Pressure metrics for deformation spaces of quasifuchsian groups with parabolics

In this paper, we produce a mapping class group invariant pressure metric on the space QF(S) of quasiconformal deformations of a co-finite area Fuchsian group uniformizing a surface S. Our pressure metric arises from an analytic pressure form on QF(S) which is degenerate only on pure bending vectors on the Fuchsian locus. Our techniques also show that the Hausdorff dimension of the limit set varies analytically over QF(S).


Introduction
We construct a pressure metric on the quasifuchsian space QF (S) of quasiconformal deformations, within PSL(2, C), of a Fuchsian group Γ in PSL(2, R) whose quotient H 2 /Γ has finite area and is homeomorphic to the interior of a compact surface S. Our pressure metric is a mapping class group invariant path metric, which is a Riemannian metric on the complement of the submanifold of Fuchsian representations.Our metric and its construction generalize work of Bridgeman [9] in the case that H 2 /Γ is a closed surface.
McMullen [31] initiated the study of pressure metrics, by constructing a pressure metric on the Teichmüller space of a closed surface.His pressure metric is one way of formalizing Thurston's notion of constructing a metric on Teichmüller space as the "Hessian of the length of a random geodesic" (see also Wolpert [49], Bonahon [4] and Fathi-Flaminio [18]) and like Thurston's metric it agrees with the classical Weil-Petersson metric (up to scalar multiplication).Subsequently, Bridgeman [9] constructed a pressure metric on quasifuchsian space, Bridgeman, Canary, Labourie and Sambarino [10] constructed pressure metrics on deformation spaces of Anosov representations, and Pollicott and Sharp [34] constructed pressure metrics on spaces of metric graphs (see also Kao [21]).The main tool in the construction of these pressure metrics is the Thermodynamic Formalism for topologically transitive, Anosov flows with compact support and their associated well-behaved finite Markov codings.
The major obstruction to extending the constructions of pressure metrics to deformation spaces of geometrically finite (rather than convex cocompact) Kleinian groups and related settings is that the support of the recurrent portion of the geodesic flow is not compact and hence there is not a well-behaved finite Markov coding.Mauldin-Urbanski [30] and Sarig [40] extended the Thermodynamical Formalism to the setting of topologically mixing Markov shifts with countable alphabet and the (BIP) property.In the case of finite area hyperbolic surfaces, Stadlbauer [43] and Ledrappier and Sarig [27] construct and study a topologically mixing countable Markov coding with the (BIP) property for the recurrent portion of the geodesic flow of the surface.In previous work, Kao [23] showed how to adapt the Thermodynamic Formalism in the setting of the Stadlbauer-Ledrappier-Sarig coding to construct pressure metrics on Teichmüller spaces of punctured surfaces.
We adapt the techniques developed by Bridgeman [9] and Kao [23] into our setting to construct a pressure metric which can again be naturally interpreted as the Hessian of the (renormalized) length of a random geodesic.
Theorem (Theorem 9.1).If S is a compact surface with non-empty boundary, the pressure form P on QF (S) induces a Mod(S)-invariant path metric, which is an analytic Riemannian metric on the complement of the Fuchsian locus.
Moreover, if v ∈ T ρ (QF (S)), then P(v, v) = 0 if and only if ρ is Fuchsian and v is a pure bending vector.
The control obtained from the Thermodynamic Formalism allows us to see that the topological entropy of the geodesic flow of the quasifuchsian hyperbolic 3-manifold varies analytically over QF (S).We recall that the topological entropy h(ρ) of ρ is the exponential growth rate of the number of closed orbits of the geodesic flow of N ρ = H 3 /ρ(Γ) of length at most T .More precisely, if where [Γ] is the collection of conjugacy classes in Γ and ℓ ρ (γ) is the translation length of the action of ρ(γ) on H 3 , then the topological entropy is given by Sullivan [46] showed that the topological entropy and the Hausdorff dimension of the limit set agree for quasifuchsian groups.So we see that the Hausdorff dimension of the limit set varies analytically over QF (S), generalizing a result of Ruelle [37] for quasifuchsian deformation spaces of closed surfaces.Schapira and Tapie [41,Thm. 6.2] previously established that the entropy is C 1 on QF (S) and computed its derivative (as a special case of a much more general result).
The pressure intersection was first defined by Burger [12] for pairs of convex cocompact Fuchsian representations.Schapira and Tapie [41] defined an intersection function for negatively curved manifolds with an entropy gap at infinity, by generalizing the geodesic stretch considered by Knieper [26] in the compact setting.Their definition applies in a much more general framework, but agrees with our notion in this setting, see [41,Prop. 2.17].Let (Σ + , σ) be the Stadlbauer-Ledrapprier-Sarig coding of a Fuchsian group Γ giving a finite area uniformization of S. If ρ ∈ QF (S) we construct a roof function τ ρ : Σ + → R whose periods are translation lengths of elements of ρ(Γ).The key technical work in the paper is a careful analysis of these roof functions.In particular, we show that they vary analytically over QF (S), see Proposition 3.1.If P is the Gurevich pressure function (on the space of all well-behaved roof functions), then the topological entropy h(ρ) of ρ is the unique solution of P (−tτ ρ ) = 0. Our actual working definition of the intersection function will be expressed in terms of equilibrium states on Σ + for the functions −h(ρ)τ ρ , but we will show in Theorem 10.3 that this thermodynamical definition agrees with the more geometric definition given above.
We use Theorem 6.1 in our proof of a rigidity result for the renormalized pressure intersection, see Corollary 7.2 , and in our proof that pressure intersection is analytic on QF (S) × QF (S), see Proposition 7.1.We also use it to obtain a rigidity theorem for weighted entropy in the spirit of the Bishop-Steger rigidity theorem for Fuchsian groups, see [3].If a, b > 0 and ρ, η ∈ QF (S), we define the weighted entropy Corollary (Corollary 6.3).If S is a compact surface with non-empty boundary, ρ, η ∈ QF (S) and a, b > 0, then with equality if and only if ρ = η.
Other viewpoints: If ρ ∈ QF (S), then N ρ = H 3 /ρ(Γ) is a geometrically finite hyperbolic 3-manifold.As such its dynamics may be analyzed using techniques from dynamics which do not rely on symbolic dynamics.For example, it naturally fits into the frameworks for geometrically finite negatively curved manifolds developed by Dal'bo-Otal-Peigné [14], negatively curved Riemannian manifolds with bounded geometry as studied by Paulin-Pollicott-Schapira [33] and negatively curved manifolds with an entropy gap at infinity as studied by Schapira-Tapie [41].In particular, the existence of equilibrium states and their continuous variation in our setting also follows from the work of Schapira and Tapie [41].
Since all the geodesic flows of manifolds in QF (S) are Hölder orbit equivalent, one should be able to think of them all as arising from an analytically varying family of Hölder potential functions on the geodesic flow of a fixed hyperbolic 3-manifold.However, for the construction of the pressure metric it will be necessary to know that the pressure function is at least twice differentiable.Results of this form do not yet seem to be available without symbolic dynamics.We have therefore chosen to develop the theory entirely from the viewpoint of the coding throughout the paper.
Iommi, Riquelme and Velozo [20] have previously used the Dal'bo-Peigné coding [16] to study negatively curved manifolds of extended Schottky type.These manifolds include the hyperbolic 3-manifolds associated to all quasiconformal deformations of finitely generated Fuchsian groups whose quotients have infinite area.In particular, they perform a phase transition analysis and show the existence and uniqueness of equilibrium states in their setting.The symbolic approach to phase transition analysis can be traced back to Iommi-Jordan [19].Riquelme and Velozo [35] work in a more general setting which includes quasifuchsian groups with parabolics, but without a coding, and obtain a phase transition analysis for the pressure function as well as the existence of equilibrium measures.

Acknowledements:
The authors would like to thank Francois Ledrappier, Mark Pollicott, Ralf Spatzier, and Dan Thompson for helpful conversations during the course of their investigation.We also thank the referees whose suggestions greatly improved the exposition.
The quasifuchsian space is given by where Hom tp (Γ, PSL(2, C)) is the space of type-preserving representations of Γ into PSL(2, C) (i.e.representations taking parabolic elements of Γ to parabolic elements of PSL(2, C)).We call X(S) the relative character variety and it has the structure of a projective variety.The space QF (S) is a smooth open subset of X(S), so is naturally a complex analytic manifold.
Suppose that {ρ z } z∈∆ is a complex analytic family of representations in QC(Γ) parameterized by the unit disk ∆. Sullivan [47,Thm. 1] showed that there is a continuous map Hartogs' Theorem then implies that ξ ρ (x) varies complex analytically over all of QC(Γ).

Countable Markov Shifts.
A two-sided countable Markov shift with countable alphabet A and transition matrix T ∈ {0, 1} A×A is the set equipped with a shift map σ : Σ → Σ which takes (x i ) i∈Z to (x i+1 ) i∈Z .Notice that the shift simply moves the letter in place i into place i − 1, i.e. it shifts every letter one place to the left.
Associated to any two-sided countable Markov shift Σ is the one-sided countable Markov shift equipped with a shift map σ : Σ + → Σ + which takes (x i ) i∈N to (x i+1 ) i∈N .In this case, the shift deletes the letter x 1 and moves every other letter one place to the left.There is a natural projection map p + : Σ → Σ + given by p + (x) = x + = (x i ) i∈N which simply forgets all the terms to the left of x 1 .Notice that p + • σ = σ • p + .We will work entirely with one-sided shifts, except in the final section.One says that (Σ + , σ) is topologically mixing if for all a, b ∈ A, there exists N = N (a, b) so that if n ≥ N , then there exists x ∈ Σ so that x 1 = a and x n = b.The shift (Σ + , σ) has the big images and pre-images property (BIP) if there exists a finite subset B ⊂ A so that if a ∈ A, then there exists b 0 , b 1 ∈ B so that t b 0 ,a = 1 = t a,b 1 .
Given a one-sided countable Markov shift (Σ + , σ) and a function g : Σ + → R, let be the n th variation of g.We say that g is locally Hölder continuous if there exists C > 0 and θ ∈ (0, 1) so that V n (g) ≤ Cθ n for all n ∈ N. We say that two locally Hölder continuous functions f : Σ + → R and g : Σ + → R are cohomologous if there exists a locally Hölder continuous function h : Sarig [38] considers the associated Gurevich pressure of a locally Hölder continuous function g : Σ + → R, given by for some (any) a ∈ A where is the ergodic sum and Fix n = {x ∈ Σ + | σ n (x) = x}.The pressure of a locally Hölder continuous function f need not be finite, but Mauldin and Urbanski [30] provide the following characterization of when P (f ) is finite.
Theorem 2.2.(Mauldin-Urbanski [30, Thm.2.1.9])Suppose that (Σ + , σ) is a one-sided countable Markov shift which has BIP and is topologically mixing.If f is locally Hölder continuous, then P (f ) is finite if and only if A Borel probability measure m on Σ + is said to be a Gibbs state for a locally Hölder continuous function g : Σ + → R if there exists a constant B > 1 and e Sng(x)−nC ≤ B for all x ∈ [a 1 , . . ., a n ]}, where [a 1 , . . ., a n ] is the cylinder consisting of all x ∈ Σ + so that x i = a i for all 1 ≤ i ≤ n.Sarig [40,Thm 4.9] shows that a locally Hölder continuous function f on a topologically mixing one-sided countable Markov shift with BIP so that P (f ) is finite admits a Gibbs state µ f .Mauldin-Urbanski [30,Thm 2.2.4] show that if a locally Hölder continuous function f on a topologically mixing one-sided countable Markov shift with BIP admits a Gibbs state, then f admits a unique shift invariant Gibbs state.We summarize their work the statement below.
Theorem 2.3.(Mauldin-Urbanski [30, Thm 2.2.4],Sarig [40, Thm 4.9]) Suppose that (Σ + , σ) is a one-sided countable Markov shift which has BIP and is topologically mixing.If f is locally Hölder continuous and P (f ) is finite, then f admits a unique shift invariant Gibbs state µ f .The transfer operator is a central tool in the Thermodynamic Formalism.Recall that the transfer operator L f : C b (Σ + ) → C b (Σ + ) of a locally Hölder continuous function f over Σ + is defined by e f (y) g(y) for all x ∈ Σ + .
If (Σ + , σ) is topologically mixing and has the BIP property, ν is a Borel probability measure for Σ + and (L f ) * (ν) = e P (f ) ν (where (L f ) * is the dual of transfer operator), then ν is a Gibbs state for f , see Mauldin-Urbanski [30, Theorem 2.

3.3].
A σ-invariant Borel probability measure m on Σ + is said to be an equilibrium measure for a locally Hölder continuous function g : Σ + → R if where h σ (m) is the measure-theoretic entropy of σ with respect to the measure m.Mauldin and Urbanski [30] give a criterion guaranteeing the existence of a unique equilibrum state.
Theorem 2.4.(Mauldin-Urbanski [30, Thm.2.2.9])Suppose that (Σ + , σ) is a one-sided countable Markov shift which has BIP and is topologically mixing.If f is locally Hölder continuous, ν f is a shift invariant Gibbs state for f and − f dν f < +∞, then ν f is the unique equilibrium measure for f .We say that {g u : Σ + → R} u∈M is a real analytic family if M is a real analytic manifold and for all x ∈ Σ + , u → g u (x) is a real analytic function on M .Mauldin and Urbanski [30, Thm.2.6.12,Prop.2.6.13 and 2.6.14],see also Sarig ([39,Cor. 4],[40, Thm 5.10 and 5.13]), prove real analyticity properties of the pressure function and evaluate its derivatives.We summarize their results in Theorem 2.5.Here the variance of a locally Hölder continuous function f : Σ + → R with respect to a probability measure m on Σ + is given by Theorem 2.5.(Mauldin-Urbanski, Sarig) Suppose that (Σ + , σ) is a one-sided countable Markov shift which has BIP and is topologically mixing.If {g u : Σ + → R} u∈M is a real analytic family of locally Hölder continuous functions such that where m gu 0 is the unique equilibrium state for g u 0 .
2.3.The Stadlbauer-Ledrappier-Sarig coding.Stadlbauer [43] and Ledrappier-Sarig [27] describe a one-sided countable Markov shift (Σ + , σ) with alphabet A which encodes the recurrent portion of the geodesic flow on T 1 (H 2 /Γ).In this section, we will sketch the construction of this coding and recall its crucial properties.They begin with the classical coding of a free group, as described by Bowen and Series [7].One begins with a fundamental domain D 0 for a free convex cocompact Fuchsian group Γ, containing the origin in the Poincaré disk model, all of whose vertices lie in ∂H 2 , so that the set S of face pairings of D 0 is a minimal symmetric generating set for Γ.One then labels any translate γ(D 0 ) by the group element γ.Any geodesic ray r z beginning at the origin and ending at z ∈ Λ(Γ) passes through an infinite sequence of translates, so we get a sequence c(z) = (γ k ) k∈N .One may then turn this into an infinite sequence in S by considering b(z) = (γ k γ −1 k−1 ) k∈N (where we adopt the convention that γ 0 = id.)If Γ is convex cocompact, this produces a well behaved one-sided Markov shift (Σ + BS , σ) with finite alphabet S. The obvious map ω : Σ + BS → Λ(Γ) which takes b(z) to z is Hölder and (Σ + BS , σ) encodes the recurrent portion of the geodesic flow of H 2 /Γ.If one attempts to implement this procedure when Γ is not convex cocompact, then one must omit all geodesic rays which end at a parabolic fixed point and there is no natural way to do this from a coding perspective.Moreover, if one simply restricts ω to the allowable words then ω will not be Hölder in this case.(To see that ω will not be Hölder, choose x, y ∈ Σ + BS , so that x i = y i = α for all 1 ≤ i ≤ n, where α is a parabolic face-pairing, and ) Roughly, the Stadlbauer-Ledrappier-Sarig begins with c(z) = (γ k ) and clumps together all terms in b(z) = (γ k γ −1 k−1 ) which lie in a subword which is a high power of a parabolic element.One must then append to our alphabet all powers of minimal word length parabolic elements and and disallow infinite words beginning or ending in infinitely repeating parabolic elements.When Γ is geometrically finite, but not co-finite area, Dal'bo and Peigné [16] implemented this process to powerful effect for geometrically finite Fuchsian groups with infinite area quotients.
However, when Γ is co-finite area, the actual description is more intricate.The states Stadlbauer-Ledrappier-Sarig use record a finite amount of information about both the past and the future of the trajectory.
Let C be the collection of all freely reduced words in S which have minimal word length in their conjugacy class and generate a maximal parabolic subgroup of Γ.Notice that the minimal word length representative of a conjugacy class of α is unique up to cyclic permutation.(One may in fact choose D 0 so that all but one pair of parabolic elements of C is conjugate to a facepairing.)Since there are only finitely many conjugacy classes of maximal parabolic subgroups of Γ, C is finite.They then choose a sufficiently large even number 2N so that the length of every element of C divides 2N and let C * be the collection of powers of elements of C of length exactly 2N .(One may assume that two elements of C * share a subword of length at least 2 if and only if they are cyclic permutations of one another.) Let So, r −1 (n) is always non-empty and there exists D so that r −1 (n) has size at most D for all n ∈ N, i.e. there are at most D states associated to each positive integer.Given a geodesic ray r z beginning at the origin and ending at a point z in the set Λ c (Γ) of points in the limit set which are not parabolic fixed points, let c(z) = (γ k ) k∈N be the sequence of elements of Γ which record the translates of D 0 which r z passes through.Let b We then associate to r z a finite collection of infinite words in S N∪{0} , by allowing b 0 to be any element of S, so that b Suppose we have a word (b k ) k∈N∪{0} arising from the previous construction.
In this case, we shift (b i ) rightward by 2N (s − 1) + k + 1 to compute x 2 .One then simply proceeds iteratively.By construction, if Examples: If Γ uniformizes a once-punctured torus, then S = {α, α −1 , β, β −1 } is a mimimal symmetric generating set for Γ and If Γ uniformizes a four times punctured sphere, then one may choose D 0 so that S = {α, The following proposition encodes crucial properties of the coding.Proposition 2.6.(Ledrappier-Sarig [27, Lemma 2.1], Stadlbauer [43]) Suppose that H 2 /Γ is a finite area hyperbolic surface, then (Σ + , σ) is topologically mixing, has the big images and preimages property (BIP), and there exists a locally Hölder continuous finite-to-one map Moreover, if γ is a hyperbolic element of Γ, then there exists x ∈ Fix n , for some n ∈ N, unique up to cyclic permutation, so that γ is conjugate to Notice that every element of A can be preceded and succeeded by some element of A 1 , so (Σ + , σ) clearly has (BIP).The topological mixing property is similarly easy to see directly from the definition, so the main claim of this proposition is that ω is locally Hölder continuous.
Another crucial property of the coding is that the translates of the origin associated to the Stadbauer-Ledrappier-Sarig coding approach points in the limit set conically (see property (1) on page 15 in Ledrappier-Sarig [27]).
Lemma 2.7.(Ledrappier-Sarig [27, Property (1) on page 15]) Given y ∈ H 2 , there exists L > 0 so that if x ∈ Σ + and n ∈ N, then Since the proof of Lemma 2.7 appears in the middle of a rather technical discussion in [27], we will sketch a proof in our language.Choose a compact subset K of H 2 /Γ so that its complement is a collection of cusp regions bounded by curves which are images of horocycles in H 2 .Without loss of generality we may assume that y is the origin in the Poincaré disk model for H 2 .Notice that if the portion of − −− → bω(x) between γ s (D 0 ) and γ s+t (D 0 ) lies entirely in the complement of the pre-image of K, and t > s, then γ s+t γ −1 s is a subword of a power of an element in C. Let K be the intersection of the pre-image of K with D 0 .Notice that we may assume that y ∈ K (by perhaps enlarging K).Suppose the last 2N + 1 letters of

Roof functions for quasifuchsian groups
If ρ ∈ QC(Γ), we define a roof function τ ρ : Σ + → R by setting where b 0 = (0, 0, 1) and B z (x, y) is the Busemann function based at z ∈ ∂H 3 which measures the signed distance between the horoballs based at z through x and y.In the Poincaré upper half space model, we write the Busemann function explicitly as where z ∈ C ⊂ ∂H 3 , p, q ∈ H 3 and h(p) is the Euclidean height of p above the complex plane and B∞ (p, q) = h(p) h(q) .
It follows from the cocycle property of the Busemann function that In particular, if x = (x 1 , . . ., x m ) ∈ Σ + , then We say that the roof function τ ρ is eventually positive if there exists C > 0 and N ∈ N so that if n ≥ N and x ∈ Σ + , then S n τ ρ (x) ≥ C.
The following lemma records crucial properties of our roof functions.It generalizes similar results of Ledrappier-Sarig [27, Lemma 2.2 and 3.1] in the Fuchsian setting.
Proof.Since ξ ρ (q) varies complex analytically in ρ for all q ∈ Λ(Γ), by Lemma 2.1, and B z (b 0 , y) is real analytic in z ∈ C and y ∈ H 3 , we see that τ ρ (x) varies analytically over QC(Γ) for all x ∈ Σ + .Recall, see Douady-Earle [17], that there exists K = K(ρ) > 1 and a ρ-equivariant Kbilipschitz map φ : H 2 → H 3 so that φ(y 0 ) = b 0 where y 0 is the origin in the disk model for H 2 .Therefore, if L is the constant from Lemma 2.7 and . The Fellow Traveller property for H 3 implies that there exist R = R(K) > 0 so that any K-bilipschitz geodesic ray lies a Hausdorff distance at most R from the geodesic ray with the same endpoints.Therefore, if M = KL + R, then, for all n ∈ N, We next obtain our claimed bounds on the roof function.If x ∈ Σ + , then Since our alphabet is infinite, our work is not done.If w ∈ C * , we may normalize so that ρ(w)(z) = z +1 and b 0 = (0, 0, b w ) in the upper half-space model for H 3 .If z ∈ C ⊂ ∂H 3 and r > 0, we let B(z, r) denote the Euclidean ball of radius r about z in C. Since g a has length at most 2N + 1 in the alphabet S, we may define where |ρ(g a )(b 0 )| is the Euclidean distance from ρ(g a )(b 0 ) to 0 = (0, 0, 0).Suppose that x ∈ Σ + , r(x 1 ) ≥ 2 and G(x 1 ) = w s g a where s = r(a) − 2. By definition, ρ(g a )(b 0 ) ∈ B(0, c w ), so ) passes through B(s, e M c w ), which implies that ξ ρ (ω(x)) ∈ B(s, e M c w ).It then follows from our formula for the Busemann function that Similarly, Since there are only finitely many choices of g a , it is easy to see that there exists C w so that 2 log(r(x 1 )) − C w ≤ τ ρ (x) ≤ 2 log(r(x 1 )) + C w whenever x ∈ Σ + , r(x 1 ) > S + 2 and G(x 1 ) = w s g a .Since there are only finitely many w in C * and only finitely many words a with r(a) ≤ S + 2, we see that there exists C ρ so that We next show that τ ρ is locally Hölder continuous.Since ω is locally Hölder continuous, there exists A and α > 0 so that if x, y ∈ Σ + and Since ξ ρ is Hölder, there exist C and β > 0 so that d(ξ ρ (z), ξ ρ (w)) ≤ Cd(z, w) β for all z, w ∈ Λ(Γ), so d(ξ ρ (ω(x)), ξ ρ (ω(y)) ≤ CA β e −αβn .
If a ∈ A, then let However, the best general estimate one can have on D a is O(r(a)), so we will have to dig a little deeper.
We again work in the upper half-space model, and assume that r(a) > S + 2, G(a) = w s g a where s = r(a) − 2 and normalize as before so that ρ(w)(z) = z + 1.We then map the limit set into the boundary of the upper-half space model by setting ξρ = T • ξ ρ where T is a conformal automorphism which takes the Poincaré ball model to the upper half-space model and takes the fixed point of ρ(w) to ∞.Notice that T is K w -bilipschitz on T −1 (B(0, e M c w )).Therefore, if x, y ∈ [a, x 2 , . . ., x n ], then Moreover, there exists D w so that 1) .
Since there are only finitely many a where r(a) ≤ S + 2 and only finitely many choices of w, our bounds are uniform over A and so τ ρ is locally Hölder continuous.
It remains to check that τ ρ is eventually positive.Since for all n ∈ N, we see that is finite, there exists N so that if γ has word length at least N (in the generators given S), then γ does not lie in B. Therefore, if n ≥ N and x ∈ Σ + , then S n τ ρ (x) > R ρ > 0. Thus, τ ρ is eventually positive and our proof is complete.
It is a standard feature of the Thermodynamic Formalism that one may replace an eventually positive roof function by a roof function which is strictly positive and cohomologous to the original roof function.(For a statement and proof which includes the current situation, see [8,Lemma 3.3].)Corollary 3.2.If ρ ∈ QC(Γ), there exists a locally Hölder continuous function τρ and c > 0 so that τρ (x) ≥ c for all x ∈ Σ + and τρ is cohomologous to τ ρ .

Phase transition analysis
We begin by extending Kao's phase transition analysis, see Kao [23,Thm. 4.1], which characterizes which linear combinations of a pair of roof functions have finite pressure.The primary use of this analysis will be in the case of a single roof function, i.e. when a = 1 and b = 0.However, we will use the full force of this result in the proof of our Manhattan curve theorem, see Theorem 6.1.

Entropy and Hausdorff dimension
Theorem 4.1 implies that if ρ ∈ QC(Γ) then there is a unique solution h(ρ) > 1  2 to P (−h(ρ)τ ρ ) = 0.This unique solution h(ρ) is the topological entropy of ρ, see the discussion in Kao [23, Section 5].Theorem 2.5 and the implicit function theorem then imply that h(ρ) varies analytically over QC(Γ), generalizing a result of Ruelle [37] in the convex cocompact case.Since the entropy h(ρ) is invariant under conjugation, we obtain analyticity of entropy over QF (S).We recall that Schapira and Tapie [41, Thm.6.2] previously established that the entropy is C 1 on QF (S).
Theorem 5.1.If S is a compact hyperbolic surface with non-empty boundary, then the topological entropy varies analytically over QF (S).
Sullivan [46] showed that the topological entropy h(ρ) agrees with the Hausdorff dimension of the limit set Λ(ρ(Γ)), so we obtain the following corollary.Theorem 5.2.(Sullivan [46,48]) If ρ ∈ QC(Γ), then its topological entropy h(ρ) is the exponential growth rate of the number of closed geodesics of length less than T in N ρ = H 3 /ρ(Γ).Moreover, h(ρ) is the Hausdorff dimension of the limit set Λ(ρ(Γ)) and the critical exponent of the Poincaré series Q ρ (s).Theorems 5.1 and 5.2 together imply that the Hausdorff dimension of the limit set varies analytically.
Remarks: 1) Sullivan [48] also showed that h(ρ) is the critical exponent of the Poincaré series 2) Bowen [6] showed that if ρ ∈ QF (S) and S is a closed surface, then h(ρ) ≥ 1 with equality if and only if ρ is Fuchsian.Sullivan [45, p. 66], see also Xie [50], observed that Bowen's rigidity result extends to the case when H 2 /Γ has finite area.
Notice that if ρ and η are conjugate in Isom(H 3 ), then τ ρ = τ η so C(ρ, η) is a straight line.We will need the following technical result in the proof of Theorem 6.1.
which converges, since 2(a + b) > 1. Theorem 2.4 then implies that dm −aτρ−bτη is the unique equilibrium state for −aτ ρ − bτ η .Proposition 3.1 implies that there exists B > 1 so that if n is large enough, then for all x ∈ Σ + so that r(x 1 ) > n. (For example, if log n > 4 max{aC ρ + bC η , C θ , 1}, then we may choose B = 8(a + b).)Since τ θ is locally Hölder continuous, it is bounded on the remainder of Σ + .Therefore, since Σ + aτ ρ + bτ η dm −aτρ−bτη < +∞, we see that Now notice that, since τ θ is cohomologous to a positive function τθ , by Corollary 3.2, Proof of Theorem 6.1: Recall that t = h(ρ) is the unique solution to the equation P (−tτ ρ ) = 0 (see the discussion at the beginning of Section 5).So, the intersection of the Manhattan curve with the boundary of D consists of the points (h(ρ), 0) and (0, h(η)).
with equality if and only if ρ and η are conjugate in Isom(H 3 ).

Pressure intersection
We define the pressure intersection on QC(Γ) × QC(Γ) given by .
We obtain the following rigidity theorem as a consequence of Theorem 6.1.The inequality portion of this result was previously established by Schapira and Tapie [41,Cor. 3.17 Proof.Recall that the slope c = c(h(ρ), 0) of C(ρ, η) at (h(ρ), 0) is given by However, by Theorem 6.1, c ≤ − h(ρ) h(η) with equality if and only if ρ and η are conjugate in Isom(H 3 ).Our corollary follows immediately.

The pressure form
We may define an analytic section s : QF (S) → QC(Γ) so that s([ρ]) is an element of the conjugacy class of ρ.Choose co-prime hyperbolic elements α and β in Γ and let s(ρ) be the unique element of [ρ] so that s(ρ)(α) has attracting fixed point 0 and repelling fixed point ∞ and s(ρ)(β) has attracting fixed point 1.This will allow us to abuse notation and regard QF (S) as a subset of QC(Γ).
Following Bridgeman [9] and McMullen [31], we define an analytic pressure form P on the tangent bundle T QF (S) of QF (S), by letting which we rewrite with our abuse of notation as: P TρQF (S) = Hess(J(ρ), •)) Corollary 7.2 implies that P is non-negative, i.e.P(v, v) ≥ 0 for all v ∈ T QF (S).
Since P is non-negative, we can define a path pseudo-metric on QF (S) by setting where the infimum is taken over all smooth paths in QF (S) joining ρ to η.
We now derive a standard criterion for when a tangent vector is degenerate with respect to P, see also [11,Cor. 2.5] and [10,Lemma 9.3].
Recall, see Sarig [40,Thm. 5.12], that Var( ψ0 , m ψ(0) ) = 0 if and only if ψ0 is cohomologous to a constant function C. On the other hand, since P (ψ t ) = 0 for all t, the formula for the derivative of the pressure function gives that so C must equal 0. However, ψ0 is cohomologous to 0 if and only if for all x ∈ Fix n , and all n, (see [40,Theorem 1.1]).Moreover, for every hyperbolic element γ ∈ Γ, there exists x ∈ Fix n (for some n) so that γ is conjugate to in every case.Therefore, ψ0 is cohomologous to 0 if and only if for all γ ∈ Γ.
We say that v ∈ T ρ QF (S) is a pure bending vector if v = ∂ ∂t ρ t , ρ = ρ 0 is Fuchsian and ρ −t is the complex conjugate of ρ t for all t.Since the Fuchsian locus F (S) is the fixed point set of the action of complex conjugation on QF (S) and the collection of pure bending vectors at a point in F (S) is half-dimensional, one gets a decomposition where B ρ is the space of pure bending vectors at ρ.If v is a pure bending vector at ρ ∈ F (S), then v is tangent to a path obtained by bending ρ by a (signed) angle t along some measured lamination λ (see Bonahon [5,Section 2] for details).
We are finally ready to show that our pressure form is degenerate only along pure bending vectors.
Theorem 9.1.If S is a compact hyperbolic surface with non-empty boundary, then the pressure form P defines an Mod(S)-invariant path metric d P on QF (S) which is an analytic Riemannian metric except on the Fuchsian locus.
Moreover, if v ∈ T ρ (QF (S)), then P(v, v) = 0 if and only if ρ is Fuchsian and v is a pure bending vector.
Proof.If v is a pure bending vector, then we may write v = ρ0 where ρ −t is the complex conjugate of ρ t for all t, so hℓ γ (ρ t ) is an even function for all γ ∈ Γ.Therefore, D v hℓ γ = 0 for all γ ∈ Γ, so Lemma 8.1 implies that P(v, v) = 0.
Our main work is the following converse: Proposition 9.2.Suppose that v ∈ T ρ QF (S).If P(v, v) = 0 and v = 0, then v is a pure bending vector.
Recall, see [10,Lemma 13.1], that if a Riemannian metric on a manifold M is non-degenerate on the complement of a submanifold N of codimension at least one and the restriction of the Riemannian metric to T N is non-degenerate, then the associated path pseudo-metric is a metric.We will see in Corollary 10.4 that the pressure metric is mapping class group invariant.Our theorem then follows from Proposition 9.2 and the fact, established by Kao [23], that P is non-degenerate on the tangent space to the Fuchsian locus.
Since v = 0, there exists α ∈ Γ so that D v tr α = 0 and and Lemma 9.3 imply that Therefore, D v ℓ γ = 0 for all γ ∈ Γ.Notice that since tr γ (ρ) 2 is real for all γ ∈ Γ, ρ(Γ) lies in a proper (real) Zariski closed subset of PSL(2, C), so is not Zariski dense.However, since the Zariski closure of ρ(Γ) is a Lie subgroup, it must be conjugate to a subgroup of either PSL(2, R) or to the index two extension of PSL(2, R) obtained by appending z → −z.Since ρ is quasifuchsian, its limit set Λ(ρ(Γ)) is a Jordan curve and no element of ρ(Γ) can exchange the two components of its complement.Therefore, ρ is Fuchsian.(We note that this is the only place where our argument differs significantly from Bridgeman's.It replaces his rather technical [9, Lemma 15].) We can then write v = v 1 + v 2 where v 1 ∈ T ρ F (S) and v 2 is a pure bending vector.Since v 2 is a pure bending vector, 0 and there are finitely many curves whose length functions provide analytic parameters for F (S), this implies that v 1 = 0. Therefore, v = v 2 is a pure bending vector.✷

Patterson-Sullivan measures
In this section, we observe that the equilibrium state m −h(ρ)τρ is a normalized pull-back of the Patterson-Sullivan measure on Λ(ρ(Γ)).We use this to give a more geometric interpretation of the pressure intersection of two quasifuchsian representations, and hence a geometric formulation of the pressure form.
We now show that H ρ is the normalization of the pull-back μρ of Patterson-Sullivan measure which gives the equilibrium measure for −h(ρ)τ ρ .Dal'bo and Peigné [16, Prop.V.3] obtain an analogous result for negatively curved manifolds whose fundamental groups "act like" geometrically finite Fuchsian groups of co-infinite area (see also Dal'bo-Peigné [15, Cor.II.5]).The quasi-invariance of Patterson-Sullivan measure implies that We first check that H ρ μρ is shift invariant.
Finally, we observe that H ρ is bounded above.If p is a vertex of D 0 , then, by construction, there exists a neighborhood U p of p, so that if ω(x) ∈ U p , then there exists w ∈ C * , so that x 1 = (b, ω s , w 1 , . . ., w k−1 , c) for some s ≥ 2. Recall that we require that b = w 2N and c = w k .Observe that w 1 is the face pairing of the edge of D 0 associated to I x and that w 2N is the inverse of the face-pairing associated to the other edge E of ∂D 0 which ends at p. So, if I is the interval in ∂H 2 − ∂D 0 bounded by E, then Λ(ρ(Γ)) x is disjoint from ξ ρ (I x ∪ I).Therefore, H ρ is uniformly bounded on ω −1 (U p ) (since e 2h(ρ) ξρ(ω(x)),z b 0 is uniformly bounded for all z ∈ Λ(ρ(Γ)) x ⊂ Λ(ρ(Γ)) − ξ ρ (I ∪ I x )).However, D 0 has finitely many vertices {p 1 , . . ., p n } and H ρ is clearly bounded above if ω(x) ∈ ∂H 2 − U p i (since again e 2h(ρ) ξρ(ω(x)),z b 0 is uniformly bounded for all z ∈ Λ(ρ(Γ)) x ⊂ Λ(ρ(Γ)) − I x ).Therefore, H ρ is bounded above on Σ + .
Since every multiple of a Gibbs state for −h(ρ)τ ρ by a continuous function which is bounded between positive constants is also a (scalar multiple of a) Gibbs state for −h(ρ)τ ρ (see [30, Remark 2.2.1]), we see that H ρ μρ is a shift invariant Gibbs state and hence an equilibrium measure for −h(ρ)τ ρ (see Theorem 2.4).(The image of κ is the complement of all flow lines which do not exit cusps of N ρ and has full measure in Ω.)The map κ conjugates the suspension flow to the geodesic flow on its image i.e. κ • φ t = φ t • κ for all t ∈ R on κ(Σ τρ ).The Bowen-Margulis-Sullivan measure m ρ BM on Ω can be described by its lift to Ω which is given by m ρ BM (z, w, t) = e 2h(ρ) z,w b 0 dµ ρ (z)dµ ρ (w)dt.The Bowen-Margulis-Sullivan measure m ρ BM is finite and ergodic (see Sullivan [46,Theorem 3]) and equidistributed on closed geodesics (see Roblin [36,  We finally obtain the promised geometric form for the pressure intersection.We may thus think of the pressure intersection, in the spirit of Thurston, as the Hessian of the length of a random geodesic.As a consequence, we obtain a geometric presentation of the pressure form which allows us to easily see that the pressure metric is mapping class group invariant.Moreover, the pressure metric is mapping class group invariant.