aa r X i v : . [ m a t h . D S ] F e b MANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS
LIEN-YUNG KAO
Abstract.
In this paper, we study an interesting curve, so-called the Manhattan curve, associatedwith a pair of boundary-preserving Fuchsian representations of a (non-compact) surface, especiallyrepresentations corresponding to Riemann surfaces with cusps. Using Thermodynamic Formalism(for countable Markov shifts), we prove the analyticity of the Manhattan curve. Moreover, wederive several dynamical and geometric rigidity results, which generalize results of Marc Burger[Bur93] and Richard Sharp [Sha98] for convex-cocompact Fuchsian representations.
Contents
1. Introduction 12. Preliminaries 52.1. Thermodynamic Formalism for Countable Markov Shifts 52.2. Thermodynamic Formalism for Suspension Flows 72.3. Hyperbolic Surfaces 73. Extended Schottky Surfaces 93.1. Coding of Closed Geodesics 103.2. Phase Transition of the Geodesic Flow 124. The Manhattan Curve 184.1. The Manhattan Curve, Critical Exponent, and Gurevich Pressure 184.2. Proof of Main Results 215. Appendix 24The Proof of Lemma 6 25The Proof of Theorem 10 30References 331.
Introduction
This paper is devoted to studying relations between Fuchsian representations of a (non-compact)surface through a dynamics tool, namely, Thermodynamic Formalism (for countable Markov shifts).Using a symbolic dynamics model associated with these representations, we investigate severalclosely related and informative geometric and dynamical objects arising from them, such as thecritical exponent, the Manhattan curve, and Thurston’s intersection number. For dynamics, weprove a version of the famous Bowen’s formula, which characterizes several geometric and dynamicsquantities via the (Gurevich) pressure. Moreover, we analyze the phase transition of the pressurefunction (of weighted geometric potentials) in detail; thus, we have a control of the analyticity ofthe pressure. In geometry, we recover and extend several rigidity results, such as Bishop-Stegerentropy rigidity and Thurston’s intersection number rigidity, to Riemann surfaces of infinite volumeand with cusps.
Kao gratefully acknowledges support from the National Science Foundation Postdoctoral Research Fellowshipunder grant DMS 1703554.
To put our results in context, we shall start from notations and definitions. Throughout thepaper, S denotes a (topological) surface with negative Euler characteristic. Let ρ , ρ be twoFuchsian (i.e., discrete and faithful) representations of G := π S into PSL(2 , R ). For short, wedenote ρ i ( G ) by Γ i , by S i = Γ i \ H the Riemann surface of ρ i for i = 1 ,
2. We write h top ( S ) and h top ( S ) for the topological entropy of the geodesic flow for S and S .The group G acts diagonally on H × H by γ · x = ( ρ ( γ ) x , ρ ( γ ) x ) where x = ( x , x ) ∈ H × H and γ ∈ G . We are interested in weighted Manhattan metrics d a,bρ ,ρ associated with S and S .More precisely, fix o = ( o , o ) ∈ H × H , d a,bρ ,ρ ( o, γo ) := a · d ( o , ρ ( γ ) o ) + b · d ( o , ρ ( γ ) o ).Moreover, we always assume that a, b ≥ a, b do not vanish at the same time, i.e., throughoutthis paper we assume that ( a, b ) ∈ D := { ( x, y ) ∈ R : x ≥ , y ≥ }\ (0 , Definition 1.
The Poincar´e series of the weighted Manhattan metric d a,bρ ,ρ is defined as Q a,bρ ,ρ ( s ) = X γ ∈ G exp( − s · d a,bρ ,ρ ( o, γo )) . Moreover, δ a,bρ ,ρ denotes the critical exponent of Q a,bρ ,ρ ( s ), that is, Q a,bρ ,ρ ( s ) diverges when s < δ a,bρ ,ρ and Q a,bρ ,ρ ( s ) converges when s > δ a,bρ ,ρ . For short, if there is no confusion, we will always dropthe subscripts ρ , ρ .Noticing that the critical exponent δ a,b , by the triangle inequality, is independent on the choiceof the reference point o = ( o , o ). We remark that when a = 0 (or b = 0), we are back to theclassical critical exponent of ρ ( G ) (or ρ ( G )), and by Sullivan’s result we know that δ , = h top ( S )and δ , = h top ( S ). Definition 2 (The Manhattan Curve) . The Manhattan curve C = C ( ρ , ρ ) of ρ , ρ is the bound-ary of the set { ( a, b ) ∈ R : Q a,bρ ,ρ (1) < ∞} . Alternatively, C can be defined as { ( a, b ) ∈ R : Q a,bρ ,ρ ( s ) has critical exponent 1 } . Our first result gives a rough picture of the Manhattan curve C ( ρ , ρ ) of ρ and ρ for any pairof Fuchsian representations. Theorem (Theorem 9) . Let S be a (topological) surface with negative Euler characteristic, and let ρ , ρ be two Fuchsian representations of G := π S into PSL(2 , R ) . We denote S = ρ ( G ) \ H and S = ρ ( G ) \ H . Then (1) ( h top ( S ) , and (0 , h top ( S )) are on C ; (2) C is convex; and (3) C is continuous. Let us briefly review the history of the Manhattan curve C ( ρ , ρ ). In [Bur93], using thePatterson-Sullivan argument, Burger proved that for ρ and ρ are convex co-compact (i.e., both ρ ( G ) and ρ ( G ) have no parabolic element), one has C is C . In [Sha98], Sharp employed Thermo-dynamic Formalism to prove that C is real analytic. In this work, we are interested in representa-tions which are not convex co-compact. The presence of parabolic elements greatly complicates theproblem. Nevertheless, thanks to recent developments on Thermodynamic Formalism for countableMarkov shifts, we are able to generlize these results to surfaces with cusps.We mainly work on representations that satisfy the following two geometric conditions, namely,being boundary-preserving isomorphic and the extended Schottky condition .Two Fuchsian representations ρ and ρ are boundary-preserving isomorphic if there exists anisomorphism ι : ρ ( G ) → ρ ( G ) so that ι is type-preserving and peripheral-structure-preserving . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 3
More precisely, ι is type-preserving if ι sends parabolic elements to parabolic elements and hyper-bolic elements to hyperbolic elements, and ι is peripheral-structure-preserving if for any element γ ∈ ρ ( G ) corresponding to a geodesic boundary of S , its image ι ( γ ) corresponds to a geodesicboundary of S and vise versa.We say a Riemann surface ρ ( G ) \ H is a extended Schottky surface or ρ satisfies the extendedSchottky condition if ρ satisfies (C1) , (C2) , (C3), and N + N ≥ classical Schottky surfaces, and they are known to be convex co-compact.One example of extended Schottky surfaces is the surface with two cusps and two funnels. Remark. (1) For ρ , ρ two convex co-compact Fuchsian representations, ρ and ρ are always type-preserving isomorphic (because they have no parabolic element). However, it does notguarantee that S and S are homeomorphic, for example, one holed torus and a pair ofpants. Therefore, the peripheral-structure-preserving condition is necessary to derive ahomeomorphism between S and S (see Theorem 7 for more details).(2) The extended Schottky condition that we use here was introduced in Dal’Bo-Peign´e [DP96].This condition is needed in our argument for some technical reasons.Now, we are ready to present our main results. Let ρ , ρ be two boundary-preserving isomor-phic Fuchsian representations satisfying the extended Schottky condition. For the convenience ofpresentation, we leave precise definitions of many dynamics and geometry terminologies in Section2. Following Dal’bo-Peign´e [DP96], there exists a symbolic coding of closed geodesics on extendedSchottky surfaces. Here we summarize relevant results in [DP96]. Proposition (Propsition 2, Propsition 3, Lemma 2) . There exists a topologically mixing countableMarkov shift (Σ + , σ ) satisfying the BIP property. Moreover, there is a function τ : Σ + → R + (resp. κ : Σ + → R + ) such that all but finitely many closed geodesics on S (resp. S ) are coded by Fix(Σ + ) the fixed points of σ and lengths of these closed geodesics are given by τ (reps. κ ). Furthermore, τ and κ are locally H¨older and bounded away from zero. Because τ and κ are constructed by the geometric potential of the corresponding Bowen-Seriesmap on the boundary of T S and T S , we will continue calling them by geometric potentials (seeSection 3 for more details).The following lemma is one of the most important result of this work. Recall that for a finiteMarkov shift, the (Gurevich) pressure P σ has no phase transition, that is, the pressure function t P σ ( tf ) is analytic for f a H¨older continuous potential. Whereas, for countable Markov shifts, Sarig[Sar99, Sar01] and Mauldin-Urba´nski [MU03] pointed out that, for f a locally H¨older continuouspotential, t P σ ( tf ) is not analytic. Inspired by the work of Iommi-Riquelme-Velozo [IRV16], westudy the phase transition in detail and give a precise picture of the pressure function of weightedgeometric potentials. Lemma (Lemma 3, Lemma 4) . Let (Σ + , σ ) be the countable Markov shift and τ , κ be the geometricpotentials given by the above proposition. We have, for a, b ≥ ,P σ ( − t ( aτ + bκ )) = ( infinite , for t < a + b ) ;real analytic , for t > a + b ) . Furthermore, similar to Bowen’s formula for hyperbolic flows over compact metric spaces, wegive a geometric interpretation of the solution for the equation P σ ( tf ) = 0 when f is a weightedgeometric potential. Namely, we prove that the critical exponent δ a,b can be realized by the growthrate of hyperbolic elements (or equivalently, closed geodesics). LIEN-YUNG KAO
Theorem (Bowen’s Formula; Lemma 5 , Theorem 8, Theorem 10) . The set { ( a, b ) ∈ D : P σ ( − aτ − bκ ) = 0 } is a real analytic curve. Moreover, for each ( a, b ) ∈ D there exists a unique t a,b such that P σ ( − t a,b ( aτ + bκ )) = 0 . Furthermore, t a,b = δ a,b . Combing the above theorems, we have the following results for the Manhattan curve C ( ρ , ρ ). Theorem (Theorem 11) . C ( ρ , ρ ) is real analytic. Moreover, using the analyticity of the Manhattan curve and the uniqueness of the equilibriumstates, we have better picture of the Manhattan curve C ( ρ , ρ ). Proposition (Proposition 4) . We have (1) C ( ρ , ρ ) is strictly convex if ρ and ρ are NOT conjugate in PSL(2 , R ) ; and (2) C ( ρ , ρ ) is a straight line if and only if ρ and ρ are conjugate in PSL(2 , R ) .Remark. Using Paulin-Pollicott-Schapira’s arguments in [PPS15], as well as Dal’Bo-Kim’s Patterson-Sullivan theory approach in [DK08], it is possible to recover some of above results without usingsymbolic dynamics. However, due to the author’s limited knowledge, without using symbolic dy-namics, there seems no clear path to proving the analyticity of the Manhattan curve C ( ρ , ρ ).Furthermore, we have the following rigidity corollaries. Corollary (Bishop-Steger’s entropy rigidity; cf. [BS93]; Corollary 2) . We have, for any o ∈ H , δ , = lim T →∞ T log { γ ∈ G : d ( o, ρ ( γ ) o ) + d ( o, ρ ( γ ) o ) ≤ T } . Moreover, δ , ≤ h top ( S ) · h top ( S ) h top ( S ) + h top ( S ) and the equality holds if and only if S and S are isometric.Remark. In Bishop-Steger’s paper [BS93], their result holds for finite volume Fuchsian representa-tions (i.e., lattices). We extend this result to some infinite volume Fuchsian representations.
Definition 3 (Thurston’s Intersection Number) . Let S and S be two Riemann surfaces. Thurston’sintersection number I ( S , S ) of S and S is given byI( S , S ) = lim n →∞ l [ γ n ] l [ γ n ]where { [ γ n ] } ∞ n =1 is a sequence of conjugacy classes for which the associated closed geodesics γ n become equidistributed on Γ \ H with respect to area. Corollary (Thurston’s Rigidity; cf. [Thu98]; Corollary 3) . Let ρ , ρ be two boundary-preservingisomorphic Fuchsian representations satisfying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Then I( S , S ) ≥ h top ( S ) h top ( S ) and the equality hold if and only if ρ and ρ are conjugatein PSL(2 , R ) . The outline of this paper is as follows. In Section 2, we briefly review necessary background ofThermodynamic Formalism (for countable Markov shifts) and hyperbolic geometry. In Section 3,we introduce extended Schottky surfaces. Moreover, we study the phase transition of the geodesicflows on them, which is one of the most important results in this work. Section 4 is devoted tothe proof of our main results. Using Paulin-Pollicott-Schapira’s arguments in [PPS15], we derivegeometric interpretations of the critical exponent δ a,b and, thus, we are able to link it with the(symbolic) suspension flow and Bowen’s formula. ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 5
Acknowledgement.
The author is extremely grateful to his Ph.D advisor Prof. Fran¸cois Ledrappier.Nothing would have been possible without Fran¸cois’ support, guidance, especially those he gave theauthor after his retirement. The author also appreciates Prof. Richard Canary for his numeroushelps on geometry parts of this work, and Prof. Godofredo Iommi for his wonderful explanationson his works. Lastly, the author would like to thank Prof. Fran¸coise Dal’Bo and Prof. Marc Peign´efor their helpful comments of this work, and Dr. Felipe Riquelme for pointing out some errors inthe earlier version. 2.
Preliminaries
Thermodynamic Formalism for Countable Markov Shifts.
Let S be a countable setand A = ( t ab ) S×S be a matrix of zeroes and ones indexed by
S × S . Definition 4.
The one-sided (countable) Markov shift (Σ + A , σ ) with the set of alphabet S is the setΣ + A = { x = ( x n ) ∈ S N : t x n x n +1 = 1 for every n ∈ N } coupled with the (left) shift map σ : Σ + A → Σ + A , ( σ ( x )) i = ( x ) i +1 .We will alway drop the subscript A of Σ + A when there is no ambiguity on the adjacency matrix A . Furthermore, we endow Σ + with the relative product topology, which is given by the base of cylinders [ a , ..., a n − ] := { x ∈ Σ + : a i = x i , for 0 ≤ i ≤ n − } . A word on an alphabet S is an element ( a , a , ..., a n − ) ∈ S n ( n ∈ N ). The length of the word( a , a , ..., a n − ) is n . A word is called admissible (w.r.t. an adjacency matrix A ) if the cylinder itdefines is non-empty.In the following, we will assume (Σ + , σ ) is topologically mixing , that is, for any a, b ∈ S , thereexists an N ∈ N such that σ − n [ a ] ∩ [ b ] is non-empty for all n > N . Noticing that under thetopologically mixing assumption and the BIP property below, the thermodynamics formalism forcountable Markov shifts is well-studied and very close to the classical thermodynamic formalismfor finite Markov shifts.The n -th variation of a function g : Σ + → R is defined by V n ( g ) = sup {| g ( x ) − g ( y ) | : x, y ∈ Σ + , x i = y i for i = 1 , , ..., n } . We say g has summable variation if P ∞ n =1 V n ( g ) < ∞ , and g is locally H¨older if there exists c > θ ∈ (0 ,
1) such that V n ( g ) ≤ cθ n for all n ≥ Definition 5 (Gurevich Pressure for Markov Shifts) . Let g : Σ + → R have summable variation.The Gurevich pressure of g is defined by P σ ( g ) = lim n →∞ n log X x ∈ Fix n e S n g ( x ) χ [ a ] ( x )where Fix n := { x ∈ Σ + : σ n x = x } and a is any element of S and S n g ( x ) := P n − i =0 g ( σ i x ).As pointed out by Sarig (cf. Theorem 1 [Sar99]) that the limit exists and is independent of thechoice of a ∈ S . Theorem 1 (Variational Principle; Theorem 3 [Sar99]) . Let (Σ + , σ ) be a topologically mixingcountable Markov shift and g have summable variation. If sup g < ∞ then P σ ( g ) = sup (cid:26) h σ ( µ ) + Z Σ + g d µ : µ ∈ M σ and − Z Σ + g d µ < ∞ (cid:27) , where M σ is the set of σ − invariant Borel probability measures on Σ + . LIEN-YUNG KAO
For µ ∈ M σ such that P σ ( g ) = h σ ( µ ) + R Σ + g d µ , we call such a measure µ an equilibrium state for the function g . Definition 6 (BIP) . A (countable) Markov shift (Σ + A , σ ) has the BIP (Big Images and Preimages)property if and only if there exists { b , b , ..., b n } ⊂ N such that for every a ∈ N there exists i, j ∈ N with t b j a t ab j = 1.The following theorem about the analyticity of pressure is found independently by Mauldin-Urba´nski [MU03] and Sarig [Sar03]. There are minor differences between their original statements;however, under the topologically mixing and the BIP assumptions their results are the same (seeRemark 1 for more details). Theorem 2 (Analyticitly of Pressure; Theorem 2.6.12, 2.6.13 [MU03], Corollary 4 [Sar03]) . Let (Σ + , σ ) be a topologically mixing countable Markov shift with the BIP property. If ∆ ⊂ R is ainterval and t → f t a real analytic family of locally H¨older continuous functions with P σ ( f t ) < ∞ ,then t → P σ ( f t ) ∈ R , t ∈ ∆ , is also real analytic. Moreover, the derivative of the pressure functionis ddt P σ ( f t ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = Z Σ + f d µ f where µ f is the equilibrium state for f .Remark . (1) We combine Proposition 2.1.9 and Theorem 2.6.12 in [MU03] in the following way to deriveTheorem 2. By Proposition 2.1.9, we know that P σ ( f t ) < ∞ implies f t are summable H¨olderfunctions (i.e., f t ∈ K sβ in [MU03] notation). The rest is a direct consequence of Theorem2.6.12.(2) A topologically mixing countable Markov shift (Σ + , σ ) with the BIP property is indeeda graph directed Markov system with a finitely irreducible adjacency matrix defined in[MU03]. Hence the definition of (Gurevich) pressure given in here (from Sarig [Sar99])matches with the one given in Mauldin-Urba´nski [MU03] (cf. Section 7 [MU01]).(3) For Corollary 4 in [Sar03], it requires f t to be positive recurrent . However, under the sameassumptions as in Theorem 2 (i.e., (Σ + , σ ) is topologically mixing with the BIP propertyand f t are functions of summable variation with P σ ( f t ) < ∞ ) then one can prove f t arepositive recurrent (cf. Corollary 2 [Sar03] or Proposition 3.8 [Sar09]). Theorem 3 (Phase Transition; [Sar99, Sar01], [MU03]) . Let (Σ + , σ ) be a countable Markov shiftwith the BIP property and g : Σ + → R be a positive locally H¨older continuous function. Then thereexists s ∞ > such that the pressure function t → P σ ( − tg ) has the following properties P σ ( − tg ) = ( ∞ if t < s ∞ , real analytic if t > s ∞ . Moreover, if t > s ∞ there exists a unique equilibrium state for − tg . Recall that two functions f, g : Σ + → R are said to be cohomologous , denoted by f ∼ g , viaa transfer function h , if f = g + h − h ◦ σ . A function which is cohomologous to zero is called a coboundary . Theorem 4 (Livˇsic Theorem; Theorem 1.1 [Sar09]) . Suppose (Σ + , σ ) is topologically mixing, and f, g : Σ + → R have summable variation. Then f and g are cohomologous if and only if for all x ∈ Σ + and n ∈ N such that σ n ( x ) = x , S n f ( x ) = S n g ( x ) . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 7
Thermodynamic Formalism for Suspension Flows.
Let (Σ + , σ ) be a topologically mix-ing (countable) Markov shift and τ : Σ + → R + be a positive function of summable variation andbounded away from zero which we call the roof function . We define the suspension space (relativeto τ ) as Σ + τ := { ( x, t ) ∈ Σ + × R : 0 ≤ t ≤ τ ( x ) } , with the identification ( x, τ ( x )) = ( σx, suspension flow φ (relative to τ ) is defined as the (vertical) translation flow on Σ + τ given by φ t ( x, s ) = ( x, s + t ) for 0 ≤ s + t ≤ τ ( x ) . Let F : Σ + τ → R be a continuous function, we define ∆ F : Σ + → R as∆ F ( x ) = Z τ ( x )0 F ( x, t )d t. The following version of the Gurevich pressure for suspension flows is given in Kempton [Kem11].
Definition 7 (Gurevich Pressure for Suspension Flows) . Suppose F : Σ + τ → R is a function suchthat ∆ F : Σ + → R has summable variation. The Gurevich pressure of F over the suspension flow(Σ + τ , φ ) is defined as P φ ( F ) := lim T →∞ T log X φ s ( x, x, ≤ s ≤ T exp (cid:18)Z s F ( φ t ( x, dt (cid:19) χ [ a ] ( x ) , where a is any element of S .Notice that as pointed out by Kempton (cf. Lemma 3.3 [Kem11]), this definition is independentwith the choice of a ∈ S . Moreover, there are several alternative ways of defining the Gurevichpressure for suspension flows such as using the variational principle. In the following, we summarizesome of them from works of Savchenko [Sav98], Barreira-Iommi [BI06], Kempton [Kem11], andJaerisch-Kesseb¨ohmer-Lamei [JKL14]. Theorem 5 (Charaterizations for the Gurevich Pressure) . Under the same assumptions as inDefinition 7, we have: P φ ( F ) = inf { t ∈ R : P σ (∆ F − tτ ) ≤ } = sup { t ∈ R : P σ (∆ F − tτ ) ≥ } = sup (cid:26) h φ ( ν ) + Z Σ + τ F d ν : ν ∈ M φ and − Z Σ + τ τ d ν < ∞ (cid:27) , where M φ is the set of φ − invariant Borel probability measures on Σ + τ . As before, we call a measure ν ∈ M φ an equilibrium state for F if P φ ( F ) = h φ ( ν ) + R F d ν .2.3. Hyperbolic Surfaces.
Let S be a surface with negative Euler characteristic. Recall thata Fuchsian representation ρ is a discrete and faithful representation from G := π S to ρ ( G ) :=Γ ≤ PSL(2 , R ) ∼ = Isom( H ). It is well-known that all hyperbolic surfaces (i.e., surfaces with constantGaussian curvature −
1) can be realized by a Fuchsian representation, and vise versa. A Fuchsianrepresentation is called geometrically finite if there exists a fundamental domain which is a finite-sided convex polygon. Recall that ∂ ∞ H the boundary of H is defined as R ∪ {∞} , and the limitset Λ(Γ) ⊂ ∂ ∞ H of Γ is the set of limit points of all Γ-orbits Γ · o for o ∈ H . We call an element γ ∈ Γ hyperbolic (reps. parabolic ), if γ has exactly two (resp. one) fixed points on ∂ ∞ H . Fora hyperbolic element γ we denote the attracting fixed point by γ + (i.e., γ + = lim n →∞ γ n o ) and repelling fixed point by γ − (i.e., γ − = lim n →∞ γ − n o ). For each hyperbolic element γ ∈ Γ, the
LIEN-YUNG KAO geodesic on H connecting γ − and γ + projects to a closed geodesic on Γ \ H . We denote this closedgeodesic on Γ \ H by λ γ . Conversely, each closed geodesic λ on Γ \ H it corresponds to a uniquehyperbolic element (up to conjugation) which is denoted by γ λ . Moreover, the length l [ λ γ ] of theclosed geodesic λ γ is exactly the translation distance l [ γ ] of γ , where l [ γ ] := min { d ( x, γx ) : x ∈ H } . Definition 8.
The
Busemann function B : ∂ ∞ H × H × H is defined as B ξ ( x, y ) := lim z → ξ d ( x, z ) − d ( x, y )where ξ ∈ ∂ ∞ H and x, y, z ∈ H .We summarize several well-known properties of the Busemann function: Proposition 1.
Let B : ∂ ∞ H × H × H → R be the Busemann function. Then for ξ ∈ ∂ ∞ H and x, y, z ∈ H (1) B ξ ( x, y ) + B ξ ( y, z ) = B ξ ( x, z ) ; (2) For γ ∈ PSL(2 , R ) , B γ ( ξ ) ( γ ( x ) , γ ( y )) = B ξ ( x, y ) ; and (3) B ξ ( x, y ) ≤ d ( x, y ) .Remark . (1) Equivalently, using the Poincar´e disk model, we can replace H by the unit disk D (throughthe map Ψ : H → D where Ψ( z ) = i z − iz + i ). We have Isom( H ) ∼ = Isom( D ) ∼ = PSL(2 , R ). In thispaper, we will alternate the use of H and D depending on the convenience of computationand presentation.(2) In the Poincar´e disk model, ∂ ∞ D is S and the Busemann function B : ∂ ∞ D × D × D → R satisfies the same properties stated above.(3) Moreover, there is a neat formula for the Busemann function: for ξ ∈ ∂ ∞ D (cid:12)(cid:12) γ ′ ( ξ ) (cid:12)(cid:12) = e B ξ ( o,γ − o ) where γ ( z ) : D → D is the M¨obius map associated with γ ∈ PSL(2 , R ) and o is the origin.2.3.1. Marked Length Spectrum.
As mentioned in the previous subsection, for a hyperbolic surface R = Γ \ H , there exists a bijection between free homotopy classes on R and conjugacy classes of Γ.Moreover, we have a bijection between closed geodesics on R and conjugacy classes of hyperbolicelements of Γ. Definition 9. A marked length spectrum function l : [ c ] l [ c ] ∈ R + which assigns to a homotopyclass [ c ] the length l [ c ]. In other words, it is also the function l : [ h ] l [ h ] which assigns to aconjugacy class of a hyperbolic element [ h ] of the length l [ h ] of the corresponding unique closedgeodesic.The following theorem shows that for each Fuchsian representation its proportional markedlength spectrum determines the surface. We remark that for convex-cocompact cases the sameresult was stated (without a proof) in Burger [Bur93]. For general Fuchsian representations, wefound it in [Kim01]. Theorem 6 (Proportional Marked Length Spectrum Rigidity; Theorem A [Kim01] ) . Let ρ , ρ : G → PSL(2 , R ) be Zariski dense Fuchsian representations having the proportional marked lengthspectrum (i.e., there exists a constant c > such that l [ ρ ( γ )] = c · l [ ρ ( γ )] for all γ ∈ G ) . Then ρ and ρ are conjugate in PSL(2 , R ) .Remark . (1) A representation ρ : G → PSL(2 , R ) is called Zariski dense if it is irreducible and non-parabolic, where non-papabolic means ρ ( G ) has no global fixed point on the boundary of H . It is clear that Fuchsian representations satisfying the extended Schottky condition (seeSection 3) are Zariski dense. ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 9 (2) Kim’s result is way more general than the version stated above. However, this version issufficient for us. Also, the stated version should be known before Kim; however, we cannotfind a proper reference earlier than this one.2.3.2.
Boundary-Preserving Isomorphic Representations.
Definition 10.
Let ρ , ρ be two geometrically finite Fuchsian representations from G (= π S ) intoPSL(2 , R ). We say ρ and ρ are boundary-preserving isomorphic if there exists an isomorphism ι : ρ ( G ) → ρ ( G ) such that(1) ι is type-preserving , i.e., ι sends hyperbolic elements to hyperbolic elements and parabolicelements to parabolic elements,(2) ι is peripheral-structure-preserving , i.e., γ ∈ ρ ( G ) corresponds to a geodesic boundary of S if and only ι ( γ ) ∈ ρ ( G ) corresponds to a geodesic boundary of S . Theorem 7 (Fenchel-Nielsen Isomorphism Theorem, cf. Theorem 5.4 [Kap09], Theorem V.H.1[Mas88]) . Let ρ , ρ be two geometrically finite Fuchsian representations and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Suppose there is a boundary-preserving isomorphism ι : ρ ( G ) → ρ ( G ) . Then thereexists an ι -equivariant bilipschitz homeomorphism f : S → S . We then lift f to their universal coverings, and, thus, derive an ι -equivariant bilipschitz homeo-morphism between universal coverings (both are H ). By abusing the notation, we still denote thishomeomorphism by f : H → H . More precisely, there exists a constant C > x, y ∈ H C d ( x, y ) ≤ d ( f ( x ) , f ( x )) ≤ Cd ( x, y ) . Remark . (1) In Theorem 5.4 [Kap09], the ι -equivariant homeomorphism f : S → S is stated to bequasiconformal. Nevertheless, it is well-known (cf. Mori’s theorem) that quasiconformalhomeomorphisms are bilipschitz maps.(2) Tukia’s isomorphism Theorem (cf. Theorem 3.3 [Tuk85]) points out that the boundariesof these two Fuchsian groups are also strongly related. More precisely, there exists an ι -equivariant H¨older continuous homeomorphism q : Λ(Γ ) → Λ(Γ ).3. Extended Schottky Surfaces
In this section, following the notations in Dal’Bo-Peign´e, we will mostly use the Poincar´e diskmodel D . Nevertheless, one can easily convert it to the upper-half plane model H . Let us fix twointegers N , N such that N + N ≥ N ≥ N hyperbolic isometries h , ..., h N and N parabolic isometries p , ..., p N satisfying the following conditions:(C1) For 1 ≤ i ≤ N there exists in ∂ ∞ D = S a compact neighborhood C h i of the attractingfixed point h + i of h i and a compact neighborhood C h − i of the repelling fixed point h − i of h i such that h i ( S \ C h − i ) ⊂ C h i . (C2) For 1 ≤ i ≤ N there exists in S a compact neighborhood C p i of the unique fixed point p ± i of p i such that for all n ∈ Z ∗ := Z \{ o } p ni ( S \ C p i ) ⊂ C p i . (C3) The 2 N + N neighborhoods introduced in (C1) and (C2) are pairwise disjoint.The group Γ = h h , ..., h N , p , ..., p N i ≤ Isom( D ) ∼ = PSL(2 , R ) is proved (cf. [DP96]) to be anon-elementary free group which acts properly discontinuously and freely on D . Definition 11.
We call Γ = h h , ..., h N , p , ..., p N i an extended Schottky group if it satisfies con-ditions (C1) , (C2) , (C3), and N + N ≥
3. Moreover, if Γ is an extended Schottky group and R is the hyperbolic surface Γ \ D , then we say that the corresponding Fuchsian representation ρ (i.e., ρ : π R → PSL(2 , R ) such that ρ ( π R ) = Γ) satisfies the extended Schottky condition . Remark . (1) If N = 0 the groups Γ is a (classical) Schottky group.(2) Hyperbolic surface satisfying (C1) , (C2) , (C3) are geometrically finite with infinite volume.(3) For a hyperbolic surface satisfying (C1) , (C2) , (C3), by the computation in Lemma 3, onehas the elementary parabolic groups h p i i for 1 ≤ i ≤ N are of divergent type.(4) The definition of extended Schottky condition here (for hyperbolic surfaces) is extractedfrom a more general definition for manifolds with pinched negative curvatures (cf. [DP96,DP98]).Let A ± = n h ± , ..., h ± N , p , ..., p N o . For a ∈ A ± denote by U a the convex hull in D ∪ ∂ ∞ D of theset C a . For extended Schottky surfaces, we have the following important and very useful lemma. Lemma 1.
Let Γ be an extended Schottky group. Fix o ∈ D , then there exists an universal constant C > (depending only on generators of Γ and the fixed point o ) such that for every a , a ∈ A ± satisfying a = a ± , and for every x ∈ U a and y ∈ U a , one has d ( x, y ) ≥ d ( x, o ) + d ( y, o ) − C. Remark . The above lemma is well-known. The version that we stated is taken from Lemma 4.4[IRV16].3.1.
Coding of Closed Geodesics.
In this subsection, we plan to present a coding of closedgeodesics on extended Schottky surfaces. This symbolic coding is given in Dal’Bo-Peign´e [DP96](the case of P = ∅ in their notation).Throughout this subsection, let S be a surface with negative Euler characteristic and ρ , ρ be two boundary-preserving isomorphic Fuchsian representations, from G = π S into PSL(2 , R ),satisfying the extended Schottky condition. For i = 1,2, we denote Γ i = ρ i ( G ), S i = Γ i \ D , andΛ(Γ i ) denotes the limit set of Γ i .Since ρ and ρ are boundary-preserving isomorphic and satisfying the extended Schottky con-dition, we write G = h h , h , ..., h N , p , p , ..., p N i where h j (resp. p k ) is called hyperbolic (resp.parabolic) and it corresponds to a hyperbolic (resp. parabolic) element ρ i ( h j ) (resp. ρ i ( p k )). Wedenote the set of generators by A = { h , h , ..., h N , p , p , ..., p N } .We first work on one fixed extended Schottky surface, say S . In the following, we recall defi-nitions and summarize several useful propositions from Dal’Bo-Peign´e [DP96] about the coding ofthe geodesics on S . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 11
Definition 12. (1) Let A = { h , h , ..., h N , p , p , ..., p N } , the countable Markov shift (Σ + , σ ) associated with S is defined asΣ + = { x = ( a n i i ) i ≥ : a i ∈ A , n i ∈ Z ∗ , and a i = a ± i +1 } where Z ∗ = Z \{ } , and the shift map σ ( a n a n a n ... ) = a n a n ... .;(2) Λ is a subset of Λ(Γ ) defined asΛ = Λ(Γ ) \{ Γ ξ : ξ is a fixed point of ρ ( α ) for α ∈ A} ; and(3) G S is the set of all closed geodesics on S except those corresponding to hyperbolic elementsin A . Proposition 2 (Coding Property and the Geometric Potential ) . (1) (p .
759 [DP96])
There exists a bijection ω : Λ → Σ + . (2) (p .
760 [DP96])
The Bowen-Series map T : Λ → Λ is given by T ( ξ ) = ω − ( σ ( ω ( ξ )) for ξ ∈ Λ . (3) (Lemma II . There exists a bijection (up to cyclic permutations) H : G S → Fix(Σ + ) where Fix(Σ + ) = ∪ n Fix n (Σ + ) is the set of fixed points of σ . (4) (p .
759 [DP96])
Let τ : Σ + → R be the geometric potential (relative to T ), that is, τ ( x ) := − log | T ′ ( ω − ( x )) | = B ω − ( x ) ( o, ρ ( a n ) o ) , where x = a n a n ... ∈ Σ + . Suppose γ ∈ Γ is a hyperbolic element and ω ( γ + ) = a n ...a n k k ∈ Fix k (Σ + ) , then l [ γ ] = S k ( τ ( ω ( γ + )) . (5) (Lemma II . There exist
K, C > such that S n τ ( x ) ≥ C for all n > K and x ∈ Σ + . (6) (Lemma V . , V . τ is locally H¨older continuous. Furthermore, the countable Markov shift (Σ + , σ ) derived above satisfies the following two favor-able conditions. Proposition 3 (Properties of the Markov Shift) . Let (Σ + ,σ ) be the countable Markov shift asso-ciated to S . Then (1) The Markov shift (Σ + , σ ) satisfies the BIP property; and (2) If N + N ≥ , then (Σ + , σ ) is topologically mixing.Proof. Taking the finite set to be A = { h , h , ..., h N , p , p , ..., p N } , then it is clear that (Σ + , σ )satisfies the BIP property (see Definition 6). The topologically mixing property for Markov shiftsis a combinatorics condition: Claim:
For every x, y ∈ { a mi : a i ∈ A , m ∈ Z } , there exists N = N ( x, y ) ∈ N such that for all k > N there is an admissible word of length k of the form xa n a n ...a n k − k − y for some n i ∈ Z ∗ and i = 2 , ..., k − + = { x = ( a n i i ) i ≥ : a i ∈ A , n i ∈ Z ∗ , and a i = a ± i +1 } . Since N + N ≥ , we have at least three distinct elements in A , say a , a , a . Pick two elements x, y in { a mi : a i ∈A , m ∈ Z } , w.l.o.g., say x = a m and y = a m .For k = 2 t + 2 for any t ∈ N , then the following word is admissible: a m ( a a ) ... ( a a ) | {z } t paris a m . For k = 2 t + 3 for any t ∈ N , then the following word is admissible: a m ( a a ) ... ( a a ) | {z } t paris a a m . (cid:3) Using a standard argument in symbolic dynamics, we observe the following handy lemma for thegeometric potential τ . Lemma 2.
There exists a locally H¨older continuous functions τ ′ such that τ ∼ τ ′ and τ ′ is boundedaway from zero.Proof. By the above proposition, we know there exist
K, C > τ + τ ◦ σ + ... + τ ◦ σ m ≥ C for all m > K . Let λ = K and consider h ′ ( x ) = K − X n =0 a n · τ ◦ σ n ( x ) where a n = 1 − nλ . Notice that a = 1, a K − = λ and a K = 0. Moreover, we have a n − a n − = − λ for n = 1 , , ..., K .Therefore, h ′ ( x ) − h ( σx ) = K − X n =0 a n · τ ◦ σ n ( x ) − K − X n =0 a n · τ ◦ σ n +1 ( x )= a · τ ( x ) − λ · ( τ ◦ σx + τ ◦ σ x + ... + τ ◦ σ K − x ) − a K − τ ◦ σ K ( x )= τ ( x ) − λ K X n =1 τ ◦ σ n x. Let τ ′ ( x ) := λ K X n =1 τ ◦ σ n x . It is clear that τ ′ ( x ) is locally H¨older; moreover, we have τ ′ ( x ) = λ K X n =1 τ ◦ σ n x ≥ CK > . (cid:3) Notice that the coding above is completely determined by the type of generators (i.e., hyperbolicor parabolic) in Γ . Because Γ and Γ are boundary-preserving isomorphic, repeating the sameconstruction as above for Γ , for S we derive the same countable Markov shift (Σ + , σ ) as for S . In other words, the same Proposition 2 holds for S . More precisely, there exists a bijection ω : Λ → Σ + and the geometric potential κ : Σ + → R given by κ ( x ) := B ω − ( x ) ( o, ρ ( a n ) o ) for x = a n a n ... ∈ Σ + . Furthermore, κ is cohomologus to a locally H¨older continuous function κ ′ which is bounded away from zero (i.e., Lemma 2). Remark . (1) Suppose ι : Γ → Γ is a type-preserving isomorphism. Then by Tukia’s isomorphismtheorem (cf. Remark 4.2) there exists an ι − equivariant homeomorphism q : Λ(Γ ) → Λ(Γ ).One can also prove that for ξ ∈ Λ we have ω ( ξ ) = ω ( q ( ξ )). Moreover, we can write κ ( x ) = B ( ω ◦ q ) − ( x ) ( o, ( ι ◦ ρ )( a n ) · o ) where a n is the first element of ω − ( x ).(2) Noticing that since τ and τ ′ (constructed in Corollary 2) are cohomologous, the thermody-namics for τ (resp. κ ) and τ ′ (resp. κ ′ ) are the same. Therefore, for brevity, we will abuseour notation and continue to denote the function τ ′ by τ and, similarly, κ ′ by κ .3.2. Phase Transition of the Geodesic Flow.
We continue this subsection with the same nota-tions and assumptions as the previous subsection. Recall D = { ( x, y ) ∈ R : x ≥ , y ≥ }\ (0 , ρ and ρ be two boundary-preserving isomorphic Fuchsian representations satis-fying the extended Schottky condition. Lemma 3.
Suppose ( a, b ) ∈ D . For any parabolic element p ∈ G (i.e., ρ ( p ) and ρ ( p ) areparabolic), we have δ a,b h p i = inf n t ∈ R : Q a,b h p i ( t ) < ∞ o = a + b ) where Q a,b h p i ( t ) = P n ∈ Z e − t ( d a,b ( o,p n )) .For h ∈ Γ is hyperbolic (i.e., ρ ( h ) and ρ ( h ) are hyperbolic), then δ a,b h h i = 0 . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 13
Proof.
Let p ∈ G be a parabolic element. Without loss generality, we can assume ρ i ( p ) : H → H tobe the M¨obius transformation ρ i ( p )( z ) = z + c i for i = 1 , c i ∈ R . Then direct computationshows that d ( i, ρ i ( p n )( i )) = d ( i, i + nc i ) = log p ( nc i ) + 4 + | nc i | p ( nc i ) + 4 − | nc i | . Notice that p ( nc i ) + 4 + | nc i | p ( nc i ) + 4 − | nc i | = 2 n c i + 4 + 2 | nc i | p ( nc i ) + 44 , so when | n | is big enough (say | n | > M p ), there exist m i and M i such that2 log | n | + m i ≤ d ( i, i + nc i ) ≤ | n | + M i . Converting the above inequalities to the disk model, we have2 log | n | + m i ≤ d ( o, p n o ) ≤ | n | + M i . Therefore, Q a,b h p i ( t ) = X n ∈ Z e − t · d a,b ( o,p n o ) = X | n |≤ M p e − t · d a,b ( o,p n o ) + X | n | >M p e − t · d a,b ( o,p n o ) , where X | n |≤ M p e − t · d a,b ( o,p n o ) < ∞ is a finite sum. Furthermore, for | n | > M one has − tad ( o, ρ ( p n )( o )) − tbd ( o, ρ ( p n )( o )) ≥ − ta (2 log | n | + M ) − tb (2 log | n | + M )= − t ( aM + bM ) | {z } C a,b ( p ) − t ( a + b ) log | n | and − tad ( o, ρ ( p n )( o )) − tbd ( o, ρ ( p n )( o )) ≤ − ta (2 log | n | + m ) − tb (2 log | n | + m )= − t ( am + bm ) | {z } C a,b ( p ) − t ( a + b ) log | n | . Hence ( 1 C a,b ( p ) ) t X | n | >M p ( 1 | n | ) t ( a + b ) ≤ X | n | >M p e − t · d a,b ( o,p n o ) ≤ ( 1 C a,b ( p ) ) t X | n | >M p ( 1 | n | ) t ( a + b ) , and, thus, δ a,b h p i = a + b ) .For each hyperbolic element h ∈ G , and Q a,b h h i ( t ) = X n ∈ Z e − t · d a,b ( o,h n o ) = X n ∈ Z e − tad ( o,ρ ( h n ) o ) − tbd ( o,ρ ( h n ) o ) =2 X n ∈ N e − tanB ρ h )+ ( o,ρ ( h ) o ) − tnbB ρ h )+ ( o,ρ ( h ) o ) =2 X n ∈ N e − tn ( aB ρ h )+ ( o,ρ ( h ) o )+ bB ρ h )+ ( o,ρ ( h ) o )) . Since B ρ i ( h ) + ( o, ρ i ( h ) o ) > i = 1 ,
2, we have δ a,b h h i = 0 . (cid:3) Recall that the Markov shift (Σ + , σ ) defined above (see Definition 12) for ρ , ρ is topologicallymixing and satisfying the BIP property. Also, the geometric potentials τ , κ defined above (seeProposition 2) are locally H¨older and bounded away from zero. Therefore, we are in the scenariothat was introduced in Section 2. Lemma 4.
Let ρ and ρ be two boundary-preserving isomorphic Fuchsian representations satisfy-ing the extended Schottky condition. Let (Σ + , σ ) be Markov shift and τ , κ be the geometric potentialsdefined in the above subsection.Then for a, b ≥ , P σ ( − t ( aτ + bκ )) = ( infinite , for t < δ a,b h p i ;analytic , for t > δ a,b h p i . Proof.
By definition, we have P σ ( − t ( aτ + bκ )) = lim n →∞ n + 1 log X x ∈ Fix n exp( − t ( aS n τ + bS n κ )) · χ [ h ] = lim n →∞ n + 1 log X x = h x ...x n +1 exp( − t ( aS n τ + bS n κ )) Notice thatFix n +1 (Σ + ) = n a m a m ....a m n +1 : a i ∈ A , a i = a ± i +1 , and m i ∈ Z ∗ for i = 1 , , .., n + 1 o . For each k ∈ N and set n + 1 = k ( N + N − B k ⊂ Fix n +1 defined as B k = ( h a m ...a m n n ∈ Fix n +1 : a i + j ( N + N − = ( h i +1 , ≤ i ≤ N − p i +1 − N , N ≤ i ≤ N + N − ) . In other words, elements b ∈ B k are in the following form: b = h h m ...h m N − N p m N ...p m N N − N | {z } ... h m ( k − N N − ...p m k ( N N − N | {z } . For brevity, let’s denote N + N − N , then we have for ξ ∈ Λ P σ ( − t ( aτ + bκ )) ≥ lim k →∞ kN log X ξ = ρ x ) ξ x ∈ Bk exp( − t ( aS kN τ + bS kN κ )) = lim k →∞ kN log X ξ = ρ x ) ξ x ∈ Bk exp( f ( a, b, t, kN )) where f ( a, b, t, n ) = − t ( n X i =1 aB ω − ( σ i x ) ( o, ρ ( x i +1 ) o ) + bB ω − ( σ i x ) ( o, ρ ( x i +1 ) o )) . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 15
Because B ξ ( x, y ) ≤ d ( x, y ) we have, P σ ( − t ( aτ + bκ )) ≥ lim k →∞ kN log X ξ = ρ ( x ) ξ x ∈ Bk exp − t ( kN X i =1 ad ( o, ρ ( x i +1 ) o ) + bd ( o, ρ ( x i +1 ) o ) ! = lim k →∞ kN log X ξ = ρ x ) ξ x ∈ Bk exp − t kN X i =1 d a,b ( o, x i +1 o ) ! Moreover, by the definition of B k one has X ξ = ρ ( x ) ξ x ∈ Bk exp − t kN X i =1 d a,b ( o, x i +1 o ) ! = e − td a.b ( o,h o ) · X ( m ,...,m kN ) ∈ ( Z ∗ ) kN exp − t kN X i =1 d a,b ( o, a m i i o ) ! . Also, notice that X ( m ,...,m kN ) ∈ ( Z ∗ ) kN exp − t kN X i =1 d a,b ( o, a m i i o ) ! = kN Y i =1 X m i ∈ Z ∗ exp − t kN X i =1 d a,b ( o, a m i i o ) ! = N Y i =2 X m ∈ Z ∗ e − td ab ( o,h mi o ) ! k N Y i =1 X m ∈ Z ∗ e − td ab ( o,p mi o ) ! k . Hence, P σ ( − t ( aτ + bκ )) ≥ lim k →∞ kN log e − td a.b ( o,h o ) N Y i =2 X m ∈ Z ∗ e − td ab ( o,h mi o ) ! k N Y i =1 X m ∈ Z ∗ e − td ab ( o,p mi o ) ! k = 1 N log N Y i =2 X m ∈ Z ∗ e − td ab ( o,h mi o ) ! N Y i =1 X m ∈ Z ∗ e − td ab ( o,p mi o ) !! = 1 N log Y g ∈A\ h (cid:16) Q a,b h g i ( t ) − (cid:17) where Q a,b h g i ( t ) = X m ∈ Z e − td ab ( o,g m o ) = 1 + X m ∈ Z ∗ e − td ab ( o,g m o ) . In the following, we derive an upper bound for P σ ( − t ( aτ + bκ )). Let ( ξ it ) be the end of thegeodesic ray [ o, ω − ( σ i +1 x )). Then by Lemma 1, we have τ ( σ i x ) = B ω − ( σ i x ) ( o, ρ ( x i ) o )= B ω − ( σ i +1 x ) ( ρ − ( x i ) o, o )= lim t →∞ d ( ξ it , ρ ( x i ) o ) − d ( ξ it , o ) ≥ (cid:0) d ( ξ it , o ) − d ( o, ρ ( x i ) o ) − C (cid:1) − d ( ξ it , o )= d ( o, ρ ( x i ) o ) − C Similarly, we have κ ( σ i x ) ≥ d ( o, ρ ( x i ) o ) − C for some constant C .Thus, e − t ( aτ ( σ i x )+ aκ ( σ i x )) ≤ e t ( aC + bC ) e − t ( d a,b ( o,x i o )) . Hence, P σ ( − taτ − tbκ ) ≤ lim n →∞ n log X a ,...,a n X m ,...,m n ∈ Z ∗ n Y i =1 e t ( aC + bC ) e − t ( d a,b ( o,a mii o )) = t ( aC + bC ) + log Y g ∈A (cid:16) Q a,b h g i ( t ) − (cid:17) . Then, by Lemma 3 we have P σ ( − t ( aτ + bκ )) = ( infinite , for t < δ a,b h p i ;finite , for t > δ a,b h p i . Finally, by Theorem 2, we know the finiteness of the pressure function implies the analyticity. (cid:3)
Remark . When a (or b ) is zero, we are back to the well-know result: P σ ( − tτ ) = ( ∞ , t ≥ finite , t < . Lemma 5.
For each ( a, b ) ∈ D there exists a unique t a,b ∈ ( a + b ) , ∞ ) such that P σ ( − t a,b ( aτ + bκ )) = 0 . Proof.
Let ( a, b ) be a point in D and f ( t ) = P σ ( − t ( aτ + bκ )). It is obvious that − t ( aτ + bκ ) is alocally H¨older continuous function. By Theorem 2, f ( t ) is real analytic on t when P σ ( − t ( aτ + bκ )) < ∞ . Let K = { t ∈ R : f ( t ) < ∞} . Then for t ∈ K one has ddt f ( t ) (cid:12)(cid:12)(cid:12)(cid:12) t = t = − Z ( aτ + bκ )d µ − t ( aτ + bκ ) < − ( ac + bc ) < τ, κ > c > µ − t ( aτ + bκ ) is the equilibrium state of − t ( aτ + bκ ).Hence, f ( t ) = P σ ( − t ( aτ + bκ )) is real analytic and strictly decreasing on K . Moreover, we know P σ ( − t ( aτ + bκ )) < t and is positive and big enough. More precisely, because κ > c >
0, weknow P σ ( − t ( aτ + bκ )) < P σ ( − taτ ) − tbc . Furthermore, we know that P σ ( − h top ( S ) τ ) = 0, so when ta > h top ( S ) we have P σ ( − taτ ) <
0. Therefore, it remains to say there exists t ′ a,b ∈ ( a + b ) , ∞ )such that 0 < P σ ( − t ′ a,b ( aτ + bκ )) < ∞ . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 17
Notice that by the computation made in Lemma 3, for a parabolic elements p ∈ G and for t > a + b ) , Q a,b h p i ( t ) − − X | n |≤ M p e − t · d a,b ( o,p n o ) + X | n | >M p e − t · d a,b ( o,p n o ) > ( 1 C a,b ( p ) ) t X | n | >M p ( 1 | n | ) t ( a + b ) > ( 1 C a,b ( p ) ) t · Z ∞ M p +1 x − t ( a + b ) d x =( 1 C a,b ( p ) ) t · · t ( a + b ) − · ( 1 M p + 1 ) t ( a + b ) − > . Moreover,log (cid:16) Q a,b h p i ( t ) − (cid:17) > − t log( C a,b ( p )) + log 2 + log( 12 t ( a + b ) − t ( a + b ) −
1) log( 1 M p + 1 ) > , when t is big enough , because log( t ( a + b ) − ) → ∞ as t → ( a + b ) ) + and other terms remain bounded when t → ( a + b ) ) + . For a hyperbolic elements h ∈ G , Q a,b h h i ( t ) − X n ∈ N e − tn · c a,b ( h ) = 2 e t · c a,b ( h ) − c a,b ( h ) = ( aB ρ ( h ) + ( o, ρ ( h ) o ) + bB ρ ( h ) + ( o, ρ ( h ) o )), one haslog (cid:16) Q a,b h h i ( t ) − (cid:17) = log 2 + log( e t · c a,b ( h ) − t → ( a + b ) ) + . By repeating the argument above for g ∈ A\ h and using the computation in Lemma 4, we canchoose t ′ a,b ∈ ( a + b ) ,
0) such that ∞ > P σ ( t ′ a,b ( aτ + bκ )) > N log Y g ∈A\ h (cid:16) Q a,b h g i ( t ) − (cid:17) > . (cid:3) Theorem 8.
The set { ( a, b ) ∈ D : P σ ( − aτ − bκ ) = 0 } is a real analytic curve.Proof. By Lemma 5, it makes sense to discuss solutions to P σ ( − aτ − bκ ) = 0. Moreover, for( a, b ) ∈ D such that f ( a, b ) = P σ ( − aτ − bκ ) < ∞ , we have f ( a, b ) is real analytic on both variables,and ∂ b f ( a, b ) | ( a,b )=( a ,b ) = − Z κ d µ − a τ − b κ < − c where τ, κ > c > µ − a τ − b κ is the equilibrium state of − a τ − b κ .Therefore, by the Implicit Function Theorem we have the solutions to P σ ( − aτ − bκ ) = 0 in D isreal analytic, i.e., one has b = b ( a ) is real analytic on a . (cid:3) The Manhattan Curve
The Manhattan Curve, Critical Exponent, and Gurevich Pressure.
For any pair ofFuchsian representations ρ , ρ , we recall that the Manhattan curve C ( ρ , ρ ) of ρ and ρ is theboundary of the convex set { ( a, b ) ∈ R : Q a,bρ ,ρ ( s ) has critical exponent 1 } where Q a,bρ ,ρ ( s ) = X γ ∈ G exp( − s · d a,bρ ,ρ ( o, γo )) is the Poincar´e series of the weighted Manhattanmetric d a,bρ ,ρ .We have a rough picture of the corresponding Manhattan curve C ( ρ , ρ ) for all Fuchsian repre-sentations. Theorem 9.
Let S be a surface with negative Euler characteristic, and let ρ , ρ be two Fuchsianrepresentations of G = π S into PSL(2 , R ) . We denote S = ρ ( G ) \ H and S = ρ ( G ) \ H . Then (1) ( h top ( S ) , and (0 , h top ( S )) are on C ( ρ , ρ ) ; (2) C ( ρ , ρ ) is convex; and (3) C ( ρ , ρ ) is a continuous curve.Proof. The first assertion is obvious. The second assertion is because that the domain { ( a, b ) : Q a,bρ ,ρ (1) < ∞} is convex. To see it is convex, by the H¨older inequality, for ( a , b ) , ( a , b ) ∈ D we have Q ta +(1 − t ) b ,ta +(1 − t ) b (1) ≤ ( Q a ,b (1)) t · ( Q a ,b (1)) − t . To see C is continuous, we notice that because C is convex, we know C is homeomorphic to thestraight line connecting ( h top ( S ) ,
0) and (0 , h top ( S )). (cid:3) In the rest of this subsection, we focus on ρ , ρ being boundary-preserving isomorphic Fuchsianrepresentations satisfying the extended Schottky condition. We will see for these representations,we have much better understanding of the Manhattan curve C ( ρ , ρ ).As it is known that for geometrically finite negatively curved manifolds, the (exponential) growthrate of closed geodesics is exactly the critical exponent (cf. [OP04]), we prove that the criticalexponent δ a,bρ ,ρ can also be realized by the growth rate of hyperbolic elements (or equivalently,closed orbits). To reach that, inspired by Paulin-Pollicott-Schapira [PPS15], we introduce severalrelated geometric growth rates. Through analyzing these growth rates, we are able to link thedynamics critical exponent t a,b (i.e., the solution to the Bowen’s formula) with the geometric criticalexponent δ a,bρ ,ρ . In result, these geometric growth rates give us the full picture of the Manhattancurve C ( ρ , ρ ).Recall that for each closed geodesic λ on S , it corresponds a unique geodesic on S , abusing thenotation, we still denote it by λ . Moreover, l i [ λ ] denotes the length of the closed geodesic λ on S i for i = 1 , Definition 13 (Geometric growth rates counted from S ) . Let S be a surface with negative Eulercharacteristic, and G := π S . Suppose ρ , ρ : G → PSL(2 , R ) are boundary-preserving isomorphicFuchsian representations satisfying the extended Schottky condition.(1) Q a,bP P S,x,y ( s ) := X γ ∈ G e − d a,b ( x,γy ) − sd ( x,ρ ( γ ) y ) is called the Paulin-Pollicott-Schapira’s (PPS)Poincar´e series. (2) δ a,bP P S is the critical exponent of Q a,bP P S,x,y ( s ), i.e., Q a,bP P S,x,y ( s ) converges when s > δ a,bP P S and Q a,bP P S,x,y ( s ) diverges when s < δ a,bP P S . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 19 (3) G a,bx,y ( s ) := X γ ∈ G ; d ( x,ρ ( γ ) y ) ≤ s e − d a,b ( x,γy ) . (4) Z W ( s ) := X λ ∩ W = φλ ∈ Per1( s ) e − al [ λ ] − bl [ λ ] where W ⊂ T S is a relatively compact open set andPer ( s ) := { λ : λ is a closed orbit on T S and l [ λ ] ≤ s } .(5) P abGur := lim sup s →∞ s log Z W ( s ) is the geometric Gurevich pressure . Lemma 6. δ a,bP P S = P abGur = lim s →∞ s log G a,bx,y ( s ) = lim s →∞ s log Z W ( s ) for any relative compact W ⊂ T S .Proof. This proof follows the (short) proof of Corollary 4.2, Corollary 4.5 and Theorem 4.7 [PPS15](also the proof of Theorem 2.4 [Pei13]). The strategy is standard but tedious. We leave the proofin the appendix. (cid:3)
Furthermore, we show in below that the geometric Gurevich pressure P abGur matches with thesymbolic Gurevich pressure (for the suspension flow).In what follows, (Σ + , σ ) stands for countable Markov shift associated with ρ , ρ defined inSection 3, and τ, κ : Σ + → R + stand for the corresponding geometric potentials. Recall that(Σ + , σ ) is topologically mixing and satisfies the BIP property, and τ, κ are locally H¨older continuousfunctions and bounded away from zero. Let Σ + τ be the suspension space relative to τ and φ : Σ + τ → Σ + τ be the suspension flow.We consider a function ψ : Σ + τ → R + given by ψ ( x, t ) := κ ( x ) τ ( x ) for x ∈ Σ + , 0 ≤ t ≤ τ ( x )and ψ ( x, τ ( x )) = ψ ( σ ( x ) , ψ , we can reparametrize the suspension flow φ : Σ + τ → Σ + τ and derive information of orbits of the geodesic flow over T S . Roughly speaking, ψ is a reparametrization function, in the symbolic sense, of the geodesic flow over T S such thatthe reparametrized flow is conjugated to the geodesic flow over T S . Lemma 7.
Suppose ψ : Σ + τ → R + is defined as ψ ( x, t ) := κ ( x ) τ ( x ) for x ∈ Σ + , ≤ t ≤ τ ( x ) and ψ ( x, τ ( x )) = ψ ( σ ( x ) , Then P φ ( − a − bψ ) = P abGur . Proof.
Notice that since S is geometrically finite, there exists a relatively compact open set W suchthat W meets every closed orbit on T S . Therefore we have for any g ∈ A = { h , ..., h N , p , ..., p N } s Z g ( s ) ≤ Z a,bW ( s ) ≤ X g ∈A Z g ( s ) + C where Z g ( T ) = X φs ( x, x, , ≤ s ≤ T e R s ( − a − bψ ) ◦ φ t ( x,t )d t χ [ g ] ( x ) for g ∈ A .The first inequality is because for a closed orbit φ t ( x,
0) = ( x, x = g x x ... , 0 ≤ t ≤ s , ofthe suspension flow, it corresponds at most s closed orbits on T S . The constant C in the secondinequality is from closed geodesics corresponding to the hyperbolic generators h i (because theseclosed geodesics are not in our coding).Recall that by definition, we have P φ ( − a − bψ ) = lim s →∞ s log Z g ( s ), and by Definition 7 P φ ( − a − bψ ) = lim s →∞ s log Z g ( s ) , for any g ∈ A ;hence P φ ( − a − bψ ) = P abGur . (cid:3) Lemma 8. δ a,bP P S = 0 if and only if δ a,b = 1 . Proof.
We first notice that the critical exponents are irrelevant with base points, therefore we canchoose d a,b ( o, γo ) = ad ( o, ρ ( γ ) o ) + bd ( f o, ρ ( γ ) f o )where f : H → H is the ι − equivalent bilipschitz given in Theorem 7 and ι : ρ ( G ) → ρ ( G ) is theboundary-preserving isomorphism. Since f : H → H is bilipschitz, there exists C > γ ∈ G and a fixed o ∈ H C d ( f o, ρ ( γ ) f o ) ≤ d ( o, ρ ( γ ) o ) ≤ Cd ( f o, ρ ( γ ) f o ) . With the inequalities above, the desire results are straightforward. To simplify the notation, inthis proof d ( o, ρ ( γ ) o ) is denoted by d ( γ ) and d ( f o, ρ ( γ ) f o ) is denoted by d ( γ ) . ( = ⇒ ) Suppose δ a,bP P S = 0. Claim: X γ ∈ G e s ( − ad ( γ ) − bd ( γ )) < ∞ for s > s = 1 + t for some t >
0. We have X γ ∈ G e s ( − ad ( γ ) − bd ( γ )) = X γ ∈ G e − ad ( γ ) − bd ( γ )+ t ( − ad ( γ ) − bd ( γ )) ≤ X γ ∈ G e − ad ( γ ) − bd ( γ )+ t ( − ad ( γ ) − b ( C d ( γ ))) = X γ ∈ G e − ad ( γ ) − bd ( γ ) − t ( a + bC ) d ( γ ) < ∞ . Similarly, we have X γ ∈ G e s ( − ad ( γ ) − bd ( γ )) = ∞ for s < δ a,b = 1 . ( ⇐ =) Suppose δ a,b = 1. Claim: X γ ∈ G e − ad ( γ ) − bd ( γ ) − td ( γ ) < ∞ for t > . pf. Recall that there exists C > C d ( γ ) < d ( γ ) < Cd ( γ ).For any t >
0, we pick s = a + bC + ta + bC >
1, and we have s = a + bC + ta + bC ⇐⇒ − as + a + ts b − b = C > d d which implies ad ( γ ) + bd ( γ ) + td ( γ ) > s ( ad ( γ ) + bd ( γ ));Because s > δ a,b , we know X γ ∈ G e − ad ( γ ) − bd ( γ ) − td ( γ ) ≤ X γ ∈ G e − s ( ad ( γ )+ bd ( γ )) < ∞ Similarly, one can show X γ ∈ G e − ad ( γ ) − bd ( γ ) − td ( γ ) = ∞ for t < . Therefore, δ a,bP P S = 1. (cid:3) ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 21
We have an immediate corollary:
Corollary 1. P φ ( − a − bψ ) = P a,bGur = 0 if and only if δ a,b = 1 . Proof of Main Results.
Throughout this subsection, ρ , ρ are boundary-preserving iso-morphic Fuchsian representations satisfying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Let (Σ + , σ ) be the topologically mixing countable Markov shift associated with ρ , ρ defined in Section 3, and τ, κ : Σ + → R + be the corresponding geometric potentials. Recallthat Σ + τ is the suspension space relative to τ and φ : Σ + τ → Σ + τ is the suspension flow, and thereparametrization function ψ : Σ + τ → R + is given by ψ ( x, t ) := κ ( x ) τ ( x ) for x ∈ Σ + , 0 ≤ t ≤ τ ( x ) and ψ ( x, τ ( x )) = ψ ( σ ( x ) , Lemma 9.
Suppose ψ : Σ + τ → R + is defined ψ ( x, t ) := κ ( x ) τ ( x ) for x ∈ Σ + , ≤ t ≤ τ ( x ) and ψ ( x, τ ( x )) = ψ ( σ ( x ) , Then P σ ( − aτ − bκ ) = 0 if and only if P φ ( − a − bψ ) = 0 . Proof. (= ⇒ ) Suppose P σ ( − aτ − bκ ) = 0 < ∞ . Then when t ∈ ( − ε, ε ), P σ ( − aτ − bκ − tτ ) is a realanalytic and is strictly decreasing, i.e., P σ ( − aτ − bκ − tτ ) < , for t > , for t = 0; > , for t < . Therefore, by Theorem 5 and ∆ − a − bψ = − aτ − bκ , we have P φ ( − a − bψ ) = 0.( ⇐ =) To see P φ ( − a − bψ ) = 0 implies P σ ( − aτ − bκ ) = 0 Notice that because τ > c > P ∞ i =0 τ ◦ σ i = ∞ , by Lemma 4.1 and Remark 4.1 in Jaerisch-Kesseb¨ohmer-Lamei [JKL14], we have0 = P φ ( − a − bψ )= sup (cid:26) h σ ( µ ) R τ d µ + R ∆ − a − bψ d µ R τ d µ : µ ∈ M σ ( τ ) with ∆ − a − bψ ∈ L ( µ ) (cid:27) = sup (cid:26) h σ ( µ ) R τ d µ + R ( − aτ − bκ )d µ R τ d µ : µ ∈ M σ ( τ ) with − aτ − bκ ∈ L ( µ ) (cid:27) where M σ ( τ ) := (cid:8) µ : µ ∈ M σ and R τ d µ < ∞ (cid:9) .For all µ ∈ M σ such that − aτ − bκ ∈ L ( µ ), we have R τ d µ > c >
0; hence,0 = sup (cid:26) h σ ( µ ) + Z ( − aτ − bκ )d µ : µ ∈ M σ ( τ ) and − aτ − bκ ∈ L ( µ ) (cid:27) . Recall that P σ ( − aτ − bκ ) = sup (cid:26) h σ ( µ ) + Z ( − aτ − bκ )d µ : µ ∈ M σ and − aτ − bκ ∈ L ( µ ) (cid:27) . Notice that for µ ∈ M σ , if − aτ − bκ ∈ L ( µ ) then R τ d µ < ∞ (i.e., µ ∈ M σ ( τ )).Moreover, it is obvious that M σ ( τ ) ⊂ M σ . Thus, we have P σ ( − aτ − bκ ) = sup (cid:26) h σ ( µ ) + Z ( − aτ − bκ )d µ : µ ∈ M σ and − aτ − bκ ∈ L ( µ ) (cid:27) = sup (cid:26) h σ ( µ ) + Z ( − aτ − bκ )d µ : µ ∈ M σ ( τ ) and − aτ − bκ ∈ L ( µ ) (cid:27) =0 (cid:3) The following theorem gives more geometric characterizations to t a,b (i.e., the solution to theequation P σ ( − t a,b ( aτ + bκ )) = 0). Without any surprise, as the famous Bowen’s formula, t a,b isindeed the critical exponent δ a,b and the growth rate of hyperbolic elements. Theorem 10 (Bowen’s formula) . For ( a, b ) ∈ D . Suppose t a,b is the solution to P σ ( − t a,b ( aτ + bκ )) =0 . Then t a,b = δ a,b = lim s →∞ s log G a,bx,y ( s ) where G a,bx,y ( s ) := { γ ∈ G : d a,b ( x, γy ) ≤ s } ;Proof. We first notice that δ a,b = 1 ⇐⇒ δ a,bP P S = 0 Lemma 8 ⇐⇒ P a,bGur = 0 Lemma 6 ⇐⇒ P φ ( − a − bψ ) = 0 Lemma 7 ⇐⇒ P σ ( − aτ − bκ ) = 0 Lemma 9 . Thus, P σ ( − t a,b ( aτ + bκ )) = 0 if and only δ t a,b a,t a,b b = 1, that is, Q t a,b a,t a,b b ( s ) = X γ ∈ G e − t ab d a,b ( o,γo ) has critical exponent 1. Hence, Q a,b ( s ) = X γ ∈ G e − sd a,b ( o,γo ) has critical exponent t a,b , i.e., δ a,b = t a,b .For the rear inequality, the prove is the same as the proof of Lemma 6 with some simplification(in other words, the proof is a modification of Lemma 3.3, Corollary 4.5, Theorem 4.7 [PPS15], orSection 2.2 [Pei13]). However, for the completeness, we put the proof in the appendix. (cid:3) Remark . Using the same argument as in Lemma 6, one can also prove that the critical exponent δ a,b is the growth rate of closed geodesics on S and S . One notices that each closed geodesic on S (and S )is corresponds to a hyperbolic element in Γ (and Γ ). In other words, δ a,b = h a,b := lim s →∞ s { γ ∈ G : γ is hyperblic and al [ γ ] + bl [ γ ] ≤ s } . Lemma 10.
The Manhattan curve C ( ρ , ρ ) is the set of solutions to P σ ( − aτ − bκ ) = 0 in D .Proof. It follows from the same argument as the above theorem:( a, b ) ∈ C ( ρ , ρ ) ⇐⇒ δ a,b = 1 by definition ⇐⇒ δ a,bP P S = 0 Lemma 8 ⇐⇒ P a,bGur = 0 Lemma 6 ⇐⇒ P φ ( − a − bψ ) = 0 Lemma 7 ⇐⇒ P σ ( − aτ − bκ ) = 0 Lemma 9 . (cid:3) Theorem 11.
The Manhattan curve C ( ρ , ρ ) is real analytic.Proof. It is a direct consequence of Theorem 8 and Lemma 10. (cid:3)
Proposition 4.
Let ρ , ρ be two boundary-preserving isomorphic Fuchsian representations satis-fying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Then (1) C is strictly convex if S and S are NOT conjugate in PSL(2 , R ) ; and (2) C is a straight line if and only if S and S are conjugate in PSL(2 , R ) . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 23
Proof.
This result is a direct consequence of Theorem 9 and Theorem 11. Indeed, the strictlyconvexity comes from the analyticity and the convexity of C .It is clear that when S and S are isometric we have C is a straight line. Conversely, suppose C is a straight line. Then the slope of the tangent line of the Manhattan curve C is a constant, i.e., b ′ = − h top ( S ) h top ( S ) = − R τ d m − aτ − b ( a ) κ R κ d m − aτ − b ( a ) κ where m − aτ − b ( a ) κ is the equilibrium state for − aτ − b ( a ) κ for all a ∈ [0 , h top ( S )]. In particular, b ′ = − R τ d m − h top ( S ) τ R κ d m − h top ( S ) τ = − R τ d m − h top ( S ) κ R κ d m − h top ( S ) κ . Claim: h top ( S ) τ and h top ( S ) κ are cohomologus.It is clear that we have the desired result after we prove the claim. Because h top ( S ) τ ∼ h top ( S ) κ means that S and S have proportional marked length spectra. Then by the proportional markedlength spectrum rigidity (i.e., Theorem 6) we are done.pf. for short, we denote m = m − h top ( S ) τ and m = m − h top ( S ) κ . We prove this claim by theuniqueness of the equilibrium states. In other words, we want to show that m is the equilibriumstate for − h top ( S ) τ , that is,0 = P σ ( − h top ( S ) τ ) = h ( m ) − h top ( S ) Z τ d m . Notice that, by definition,0 = P σ ( − h top ( S ) κ ) = h ( m ) − h top ( S ) Z κ d m and, by the above observation, h top ( S ) h top ( S ) = R κ d m R τ d m . Thus, we have h ( m ) − h top ( S ) Z τ d m = h top ( S ) Z κ d m − h top ( S ) Z τ d m = 0= P σ ( − h top ( S ) τ ) . By the uniqueness of the equilibrium states (cf. Theorem 3), we know m = m . Moreover,Theorem 4.8 [Sar09] showed that this only happens when − h top ( S ) τ and − h top ( S ) κ are cohomol-ogous. (cid:3) Corollary 2 (Bishop-Steger’s entropy rigidity [BS93]) . Let ρ , ρ be two boundary-preserving iso-morphic Fuchsian representations satisfying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Then, for any fixed o ∈ H , δ , = lim T →∞ T log { γ ∈ G : d ( o, ρ ( γ ) o ) + d ( o, ρ ( γ ) o ) ≤ T } . Moreover, δ , ≤ h top ( S ) · h top ( S ) h top ( S ) + h top ( S ) and the equality holds if and only if S and S are isometric.Proof. By Theorem 10, we know δ , (1 , ∈ C is the intersection of C and the line a = b . By theconvexity of C , we know that the intersection of the line a = b and b = − h top ( S ) h top ( S ) a + h top ( S ) liesabove δ , (1 , (1 ; h top ( S ) h top ( S ) b a δ ; · (1 ; h top ( S ) h top ( S ) h top ( S )+ h top ( S ) · (1 ; Therefore, we have δ , ≤ h top ( S ) · h top ( S ) h top ( S ) + h top ( S ) . Moreover, when the equality holds, we have C isa straight line. By Proposition 4, we are done. (cid:3) Definition (Thurston’s intersection number, Definition 3) . Let S and S be two Riemann surfaces.Thurston’s intersection number I( S , S ) of S and S is given byI( S , S ) = lim n →∞ l [ γ n ] l [ γ n ]where { [ γ n ] } ∞ n =1 is a sequence of conjugacy classes for which the associated closed geodesics γ n become equidistributed on Γ \ H with respect to area. Corollary 3 (Thurston’s rigidity) . Let ρ , ρ be two boundary-preserving isomorphic Fuchsianrepresentations satisfying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H .Then I( S , S ) ≥ h top ( S ) h top ( S ) and the equality hold if and only if ρ and ρ are conjugate in PSL(2 , R ) .Proof. It is enough to show that the normal of the tangent of C ( S , S ) at ( h top ( S ) ,
0) is I ( S , S ).Recall that b ′ ( a ) = − R τ d m R κ d m where m = m − aτ − bκ is the equilibrium state of − aτ − bκ . So, for a = h top ( S ), b = 0 we have b ′ ( − h top ( S )) = − R τ d m − h top ( S ) τ R κ d m − h top ( S ) τ . Thus, it is sufficient to showI( S , S ) := lim T →∞ X λ ∈ Per ( T ) l [ λ ] X λ ∈ Per ( T ) l [ λ ] = R κ d m − h top ( S ) τ R τ d m − h top ( S ) τ . Because m − h top ( S ) τ is the Bowen-Margulis measure for the geodesic flow on T S , and S isgeometrically finite, we know the Bowen-Margulis measure is equidistributed with respect to closedorbits (see, for example, Theorem 4.1.1 [Rob03]). Therefore, the above equation is true. (cid:3) Appendix
Recall our notation that ρ , ρ are two boundary-preserving isomorphic Fuchsian representationssatisfying the extended Schottky condition, and S = ρ ( G ) \ H , S = ρ ( G ) \ H . Let d a,bρ ,ρ bethe weighted Manhattan metric. Recall that δ a,b is the critical exponent of the Poincar´e seriesassociated with d a,bρ ,ρ . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 25
The Proof of Lemma 6.
We first recall three useful lemmas.
Lemma 11 (Lemma 2.2 [Sch04]) . Suppose a, b, c ∈ H and d ( a, b ) + d ( a, c ) − d ( b, c ) ≤ C for some C > , then a is in a D − neighborhood of the geodesic segment [ b, c ] where D is a constant onlydepending on C . Lemma 12 (Lemma 4.4 [PPS15]) . Let b n ≥ such that there exist C > and N ∈ N such thatfor all n, m ∈ N , we have b n b m ≤ C i = N X i = − N b n + m + i , then with a n = P n − k =0 b n , the limit of a n n as n → ∞ exists (and hence is equal to its limit-sup). Recall that δ a,bP P S is the critical exponent of Q a,bP P S,x,y ( s ), i.e., Q a,bP P S,x,y ( s ) converges when s >δ a,bP P S and Q a,bP P S,x,y ( s ) diverges when s < δ a,bP P S where Q a,bP P S,x,y ( s ) = X γ ∈ G e − d a,b ( x,γy ) − sd ( x,ρ ( γ ) y ) .W.l.o.g., we can write d a,b ( x, γy ) = ad ( x, γy ) + bd ( f x, ι ( γ ) f y ) for γ ∈ Γ and ι : Γ → Γ is a boundary-preserving isomorphism and f : H → H is the bilipschitz map given by Theorem7. To simply our notation, we denote d ( x, γy ) := d ( x, γy ) and d ( x, γy ) := d ( f x, ι ( ρ ( γ )) · f y ).Therefore, G a,bx,y ( s ) can be equivalently defined as: G a,bx,y ( s ) := X γ ∈ Γ ; d ( x,γy ) ≤ s e − d a,b ( x,γy ) . Similarly, the PPS Poincar´e series Q a,bP P S,x,y ( s ) can be rewrite as Q a,bP P S,x,y ( s ) = X γ ∈ Γ e − d a,b ( x,γy ) − sd ( x,γy ) . Let us first define several useful growth rates. • G a,bx,y, ( s ) := X γ ∈ Γ ; s − Lemma 13. lim inf s →∞ s log G a,bx,y, ( s ) ≤ δ a,bP P S ≤ lim sup s →∞ s log G a,bx,y, ( s ) Proof. The proof is elementary. However, for the completeness, we give a proof here. We firstnotice that Q a,bP P S,x,y ( t ) = ∞ X n =0 X γ ∈ E n e − d a,b ( x,γy ) − td ( x,γy ) where E n := { γ ∈ Γ : n − < d ( x, γy ) ≤ n } . Therefore, we have X n e − tn G a,bx,y, ( n ) ≤ Q a,bP P S,x,y ( t ) ≤ X n e − t ( n − G a,bx,y, ( n ) . Claim: If t > ∆ := lim sup s →∞ s log G a,bx,y, ( s ) then Q a,bP P S,x,y ( t ) converges (i.e., δ a,b ≤ ∆) . pf. For ε = t − ∆2 , there exists N > n > N we have ∆ + ε > n log G a,bx,y, ( n ).Therefore, we have Q a,bP P S,x,y ( t ) ≤ X n e − t ( n − G a,bx,y, ( n ) < C + ∞ X n = N e − t ( n − e n (∆+ ε ) = C + e t ∞ X n = N e n ( − t +∆+ ε ) < C + e t ∞ X n = N e n ( − t +∆2 ) < ∞ where C = N − X n =0 e − t ( n − G a,bx,y, ( n ) < ∞ . Claim: If t < ∆ := lim inf s →∞ s log G a,bx,y, ( s ) then Q a,bP P S,x,y ( t ) diverges (i.e., δ a,b ≥ ∆).For ε = ∆ − t , there exists N ′ > n > N ′ we have ∆ − ε < n log G a,bx,y, ( n ).Therefore, we have Q a,bP P S,x,y ( t ) ≥ X n e − tn G a,bx,y, ( n ) ≥ ∞ X n = N ′ e − tn e n (∆ − ε ) = ∞ X n = N ′ e n ( − t +∆ − ε ) > X n e n ( − t +∆2 ) = ∞ . (cid:3) Lemma 14. The inequalities of the above lemma are indeed equalities. Moreover, δ a,bP P S = lim n →∞ n log G a,bx,y ( s ) . Proof. The proof of this Lemma follows the idea of the (short) proof of Lemma 4.2 (see also theproof of Theorem 2.4 [Pei13]).We notice that by the triangle inequality, it is obvious that the lim sup s →∞ s log G a,bx,y ( s ) doesnot depend on the reference point x and y . W.l.o.g, we pick x = y = o . Recall the generating setof the extended Schottky group G = π S is A ± = { h ± , .., h ± N , p , ..., p N } with N + N ≥ ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 27 • E n := { γ ∈ Γ : n − < d ( o, γo ) ≤ n } . • b n := G a,bx,y, ( n ) = X γ ∈ E n e − d a,b ( o,γo ) . By Lemma 12, it is enough prove that there exist M > N ∈ N such that for all n, m ∈ N ,we have b n b m ≤ M i = N X i = − N b n + m + i . Claim: There exist N ∈ N and M > E n × E m ≤ M · P i = Ni = − N E n + m + i pf. Let γ n ∈ E n and γ m ∈ E m , by Lemma 1, there exists α ∈ A ± (more precisely, if γ n = g i ... and γ m = g j ... for g i , g j ∈ A then we take α = g k for g k ∈ A ± \{ g ± i , g ± j } ) such that | d ( o, γ n ρ ( α ) γ m o ) − d ( o, γ n o ) − d ( o, γ m o ) | < C and | d ( o, ( ι ◦ γ n ) ρ ( α )( ι ◦ γ m ) o ) − d ( o, ( ι ◦ γ n ) o ) − d ( o, ( ι ◦ γ m ) o ) | < C where C only depending on ρ and C only depending on ρ .Thus, n + m − C − < d ( o, γ n ρ ( α ) γ m o ) ≤ n + m + C + 2 . Let us consider the map Ψ : E n × E m → i = C +2 X i = − C − E n + m + i ( γ n , γ m ) γ n ρ ( α ) γ m This maps is obvious not one-to-one. Nevertheless, we claim − ( γ n ρ ( α ) γ m ) is finite. ByLemma 11, we know that d ( γ n o, [ o, γ n ρ ( α ) γ m o ]) ≤ D (where D only depends on C ), and whichimplies if there exist γ ′ n ∈ E n and γ ′ m ∈ E m such that γ ′ n ρ ( α ) γ ′ m = γ n ρ ( α ) γ m = γ then d ( γ n o, γ ′ n o ) ≤ D + 1) (because n − < d ( γ n o, o ) , d ( γ ′ n o, o ) ≤ n , and γ n o , γ ′ n o are in a D -neighborhood of [ o, γo ]). Moreover, by the discreteness of Γ , the set { γ ∈ Γ : d ( γo, o ) ≤ D + 1) } is finite (say, smaller than or equal to M ). Hence − ( γ n ρ ( α ) γ m ) ≤ M .Therefore, E n × E m ≤ (2 N + N ) M · i = C +2 X i = − C − E n + m + i where 2 N + N is the cardinality of A ± .Moreover, we know (cid:12)(cid:12)(cid:12) d a,b ( o, γ n ρ ( α ) γ m o ) − d a,b ( o, γ n o ) − d a,b ( o, γ m o ) (cid:12)(cid:12)(cid:12) ≤ aC + bC , thus we have the lemma, more precisely, b n b m ≤ ( N + N ) M · e aC + bC i = C +2 X i = − ( C +2) b n + m + i . (cid:3) Lemma 15. lim s →∞ s log a a,bx,y,U ′ ( s ) = δ a,bP P S . Proof. It is obvious that a a,bx,y,U ′ ( s ) ≤ G a,bx,y ( s ), so it is enough to provelim inf s →∞ s log a a,bx,y,U ′ ( s ) ≥ δ a,bP P S . By the compactness of Λ , there exist γ , ..., γ k ∈ Γ such that Λ ⊂ k [ i =1 γ k U ′ . Since Conv(Λ ) \ S ki =1 γ k U ′ is compact, there exists a constant c ′ ≥ G a,bx,y ( s ) ≤ c ′ + k X i =1 a x,y,γ i U ′ ( s ) . Claim: There exists a constant c > a x,y,γ i U ′ ( s − r ) ≤ e c a x,y,U ′ ( s ) for all s > r where r = max { d ( x, γ i x ) : i ∈ , .., k } .It is clear that using this claim and (5.1), we have a x,y,U ′ ( s ) ≥ ke c ( G a,bx,y ( s − r ) − c ′ ) , and, thus, the lemma.pf of the claim: We first notice that by definition we have a x,y,U ′ ( s ) = a γx,y,γU ′ ( s ) . To be more precise, it is because γ − (cid:0) A γx,y,γU ′ ( s ) (cid:1) = A x,y,U ′ ( s ), and also d a,b ( x, gy ) = d a,b ( γx, γgy )for all g ∈ A x,y,U ′ ( s ).Therefore, it is enough to show there exists c > a x,y,γ i U ′ ( s − r ) = a γ − i x,y,U ′ ( s − r ) ≤ e c a x,y,U ′ ( s ).To see that, we notice that by the triangle inequality we have if d ( γ − i x, γy ) ≤ s − r then d ( x, γy ) ≤ s . Thus, A γ − i x,y,U ′ ( s − r ) = { g ∈ Γ : d ( γ − i x, gy ) ≤ s − r and gy ∈ U ′ }⊂ A x,y,U ′ ( s ) = { g ∈ Γ : d ( x, gy ) ≤ s and gy ∈ U ′ } . Furthermore, since d a,b satisfies the triangle inequality, we have (cid:12)(cid:12) − ad ( x, γy ) − bd ( x, γy ) − ( − ad ( γ − i x, γy ) − bd ( γ − i x, γy )) (cid:12)(cid:12) ≤ ad ( x, γ − i x ) + bd ( x, γ − i x ) ≤ c ′ ( x, a, b ) . Hence a x,y,γ i U ′ ( s − r ) ≤ e c ′ a x,y,U ′ ( s ). (cid:3) Lemma 16. lim s →∞ s log b a,bx,y,U ′ ,V ′ ( s ) = δ a,bP P S . Proof. Similar to proof of the previous lemma. There exist α , ..., α l ∈ Γ such that Λ ⊂ k [ i =1 α i V ′ .Therefore, there exists a constant c ′ ≥ s > 0, we have a x,y,U ′ ( s ) ≤ c ′ + l X i =1 b x,y,U ′ ,α i V ′ ( s ) . We first notice that by definition we have b x,y,U ′ ,γV ′ ( s ) = b x,γ − y,U ′ ,V ′ ( s ) . ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 29 It is because (cid:0) B x,γ − y,U ′ ,V ( s ) (cid:1) γ − = B x,y,U ′ ,γV ′ ( s ) and d a,b ( x, g · γ − y ) = d a,b ( x, gγ − · y ) for all g ∈ B x,γ − y,U ′ ,V ( s ).Pick r = max { d ( y, α i y ) : 1 ≤ i ≤ l } , we notice that by the triangle inequality of d , we knowif d ( x, γα − i y ) ≤ s − r, then d ( x, γy ) ≤ s. So, we have B x,α − i y,U ′ ,V ′ ( s ) ⊂ B x,y,B r U ′ ,V ′ where B r U ′ is the r − neighborhood of U ′ .Moreover, again by the triangle equality of d a,b , we know (cid:12)(cid:12) − ad ( x, γy ) − bd ( x, γy ) − ( − ad ( x, γα − i x ) − bd ( x, γα − i x )) (cid:12)(cid:12) ≤ ad ( y, α − i x ) + bd ( y, α − i x )) ≤ c ′ ( x, y, a, b ) . Therefore, for 1 ≤ i ≤ l and s > r , b x,α − i y,U ′ ,V ′ ( s − r ) ≤ e c ′ b x,y,B r U ′ ,V ′ ( s ) . Lastly, pick U ′′ such that B r U ′′ ⊂ U ′ then for s > ra x,y,U ′′ ,V ′ ( s − r ) ≤ c ′ + l X i =1 b x,y,U ′′ ,α i V ′ ( s − r ) ≤ c ′ + le c ′ · b x,y,B r U ′′ ,V ′ ( s ) ≤ c ′ + le c ′ · b x,y,U ′ ,V ′ ( s ) ≤ c ′ + le c ′ · a x,y,U ′ ( s ) . Taking limit of the both side and using the above lemma, we have completed the proof. (cid:3) Lemma 17. For any relatively compact open set W ⊂ T S , we have lim sup s →∞ s log Z a,bW ( s ) ≤ δ a,bP P S . Proof. Let T e p : T H → T S = Γ \ H be the projection. Since W is relative compact, there exista compact set K ⊂ H such that W ⊂ T e p ( π − ( K )). Claim: For a fixed x ∈ K , there exists a constant c > Z a,bW ( s ) ≤ e c G x,x ( s + 2 r )where r is the diameter of K .pf. To see that, first we notice that for all s ≥ λ ∈ Per ( s ) such that λ ∩ W = ∅ , thereexists a hyperbolic element γ λ ∈ Γ such that its translation axis Axe γ λ meets K , it has translationlength l [ λ ], and ∀ y ∈ Axe γ λ , the image by T e p of the unit tangent vector at y pointing towards γ λ y belongs to λ .We remark that the number of these elements γ λ is at least equal to the cardinality of thepointwise stabilizer of Axe γ λ (i.e., the multiplicity of λ ).Let x λ be the closest point to x on Axe γ λ . We have d ( x, x λ ) ≤ r , because x ∈ K and Axe γ λ ∩ K = ∅ . Thus by the triangle inequality, we know l [ λ ] ≤ d ( x, γ λ x ) ≤ d ( x, x λ ) + d ( x λ , γ λ x λ ) + d ( γ λ x, γ λ x λ ) ≤ l [ λ ] + 2 r ≤ s + 2 r. Moreover, |− ad ( x, γ λ x ) − bd ( x, γ λ x ) − al [ λ ] − bl [ λ ] | ≤ (cid:12)(cid:12)(cid:12) − d a,b ( x, γ λ x ) − ( − d a,b ( x λ , γ λ x λ ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) d a,b ( x, x λ ) (cid:12)(cid:12)(cid:12) < c ′ . Hence, Z a,bW ( s ) ≤ e c ′ G x,x ( s + 2 r ) . (cid:3) Lemma 18. For any relatively compact open set W ⊂ T S , we have lim inf s →∞ s log Z a,bW ( s ) ≥ δ a,bP P S . Proof. Let v ∈ T H such that ( v − , v + ) ∈ Λ × Λ \ diagonal and T e p ( v ) ∈ W , and let x = π ( v ). Claim: There exists a constant c > Z a,bW ( s ) ≥ c · b x,x,U ′ ,V ′ ( s ) . pf. Firstly, using the standard arguments (cf. Lemma 2.8 [PPS15] or P.150-151 [GdlH90]), thereexist small neighborhoods U ′ and V ′ in H ∪ ∂ ∞ H of v + and v − , respectively, such that if γ ∈ Γ satisfying γx ∈ U ′ and γ − y ∈ V ′ , then γ is a hyperbolic element and v is close to the translationsaxis Axe γ . Also, if v γ is the unit tangent vector at x γ pointing γx γ , then we know p ( v γ ) ∈ W (recall W is open). Note that U ′ ∩ Λ = ∅ and V ′ ∩ Λ = ∅ .Let γ ∈ Γ such that d ( x, γx ) ≤ s , γx ∈ U ′ and γ − x ∈ V ′ . Since γ is hyperbolic, thecorresponding orbit λ γ is a closed orbit, and its length satisfies d ( x, γx ) − ≤ l [ λ γ ] ≤ d ( x, γx ) ≤ s. Similarly, there exists a constant c ′′ > d ( x, γx ) − c ′′ ≤ l [ λ γ ] ≤ d ( x, γx ).Thus, there exists c ′ ≥ (cid:12)(cid:12)(cid:12) al [ λ γ ] + bl [ λ γ ] − d a,b ( x, γx ) (cid:12)(cid:12)(cid:12) = | a ( d ( x γ , γx γ ) − d ( x, γx )) + b ( d ( x γ , γx γ ) − d ( x, γx )) |≤ a + bc ′′ = c ′ Notice that because the number of times a closed geodesic passes close to a given point is atmost linear in its length, we know the cardinality of the fibers of the map γ → λ γ is at most c ′ s for some constant c ′ > 0. Hence, we have Z a,bW ( s ) ≥ e − c ′ c ′ b x,x,U ′ ,V ′ ( s ) . (cid:3) The Proof of Theorem 10. We continue using the same assumption as in the above subsection. Definition. (1) G a,bx,y ( s ) := { γ ∈ G : d a,b ( x, γ · y ) ≤ s } ;(2) G a,bx,y, ( s ) := { γ ∈ G : s − < d a,b ( x, γ · y ) ≤ s } ;(3) A x,y,U ′ ,s := { γ ∈ G : d a,b ( x, γy ) ≤ s and ρ ( γ ) y ∈ U ′ } where U ′ is an open set in ∂ ∞ H × H ;(4) a x,y,U ′ ( s ) := A x,y,U ′ ,s ;(5) B x,y,U ′ ,V ′ ,s := { γ ∈ G : d a,b ( x, γy ) ≤ s, ρ ( γ ) y ∈ U ′ and ρ ( γ − ) x ∈ V ′ } where U ′ , V ′ areopen sets in ∂ ∞ H × H ; and(6) b x,y,U ′ ,V ′ ( s ) := B x,y,U ′ ,V ′ ,s . Recall that δ a,b is the critical exponent of Q a,b ( s ) = X γ ∈ G exp( − s · d a,b ( x, γy )), i.e., Q a,b ( s ) con-verges when s > δ a,b and diverges when s < δ a,b . Theorem (Theorem 10) . we have δ a,b = lim s →∞ s log G a,bx,y ( s ) = lim s →∞ s log a x,y,U ′ ( s ) = lim s →∞ s log b x,y,U ′ ,V ′ ( s ) ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 31 Let ι : ρ ( G ) → ρ ( G ) be the boundary-preserving isomorphism. For γ ∈ ρ ( G ) := Γ theweighted distance can be written as d a,b ( x, γy ) = d ( x, γy ) + d ( x, ι ( ρ ( γ )) y ). Therefore, the abovegrowth rates can be equivalently defined as: • G a,bx,y ( s ) := { γ ∈ Γ : d a,b ( x, γ · y ) ≤ s } . • G a,bx,y, ( s ) := { γ ∈ Γ : s − < d a,b ( x, γ · y ) ≤ s } . The proof is given by the following lemmas. Lemma 19. lim inf s →∞ s log G a,bx,y, ( s ) ≤ δ a,b ≤ lim sup s →∞ s log G a,bx,y, ( s ) . Proof. The proof of Lemma 13 works here. One just need to replace Q a,bP P S,x,y ( s ) by Q a,b ( s ), G a,bx,y, ( s )by G a,bx,y, ( s ) and δ a,bP P S by δ a,b . (cid:3) Lemma 20. The above inequalities are indeed equalities. Moreover, δ a,b = lim n →∞ n log G a,bx,y ( s ) . Proof. This proof is identical with the proof of Lemma 14.We notice that by the triangle inequality, it is obvious that the lim sup s →∞ s log G a,bx,y ( s ) doesnot depend on the reference point x and y . W.l.o.g., we pick x = y = o . Recall the generating setof the extended Schottky group G = π S is A = { h , .., h N , p , ..., p N } with N + N ≥ • E n := { γ ∈ G : n − < d a,b ( o, γo ) ≤ n } . • b n := G a,bx,y, ( n ) = X γ ∈ E n e − d a,b ( o,γo ) . By Lemma 12, it is enough prove that there exist M > N ∈ N such that for all n, m ∈ N ,we have b n b m ≤ M i = N X i = − N b n + m + i . Claim: There exist N ∈ N and M > o such that E n × E m ≤ M · P i = Ni = − N E n + m + i pf. Let γ n ∈ E n and γ m ∈ E m , by Lemma 1, there exists α ∈ A such that (cid:12)(cid:12)(cid:12) d a,b ( o, γ n αγ m o ) − d a,b ( o, γ n o ) − d a,b ( o, γ m o ) (cid:12)(cid:12)(cid:12) < aC + bC where C only depending on ρ and C only depending on ρ (same C , C in the proof of Lemma14).Thus, n + m − aC + bC − < d a,b ( o, γ n αγ m o ) ≤ n + m + aC + bC + 2 , Let us consider the mapΨ : E n × E m → i = C + aC + bC +2 X i = − ( C + aC + bC +2) E n + m + i ( γ n , γ m ) γ n αγ m This maps is obvious not one-to-one. Nevertheless, we claim − ( γ n αγ m ) is finite. By Lemma11, we know that for i = 1 , d ( ρ i ( γ n ) o, [ o, ρ i ( γ n αγ m ) o ]) ≤ D ′ , which implies if there exist γ ′ n ∈ E n and γ ′ m ∈ E m such that γ ′ n αγ ′ m = γ n αγ m then d a,b ( γ n o, γ ′ n o ) ≤ D ′′ where D ′ and D ′′ are constantsonly depending on a, b, ρ and ρ . Moreover, by the discreteness of ρ i (Γ), the set { γ ∈ G : d a,b ( γo, o ) ≤ D ′′ + 1) } is finite (say, smaller than or equal to M ). Hence − ( γ n αγ m ) ≤ M .Therefore, E n × E m ≤ ( N + N ) M · i = aC + bC +2 X i = − ( aC + bC +2) E n + m + i where N + N is the cardinality of A .Moreover, we know (cid:12)(cid:12)(cid:12) d a,b ( o, γ n αγ m o ) − d a,b ( o, γ n o ) − d a,b ( o, γ m o ) (cid:12)(cid:12)(cid:12) ≤ aC + bC , thus we have the lemma, more precisely, b n b m ≤ ( N + N ) M · e aC + bC i = aC + bC +2 X i = − ( aC + bC +2) b n + m + i . (cid:3) As presented in the proof of Lemma 6, using the following lemma one could prove e δ a,b is alsothe growth rate of closed geodesics. More precisely, one could prove that the growth rate of b a,b equal to the growth rate of closed geodesics on S and S . Lemma 21. δ a,b = lim s →∞ s log a x,y,U ′ ( s ) . Proof. This proof is identical with the proof of Lemma 15.It is clear that G a,bx,y ( s ) ≥ a x,y,U ′ ( s ), so it is enough the prove thatlim inf s →∞ s log a x,y,U ′ ( s ) ≥ δ a,b . For fixed ( x, y ) ∈ H and U ′ an open set in ∂ ∞ H ∪ H , since Γ is non-elementary every orbit inΛ is dense. Thus, by the compactness of Λ , there exist γ , γ , ..., γ k such that Λ ⊂ S ki =1 γ i U ′ .Furthermore, since Conv(Λ ) \ S ki =1 γ i U ′ is compact, there exists a constant c such that G a,bx,y ( s ) ≤ k X i =1 a x,y,γ i U ′ ( s ) + c . Claim: a x,y,γ i U ′ ( s − r ) ≤ a x,y,U ′ ( s ) where r = max { d a,b ( γ i x, x ) } .pf. First we notice that by definition we have a x,y,U ′ ( s ) = a γx,y,γU ′ ( s ) . Therefore, we have a x,y,γU ′ ( s − r ) = a γ − i x,y,U ′ ( s − r ) := { γ ∈ Γ : d a,b ( γ − i x, γy ) ≤ s − r, and γy ∈ U ′ } . Notice that since d a,b satisfies the triangle inequality, we know if d a,b ( γ − i x, γy ) ≤ s − r for some s ∈ ( r, ∞ ), then d a,b ( γy, x ) < s . So, we have A γ − x,y,U ′ ,s − r = { γ ∈ Γ : d a,b ( γ − i x, γy ) ≤ s − r, and γy ∈ U ′ }⊂{ γ ∈ Γ : d a,b ( x, γy ) ≤ s, and γy ∈ U ′ } = A x,y,U ′ ,s, . Hence, we have the claim.Using this claim, we have G a,bx.x ( s − r ) ≤ k · a x,y,U ′ ( s ) + c , and the result follows. (cid:3) ANHATTAN CURVES FOR HYPERBOLIC SURFACES WITH CUSPS 33 Lemma 22. δ a,b = lim s →∞ s log b x,y,U ′ ,V ′ ( s ) . Proof. This proof is the same one as the proof of Lemma 16.We first notice that there exist α , ..., α l such that Λ ⊂ S li =1 α i V ′ , and because Conv(Λ ) \ S li =1 α i V ′ is compact, we have a x,y,U ′ ( s ) ≤ c + l X i =1 b x,y,U ′ ,α i V ′ ( s ) . We first notice that by definition we have b x,y,U ′ ,γV ′ ( s ) = b x,γ − y,U ′ ,V ′ ( s ) . Then we pick r = max { d a,b ( y, α i y ) : 1 ≤ i ≤ l } . Notice that by the triangle inequality of d a,b , weknow if d a,b ( x, γα − i y ) ≤ s − r, then d a,b ( x, γy ) ≤ s. So, we have B x,α − i y,U ′ ,V ′ ( s ) ⊂ B x,y,B r U ′ ,V ′ where B r U ′ is the r − neighborhood of U ′ .Moreover, again by the triangle equality of d a,b , we know (cid:12)(cid:12) − ad ( x, γy ) − bd ( x, γy ) − ( − ad ( x, γα − i x ) − bd ( x, γα − i x )) (cid:12)(cid:12) ≤ ad ( y, α − i x ) + bd ( y, α − i x )) ≤ c ′ ( x, y, a, b ) . Therefore, for 1 ≤ i ≤ l and s > r , b x,α − i y,U ′ ,V ′ ( s − r ) ≤ e c ′ b x,y,B r U ′ ,V ′ ( s ) . 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