symbolic level we have htop $ inf t : P! t' 0 . In partic-ular, we have that

htop g !# htop $ .

The existence of a measure of maximal entropy for the flow g is related to conver-gence properties of the Poincare series at the critical exponent. Indeed, Patterson[Pa] and Sullivan [Su2] constructed an invariant measure that when finite is themeasure of maximal entropy (see [OP, Theorem 1]). It turns out that: if the group" is of convergence type then the Patterson-Sullivan measure is infinite and dissi-pative, hence the geodesic flow does not have a measure of maximal entropy. Onthe other hand, if the group " is of divergence type then the Patterson-Sullivanmeasure is conservative. Thus, if finite it is the measure of maximal entropy.

It is, therefore, of interest to determine conditions that will ensure that the groupis of divergence type and that the Patterson-Sullivan measure is finite. It is alongthese lines that Dal’bo, Otal and Peigne [DOP] introduced the following:

Definition 4.5.11. The geometrically finite group " satisfies the parabolic gapcondition (PGC) if its critical exponent !# is strictly greater than the one of eachof its parabolic subgropus.

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

In our context, an extended Schottky group is a geometrically finite group andit satisfies the PGC-condition (see [DP]). Thus, the following property is a directconsequence of Theorem 4.4.1 .

Proposition 4.5.12. Let " be an extended Schottky group and Y,$ its symbolicrepresentation. Then s htop $ .

4.6. Entropy in the cusp. In this sub-section we prove that there is a uniformbound, depending only in the entropy of a measure, for the amount of mass ameasure can give to the cusps. We also characterise the amount of entropy thatthe cusp can have. These results are similar in spirit to those obtained in [EKP,ELMV, KLKM] for other types of flows.


Theorem 4.6.1. Let " be an extended Schottky group with property . For everyc !p,max and & M$ with h & c there exists a C 0 such that

'dµ C,

where & has a symbolic representation as µ m µ m Y .

The above theorem shows that measures of su%ciently high entropy, i.e. strictlybigger than !p,max can not give too much weight to the cusps. Also note thatthe optimal constant C can be computed as in Corollary 3.0.10. It is a directconsequence of Theorem 3.0.11 and of the symbolic model for the geodesic flowthat the following bound for the entropy in the cusp is obtained.

Theorem 4.6.2. Let " be an extended Schottky group with property . If &nM$ is a sequence of g invariant probability measures such that limn 'dµn

. Then

lim supn

h &n !p,max.

Moreover, there exists a sequence &n M$ of ( invariant probability measuressuch that limn 'dµn and

limn

h &n !p,max.

We proceed to show an explicit family of measures satisfying the second part ofTheorem 4.6.2.

Example 4.6.3. Let " be a Schottky group with property . Denote by p theparabolic isometry in the generator set A with maximal critical exponent, that is!p,max !p. Assume also N1 1. Let h be any hyperbolic isometry in A and forn 1 define Hn p, hn . Observe Hn is an extended Schottky group. Let mPS

n

be the Patterson-Sullivan measure on T 1X Hn. Since any extended Schottky grouphas finite Patterson-Sullivan measure, mPS

n is finite. Moreover, it maximises theentropy of the geodesic flow on T 1X Hn, in other words hmPS

ngt equals !Hn .

The groups Hn satisfies the approximation properties discussed in Theorem 4.1.5,in particular, the critical exponent !Hn converges to !p,max as n goes to infinity.

Using the coding property, we know that T 1X Hn (except geodesic limiting to apoint in the Hn-orbit of the fixed points of h and p) is identified with Yn x, t#n R : 0 t ' x , where

#n amii i Z : ai p, hn ,mi Z ,

and the geodesic flow is conjugated to the suspension flow on Yn, ' (same ' asbefore, but for this coding). It is convenient to think #n ," as a sub-shift of# ," . Since the Patterson-Sullivan measure mPS

n is ergodic and has positiveentropy it need to be supported in Yn, i.e. in the space of geodesics modeled by thesuspension flow. In particular we can consider mPS

n as supported in some invariantsubset of Y , where Y is the total space for the suspension # , ' . Let us call &PS

n

the image measure of mPSn given by the inclusion Yn Y , and µPS

n the invariantmeasure for # ," associated to it. We will prove that

limn

'dµPSn .


Let x #n . For k 1 an even integer, we have

k

i 0

' "ixk

i 0

B( 1 !ix o, amii o

k

i 0

B( 1 a

mii !ix

a mii o, o

k 2

i 0

B( 1 h nm2i!2ix h nm2io, o

k

2d h nm2io, o C

k

2d h no, o C .

Thus, by Birkho! Ergodic Theorem, it follows

'dµPSn

1

2d h no, o C .

In particular we have limn 'dµPSn and limn h &PS

n !p,max.

4.7. Phase transitions. This sub-section is devoted to study the regularity prop-erties of the pressure function t Pg tf for a certain class of functions f . Wesay that the pressure function t Pg tf has a phase transition at t t0 if thepressure function is not real analytic at t t0. The set of points at which thepressure function exhibits phase transitions might be very large set. However, sincethe pressure is a convex function it can only have a countable set of points where itis not 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 P &f t' is real analytic andthat P &f P!' 0. The result then follows from the implicit function theoremnoticing that the non-degeneracy condition is fulfilled:

tP! &f t' 'dµ 0,

where µ is the equilibrium measure corresponding to &f t' . This, together withthe coding properties established in [Bo1, BR, Ra], allow us to establish that thepressure is real analytic for regular potentials in the Axiom A setting. In the non-compact case the situation can be di!erent. However, there doesn’t seem to bemuch work in this direction. The only results involving the regularity properties ofthe pressure function for geodesic flows defined on non-compact manifolds, that weare aware of, are those concerning the modular surface (see [IJ, Section 6]). In thissection we establish regularity results for the pressure for geodesic flows defined onextended Schottky groups. We begin by defining conditions (F1), (F2) and (F3)on the potentials.

Definition 4.7.1. Consider a positive function f : T 1X " R. We will say fsatisfies Condition (F1), (F2) or (F3) if the corresponding property below is true.

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

(F2) There exists a sequence of invariant measures &n n Mg such that

limn T 1X #

fd&n 0.


(F3) Consider any indexation Cn n N of the cylinders of the form Cam . Then,

limn

sup &f x : x Cn

inf ' x : x Cn0.

We say f belong to the class F if it satisfies (F1), (F2) and (F3).

Remark 4.7.2. Condition (F3) implies (F2). It is enough to consider a family ofinvariant measures &n n N with the corresponding invariant measures in the baseµn n N satisfying 'dµn , then condition (F3) gives the desired limit. Notesuch a family of measures has been presented in Example 4.6.3.

In the following Lemma we establish a property of potentials in F that will beused in the sequel.

Lemma 4.7.3. Let f be a potential satisfying (F1). If there exits a sequence&n n N of measures in M$ such that limn fd&n 0. Then, limn 'dµn

, where &n and µn are related by the formula &nµn m

µn m Y .

Proof. We will argue by contradiction. Assume, passing to a sub-sequence if nec-essary, that

limn

'dµn C.

Let * 0, there exists N N such that for every n N we have that

'dµn C *.

Lemma 4.7.4. Let r 1 then for every n N we have that

µn x : ' x r 1C *

r.

Proof of Lemma 4.7.4. Since the function ' is positive we have

'dµn rµn x : ' x rx:" x r

'dµn.

Thus,C * rµn x : ' x r .

C *

rµn x : ' x r .

Finally

µn x : ' x r 1C *

r.

!Note that the set x : ' x r is contained in a finite union of cylinders on

#. Indeed, this follows from the inequality d o, am, o C ' x . Since &f isbounded away from zero in every one of them, there exist a constant G r 0 suchthat

&f x G r ,

on x : ' x r . Thus

fd&n" &f x dµn

" 'dµn

x:" x r &f x dµn

" 'dµn

G r 1 C $r

C *.


If we choose r large enough so that 1 C $r 0 we obtain the desired contradiction.

!

Combining Theorem 4.6.2 and Lemma 4.7.3 we obtain the following

Lemma 4.7.5. Let " be an extended Schottky group with property and let fa function satisfying property (F1). If &n n Mg is a sequence of invariantprobability measure for the geodesic flow such that

limn T 1X #

fd&n 0.

Then

lim supn

h &n !p,max.

Proof. If &n M$ the Lemma follows directly from Lemma 4.7.3 and Theorem4.6.2. If &n Mg M$ 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 tohyperbolic generator hi 0. Let Cn &n X " . If Cn 0 then h &n 0 andthere is not contribution to the desired lim suph &n , otherwise define yn C 1

n &n.By definition yn is a probability measure in M$, we claim limn fdyn 0.

Observe fd&n fd&nN1

i 1 cni &

hi f has non-negative summands and it isconverging to zero, it follows fd&n 0, cni 0 as n . By definition

Cn 1 N1

i 1 cni , therefore Cn 1. Recalling yn C 1

n &n we get lim fdyn 0.

Because &n Cnyn 1 Cn 1 Cn1 N1

i 1 cni &

hi , we have h &n Cnh yn .Finally since Cn 1 and lim supn h yn !p,max (because yn M$) we getlim supn h &n !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 due to Theorem 4.4.1.

Theorem 4.7.6. Let " be an extended Schottky group with property and f F .Then

(1) For every t R we have that Pg tf !p,max.(2) We have that limt Pg tf !p,max.(3) Let t : sup t R : Pg tf !p,max , then

Pg tf!p,max if t t ;

real analytic, strictly convex, strictly increasing if t t .

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

Note that Theorem 4.7.6 shows that when t is finite then the pressure functionexhibits a phase transition at t t and when t the pressure function is realanalytic where defined (see Figure below). Recall that !p,max s .


No phase transitions Phase transition at t t

t t

Pg tf Pg tf

!! !!

!p,max !p,max

t

Proof of (1). The first claim follows from the variational principle. Consider asequence of measures &n n N as in Condition (F2). For every t R we have

s !p,max limn

h &n t$fd&n

sup h & t$fd& : & Mg Pg tf .

!Proof of (2). Since t Pg tf is non-decreasing and bounded below, the followinglimit limt P tf exists. Assume by way of contradiction that limt Pg tfA s . Therefore, there exists a sequence of measures &n n in Mg for which thesequence of straight lines

Ln t : h &n t fd&n,

approximates the asymptote of Pg tf . By Condition F2 the asymptotic deriva-tive is given by

inf fd& : & Mg 0,

we have that

limn

h &n t fd&n A.

In particular, limn fd&n 0. Hence, from Lemma 4.7.5 we obtain that

lim supn

h &n s !p,max A.

But we have thatlimn

Ln 0 A s .

This contradiction proves the statement. !Proof of (3): Real analyticity. We first prove Pg tf P$ tf . After this is donewe can proceed with standard regularity arguments in the symbolic picture. Ob-serve that for t 0 our claim is evident. Indeed, the pressure P$ tf is alwayspositive while the contribution of the pressure on T 1X " ( is negative. Considernow t 0. Pick &hi as above Proposition 4.3.5 (see also Definition 4.5.6), denotex (resp. x ) the repulsor (attractor) of hi and (hi the geodesic defined by those


points. Consider p a parabolic element in A and define the geodesic (n connectingthe points + and + where - + p 1h n and - + hnp. Denote ( thegeodesic connecting p 1x and x . Observe (n descends to a closed geodesic inT 1X ". By comparing (n and ( we can justify that for any * 0, the amount oftime (n leaves a *-neighborhood of (hi is uniformly bounded for big enough n. Let&n to be the invariant probability measure defined by the closed geodesic (n, thenwe get the weak convergence &n &hi . Then

t fd& limn

t fd&n limn

h*n t fd&n P$ tf .

This give us Pg tf P$ tf . The pressure function, t Pg tf , is convex, non-decreasing and bounded below by s . We now prove that for t t it is realanalytic. Note that since t t we have that

P t&f s ' 0,

possibly infinity and that there exists p s such that 0 P t&f p' (see[IJ, Lemma 4.2]). Moreover, the condition F3 implies that Pg tf for everyt t , hence

P t&f Pg tf ' 0.

Since ' is a positive, the function p P t&f p' is decreasing and limp P t&f

p' . Moreover, since the base map of the symbolic model satisfies the BIPcondition, it is real analytic in both variables. Hence, there exists a unique realnumber pf s such that P t&f pf' 0 and

pP t&f p'

p pf

0.

Therefore, Pg tf pf and by the implicit function theorem the function tPg tf is real analytic in t , t . !

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 obtain thatfor t t , we have that P t&f Pg tf ' 0. Since the system satisfies theBIP condition there exists an equilibrium measure µf M! for t&f Pg tf 'such that 'dµf . Therefore it follows from Proposition 4.5.5 that tf has anequilibrium measure.

In order to prove the last claim, assume by way of contradiction that for somet1 t the potential t1f has an equilibrium measure &t1 . Then s P t1fh &t1 t1 $ fd&t1 . Since f 0 we have that $ fd&t1 : B 0. Thus the straightline r h µt1 r $ fd&t1 is increasing with r, therefore for t t1, t we havethat

h µt1 t$fd&t1 s P tf .

This contradiction proves the statement. !

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

Proposition 4.8.1. Let f F . Then

(1) If there exist t0 R such that P t0&f s ' . Then there exists t t0such that for every t t we have Pg t f s .


(2) Suppose we have an interval I such that P t&f s ' for every t I.Then t Pg tf is real analytic in I. In particular, if for every t R wehave P t&f s ' . Then t Pg tf is real analytic in 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 4.8.3) and then one with pressurereal analytic everywhere (Example 4.8.4). A useful lemma in order to show anexample of a phase transition, is the following

Lemma 4.8.2. Let an n be a sequence of positive real numbers such that n 1 atn

converges for every t t and diverges at t t . Then there exists a sequence.n n of positive numbers such that limn .n 0 and

n 1

at +nn .

Proof of Lemma 4.8.2. Let /m m any sequence of real numbers in 0, 1 convergingto zero. Note that, for every m 1 we have

n 1

at &mn .

Then, there exists an integer Km 1 such that

n Km

at &mn 1 m2.

We can suppose without loss of generality that Km Km 1. Define .n for everyKm n Km 1 as .n /m, and .n 1 for 1 n K1. Thus

n 1

at +nn

K1 1

n 1

at +nn

m 1

Km 1 1

n Km

at +nn

K1 1

n 1

at 1n

m 1

Km 1 1

n Km

at &mn

K1 1

n 1

at 1n

m 1 n Km

at &mn

K1 1

n 1

at 1n

m 1

1 m2

.

!Example 4.8.3. (Phase transition) Let " be a Schottky group with propertyand N2 2. Moreover assume there exists a unique parabolic generator p with!p !p,max. Let Kpm be the sequence given in the second part of Theorem 4.4.1.

We recall that m Z K)ppm diverges since p is a parabolic isometry of divergence

type (see Theorem 4.4.1). Take a decreasing sequence of real numbers .m 0,

as in Lemma 4.8.2, such that limm .m 0 and m Z K)p +mpm . Define a

function f0 : # R by

(1) f0 x .m' x if the first symbol of x is pm for certain m Z.


(2) f0 x 1 otherwise.

Observe that since ' is locally Holder, the function f0 is also locally Holder. Asshown in [BRW, Section 2], we can construct a continuous function f : Y R with&f f0. We define t : # R by t x s .m if the first symbol of x is pm,and s otherwise. By simplicity we will denote s am t x if the first symbol ofx is am. Following notations and ideas of the second part of Theorem 4.4.1.

P &f s ' limn

1

n 1log

x:!n 1x x

expn

i 0

&f "ix s ' "ix %Ch1x

limn

1

n 1log

x:!n 1x x

expn

i 0

' "ix t "ix %Ch1x

limn

1

nlog

a1,...,an m1,...,mn

n

i 1

Cs amii K

s amii

amii

limn

1

nlogCn s 1

a1,...,an m1,...,mn

n

i 1

Ks a

mii

amii

limn

1

nlogCn s 1

a Am ZKs am

am

n

logCs 1

a Am ZKs am

am .

Since m Ksam converges for s ! a for every a p we have m Z K

sam .

On the other hand, the series m Z K)p +mpm is finite by construction. In particular

P &f s ' is finite. Observe that f is a potential belonging to the family F ,then from Proposition 4.8.1 it follows that t Pg tf exhibits a phase transition.

Example 4.8.4. (No phase transition) Let " be a Schottky group with property .Define f0 : # R to be constant of value 1 and construct a continuous functionf : Y R with &f f0. Observe

P t s ' t P s ' .

We recall P s ' , because the maximal parabolic generator is of divergencetype (see the first part of Theorem 4.4.1). Observe that since ' is unbounded and f0

is constant, we can apply Proposition 4.8.1 to show that t Pg tf is real analyticin R. In particular, it never attains the lower bound s .

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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]

http://http://www.mat.uc.cl/~giommi/

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