The pressure metric for convex representations

arXiv:1301.7459v4 [math.DG] 1 Feb 2015 THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS MARTIN BRIDGEMAN, RICHARD CANARY, FRANÇOIS LABOURIE, AND ANDRES SAMBARINO Abstract. Using the thermodynamic formalism, we introduce a notion of intersection for projective Anosov representations, show analyticity results for the intersection and the entropy, and rigidity results for the intersection. We use the renormalized intersection to produce an Out(Γ)-invariant Riemannian metric on the smooth points of the deformation space of irreducible, generic, projective Anosov representations of a word hyperbolic group Γ into SLm (R). In particular, we produce mapping class group invariant Riemannian metrics on Hitchin components which restrict to the Weil–Petersson metric on the Fuchsian loci. Moreover, we produce Out(Γ)-invariant metrics on deformation spaces of convex cocompact representations into PSL2 (C) and show that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into any rank 1 semi-simple Lie group. 1. Introduction In this paper we produce a mapping class group invariant Riemannian metric on a Hitchin component of the character variety of representations of a closed surface group into SLm (R) whose restriction to the Fuchsian locus is a multiple of the Weil-Petersson metric. More generally, we produce a Out(Γ)-invariant Riemannian metric on the smooth generic points of the deformation space of irreducible, projective Anosov representations of a word hyperbolic group Γ into SLm (R). We use Plücker representations to produce metrics on deformation spaces of convex cocompact representations into PSL2 (C) and on the smooth points of deformation spaces of Zariski dense Anosov representations into an arbitrary semi-simple Lie group. Our metric is produced using the thermodynamic formalism developed by Bowen [12, 13], Parry–Pollicott [55], Ruelle [61] and others. It generalizes earlier work done in the Fuchsian and quasifuchsian cases by McMullen [53] and Bridgeman [9]. In order to use the thermodynamic formalism, we associate a natural flow Uρ Γ to any projective Anosov representation ρ, and show that it is a topologically transitive metric Anosov flow and is a Hölder reparameterization of the geodesic flow U0 Γ of Γ as defined by Gromov. We then see that entropy varies analytically over any smooth analytic family of projective Anosov homomorphisms of Γ into SLm (R). As a consequence, again using the Plücker embedding, we see that the Hausdorff Canary was partially supported by NSF grant DMS - 1006298. Labourie and Sambarino were partially supported by the European Research Council under the European Community’s seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no FP7-246918, as well as by the ANR program ETTT (ANR-09-BLAN-0116-01) The authors also acknowledge support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 ”RNMS: GEometric structures And Representation varieties” (the GEAR Network). 1 2 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO dimension of the limit set varies analytically over analytic families of convex cocompact representations into a rank one semi-simple Lie group. We also introduce a renormalized intersection J on the space of projective Anosov representations. Our metric is given by the Hessian of this renormalised intersection J. We now introduce the notation necessary to give more careful statements of our results. Let Γ be a word hyperbolic group with Gromov boundary ∂∞ Γ. Loosely speaking, a representation ρ : Γ → SLm (R) is projective Anosov if it has transverse projective limit maps, the image of every infinite order element is proximal, and the proximality “spreads uniformly” (see Section 2.1 for a careful definition). An element A ∈ SLm (R) is proximal if its action on RP(m) has an attracting fixed point. A representation ρ : Γ → SLm (R) is said to have transverse projective limit maps if there exist continuous ρ-equivariant maps ξ : ∂∞ Γ → RP(m) and θ : ∂∞ Γ → RP(m)∗ such that if x and y are distinct points in ∂∞ Γ, then ξ(x) ⊕ θ(y) = Rm (where we identify RP(m)∗ with the Grassmanian of (m − 1)-dimensional vector subspaces of Rm ). If γ ∈ Γ has infinite order, ρ is projective Anosov and γ + is the attracting fixed point of the action of γ on ∂Γ, then ξ(γ + ) is the attracting fixed point for the action of ρ(γ) on RP(m). Moreover, Guichard and Wienhard [26, Proposition 4.10] proved that every irreducible representation ρ : Γ → SLm (R) with transverse projective limit maps is projective Anosov. If ρ is a projective Anosov representation, we can associate to every conjugacy class [γ] of γ ∈ Γ its spectral radius Λ(γ)(ρ). The collection of these radii form the radius spectrum of ρ. For every positive real number T we define RT (ρ) = {[γ] | log(Λ(γ)(ρ)) 6 T }. We will see that RT (ρ) is finite (Proposition 2.8). We also define the entropy of a representation by 1 log ♯(RT (ρ)). h(ρ) = lim T →∞ T If ρ1 and ρ2 are two projective Anosov representations, we define their intersection by   X log(Λ(γ)(ρ )) 1 2  . I(ρ1 , ρ2 ) = lim  T →∞ ♯(RT (ρ1 )) log(Λ(γ)(ρ1 )) [γ]∈RT (ρ0 ) We also define the renormalised intersection by J(ρ1 , ρ2 ) = h(ρ2 ) I(ρ1 , ρ2 ). h(ρ1 ) We prove, see Theorem 1.3, that all these quantities are well defined and obtain the following inequality and rigidity result for the renormalised intersection. Let πm : SLm (R) → PSLm (R) be the projection map. If ρ : Γ → SLm (R) is a representation, let Gρ be the Zariski closure of ρ(Γ). Theorem 1.1. [Intersection] If Γ is a word hyperbolic group and ρ1 : Γ → SLm1 (R) and ρ2 : Γ → SLm2 (R) are projective Anosov representations, then J(ρ1 , ρ2 ) > 1. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 3 Moreover, if ρ1 and ρ2 are irreducible, Gρ1 and Gρ2 are connected and J(ρ1 , ρ2 ) = 1, then there exists an isomorphism φ : πm1 (Gρ1 ) → πm2 (Gρ2 ) such that φ ◦ πm1 ◦ ρ1 = πm2 ◦ ρ2 . We also establish a spectral rigidity result. If ρ : Γ → SLm (R) is projective Anosov and γ ∈ Γ, then let L(γ)(ρ) denote the eigenvalue of maximal absolute value of ρ(γ), so Λ(γ)(ρ) = |L(γ)(ρ)|. Theorem 1.2. [Spectral rigidity] Let Γ be a word hyperbolic group and let ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) be projective Anosov representations with limit maps ξ1 and ξ2 such that L(γ)(ρ1 ) = L(γ)(ρ2 ) for every γ in Γ. Then there exists g ∈ GLm (R) such that gξ1 = ξ2 . Moreover, if ρ1 is irreducible, then gρ1 g −1 = ρ2 . We now introduce the deformation spaces which occur in our work. In section 7, we will see that each of these deformation spaces is a real analytic manifold. Let us introduce some terminology. If G is a reductive subgroup of SLm (R), we say that an element of G is generic if its centralizer is a maximal torus in G. For example, an element of SLm (R) is generic if and only if it is diagonalizable over C with distinct eigenvalues. We say that a representation ρ : Γ → G is G-generic if the Zariski closure of ρ(Γ) contains a generic element of G. Finally, we say that ρ ∈ Hom(Γ, G) is regular if it is a smooth point of the algebraic variety Hom(Γ, G). • Let C(Γ, m) denote the space of (conjugacy classes of) regular, irreducible, projective Anosov representations of Γ into SLm (R). • Let Cg (Γ, G) denote the space of (conjugacy classes of) G-generic, regular, irreducible, projective Anosov representations. We show that the entropy and the renormalised intersection vary analytically over our deformation spaces. Moreover, we obtain analyticity on analytic families of projective Anosov homomorphisms. An analytic family of projective Anosov homomorphisms is a continuous map β : M → Hom(Γ, SLm (R)) such that M is an analytic manifold, βm = β(m) is projective Anosov for all m ∈ M , and m → βm (γ) is an analytic map of M into SLm (R) for all γ ∈ Γ. Theorem 1.3. [Analyticity] If Γ is a word hyperbolic group, then the entropy h and the renormalised intersection J are well-defined positive, Out(Γ)-invariant analytic functions on the spaces C(Γ, m) and C(Γ, m) × C(Γ, m) respectively. More generally, they are analytic functions on any analytic family of projective Anosov homomorphisms. Moreover, let γ : (−1, 1) → C(Γ, m) be any analytic path with values in the deformation space, let Jγ (t) = J(γ(0), γ(t)) then d dt Jγ = 0 and t=0 d2 dt2 Jγ > 0. (1) t=0 Theorem 1.3 allows us to define a non-negative analytic 2-tensor on Cg (Γ, G). The pressure form is defined to be the Hessian of the restriction of the renormalised intersection J. Our main result is the following. 4 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Theorem 1.4. [Pressure metric] Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). The pressure form is an analytic Out(Γ)-invariant Riemannian metric on Cg (Γ, G). If S is a closed, connected, orientable, hyperbolic surface, Hitchin [30] exhibited a component Hm (S) of Hom(π1 (S), PSLm (R))/PGLm (R) now called the Hitchin component, which is an analytic manifold diffeomorphic to a ball. Each Hitchin component contains a Fuchsian locus which consists of representations obtained by composing Fuchsian representations of π1 (S) into PSL2 (R) with the irreducible representation τm : PSL2 (R) → PSLm (R). The representations in a Hitchin component are called Hitchin representations and can be lifted to representations into SLm (R). Labourie [41] showed that lifts of Hitchin representations are projective Anosov, irreducible and SLm (R)-generic. In particular, if ρi : π1 (S) → PSLm (R) are Hitchin representations, then one can define h(ρi ), I(ρ1 , ρ2 ) and J(ρ1 , ρ2 ) just as for projective Anosov representations. Guichard has recently announced a classification of the possible Zariski closures of Hitchin representations, see Section 11.3 for a statement. As a corollary of Theorem 1.1 and Guichard’s work we obtain a stronger rigidity result for Hitchin representations. Corollary 1.5. [Hitchin rigidity] Let S be a closed, orientable surface and let ρ1 ∈ Hm1 (S) and ρ2 ∈ Hm2 (S) be two Hitchin representations such that J(ρ1 , ρ2 ) = 1. Then, either • m1 = m2 and ρ1 = ρ2 in Hm1 (S), or • there exists an element ρ of the Teichmüller space T (S) so that ρ1 = τm1 (ρ) and ρ2 = τm2 (ρ). In section 11.4 we use work of Benoist [5, 6] to obtain a similar rigidity result for representations which arise as monodromies of strictly convex projective structures on compact manifolds with word hyperbolic fundamental group. We will call such representations Benoist representations. Each Hitchin component lifts to a component of Cg (π1 (S), SLm (R)). As a corollary of Theorem 1.4 and work of Wolpert [68] we obtain: Corollary 1.6. [Hitchin component] The pressure form on the Hitchin component is an analytic Riemannian metric which is invariant under the mapping class group and restricts to the Weil-Petersson metric on the Fuchsian locus. The same naturally holds for Hitchin components of representations into PSp(n, R), S0(n, n + 1) and G2,0 , since they embed in Hitchin components of representations into PSL(n, R). Labourie and Wentworth [46] have announced an explicit formula (in term of the Hitchin parametrisation) for the pressure metric along the Fuchsian locus. Li [48] has used the work of Loftin [50] and Labourie [43] to exhibit a metric on H3 (S), which she calls the Loftin metric, which is invariant with respect to the mapping class group, restricts to a multiple of the Weil-Petersson metric on the Fuchsian locus and such that the Fuchsian locus is totally geodesic. She further shows that a metric on H3 (S) constructed earlier by Darvishzadeh and Goldman [24] restricts to a multiple of the Weil-Petersson metric on the Fuchsian locus. Kim and Zhang [39] introduced a mapping class group invariant Kähler metric on THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 5 the Hitchin component H3 (S) for SL(3, R), which Labourie [45] generalized to the Hitchin components associated to all real split simple Lie groups of rank 2. If Γ is a word hyperbolic group, we let Cc (Γ, PSL2 (C)) denote the space of (conjugacy classes of) convex cocompact representations of Γ into PSL2 (C). In Section 2.3 we produce a representation, called the Plücker representation, α : PSL2 (C) → SLm (R) (for some m), so that if ρ ∈ Cc (Γ, PSL2 (C)), then α ◦ ρ is projective Anosov. The deformation space Cc (Γ, PSL2 (C)) is an analytic manifold and we may define a renormalised intersection J and thus a pressure form on Cc (Γ, PSL2 (C)). The following corollary is a direct generalization of Bridgeman’s pressure metric on quasifuchsian space (see [9]). Corollary 1.7. [Kleinian groups] Let Γ be a torsion-free word hyperbolic group. The pressure form gives rise to a Out(Γ)-invariant metric on the analytic manifold Cc (Γ, PSL2 (C)) which is Riemannian on the open subset consisting of Zariski dense representations. Moreover, (1) If Γ does not have a finite index subgroup which is either a free group or a surface group, then the metric is Riemannian at all points in Cc (Γ, PSL2 (C)). (2) If Γ is the fundamental group of a closed, connected, orientable surface, then the metric is Riemannian off of the Fuchsian locus in Cc (Γ, PSL2 (C)) and restricts to a multiple of the Weil-Petersson metric on the Fuchsian locus. If G is a rank one semi-simple Lie group, then work of Patterson [56], Sullivan [66], Yue [69] and Corlette-Iozzi [20] shows that the entropy of a convex cocompact representation ρ : Γ → G agrees with the Hausdorff dimension of the limit set of ρ(Γ). We may then apply Theorem 1.3 and the Plücker representation to conclude that that the Hausdorff dimension of the limit set varies analytically over analytic families of convex cocompact representations into rank one semi-simple Lie groups. Corollary 1.8. [Analyticity of Hausdorff Dimension] If Γ is a finitely generated group and G is a rank one semi-simple Lie group, then the Hausdorff dimension of the limit set varies analytically on any analytic family of convex cocompact representations of Γ into G. In particular, the Hausdorff dimension varies analytically over Cc (Γ, PSL2 (C)) One may further generalize our construction into the setting of virtually Zariski dense Anosov representations into an arbitrary semi-simple Lie group G. A representation ρ : Γ → G is virtually Zariski dense if the Zariski closure of ρ(Γ) is a finite index subgroup of G. If Γ is a word hyperbolic group, G is a semi-simple Lie group with finite center and P is a non-degenerate parabolic subgroup, then we let Z(Γ; G, P) denote the space of (conjugacy classes of) regular virtually Zariski dense (G, P)-Anosov representations of Γ into G. The space Z(Γ; G, P) is an analytic orbifold, see Proposition 7.3, and we can again use a Plücker representation to define a pressure metric on Z(Γ; G, P). If G is connected, then Z(Γ; G, P) is an analytic manifold. Corollary 1.9. [Anosov representations] Suppose that Γ is a word hyperbolic group, G is a semi-simple Lie group with finite center and P is a non-degenerate parabolic subgroup of G. Then there exists an Out(Γ)-invariant analytic Riemannian metric on the orbifold Z(Γ; G, P). 6 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO A key tool in our proof is the introduction of a flow Uρ Γ associated to a projective Anosov representation ρ. Let ρ : Γ → SLm (R) be a projective Anosov representation with limit maps ξ and θ. Let F be the total space of the principal R-bundle over RP(m) × RP(m)∗ whose fiber at the point (x, y) is the space of norms on the line ξ(x). There is a natural R-action on F which takes a norm u on x to the norm e−t u. Let Fρ be R-principal bundle over ∂∞ Γ(2) = ∂∞ Γ × ∂∞ Γ \ {(x, x) | x ∈ ∂∞ Γ}. which is the pull back of F by (ξ, θ). The R-action on F gives rise to a flow on Fρ . (An analogue of this flow was first introduced by Sambarino [63, 62] in the setting of projective Anosov irreducible representations of fundamental groups of closed negatively curved manifolds.) We then show that this flow is metric Anosov and is a Hölder reparameterization of the Gromov geodesic flow U0 Γ of Γ. Moreover, this flow encodes the spectral radii of elements of ρ(Γ), i.e. the period of the flow associated to (the conjugacy class of ) an element γ ∈ Γ is log Λ(γ)(ρ). (Metric Anosov flows are a natural generalization of Anosov flows in the setting of compact metric spaces and were studied by Pollicott [57].) Theorem 1.10. [geodesic flow] The action of Γ on Fρ is proper and cocompact. Moreover, the R action on Uρ Γ = Fρ /Γ is a topologically transitive metric Anosov flow which is Hölder orbit equivalent to the geodesic flow U0 Γ. Theorem 1.10 allows us to make use of the thermodynamic formalism. We show that if fρ is the Hölder function regulating the change of speed of Uρ Γ and U0 Γ, then Φρ = −h(ρ)fρ is a pressure zero function on U0 Γ. Therefore, we get a mapping T : C(Γ, m) → H(U0 Γ), called the thermodynamic mapping, from C(Γ, m) into the space H(U0 Γ) of Livšic cohomology classes of pressure zero Hölder functions on U0 Γ. Given any [ρ] ∈ C(Γ, m), there exists an open neighborhood U of [ρ] and a lift of T|U to an analytic map of U into the space P(U0 Γ) of pressure zero Hölder functions on U0 Γ. Our pressure form is obtained as a pullback of the pressure 2-tensor on P(U0 Γ) with respect to this lift. Remarks and references: Anosov representations were introduced by Labourie [41] in his study of Hitchin representations, and their theory was further developed by Guichard and Wienhard [26]. Benoist [5, 6, 7] studied holonomy maps of strictly convex projective structures on closed manifolds which he showed were irreducible representations with transverse projective limit maps, hence projective Anosov. Sambarino [62, 63, 64] introduced a flow, closely related to our flow, associated to a representation with transverse projective limit maps and used it to prove the continuity of the associated entropy on a Hitchin component. Pollicott and Sharp [58] applied the thermodynamic formalism and work of Dreyer [23] to show that a closely related entropy gives rise to an analytic function on any Hitchin component. Our metric generalizes Thurston’s Riemannian metric on Teichmüller space which he defined to be the Hessian of the length of a random geodesic. Wolpert [68] proved that Thurston’s Riemannian metric was a multiple of the more classical Weil-Petersson metric. Bonahon [11] gave an interpretation of Thurston’s metric in terms of the Hessian of an intersection function. Burger [16] previously studied the intersection number for convex cocompact subgroups of rank 1 simple Lie groups THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 7 and proved a strong version of Theorem 1.1 in this setting (see also Kim [37]). The study of geometric properties of surfaces using the thermodynamic formalism originated in Bowen [14]. Using a Bowen-Series coding and building on work of Bridgeman and Taylor [10], McMullen [53] gave a pressure metric formulation of the Weil–Petersson metric on Teichmüller space. Bridgeman [9] developed a pressure metric on quasifuchsian space which restricts to the Weil–Petersson metric on the Fuchsian locus. Our Theorem 1.4 is a natural generalization of Bridgeman’s work into the setting of projective Anosov representations, while Corollary 1.7 is a generalization into the setting of general deformation spaces of convex cocompact representations into PSL2 (C). Corollary 1.8 was established by Ruelle [60] for quasifuchsian representations, i.e. when Γ = π1 (S) and G = PSL2 (C), and by Anderson and Rocha [2] for function groups, i.e. when Γ is a free product of surface groups and free groups and G = PSL2 (C). Previous work of Tapie [67] implies that the Hausdorff dimension of the limit set is a C 1 function on C 1 -families of convex cocompact representations of Γ into a rank one Lie group G. Tapie’s work was inspired by work of Katok, Knieper, Pollicott and Weiss [35, 36] who established analytic variation of the entropy for analytically varying families of Anosov flows on closed Riemannian manifolds. Our Theorem 1.2 is related to the marked length spectrum rigidity theorem of Dal’BoKim [21]. Coornaert–Papadopoulos [19] showed that if Γ is word hyperbolic, then there is a symbolic coding of its geodesic flow U0 Γ. However, this coding is not necessarily one-to-one on a large enough set to apply the thermodynamic formalism. Therefore, word hyperbolic groups admitting projective Anosov representations represent an interesting class of groups from the point of view of symbolic dynamics. Acknowledgements: We thank Bill Goldman, Alex Lubotzky, François Ledrappier, Olivier Guichard, Frédéric Paulin, Jean-François Quint, Hans-Henrik Rugh, Ralf Spatzier, Matthew Stover and Amie Wilkinson for helpful discussions. We thank the referee for many useful comments which improved the exposition. This research was begun while the authors were participating in the program on Geometry and Analysis of Surface Group Representations held at the Institut Henri Poincaré in Winter 2012. Contents 1. Introduction 2. Anosov representations 2.1. Projective Anosov representations 2.2. Anosov representations 2.3. Plücker representations 2.4. Irreducible representations 2.5. G-generic representations 3. Thermodynamic formalism 3.1. Hölder flows on compact spaces 3.2. Metric Anosov flows 3.3. Entropy and pressure for Anosov flows 3.4. Intersection and renormalised intersection 3.5. Variation of the pressure and the pressure form 1 8 9 12 13 14 16 16 17 18 20 21 22 8 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO 3.6. Analyticity of entropy, pressure and intersection 4. The geodesic flow of a projective Anosov representation 5. The geodesic flow is a metric Anosov flow 5.1. The geodesic flow as a metric space 5.2. Stable and unstable leaves 5.3. The leaf lift and the distance 5.4. The geodesic flow is Anosov 6. Analytic variation of the dynamics 6.1. Transverse regularity 6.2. Analytic variation of the limit maps 6.3. Analytic variation of the reparameterization 7. Deformation spaces of projective Anosov representations 7.1. Irreducible projective Anosov representations 7.2. Virtually Zariski dense representations 7.3. Kleinian groups 7.4. Hitchin components 8. Thermodynamic formalism on the deformation space of projective Anosov representations 8.1. Analyticity of entropy and intersection 8.2. The thermodynamic mapping and the pressure form 9. Degenerate vectors for the pressure metric 9.1. Log-type functions 9.2. Trace functions 9.3. Technical lemmas 9.4. Degenerate vectors have log-type zero 10. Variation of length and cohomology classes 10.1. Invariance of the cross-ratio 10.2. An useful immersion 10.3. Vectors with log type zero 11. Rigidity results 11.1. Spectral rigidity 11.2. Renormalized intersection rigidity 11.3. Rigidity for Hitchin representations 11.4. Benoist representations 12. Proofs of main results 13. Appendix References 24 25 28 28 30 31 32 34 35 40 42 44 44 46 48 48 49 49 50 51 51 52 54 58 59 60 61 62 63 64 66 67 69 70 72 72 2. Anosov representations In this section, we recall the theory of Anosov representations. We begin by defining projective Anosov representations and developing their basic properties. In section 2.3, we will see that any Anosov representation can be transformed, via post-composition with a Plücker representation, into a projective Anosov representation, while in section 2.4 we will study properties of irreducible projective Anosov representations. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 9 2.1. Projective Anosov representations. A representation ρ : Γ → SLm (R) is projective Anosov if it has transverse projective limit maps and the associated flat bundle over its Gromov geodesic flow has a contraction property we will define carefully below. Definition 2.1. Let Γ be a word hyperbolic group and ρ be a representation of Γ in SLm (R). We say ρ has transverse projective limit maps if there exist ρ-equivariant continuous maps ξ : ∂∞ Γ → RP(m) and θ : ∂∞ Γ → RP(m)∗ such that if x 6= y, then ξ(x) ⊕ θ(y) = Rm . Conventions: Denote by RP(m) the projective space of Rm . We will often identify RP(m)∗ with the Grassmannian Grm−1 (Rm ) of (m − 1)-dimensional subspaces of Rm , via ϕ 7→ ker ϕ. The action of SLm (R) on RP(m)∗ consistent wth this identification is g · ϕ = ϕ ◦ g −1 . We will also assume throughout this paper that our word hyperbolic group does not have a finite index cyclic subgroup. Since all the word hyperbolic groups we study are linear, Selberg’s Lemma implies that they contain finite index torsion-free subgroups. Gromov [25] defined a geodesic flow U0 Γ for a word hyperbolic group – that we shall call the Gromov geodesic flow – (see Champetier [17] and Mineyev [54] for details). He defines a proper cocompact action of Γ on ∂∞ Γ(2) × R which commutes with the action of R by translation on the final factor. The action of Γ restricted to ∂∞ Γ(2) is the diagonal action arising from the standard action of Γ on ∂∞ Γ. There is a metric on ∂∞ Γ(2) × R, well-defined up to Hölder equivalence, so that Γ acts by isometries, every orbit of the R action gives a quasi-isometric embedding and the geodesic flow acts by Lipschitz homeomorphisms. The flow on (2) g U ×R 0 Γ = ∂∞ Γ descends to a flow on the quotient U0 Γ = ∂∞ Γ(2) × R/Γ. In the case that M is a closed negatively curved manifold and Γ = π1 (M ), U0 Γ may be identified with T1 M in such a way that the flow on U0 Γ is identified with the geodesic flow on T1 M . Since the action of Γ on ∂∞ Γ2 is topologically transitive, the Gromov geodesic flow is topologically transitive. If ρ : Γ → SLm (R) is a representation, we let Eρ be the associated flat bundle over the geodesic flow of the word hyperbolic group U0 Γ. Recall that m g Eρ = U 0 Γ × R /Γ where the action of γ ∈ Γ on Rm is given by ρ(γ). If ρ has transverse projective limit maps ξ and θ, there is an induced splitting of Eρ as Eρ = Ξ ⊕ Θ where Ξ and Θ are sub-bundles, parallel along the geodesic flow, of rank 1 and m − 1 respectively. Explicitly, if we lift Ξ and Θ to sub-bundles Ξ̃ and Θ̃ of the m g g bundle U over U 0Γ × R 0 Γ, then the fiber of Ξ̃ above (x, y, t) is simply ξ(x) and the fiber of Θ̃ is θ(y). 10 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO m g g The R-action on U 0 Γ extends to a flow {ψ̃t }t∈R on U0 Γ × R (which acts trivially m on the R factor). The flow {ψ̃t }t∈R descends to a flow {ψt }t∈R on Eρ which is a lift of the geodesic flow on U0 Γ. In particular, the flow respects the splitting Eρ = Ξ ⊕ Θ. In general, we say that a vector bundle E over a compact topological space whose total space is equipped with a flow {φt }t∈R of bundle automorphisms is contracted by the flow if for any metric k.k on E, there exists t0 > 0 such that if v ∈ E, then 1 kvk. 2 Observe that if bundle is contracted by a flow, its dual is contracted by the inverse flow. Moreover, if the flow is contracting, it is also uniformly contracting, i.e. given any metric, there exists positive constants A and c such that kφt0 (v)k 6 kφt (v)k 6 Ae−ct kvk for any v ∈ E. Definition 2.2. A representation ρ : Γ → SLm (R) with transverse projective limit maps is projective Anosov if the bundle Hom(Θ, Ξ) is contracted by the flow {ψt }t∈R . In the sequel, we will use the notation Θ∗ = Hom(Θ, R). The following alternative description will be useful. Proposition 2.3. A representation ρ : Γ → SLm (R) with transverse projective limit maps ξ and θ is projective Anosov if and only if there exists t0 > 0 such that for all Z ∈ U0 Γ, v ∈ ΞZ \ {0} and w ∈ ΘZ \ {0}, 1 kvk kψt0 (v)k ≤ . kψt0 (w)k 2 kwk (2) Proof. Given a projective Anosov representation ρ : Γ → SLm (R) and a metric k.k on Eρ , let t0 > 0 be chosen so that 1 kηk. 2 for all η ∈ Ξ ⊗ Θ∗ . If Z ∈ U0 Γ, v ∈ ΞZ \ {0} and w ∈ ΘZ \ {0}, then there exists η ∈ Hom(ΘZ , ΞZ ) = (Ξ ⊗ Θ∗ )Z such that η(w) = v and kηk = kvk/kwk. Then, kψt0 (η)k 6 1 kvk kψt0 (v)k ≤ kψt0 (η)k 6 kηk = . kψt0 (w)k 2 kwk The converse is immediate.  Furthermore, projective Anosov representations are contracting on Ξ. Lemma 2.4. If ρ : Γ → SLm (R) is projective Anosov, then {ψt }t∈R is contracting on Ξ. Proof. Since the bundle Ξ ⊗ Θ∗ is contracted, so is Ω = det(Ξ ⊗ Θ∗ ) = Ξ⊗(m−1) ⊗ det(Θ∗ ). One may define an isomorphism from Ξ to det(Θ)∗ by taking u to the map α → Vol(u ∧ α). Since det(Θ)∗ is isomorphic to det(Θ∗ ), it follows that Ω is isomorphic to Ξ⊗m . Thus Ξ is contracted.  THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 11 It follows from standard techniques in hyperbolic dynamics that our limit maps are Hölder. We will give a proof of a more general statement in Section 6 (see [41, Proposition 3.2] for a proof in a special case). Lemma 2.5. Let ρ be a projective Anosov representation, then the limit maps ξ and θ are Hölder. If γ is an infinite order element of Γ, then there is a periodic orbit of U0 Γ associated to γ. If γ + is the attracting fixed point of γ on ∂∞ Γ and γ − is its other fixed point, then this periodic orbit is the image of (γ + , γ − ) × R. Inequality (2) and Lemma 2.4 applied to the periodic orbit of U0 Γ associated to γ imply that ρ(γ) is proximal and that ξ(γ + ) is the eigenspace associated to the largest modulus eigenvalue of ρ(γ). Similarly, ξ(γ − ) is the repelling hyperplane of ρ(γ). It follows that the limit maps ξ and θ are uniquely determined by ρ (see also [26, Lemmas 3.1 and 3.3]). Let L(γ)(ρ) denote the eigenvalue of ρ(γ) of maximal absolute value and let Λ(γ)(ρ) denote the spectral radius of ρ(γ), so Λ(γ)(ρ) = |L(γ)(ρ)|. If S is a fixed generating set for Γ and γ ∈ Γ, then we let l(γ) denote the translation length of the action of γ on the Cayley graph of Γ with respect to S; more explicitly, l(γ) is the minimal word length of any element conjugate to γ. Since the contraction is uniform and the length of the periodic orbit of U0 Γ associated to γ is comparable to l(γ), we obtain the following uniform estimates: Proposition 2.6. If ρ : Γ → SLm (R) is a projective Anosov representation, then there exists δ ∈ (0, 1) such that if γ ∈ Γ has infinite order, then L(γ)(ρ) and (L(γ −1 )(ρ))−1 are both eigenvalues of ρ(γ) of multiplicity one and ρ(γ) = L(γ)(ρ)pγ + mγ + 1 qγ L(γ −1 )(ρ) where • pγ is the projection on ξ(γ + ) parallel to θ(γ − ), • qγ = pγ −1 , • mγ = A ◦ (1 − qγ − pγ ) and A is an endomorphism of θ(γ − ) ∩ θ(γ+) whose spectral radius is less than δ ℓ(γ) Λ(γ)(ρ). Moreover, we see that ρ is well-displacing in the following sense: Proposition 2.7. [Displacing property] If ρ : Γ → SLm (R) is a projective Anosov representation, then there exists constants K > 0 and C > 0, and a neighborhood U of ρ0 in Hom(Γ, SLm (R)) such that that for every γ ∈ Γ and ρ ∈ U we have 1 ℓ(γ) − C 6 log(Λ(γ)(ρ)) 6 Kℓ(γ) + C, (3) K Proposition 2.7 immmediately implies: Proposition 2.8. For every real number T , the set RT (ρ) = {[γ] | log(Λ(γ)(ρ)) 6 T } is finite. 12 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Remark: Proposition 2.6 is a generalization of results of Labourie [41, Proposition 3.4], Sambarino [63, Lemma 5.1] and Guichard-Wienhard [26, Lemma 3.1]. Proposition 2.7 is a generalization of a result of Labourie [44, Theorem 1.0.1] and a special case of a result of Guichard-Wienhard [26, Theorem 5.14]. See [22] for a discussion of well-displacing representations and their relationship with quasi-isometric embeddings. 2.2. Anosov representations. We now recall the general definition of an Anosov representation and note that projective Anosov representations are examples of Anosov representations. We first recall some notation and definitions. Let G be a semi-simple Lie group with finite center and Lie algebra g. Let K be a maximal compact subgroup of G and let τ be the Cartan involution on g whose fixed point set is the Lie algebra of K. Let a = aG be a maximal abelian subspace contained in {v ∈ g : τ v = −v}. For a ∈ a, let M be the connected component of the centralizer of exp a which contains the identity, and let m denote its Lie algebra. Let Eλ be the eigenspace of the action of a on g with eigenvalue λ and consider M n+ = Eλ , λ>0 n− = M Eλ , λ<0 so that g = m ⊕ n+ ⊕ n− . + − (4) ± Then n and n are Lie algebras normalized by M. Let P the connected Lie subgroups of G whose Lie algebras are p± = m ⊕ n±. Then P+ and P− are opposite parabolic subgroups. We will say that P+ is non-degenerate if p+ does not contain a simple factor of g. We may identify a point ([X], [Y ]) in G/P+ ×G/P− with the pair (Ad(X)P+ , Ad(Y )P− ) of parabolic subgroups. The pair (Ad(X)P+ , Ad(Y )P− ) is transverse if their intersection Ad(X)P+ ∩ Ad(Y )P− is conjugate to M. We now suppose that ρ : Γ → G is a representation of word hyperbolic group Γ and ξ + : ∂∞ Γ → G/P+ and ξ − : Γ → G/P− are continuous ρ-equivariant maps. We say that ξ + and ξ − are transverse if given any two distinct points x, y ∈ ∂∞ Γ, ξ + (x) and ξ − (y) are transverse. The G-invariant splitting described by Equation g e + and N e − be the bundles over U (4) then gives rise to bundles over U0 Γ. Let N 0Γ ρ ρ whose fibers over the point (x, y, t) are Ad(ξ − (y))n+ and Ad(ξ + (x))n− . e + and N e − , where the action on the fiber is There is a natural action of Γ on N ρ ρ given by ρ(Γ), and we denote the quotient bundles over U0 Γ by Nρ+ and Nρ− . We may lift the geodesic flow to a flow on the bundles Nρ+ and Nρ− which acts trivially on the fibers. Definition 2.9. Suppose that G is a semi-simple Lie group with finite center, P+ is a parabolic subgroup of G and Γ is a word hyperbolic group. A representation ρ : Γ → G is (G, P+ )-Anosov if there exist transverse ρ-equivariant maps ξ + : ∂∞ Γ → G/P+ and ξ − : ∂∞ Γ → G/P− THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 13 so that the geodesic flow is contracting on the associated bundle Nρ+ and the inverse flow is contracting on the bundle Nρ− . We now recall some basic properties of Anosov representations which were established by Labourie, [41, Proposition 3.4] and [44, Theorem 6.1.3], and GuichardWienhard [26, Theorem 5.3 and Lemma 3.1]. We recall that an element g ∈ G is proximal relative to P+ if g has fixed points x+ ∈ G/P+ and x− ∈ G/P− so that x+ is transverse to x− and if x ∈ G/P+ is transverse to x− then limn→∞ g n (x) = x+ . Theorem 2.10. Let G be a semi-simple Lie group, P+ a parabolic subgroup, Γ a word hyperbolic group and ρ : Γ → G a (G, P+ )-Anosov representation. (1) ρ has finite kernel, so Γ is virtually torsion-free. (2) ρ is well-displacing, so ρ(Γ) is discrete. (3) If γ ∈ Γ has infinite order, then ρ(γ) is proximal relative to P+ In this language, projective Anosov representations are exactly the same as (SLm (R), P+ )-Anosov representations where P+ is the stabilizer of a line in Rm . Proposition 2.11. Let P+ be the stabilizer of a line in Rm . A representation ρ : Γ → SLm (R) is projective Anosov if and only if it is (SLm (R), P+ )-Anosov. Moreover, the limit maps ξ and θ in the definition of projective Anosov representation agree with the limit maps ξ + and ξ − in the definition of a (SLm (R), P+ )-Anosov representation. Proof. If ρ is projective Anosov with limit maps ξ and θ, one may identify SLm (R)/P + with RP(m) and SLm (R)/P − with RP(m)∗ so that, after letting ξ + = ξ and ξ − = θ, Nρ+ is identified with Hom(Θ, Ξ) and Nρ− is identified with Hom(Ξ, Θ). The same identification holds if ρ is (SLm (R), P+ )-Anosov with limit maps ξ + and ξ − .  2.3. Plücker representations. Guichard and Wienhard [26] showed how to obtain a projective Anosov representation from any Anosov representation by postcomposing with a Plücker representation. We first recall the following general result. Theorem 2.12. [Guichard-Wienhard [26, Prop. 4.3]] Let φ : G → SL(V ) be a finite dimensional irreducible representation. Let x ∈ P(V ) and assume that P = {g ∈ G : φ(g)(x) = x} is a parabolic subgroup of G with opposite parabolic Q. If Γ is a word hyperbolic group, then a representation ρ : Γ → G is (G, P)-Anosov if and only if φ ◦ ρ is projective Anosov. Furthermore, if ρ is (G, P)-Anosov with limit maps ξ + and ξ − , then the limit maps of φ ◦ ρ are given by ξ = β ◦ ξ + and θ = β ∗ ◦ ξ − where β : G/P → P(V ) and β ∗ : G/Q → P(V ∗ ) are the maps induced by φ. The following corollary is observed by Guichard-Wienhard [26, Remark 4.12]. We provide a proof here for the reader’s convenience. The representation given in the proof will be called the Plücker representation of G with respect to P. Corollary 2.13. [Guichard-Wienhard] For any parabolic subgroup P of a semisimple Lie group G with finite center, there exists a finite dimensional irreducible representation α : G → SL(V ) such that if Γ is a word hyperbolic group and ρ : Γ → G is a (G, P)-Anosov representation, then α ◦ ρ is projective Anosov. 14 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Moreover, if P is non-degenerate, then ker(α) = Z(G) and α is an immersion. Proof. In view of Theorem 2.12 it suffices to find a finite dimensional irreducible representation α : G → SL(V ) such that α(P) is the stabilizer (in α(G)) of a line in V. Let Λk W denote the k-th exterior power of the vector space W. Let n = dim n+ = dim n− and consider α : G → SL(Λn g) given by α(g) = Λn Ad(g). One may readily check that the restriction of α to V = hα(G) · Λn n+ i works. If P is non-degenerate, then ker(α|V ) is a normal subgroup of G which is contained in P, so ker(α|V ) is contained in Z(G) (see [59]). Since Z(G) is in the kernel of the adjoint representation, we see that ker(α|V ) = Z(G). Since α|V is algebraic and Z(G) is finite, it follows that α|V is an immersion.  If G has rank one, then it contains a unique conjugacy class of parabolic subgroups. A representation ρ : Γ → G is Anosov if and only if it is convex cocompact (see [26, Theorem 5.15]). We then get the following. We recall that the topological entropy of a convex cocompact representation ρ : Γ → G of a word hyperbolic group into a rank one semi-simple Lie group is given by 1 log (♯{[γ] |d(ρ(γ)) 6 T }) , h(ρ) = lim T →∞ T where d(ρ(γ)) denotes the translation length of ρ(γ). We obtain the following immediate corollary. Corollary 2.14. Let G be a rank one semi-simple Lie group, let Γ be a word hyperbolic group and let α : G → SL(V ) be the Plücker representation. There exists K > 0, such that if ρ : Γ → G is convex cocompact, then α ◦ ρ is projective Anosov and h(ρ) . h(α ◦ ρ) = K Proof. Let λG : G → aG be the Jordan projection of G. Since aG is one dimensional, we can identify it with R by setting λG (g) = d(g). Denote by χα ∈ aG the highest (restricted) weight of the representation α (see, for example, Humphreys [32]). By definition, one has Λ(α(g)) = χα (d(g)), for every g ∈ G. Hence, since aG is one dimensional, one has Λ(α(ρ(γ))) = Kd(ρ(γ)) (5) for every γ ∈ Γ. It follows immediately that h(α ◦ ρ) = h(ρ) . K  2.4. Irreducible representations. Guichard and Wienhard [26, Proposition 4.10] proved that irreducible representations with transverse projective limit maps are projective Anosov (see also [41] for hyperconvex representations). Proposition 2.15. [Guichard–Wienhard] If Γ is a word hyperbolic group, then every irreducible representation ρ : Γ → SLm (R) with transverse projective limit maps is projective Anosov. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 15 It will be useful to note that if ρ : Γ → SLm (R) is projective Anosov and irreducible, then ξ(∂∞ Γ) contains a projective frame for RP(m). We recall that a collection of m+1 elements in RP(m) is a projective frame if every subset containing m elements spans Rm . We first prove the following lemma. Lemma 2.16. Let ρ : Γ → SLm (R) be a representation with a continuous ρ-equivariant map ξ : ∂∞ Γ → RP(m), then the preimage ξ −1 (V ) of a vector subspace V ⊂ Rm is either ∂∞ Γ or has empty interior on ∂∞ Γ. Proof. Choose {x1 , . . . , xp } ⊂ ∂∞ Γ so that {ξ(x1 ), . . . , ξ(xp )} spans the vector subspace hξ(∂∞ Γ)i spanned by ξ(∂∞ Γ). Suppose that ξ −1 (V ) = {x ∈ ∂∞ Γ : ξ(x) ∈ V } has non-empty interior in ∂∞ Γ. Choose γ ∈ Γ so that γ − ∈ / {x1 , . . . , xp } and γ + belongs to the interior of ξ −1 (V ). n + Since γ (xi ) → γ for every i ∈ {1, . . . , p}, if we choose n large enough, then γ n (xi ) is contained in the interior of ξ −1 (V ), so ξ(γ n xi ) ∈ V. Since {ξ(γ n (x1 )), . . . , ξ(γ n (xp ))} still spans hξ(∂∞ Γ)i, we see that hξ(∂∞ Γ)i ⊂ V , in which case ξ −1 (V ) = ∂∞ Γ.  The following generalization of the fact that every irreducible projective Anosov representation admits a projective frame will be useful in Section 11. Lemma 2.17. Let ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) be representations with continuous equivariant limit maps ξ1 and ξ2 such that dim hξ1 (∂∞ Γ)i = dim hξ2 (∂∞ Γ)i = p. Then there exist p + 1 distinct points {x0 , . . . , xp } in ∂∞ Γ such that {ξ1 (x0 ), . . . , ξ1 (xp )} and {ξ2 (x0 ), . . . , ξ2 (xp )} are projective frames of hξ1 (∂∞ Γ)i and hξ2 (∂∞ Γ)i respectively. Proof. We first proceed by iteration to produce {x1 , . . . , xp } so that {ξ1 (x1 ), . . . , ξ1 (xp )} and {ξ2 (x1 ), . . . , ξ2 (xp )} generate V = hξ1 (∂∞ Γ)i and W = hξ2 (∂∞ Γ)i . Assume we have found {x1 , . . . , xk } so that {ξ1 (x1 ), . . . , ξ1 (xk )} and {ξ2 (x1 ), . . . , ξ2 (xk )} are both linearly independent. Define Vk = h{ξ1 (x1 ), . . . , ξ1 (xk )}i and Wk = h{ξ2 (x1 ), . . . , ξ2 (xk )}i . By the previous lemma, if k < p, then ξ1−1 (Vk ) and ξ2−1 (Wk ) have empty interior, so their complements must intersect. Pick xk+1 ∈ ξ1−1 (Vk )c ∩ ξ2−1 (Wk )c . This process is complete when k = p. It remains to find x0 . For each i = 1, . . . , p, let Ui1 = h{ξ1 (x1 ), . . . , ξ1 (xp )} \ {ξ1 (xi )}i and Ui2 = h{ξ2 (x1 ), . . . , ξ2 (xp )} \ {ξ2 (xi )}i . Then, choose x0 ∈ \ ξ1−1 (Ui1 )c ∩ ξ2−1 (Ui2 )c . i One easily sees that {x0 , . . . , xp } has the claimed properties.  If ρ : Γ → SLm (R) is projective Anosov and irreducible, then hξ(∂∞ Γ)i = Rm (since hξ(∂∞ Γ)i is ρ(Γ)-invariant), so Lemma 2.17 immediately gives: 16 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Lemma 2.18. If ρ : Γ → SLm (R) is an irreducible projective Anosov representation with limit maps ξ and θ, then then there exist {x0 , . . . , xm } ⊂ ∂∞ Γ so that {ξ(x0 ), . . . , ξ(xm )} is a projective frame for RP(m). We will also need the following lemma which was explained to us by J.-F. Quint. Lemma 2.19. [Quint] If ∆ is an irreducible subgroup of SLm (R) that contains a proximal element, then the Zariski closure G of ∆ is a semi-simple Lie group without compact factors whose center Z(G) ⊂ {±I}. Proof. Since G acts irreducibly on Rm , it is a reductive group. Moreover, since G contains a proximal matrix, one easily sees that attracting lines of proximal matrices in G span Rm , and that each attracting line of a proximal matrix in G is invariant under Z(G). Therefore, Z(G) ⊂ {±I}, so G is a semi-simple Lie group. Let K be the maximal normal connected compact subgroup of G, and let H be the product of the non-compact Zariski connected, simple factors of G. Then H and K commute and HK has finite index in G. Consider now a proximal element g ∈ G. Replacing g by a large enough power, we can assume that g = hk for some h ∈ H and k ∈ K. Since all eigenvalues of k have modulus 1 and k and h commute, we conclude that h is proximal. So we can assume that g ∈ H. Since g and K commute, the attracting line of g is fixed by K, and, since K is connected, each vector of this attracting line is fixed by K. Let W be the vector space of K-fixed vectors on Rm , then W is G-invariant , since K is normal in G, and nonzero. Since G is irreducible, W = Rm and so K = {I}.  Proposition 2.6 and Lemma 2.19 together have the following immediate consequence. Corollary 2.20. Let ρ : Γ → SLm (R) be an irreducible projective Anosov representation, then the Zariski closure Gρ of ρ(Γ) is a semi-simple Lie group without compact factors such that Z(Gρ ) ⊂ {±I}. 2.5. G-generic representations. Let G be a reductive subgroup of SLm (R). We recall that an element in G is generic if its centralizer is a maximal torus in G. We say that a representation ρ : Γ → SLm (R) of Γ is G-generic if ρ(Γ) ⊂ G and the Z Zariski closure ρ(Γ) of ρ(Γ) contains a G-generic element. We will need the following observation. Lemma 2.21. If G is a reductive subgroup of SLm (R) and ρ : Γ → G is a G-generic representation, then there exists γ ∈ Γ such that ρ(γ) is a generic element of G. Proof. We first note that the set of non-generic elements of G is Zariski closed in G, so the set of generic elements is Zariski open in G. Therefore, if the Zariski closure of ρ(Γ) contains generic elements of G, then ρ(Γ) must itself contain generic elements of G.  3. Thermodynamic formalism In this section, we recall facts from the thermodynamic formalism, as developed by developed by Bowen [12, 13], Parry–Pollicott [55], Ruelle [61] and others, which we will need in our work. In section 3.5, we will describe a variation of a construction of McMullen [53], which produces a pressure form on the space of pressure zero functions on a flow space. Our pressure metric will be a pull-back of this form. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 17 3.1. Hölder flows on compact spaces. Let X be a compact metric space with a Hölder continuous flow φ = {φt }t∈R without fixed points. 3.1.1. Flows and parametrisations. Let f : X → R be a positive Hölder continuous function. Then, since X is compact, f has a positive minimum and for every x ∈ X, Rt the function κf : X × R → R, defined by κf (x, t) = 0 f (φs x)ds, is an increasing homeomorphism of R. We then have a map αf : X × R → R that verifies αf (x, κf (x, t)) = κf (x, αf (x, t)) = t, (6) for every (x, t) ∈ X × R. The reparametrization of φ by f, is the flow φf = {φft }t∈R on X, defined by f φt (x) = φαf (x,t) (x), for all t ∈ R and x ∈ X. 3.1.2. Livšic-cohomology classes. Two Hölder functions f, g : X → R are Livšiccohomologous if there exists V : X → R of class C1 in the flow’s direction such that ∂ V (φt (x)). f (x) − g(x) = ∂t t=0 Then one easily notices that: (1) If f and g are Livšic cohomologous then they have the same integral over any φ-invariant measure, and (2) If f and g are both positive and Livšic cohomologous, then the flows φf and φg are Hölder conjugate. 3.1.3. Periods and measures. Let O be the set of periodic orbits of φ. If a ∈ O then its period as a {φft } periodic orbit is Z p(a) f (φs (x))ds 0 where p(a) is the period of a for φ and x ∈ a. In particular, if δba is the probability measure invariant by the flow and supported by the orbit a, and if then hδa |f i = Z δba = δa , hδa |1i p(a) f (φs (x))ds and p(a) = hδa |1i . 0 In general, if µ is a φ-invariant measure on X and f : X → R is a Hölder function, we will use the notation Z f dµ. hµ|f i = X Let µ be a φ-invariant probability measure on X and let φf be the reparametrizac by tion of φ by f . We define f.µ c = f.µ 1 f.µ. hµ|f i 18 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO c induces a bijection between φ-invariant probability measures The map µ 7→ f.µ f and φ -invariant probability measures. If δbaf is the unique φf invariant probability da . In particular, we have measure supported by a, then δbaf = f.δ hδbaf |gi = hδa |f.gi hδa |f i (7) 3.1.4. Entropy, pressure and equilibrium states. If µ is a φ-invariant probability measure on X, then we denote by h(φ, µ), its metric entropy. The Abramov formula [1] relates the metric entropies of a flow and its reparameterization: c = R 1 h(φ, µ). h(φf , f.µ) f dµ (8) Let Mφ denote the set of φ-invariant probability measures. The pressure of a function f : X → R is defined by   Z f dm . (9) P(φ, f ) = sup h(φ, m) + m∈Mφ X In particular, htop (φ) = P(φ, 0) is the topological entropy of the flow φ. A measure m ∈ Mφ on X such that P(φ, f ) = h(φ, m) + Z f dm, X is called an equilibrium state of f . An equilibrium state for the function f ≡ 0 is called a measure of maximal entropy. Remark: The pressure P(φ, f ) only depends on the Livšic cohomology class of f. The following lemma from Sambarino [63] is a consequence of the definition and the Abramov formula. Lemma 3.1. (Sambarino [63, Lemma 2.4]) If φ is a Hölder continuous flow on a compact metric space X and f : X → R is a positive Hölder continuous function, then P (φ, −hf ) = 0 if and only if h = htop (φf ). d is a Moreover, if h = htop (φf ) and m is an equilibrium state of −hf , then f.m measure of maximal entropy for the reparameterized flow φf . 3.2. Metric Anosov flows. We shall assume from now on that the flow {φt }t∈R is a topologically transitive metric Anosov flow on X. We recall that a flow {φ}t∈R on a metric space X is topologically transitive if given any two open sets U and V in X, there exists t ∈ R so that φt (U ) ∩ V is non-empty. Let X be metric space. Let L be an equivalence relation on X. We denote by Lx the equivalence class of x and call it the leaf through x, so that we have a partition THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS of X into leaves X= G 19 Ly , y∈Y where Y is the collection of equivalence classes of L. Such a partition is a lamination if we can for every x in X, an open neighbourhood Ox of x, two topological spaces U and K, a homeomorphism νx = (νx1 , νx2 ) called a chart from Ox to U × K satisfying the following conditions • for all z, w ∈ Ox ∩ Oy , νx1 (w) = νx1 (z) ⇐⇒ νy1 (w) = νy1 (z), • we have that w L z if and only if there exists a sequence wi , i ∈ {1, . . . n} 1 1 (wi+1 ). (wi ) = νw with w1 = w and wn = z, such that wi+1 ∈ Owi and νw i i A plaque open set in the chart corresponding to ν is a set of the form ν (O × {z0 }) where x = ν(y0 , z0 ) and O is an open set in U containing y0 . The plaque topology on Lx is the topology generated by the plaque open sets. A plaque neighborhood of x is a neighborhood for the plaque topology on Lx . We say that two laminations L and L′ define a local product structure, if for any point x in X there exist plaque neighborhoods U and U ′ of x in L and L′ respectively, and a map ν : U × U ′ → X, which is an homeomorphism onto an open set of X, such that ν is both a chart for L and for L′ . Assume now we have a flow {φt }t∈R on X. If L is a lamination invariant by {φt }, we say that L is transverse to the flow, if for every x in X, there exists a plaque neighborhood U of x in Lx , a topological space K, ǫ > 0, and a chart ν : U × K × (−ǫ, ǫ) → X, such that φt (ν(u, k, s)) = ν(u, k, s + t). If L is tranverse to the flow, we define a new lamination, called the central lamination with respect to L, denoted by Lc , by letting x Lc y if and only if there exists s such that φs (x) L y. Finally, a {φt } invariant lamination L is contracted by the flow, if there exists t0 > 0 such that for all x ∈ X, there exists a chart νx : U × K → V of an open neighborhood V of x, such that if z = νx (u, k), and y = νx (v, k), then for all t > t0 d(φt (z), φt (y)) < 1 d(z, y). 2 Definition 3.2. [Metric Anosov flow] A flow {φt }t∈R on a compact metric space X is metric Anosov, if there exist two laminations, L+ and L− , transverse to the flow, such that (1) (L+ , L−,c ) defines a local product structure, (2) (L− , L+,c ) defines a local product structure, (3) L+ is contracted by the flow, and (4) L− is contracted by the inverse flow. Then L+ , L− , L+,c , L−,c are respectively called the stable, unstable, central stable and central unstable laminations. 20 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Remark: In the language of Pollicott [57], a metric Anosov flow is a Smale flow: the local product structure of (L+ , L−,c ) is what he calls the map h·, ·i : {(x, y) ∈ X × X : d(x, y) < ε} → X. 3.2.1. Livšic’s Theorem. Livšic [49] shows that the Livšic cohomology class of a Hölder function f : X → R is determined by its periods: Theorem 3.3. Let f : X → R be a Hölder continuous function, then hδa |f i = 0 for every a ∈ O if and only if f is Livšic cohomologous to zero. 3.2.2. Coding. We shall say that the triple (Σ, π, r) is a Markov coding for φ if Σ is an irreducible two-sided subshift of finite type, the maps π : Σ → X and r : Σ → R∗+ are Hölder-continuous and verify the following conditions: Let σ : Σ → Σ be the shift, and let r̂ : Σ × R → Σ × R be the homeomorphism defined by r̂(x, t) = (σx, t − r(x)), then i) the map Π : Σ × R → X defined by Π(x, t) = φt (π(x)) is surjective and r̂-invariant, ii) consider the suspension flow σ r = {σtr }t∈R on (Σ × R)/r̂, then the induced map Π : (Σ × R)/r̂ → X is bounded-to-one and, injective on a residual set which is of full measure for every ergodic invariant measure of total support of σ r . Remark: If a flow φ admits a Markov coding, then every reparametrization φf of φ also admits a Markov coding, simply by changing the roof function r. We recall, see Remark 3.2, that a metric Anosov flow is a Smale flow. One then has the following theorem of Bowen [12, 13] and Pollicott [57]. Theorem 3.4. A topologically transitive metric Anosov flow on a compact metric space admits a Markov coding. 3.3. Entropy and pressure for Anosov flows. The thermodynamic formalism of suspensions of subshifts of finite type extends thus to topologically transitive metric Anosov flows. For a positive Hölder function f : X → R+ and T ∈ R, we define RT (f ) = {a ∈ O | hδa |f i 6 T }. Observe that RT (f ) only depends on the cohomology class of f . 3.3.1. Entropy. For a topologically transitive metric Anosov flow Bowen [12] (see also Pollicott [57]) showed: Proposition 3.5. The topological entropy of a topologically transitive metric Anosov flow φ = {φt }t∈R on a compact metric space X is finite and positive. Moreover, 1 log ♯ {a ∈ O | p(a) 6 T } . htop (φ) = lim T →∞ T In particular, for a nowhere vanishing Hölder continuous function f , 1 hf = lim log ♯ (RT (f )) = htop (φf ) T →∞ T is finite and positive. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 21 3.3.2. Pressure. The Markov coding may be used to show the pressure of a Hölder function g : X → R is finite and that there is a unique equilibrium state of g. We shall denote this equilibrium state as mg . Theorem 3.6. [Bowen–Ruelle [15],Pollicott [57]] Let φ = {φt }t∈R be a topologically transitive metric Anosov flow on a compact metric space X and let g : X → R be a Hölder function, then there exists a unique equilibrium state mg for g. Moreover, if f : X → R is a Hölder function such that mf = mg , then f − g is Livšic cohomologous to a constant. The pressure function has the following alternative formulation in this setting (see Bowen–Ruelle [15]):   X 1 ehδa |gi  . log  P(φ, g) = lim (10) T →∞ T a∈RT (1) 3.3.3. Measure of maximal entropy. We have the following equidistribution result of Bowen [12] (see also Pollicott [57]). Theorem 3.7. A topologically transitive metric Anosov flow φ = {φt }t∈R on a compact metric space X has a unique probability measure µφ of maximal entropy. Moreover,   µφ = lim  T →∞ 1 ♯RT (1) X a∈RT (1) δba  . (11) The probability measure of maximal entropy for φ is called the Bowen–Margulis measure of φ. 3.4. Intersection and renormalised intersection. 3.4.1. Intersection. Let φ = {φt }t∈R be a topologically transitive metric Anosov flow on a compact metric space X. Consider a positive Hölder function f : X → R+ and a continuous function g : X → R. We define the intersection of f and g as Z g I(f, g) = dµ f , f φ where µφf is the Bowen–Margulis measure of the flow φf . We also have the following two alternative ways to define the intersection   X hδa |gi 1  (12) I(f, g) = lim  T →∞ ♯RT (f ) hδa |f i a∈RT (f ) R g dm−hf f R I(f, g) = (13) f dm−hf f where hf is the topological entropy of φf , and m−hf .f is the equilibrium state of −hf .f . The first equality follows from Theorem 3.7 and Equation (7), the second equality follows from the second part of Lemma 3.1. Since hδa |f i depends only on the Livšic cohomology class of f and hδa |gi depends only on the Livšic cohomology class of g, the intersection I(f, g) depends only on the Livšic cohomology classes of f and g. 22 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO 3.4.2. A lower bound on the renormalized intersection. For two positive Hölder functions f, g : X → R+ define the renormalized intersection as J(f, g) = hg I(f, g), hf where hf and hg are the topological entropies of φf and φg . Uniqueness of equilibrium states together with the definition of the pressure imply the following proposition. Proposition 3.8. If φ = {φt }t∈R is a topologically transitive metric Anosov flow on a compact metric space X, and f : X → R+ and g : X → R+ are positive Hölder functions, then J(f, g) > 1. Moreover, J(f, g) = 1 if and only if hf f and hg g are Livšic cohomologous. Proof. Since P(φ, −hg g) = 0, hg Z g dm > h(φ, m) for all m ∈ Mφ and, by Theorem 3.6, equality holds only for m = m−hg g , the equilibrium state of −hg g. Applying the analogous inequality for m−hf f , together with Abramov’s formula (8) and Lemma 3.1, one sees that Z Z hg g dm−hf .f > h(φ, m−hf .f ) = hf f dm−hf .f , which implies that J(f, g) > 1. If J(f, g) = 1, then m−hg g = m−hf f and thus, applying theorem 3.6, one sees that hg g − hf f is Livšic cohomologous to a constant c. Thus, 0 = P(φ, −hg g) = P(φ, −hf f − c) = P(φ, −hf f ) − c = −c. Therefore, hg g and hf f are Livšic cohomologous.  3.5. Variation of the pressure and the pressure form. McMullen [53] introduced a pressure metric on the space of Livšic cohomology classes of pressure zero Hölder functions on a shift space Σ. In this section, we use his construction to produce a pressure form, and associated semi-norm, on the space of pressure zero Hölder functions on our flow space X. 3.5.1. First R and second derivatives. For g a Hölder continuous function with mean zero (i.e. g dmf = 0), we define the variance of g with respect to f as !2 Z Z T 1 g(φs (x))ds dmf (x), Var(g, mf ) = lim T →∞ T 0 where mf is the equilibrium state of f. Similarly, for two mean zero Hölder continuous functions g and h, we define the covariance of g and h with respect to f as ! Z ! Z Z T T 1 g(φs (x))ds Cov(g, h, mf ) = lim h(φs (x))ds dmf (x). T →∞ T 0 0 THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 23 Since mf is invariant with respect to flow φt , we may rewrite this as ! Z Z T Cov(g, h, mf ) = lim g(x) h(φs (x))ds dmf (x). T →∞ −T We shall omit the background flow in the notation of the pressure function and simply write P(·) = P(φ, ·). Proposition 3.9. (Parry-Pollicott [55, Prop. 4.10,4.11], Ruelle [61]) Suppose that φ = {φt }t∈R is a topologically transitive metric Anosov flow on a compact metric space X, and f : X → R and g : X → R are Hölder functions. If mf is the equilibrium state of f , then (1) The function t 7→ P(f + tg) is analytic, (2) The first derivative is given by Z ∂P(f + tg) = g dmf , ∂t t=0 R (3) If g dmf = 0 then ∂ 2 P(f + tg) ∂t2 = Var(g, mf ), t=0 (4) If Var(g, mf ) = 0 then g is Livšic cohomologous to zero. 3.5.2. The pressure form. Let C h (X) be the set of real valued Hölder continuous functions on X. Define P(X) to be the set of pressure zero Hölder functions on X, i.e.  P(X) = Φ ∈ C h (X) : P(Φ) = 0 . The tangent space of P(X) at Φ is the set   Z TΦ P(X) = ker dΦ P = g ∈ C h (X) | g dmΦ = 0 where mΦ is the equilibrium state of Φ. Define the pressure semi-norm of g ∈ TΦ P(X) as Var(g, mΦ ) . kgk2P = − R Φ dmΦ One has the following computation. Lemma 3.10. Let φ = {φt }t∈R be a topologically transitive metric Anosov flow on a compact metric space X. If {Φt }t∈(−1,1) is a smooth one parameter family contained in P(X), then R Φ̈0 dmΦ0 2 kΦ̇0 kP = R . Φ0 dmΦ0 Proof. As P(Φt ) = 0 by differentiating twice we get the equation Z D2 P(Φ0 )(Φ̇0 , Φ̇0 ) + DP(Φ0 )(Φ̈0 ) = 0 = Var(Φ̇0 , mΦ0 ) + Φ̈0 dmΦ0 . Thus kΦ̇0 k2P R Φ̈0 dmΦ0 Var(Φ̇0 , mΦ ) = R . =− R Φ0 dmΦ0 Φ0 dmΦ0  24 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO We then have the following relation, generalizing Bonahon [11], between the renormalized intersection and the pressure metric. Proposition 3.11. Let φ = {φt }t∈R be a topologically transitive metric Anosov flow on a compact metric space X. If {ft : X → R+ }t∈(−1,1) is a one-parameter family of positive Hölder functions and Φt = −hft ft for all t ∈ (−1, 1), then ∂2 ∂t2 J(f0 , ft ) = kΦ̇0 k2P . t=0 Proof. By Equation (13) and the definition of the renormalised intersection, we see that R Φt dmΦ0 . J(f0 , ft ) = R Φ0 dmΦ0 Differentiating twice and applying the previous lemma, one obtains R Φ̈0 dmΦ0 ∂2 = kΦ̇0 k2P J(f0 , ft ) = R 2 ∂t t=0 Φ0 dmΦ0 which completes the proof.  So, the pressure semi-norm arises naturally from the pressure form p which is the symmetric 2-tensor on TΦ P(X) given by the Hessian of JΦ = J(Φ, ·). One may compute that if f, g ∈ TΦ P(X), then p(f, g) = − Cov(f, g, mΦ ) R . Φ dmΦ 3.6. Analyticity of entropy, pressure and intersection. We now show that pressure, entropy and intersection vary analytically for analytic families of positive Hölder functions. Proposition 3.12. Let φ = {φt }t∈R be a topologically transitive metric Anosov flow on a compact metric space X. Let {fu : X → R}u∈D and {gv : X → R}v∈D be two analytic families of Hölder functions. Then the function u 7→ P(fu ) is analytic. Moreover, if the family {fu }u∈D consists of positive functions then the functions u 7→ hu = hfu , (u, v) 7→ I(fu , gv ). (14) (15) are both analytic. Proof. Since the pressure function is analytic on the space of Hölder functions (see Parry-Pollicott [55, Prop. 4.7] or Ruelle [61, Cor. 5.27]) the function u 7→ P(fu ) is analytic. Since the family {fu }u∈D consists of positive functions, Proposition 3.9 implies that Z d d P(−tfu ) = P(−hu fu − (t − hu )fu ) = − fu dm−hu fu < 0. dt t=hu dt t=hu Thus an application of the Implicit Function Theorem yields that u 7→ hu is analytic. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 25 We also get that d P(−hu fu + tgv ), dt t=0 is analytic. But, applying Proposition 3.9 again, Z d P(−hu fu + tgv ) = gv dm−hu fu . dt t=0 R Thus the functionR (u, v) 7→ gv dm−hu fu is analytic. Similarly (taking gv = fu ), the function u 7→ fu dm−hu fu is analytic. Thus, we get, by Equation (13) that R gv dm−hu fu R (u, v) 7→ I(fu , gv ) = , fu dm−hu fu (u, v, t) 7→ is analytic.  4. The geodesic flow of a projective Anosov representation In this section, we define a flow (Uρ Γ, {φt }t∈R ) associated to a projective Anosov representation ρ : Γ → SLm (R). We will show that Uρ Γ is a Hölder reparameterization of the geodesic flow U0 Γ of the domain group Γ, so it will make sense to refer to Uρ Γ as the geodesic flow of the representation. Let F be the total space of the bundle over RP(m)(2) = RP(m) × RP(m)∗ \ {(U, V ) | U 6⊂ V }, whose fiber at the point (U, V ) is the space M(U, V ) = {(u, v) | u ∈ U, v ∈ V, hv|ui = 1}/ ∼, where (u, v) ∼ (−u, −v) and RP(m)∗ is identified with the projective space of the dual space (Rm )∗ . Notice that u determines v, so that F is an R-bundle. One may also identify M(U, V ) with the space of norms on U . Then F is equipped with a natural R-action, given by φt (U, V, (u, v)) = (U, V, (et u, e−t v)). If ρ : Γ → SLm (R) is a projective Anosov representation and ξ and θ are the associated limit maps, we consider the associated pullback bundle Fρ = (ξ, θ)∗ F over ∂∞ Γ(2) which inherits an R action from the action on F . The action of Γ on ∂∞ Γ(2) extends to an action on Fρ . If we let Uρ Γ = Fρ /Γ, then the R-action on Fρ descends to a flow {φt }t∈R on Uρ Γ, which we call the geodesic flow of the representation. Proposition 4.1. [The geodesic flow] If ρ : Γ → SLm (R) is a projective Anosov representation, then the action of Γ on Fρ is proper and cocompact. Moreover, the flow {φt }t∈R on Uρ Γ is Hölder conjugate to a Hölder reparameterization of the Gromov geodesic flow on U0 Γ and the orbit associated to [γ], for any infinite order primitive element γ ∈ Γ, has period Λ(ρ)(γ). 26 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO g We produce a Γ-invariant Hölder orbit equivalence between U 0 Γ and Fρ which (2) g g is a homeomorphism. Recall that U0 Γ = ∂∞ Γ × R and that U0 Γ/Γ = U0 Γ. Since g the action of Γ on U 0 Γ is proper and cocompact, it follows immediately that Uρ Γ is Hölder conjugate to a Hölder reparameterization of the Gromov geodesic flow on U0 Γ. Proposition 4.2. If ρ : Γ → SLm (R) is a projective Anosov representation, there exists a Γ-equivariant Hölder orbit equivalence g ν̃ : U 0 Γ → Fρ which is a homeomorphism. Let Eρ be the flat bundle associated to ρ on U0 Γ. Recall that Eρ splits as Eρ = Ξ ⊕ Θ. Let {ψt }t∈R be the lift of the geodesic flow on U0 Γ to a flow on Eρ . We first observe that we may produce a Hölder metric on the bundle Ξ which is contracting on all scales. Lemma 4.3. There exists a Hölder metric τ 0 on the bundle Ξ and β > 0 such that for all t > 0 we have, ψt∗ (τ 0 ) < e−βt τ 0 . Proof. Let τ be any Hölder metric on Ξ. Since ρ is projective Anosov, Lemma 2.4 implies that there exists t0 > 0 such that 1 ψt∗0 (τ ) 6 τ. 4 Choose β > 0 so that 2 < eβt0 < 4 and, for all s, let τs = ψs∗ (τ ). Let Z t0 0 eβs τs ds. τ = 0 Notice that τ 0 has the same regularity as τ . If t > 0, then Z t0 eβs τt+s ds ψt∗ (τ 0 ) = 0 Z t+t0 −βt eβu τu du. = e (16) t Now observe that Z t+t0 e βu Gu du 0 = τ + t = τ0 + Z t+t0 e Zt0t 0 But eβt0 ψt∗0 (τ ) 6 Thus Z t+t0 βu τu du − Z t eβu τu du 0
eβu ψu∗ eβt0 ψt∗0 (τ ) − τ du. (17) eβt0 τ < τ. 4 eβu τu du < τ 0 . t and the result follows from Inequality (16).  THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 27 Proof of Proposition 4.2 Let τ 0 be the metric provided by Lemma 4.3 and let β be the associated positive number. Let Ξ̃ denote the line bundle over ∂∞ Γ(2) × R which is the lift of Ξ. Notice that τ 0 lifts to a Hölder metric τ̃ 0 on Ξ̃. Our Hölder orbit equivalence ν̃ : ∂∞ Γ(2) × R → Fρ will be given by ν̃(x, y, t) = (x, y, (u(x, y, t), v(x, y, t))) , 0 0 where τ̃(x,y,t) (u(x, y, t)) = 1 and τ̃(x,y,t) is the metric on the line ξ(x) induced by the metric G̃0 by regarding ξ(x) as the fiber of Ξ̃ over the point (x, y, t). The fact that ψt∗ τ 0 < τ 0 for all t > 0 implies that ν̃ is injective. Since τ̃ 0 is Hölder and Γ-equivariant, ν̃ is also Hölder and Γ-equivariant. It remains to prove that ν̃ is proper. We will argue by contradiction. If ν̃ is not proper, then there exists a sequence {(xn , yn , tn )}n∈N leaving every compact subset of ∂∞ Γ(2) × R, such that {ν̃(xn , yn , tn )}n∈N converges to (x, y, (u, v)) in Fρ . Letting ν̃(xn , yn , tn ) = (xn , yn , (un , vn )), we see immediately that lim xn = x, lim yn = y, and lim (un , vn ) = (u, v). n→∞ n→∞ n→∞ Writing ν̃(xn , yn , 0) = (xn , yn , (ûn , v̂n )) and ν̃(x, y, 0) = (x, y, (û, v̂)), we obtain, by the continuity of the map ν̃, lim (ûn , v̂n ) = (û, v̂). n→∞ If t > 0, then 0 τ̃(x,y,t) 0 τ̃(x,y,0) =   0 ψt∗ τ̃(x,y,0) 0 τ̃(x,y,0) < e−βt . In particular, hv | un i (18) < e−βtn . hv | ûn i Without loss of generality, either tn → ∞ or tn → −∞. If tn → ∞, then by Inequality (18), hv | un i , 0 = lim n→∞ hv | ûn i on the other hand, hv | un i hv | ui lim = 6= 0. t→∞ hv | ûn i hv | ûi We have thus obtained a contradiction. Symmetrically, if tn → −∞, then hv | ûi hv | ûn i 0 = lim = 6= 0, n→∞ hv | un i hv | ui which is again a contradiction. The restriction of ν̃ to each orbit {(x, y)} × R is a proper, continuous, injection into the fiber of Fρ over (x, y) (which is also homeomorphic to R). It follows that the restriction of ν̃ to each orbit is a homeomorphism onto the image fiber. We conclude that ν̃ is surjective and hence a proper, continuous, bijection. Therefore, ν̃ is a homeomorphism. This completes the proof of Proposition 4.2. In order to complete the proof of Proposition 4.1, it only remains to compute the period of the orbit associated to [γ] for an infinite order primitive element γ ∈ Γ. Since ρ is projective Anosov, Proposition 2.6 implies that ρ(γ) is proximal, ξ(γ + ) 28 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO is the attracting line and θ(γ − ) is the repelling hyperplane. If u ∈ ξ(γ + ) and v ∈ θ(γ − ) one sees that ρ(γ)(u) = L(γ)(ρ) u and ρ(γ)(v) = 1 v. L(γ)(ρ) Thus, (γ + , γ − , (u, v)) and (γ + , γ − , L(γ)(ρ)u, 1 v) = φlog(Λ(γ)(ρ)) (γ + , γ − , (u, v)) L(γ)(ρ) project to the same point on Uρ Γ. (Recall that (L(γ)(ρ)u, 1 −1 v) ∼ (−L(γ)(ρ)u, v) L(γ)(ρ) L(γ)(ρ) in M (ξ(γ + ), θ(γ − )).) Since γ is primitive, this finishes the proof.  5. The geodesic flow is a metric Anosov flow In this section, we prove that the geodesic flow of a projective Anosov representation is a metric Anosov flow: Proposition 5.1. [Anosov] If ρ : Γ → SLm (R) is a projective Anosov representation, then the geodesic flow (Uρ Γ, {φt }t∈R ) is a topologically transitive metric Anosov flow. The reader with a background in hyperbolic dynamics may be convinced by the following heuristic argument: essentially the splitting of an Anosov representation yields a section of some (product of) flag manifolds and the graph of this section should be thought as a Smale locally maximal hyperbolic set; then the result follows from the “fact” that the restriction of the flow on such a set is a metric Anosov flow. However, the above idea does not exactly work, and moreover it is not easy to extricate it from the existing literature in the present framework. Therefore, we give a detailed and ad-hoc construction, although the result should be true in a rather general setting. The topological transitivity of (Uρ Γ, {φt }t∈R ) follows immediately from the topological transitivity of the action of Γ on ∂∞ Γ2 . We define a metric on the geodesic flow in Section 5.1, introduce the stable and unstable leaves in Section 5.2, explain how to control the metric along the unstable leaves in Section 5.3 and finally proceed to the proof in Section 5.4. A more precise version of Proposition 5.1 is given by Proposition 5.7. 5.1. The geodesic flow as a metric space. Recall that F is the total space of an R-bundle over RP(m)(2) whose fiber at the point (U, V ) is the space M(U, V ) = {(u, v) | u ∈ U, v ∈ V, hv|ui = 1}/ ∼ . (2) Since RP(m) ⊂ RP(m) × RP(m)∗ , any Euclidean metric on Rm gives rise to a metric on F which is a subset of
∗
RP(m) × RP(m)∗ × Rm × (Rm ) / ± 1 . The metric on F pulls back to a metric on Fρ . A metric on Fρ obtained by this procedure is called a linear metric. Any two linear metrics are bilipschitz equivalent. The following lemma allows us to use a linear metric to study Fρ . THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 29 Lemma 5.2. There exists a Γ-invariant metric d0 on Fρ which is locally bilipschitz equivalent to any linear metric. The Γ-invariant metric d0 descends to a metric on Uρ Γ which we will also call d0 and is defined for every x and y in Fρ by d0 (π(x), π(y)) = inf (x, γ(y)), γ∈Γ where π is the projection Fρ → Uρ Γ. Proof. We first notice that all linear metrics on Fρ are bilipschitz to one another, so that it suffices to construct a metric which is locally bilipschitz to a fixed linear metric d. Let V be an open subset of Fρ with compact closure which contains a closed fundamental domain for the action of Γ on Fρ . Since the action of Γ on Fρ is proper, {Vγ = γ(V )}γ∈Γ is a locally finite cover of Fρ . Let {dγ = γ ∗ d}γ∈Γ be the associated family of metrics on Fρ . Since each element of Γ acts as a bilipschitz automorphism with respect to any linear metric, any two metrics in the family {dγ = γ ∗ d}γ∈Γ are bilipschitz equivalent. We will use this cover and the associated family of metrics to construct a Γ-invariant metric on Fρ . A path joining two points x and y in Fρ is a pair of tuples P = ((z0 , . . . , zn ), (γ0 , . . . , γn )), where (z0 , . . . , zn ) is an n-tuple of points in Fρ and (γ0 , . . . , γn ) is an n-tuple of elements of Γ such that • x = z0 ∈ Vγ0 and y = zn ∈ Vγn , • for all n > i > 0, zi ∈ Vγi−1 ∩ Vγi . The length of a path is given by ! n−1 1 X dγ (zi , zi+1 ) + dγi+1 (zi , zi+1 ) ℓ(P) = 2 i=0 i We then define d0 (x, y) = inf{ℓ(P) | P joins x and y}. It is clear that d0 is a Γ-invariant pseudo metric. It remains to show that d0 is a metric which is locally bilipschitz to d. Let z be a point in Fρ . Then there exists a neighborhood Z of z so that A = {γ | Vγ ∩ Z 6= ∅}, is a finite set. Choose α > 0 so that [ {x | dγ (z, x) 6 α} ⊂ Z. γ∈A Let K be chosen so that if α, β ∈ A, then dα and dβ are K-bilipschitz. Finally, let \ n α o W = x | dγ (z, x) 6 . 10K γ∈A By construction, if x and y belong to W , then for all γ ∈ A, α . dγ (x, y) 6 5K (19) 30 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Let x be a point in W . Let P = ((z0 , . . . , zn ), (γ0 , . . . , γn )) be a path joining x to a point y. If there exists j such that γj 6∈ A, then ℓ(P) > i=j−1 1 X dγi (zi−1 , zi ) 2 i=0 ! i=j−1 X 1 dγj−1 (zi , zi+1 ) > 2K i=0
1 1 > dγj−1 (z, zj )) − dγj−1 (z0 , z) dγj−1 (z0 , zj ) > 2K  2K 1 α  α > α− > . (20) 2K 10K 5K If γj ∈ A for all j, then the triangle inequality and the definition of K immediately imply that for all γ ∈ A, 1 ℓ(P) > dγ (x, y). (21) K Inequalities (20) and (21) imply that α  1 inf , dγ (x, y) > 0, (22) d0 (x, y) > K 5 hence d0 is a metric. Moreover, if x, y ∈ W , then by inequalities (22) and (19), 1 (23) d0 (x, y) > dγ (x, y). K By construction, and taking the path P0 = ((x, y), (γ, γ)) with γ in A, we also get d0 (x, y) 6 ℓ(P0 ) = dγ (x, y). (24) As consequence of inequalities (23) and (24), d0 is bilipschitz on W to any dγ with γ ∈ A. Since d is bilipschitz to dγ for any γ ∈ A, we see that d0 is bilipschitz to d on W . Since z was arbitrary, it follows that d0 is locally bilipschitz to d.  5.2. Stable and unstable leaves. In this section, we define the stable and unstable laminations of the geodesic flow Fρ . Let Z = (x0 , y0 , (u0 , v0 )) be a point in Fρ . (1) The unstable leaf through Z is L− Z = {(x, y0 , (u, v0 )) | x ∈ ∂∞ Γ, u ∈ ξ(x), hv0 |ui = 1}. The central unstable leaf through Z is L−,c Z = = {(x, [ y0 , (u, v)) | x ∈ ∂∞ Γ, (u, v) ∈ M(ξ(x), θ(y0 ))} φt (LZ +). t∈R (2) The stable leaf through Z is L+ Z = {(x0 , y, (u0 , v)) | y ∈ ∂∞ Γ, v ∈ θ(y), hv|u0 i = 1}. The central stable leaf through Z is L+,c Z = {(x0 , y, (u, v)) | y ∈ ∂∞ Γ, (u, v) ∈ M(ξ(x0 ), θ(y))} THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS = [ 31 φt (L− Z ). t∈R L+ Z Observe that is homeomorphic to ∂∞ Γ \ {x0 } and L− Z is homeomorphic to ∂∞ Γ \ {y0 }. The following two propositions are immediate from our construction. Proposition 5.3. [Invariance] If γ ∈ Γ and t ∈ R, then

± ± L± and L± γ(Z) = γ LZ φt (Z) = φt LZ . Proposition 5.4. [product structure] The (two) pairs of lamination (L± , L∓,c ) define a local product structure on Fρ , and hence on Uρ Γ. Remark: Throughout this section, we abuse notation by allowing {φt }t∈R to denote both the flow on Uρ Γ and the flow on Fρ which covers it and letting L± denote both the lamination on Fρ and the induced lamination on Uρ Γ. 5.3. The leaf lift and the distance. In this section we introduce the leaf lift and show that it helps in controlling distances in Fρ . We first define the leaf lift for points in the bundle F . Let A = (U, V, (u0 , v0 )) be a point in F . We observe that there exists a unique continuous map, called the leaf lift from OA = {w ∈ RP(m)∗ | U ∩ ker(w) = {0}}. m ∗ to ((R ) \ {0}) / ± 1 such that w is taken to Ωw,A such that Ωw,A ∈ w, hΩw,A |u0 i = 1. (25) In particular, Ωv0 ,A = v0 . Observe that at this stage the leaf lift coincides with the classical notion of an affine chart. The following lemma records immediate properties of the leaf lift . Lemma 5.5. Let k.k1 be a Euclidean norm on Rn and d1 the associated metric on RP(m)∗ . If A = (x, y, (u, v)) ∈ F , then there exist constants K1 > 0 and α1 > 0 such that for w0 , w1 ∈ RP(m)∗ • If d1 (wi , y) 6 α1 , for i = 0, 1, then kΩw0 ,A − Ωw1 ,A k1 6 K1 d1 (w0 , w1 ) , • If kΩwi ,A − Ωy,A k1 6 α1 for i = 0, 1, then d1 (w0 , w1 ) 6 K1 kΩw0 ,A − Ωw1 ,A k1 . If Z = (x, y, (u0 , v0 )) ∈ Fρ and W = (x, w, (u0 , v)) ∈ L+ Z , then we define the leaf lift ωW,Z = Ωξ∗ (w),(ξ(x),ξ∗ (y),(u0 ,v0 )) = v. The following result allows us to use the leaf lift to bound distances in Fρ Proposition 5.6. Let d0 be a Γ-invariant metric on Fρ which is locally bilipschitz equivalent to a linear metric and let Z → k.kZ be a Γ-invariant map from Fρ into the space of Euclidean metrics on Rm . There exist positive constants K and α such that for any Z ∈ Fρ and any W ∈ L− Z, • if d0 (W, Z) 6 α, then ωW,Z − ωZ,Z Z 6 Kd0 (W, Z) , (26) 32 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO • if ωW,Z − ωZ,Z Z 6 α then d0 (W, Z) 6 K ωW,Z − ωZ,Z Z . (27) Proof. Since Γ acts cocompactly on Fρ and both d0 and the section k.k are Γ-invariant, it suffices to prove the previous assertion for Z in a compact subset R of Fρ . Observe first that d0 is uniformly C-bilipschitz on R to any of the linear metrics dZ coming from k.kZ for Z in R for some constant C. Lemma 5.5 implies that, for all Z ∈ R, there exist positive constants KZ and αZ such that if W0 , W1 ∈ L− Z ∩ O, then • If d0 (Wi , Z) 6 αZ for i = 0, 1, then ωW0 ,Z − ωW1 ,Z • If ωWi ,Z − ωZ,Z Z Z 6 KZ d0 (W0 , W1 ) , 6 αZ for i = 0, 1, then d0 (W0 , W1 ) 6 KZ ωW0 ,Z − ωW1 ,Z Z . Since R is compact, one may apply the classical argument which establishes that continuous functions are uniformly continuous on compact sets, to show that there are positive constants K and α which work for all Z ∈ R.  5.4. The geodesic flow is Anosov. The following result completes the proof of Proposition 5.1 Proposition 5.7. [Anosov property] Let ρ : Γ → SLm (R) be a projective Anosov representation, and let L± be the laminations on Uρ Γ defined above. Then there exists a metric on Uρ Γ, Hölder equivalent to the Hölder structure on Uρ Γ, such that (1) L+ is contracted by the flow, (2) L− is contracted by the inverse flow, We first show that the leaf lift is contracted by the flow. Lemma 5.8. There exists a Γ-invariant map Z 7→ k.kZ from Fρ into the space of Euclidean metrics on Rm , such that for every positive integer n, there exists t0 > 0 such that if t > t0 , Z ∈ Fρ , and W ∈ L+ Z then ωφt (W ),φt (Z) − ωφt (Z),φt (Z) φt (Z) 6 1 ωW,Z − ωZ,Z 2n Z . (28) The following notation will be used in the proof. • For a vector space A and a subspace B ⊂ A, let B ⊥ = {ω ∈ A∗ | B ⊂ ker(ω)}. • We consider the Γ-invariant splitting of the trivial Rm -bundle Fρ × Rm = Ξ̂ ⊕ Θ̂ – where Ξ̂ is the line bundle over Fρ such that the fiber above (x, y, (u, v)) is given by ξ(x) and – Θ̂ is a hyperplane bundle over Fρ with fiber θ(y) above the point (x, y, (u, v)) ∈ Fρ . THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 33 Proof. Suppose that Z = (x, y, (u0 , v0 )) and W = (x, w, (u0 , v)) ∈ L+ Z , then by the definition of the leaf lift (ωW,Z − ωZ,Z )(u0 ) = 0, and thus ωW,Z = αW,Z + ωZ,Z , where αW,Z ∈ ξ(x)⊥ . Then φt (ωW,Z ) = φt (αW,Z ) + φt (ωZ,Z ). We choose a Γ-invariant map from Fρ into the space of Euclidean metrics on Rm so that for all Y ∈ Fρ ωY,Y Y = 1. Then 1 ωφt (Z),φt (Z) = φt (ωZ,Z ), φt (ωZ,Z ) φt (Z) hence ωφt (W ),φt (Z) = φt (αW,Z ) + ωφt (Z),φt (Z) . φt (ωZ,Z ) φt (Z) It follows that ωφt (W ),φt (Z) − ωφt (Z),φt (Z) φt (Z) = φt (αW,Z ) φt (Z) φt (ωZ,Z ) φt (Z) Since ρ is projective Anosov, and (Uρ Γ, {φt }t∈R ) is a Hölder reparameterization of (U0 Γ, {ψt }t∈R ), there exists t1 > 0 so that for all Z ∈ Fρ and for all t > t1 , if ⊥ v ∈ Ξ̂⊥ Z and w ∈ Θ̂Z , then φt (v) φt (Z) φt (w) φt (Z) 6 1 v 2 w Z . Z ⊥ Thus, since αW,Z ∈ Ξ̂⊥ Z and ωZ,Z ∈ Θ̂Z , for all n ∈ N and t > nt1 , we have ωφt (W ),φt (Z) − ωφt (Z),φt (Z) Since αW,Z = ωW,Z − ωZ,Z and ωZ,Z result with t0 = nt1 . Z φt (Z) 6 1 αW,Z 2n ωZ,Z Z . Z = 1, the previous assertion yields the  We are now ready to establish Proposition 5.7. Proof of Proposition 5.7: Let K and α be as in Proposition 5.6. Choose n ∈ N so that K2 1 K 6 1 and 6 . (29) 2n 2n 2 Let t0 be the constant from Lemma 5.8 with our choice of n. Suppose that Z ∈ Fρ , W ∈ L+ Z , t > t0 and d0 (W, Z) 6 α. Then, by Inequality (26), (30) ωW,Z − ωZ,Z 6 Kd0 (W, Z). By Lemma 5.8, 1 (31) ωφt (W ),φt (Z) − ωφt (Z),φt (Z) φt (Z) 6 n ωW,Z − ωZ,Z Z . 2 34 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO In particular, combining Equations (30), (31) and (29), 1 ωφt (W ),φt (Z) − ωφt (Z),φt (Z) φt (Z) 6 n Kα 6 α. 2 Thus, using Inequality (27), d0 (φt (W ), φt (Z)) 6 K ωφt (W ),φt (Z) − ωφt (Z),φt (Z) φt (Z) (32) . (33) Combining finally Equations (30), (31), (33) and (29), we get that d0 (φt (W ), φt (Z)) 6 K2 1 d0 (W, Z) 6 d0 (W, Z) 2n 2 (34) for all t > t0 . Therefore L+ is contracted by the flow on Fρ . Let us now consider what happens in the quotient Uρ Γ = Fρ /Γ. For any Z ∈ Fρ and ǫ > 0, let ± L± ǫ (Z) = LZ ∩ B(Z, ǫ). and let
− Kǫ (Z) = ΠZ L+ ǫ (Z) × Lǫ (Z) × (−ǫ, ǫ) , where ΠZ is the product structure of Proposition 5.4. By Proposition 4.1, there exists ǫ0 > 0 such that for all γ ∈ Γ \ {1} and Z ∈ Fρ , γ(Kǫ0 (X)) ∩ Kǫ0 = ∅. Let ǫ ∈ (0, min{ǫ0 /2, α}) and Ẑ ∈ Uρ Γ. Choose Z ∈ Fρ in the pre image of Ẑ, then inequality (34) holds for the flow on Uρ Γ for points in the chart which is the projection of Kǫ (Z). Therefore, L+ is contracted by the flow on Uρ Γ. A symmetric proof holds for the central unstable leaf. 6. Analytic variation of the dynamics In order to apply the thermodynamic formalism we need to check that if {ρu }u∈M is an analytic family of projective Anosov representations, then the associated limit maps and reparameterizations of the Gromov geodesic flow may be chosen to vary analytically, at least locally. Our proofs generalize earlier proofs of the stability of Anosov representations, see Labourie [41, Proposition 2.1] and Guichard-Wienhard [26, Theorem 5.13], and that the limit maps vary continuously, see GuichardWienhard [26, Theorem 5.13]. In the process, we also see that our limit maps are Hölder. We will make use of the following concrete description of the analytic structure of Hom(Γ, G). Suppose that Γ is a word hyperbolic group, hence finitely presented, and let {g1 , . . . , gm } be a generating set for Γ. If G is a real semi-simple Lie group, then Hom(Γ, G) has the structure of a real algebraic variety. An analytic family β : M → Hom(Γ, G) of homomorphisms of Γ into G is a map with domain an analytic manifold M so that, for each i, the map βi : M → G given by βi (u) = β(u)(gi ) is real analytic. If G is a complex Lie group, we may similarly define complex analytic families of homomorphisms of a complex analytic manifold into Hom(Γ, G). We first show that the limit maps of an analytic family of Anosov homomorphisms vary analytically. We begin by setting our notation. If α > 0, X is a compact metric space and D and M are real-analytic manifolds, then we let C α (X, M ) denote the space of α-Hölder maps of X into M and let C ω (D, M ) denote the space of real analytic maps of D into M . If D and M are complex analytic manifolds, THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 35 we will abuse notation by letting C ω (D, M ) denote the space of complex analytic maps. Theorem 6.1. Let G be a real (or complex) semi-simple Lie group and let P be a parabolic subgroup of G. Let {ρu }u∈D be a real (or complex) analytic family of homomorphisms of Γ into G parameterized by a real (or complex) disk D about 0. If ρ0 is a (G, P)-Anosov homomorphism with limit map ξ0 : ∂∞ Γ → G/P, then there exists a sub-disk D0 of D (containing 0), α > 0 and a continuous map ξ : D0 × ∂∞ Γ → G/P with the following properties: (1) If u ∈ D0 , then ρu is a (G, P)-Anosov homomorphism with α-Hölder limit map ξu : ∂∞ Γ → G/P given by ξu (·) = ξ(u, ·). (2) If x ∈ ∂∞ Γ, then ξx : D0 → G/P given by ξx = ξ(·, x) is real (or complex) analytic (3) The map from ∂∞ Γ to C ω (D0 , G/P) given by x 7→ ξx is α-Hölder. (4) The map from D0 to C α (∂∞ Γ, G/P) given by u → ξu is real (or complex) analytic. Given a projective Anosov representation ρ : Γ → SLm (R), we constructed a geodesic flow Uρ Γ which is a reparameterization of the Gromov geodesic flow U0 Γ. In Section 6.3, we show that given a real analytic family of projective Anosov representions, one may choose the parameterizing functions to vary analytically. Proposition 6.2. Let {ρu }u∈D be a real analytic family of projective Anosov homomorphisms of Γ into SLm (R) parameterized by a disk about 0. Then, there exists a sub-disk D0 about 0 and a real analytic family {fu : U0 Γ → R}u∈D0 of positive Hölder functions such that the reparametrization of U0 Γ by fu is Hölder conjugate to Uρu Γ for all u ∈ D0 . We first observe that the real analytic case of Theorem 6.1 follows from the complex analytic case, which we will establish in Section 6.2. If G is a real semisimple Lie group and P is a parabolic subgroup of G, we let GC and P C be the complexification of G and P. Observe that a (G, P)-Anosov representation is automatically a (GC , PC )-Anosov representation. On a sub-disk D1 of D, containing 0, one may extend {ρu }u∈D1 to a complex analytic family {ρu }u∈D1C of homomorphisms of Γ into GC defined on the complexification D1C of D1 . The map ξ : D0C × ∂∞ Γ → GC /PC provided by the complex analytic case of Theorem 6.1 restricts to a map ξ|D0 : D0 × ∂∞ Γ → G/P with the desired properties. Notice that the real analyticity in properties (2) and (4) follows from the fact that restrictions of complex analytic functions to real analytic submanifolds are real analytic. 6.1. Transverse regularity. In this section, we set up our notation and establish a version of the C r -section Theorem of Hirsch-Pugh-Shub [29, Theorem 3.8] which keeps track of the transverse regularity of the resulting section. Our version of Hirsch, Pugh and Shub’s result will be the main tool in the proof of Theorem 6.1. Definition 6.3. [transversely regular functions] Let D be a real (or complex) disk, let X be a compact metric space and let M be a real (or complex) analytic manifold. A continuous function f : D × X → M is transversely real (or complex) analytic if 36 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO (1) For every x ∈ X, the function fx : D → M given by fx (·) = f (·, x) is real (or complex) analytic, and (2) The function from X to C ω (D, M ) given by x → fx is continuous. Furthermore, we say that f is α-Hölder (or Lipschitz) transversely real (or complex) analytic if the map in (2) is α-Hölder (or Lipschitz). If we replace M with a C k manifold, we can similarly define α-Hölder (or Lipschitz) transversely C k functions by requiring that the maps in (1) are C k and that the map in (2) from X to C p (D, M ) is α-Hölder (or Lipschitz) for all p ≤ k, Similarly, we define transverse regularity of bundles in terms of the transverse regularity of their trivializations. Definition 6.4. [tranversally regular bundles] Suppose that the fiber of a bundle π : E → D × X is a real (or complex) analytic manifold M and that D is a real (or complex) disk. We say that E is transversely real (or complex) analytic if it admits a family of trivializations of the form {D × Uα × M } (where {Uα } is an open cover of X) so that the the corresponding change of coordinate functions are transversely real (or complex) analytic. We similarly say π : E → D × X is α-Hölder (or Lipschitz) transversely real (or complex) analytic if it admits a family of trivializations which are α-Hölder (or Lipschitz) transversely real (or complex) analytic. In this case, a section of E is α-Hölder (or Lipschitz) transversely real (or complex) analytic, if in any of the trivializations the corresponding map to M is αHölder (or Lipschitz) transversely real (or complex) analytic. Clearly, if M is a C k -manifold, we can similarly define α-Hölder (or Lipschitz) transversely C k bundles and sections. We are now ready to state our version of the C r -Section Theorem. Theorem 6.5. Let X be a compact metric space and let M be a complex analytic (or C k ) manifold. Suppose that π : E → D × X is a Lipschitz transversely complex analytic (or C k ) bundle with fibre M and D is a complex (or real) disk. Let f : X → X be a Lipschitz homeomorphism and let F be a Lipschitz transversely complex analytic (or C k ) bundle automorphism of E lifting id × f . Suppose that σ0 is a section of the restriction of E over {0} × X which is fixed by F and that F contracts along σ0 . Then there exists a neighborhood U of 0 in D, a positive number α > 0, an α-Hölder transversely complex analytic (or C k ) section η over D0 × X and a neighborhood B of η(U × X) in π −1 (U × X) such that (1) (2) (3) (4) F fixes η, F contracts E along η, η|{0}×X = σ0 , and if ν : U × X → E is a section so that ν(U × X) ⊂ B and ν is fixed by F , then ν = η. We recall that if U is a subset of D, then a section σ over U × X is fixed by F if F (σ(u, x)) = σ(u, f (x)). In such a case, we further say that F contracts along σ if there exists a continuously varying fibrewise Riemannian metric k · k on the bundle E such that if Df Fσ(u,x) : Tσ(u,x) π −1 (u, x) → Tσ(u,f (x)) π −1 (u, f (x)) THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 37 is the fibrewise tangent map, then kDf Fσ(u,x) k < 1. We will derive Theorem 6.5 from the following version of the C r -section theorem which is a natural generalization of the ball bundle version of the C r -section theorem in Shub [65, Theorem 5.18]. Theorem 6.6. [Fixed sections] Let X be a compact metric space equipped with a Lipschitz homeomorphism f : X → X. Suppose that π : W → D × X is a Lipschitz transversely complex analytic (or C k ) Banach space bundle, D is a complex (or real) disk, B ⊂ W is the closed ball sub-bundle of radius r, and F is a Lipschitz transversely complex analytic (or C k ) bundle morphism of B lifting the homeomorphism id × f : D × X → X. If F contracts B, then there exists a unique α-Hölder transversely complex analytic (or C k ) section η of B which is fixed by F (for some α > 0). Notice that we have not assumed that F is either linear or bijective. Proof. Let σ be the zero section of B. Observe that σ has the same regularity as W and is thus transversally complex analytic (or C k ). We first assume that π : W → D × M is a Lipschitz transversely C k -bundle. The existence of a unique continuous fixed section η is a standard application of the contraction mapping theorem. Explicitly, for all (u, x) ∈ D × X, we let η(u, x) = lim F n (σ(u, f −n (x)). n→∞ (35) We must work harder to show that η is α-Hölder transversely complex analytic (or C k ). We first assume that W is transversely C k –and so is σ– and obtain the C k -regularity of η. For any p ∈ N, let Γp be the Lipschitz Banach bundle over X whose fibers over a point x ∈ X is the Banach space Γpx of C p -sections of the restriction of W to D × {x}. Let B p be the sub-bundle whose fiber Bxp over x is the set of those sections with values in B. Notice that each fiber Bxp can be identified with C p (D × {x}, B0 ) where B0 is a closed ball of radius r in the fiber Banach space. Let F∗p be the bundle automorphism of Γp given by [F∗p (ν)](u, x) = F (ν(u, f −1 (x))). We can renormalise the metric on D, so that all the derivatives of F of order n (with p > n > 1) along D are arbitrarily small. Thus after this renormalisation the metric on D, F∗p is contracting, since F is contracting. We now apply Theorem 3.8 of Hirsch-Pugh-Shub [29] (see also Shub [65, Theorem 5.18]) to obtain an invariant α-Hölder section ω. By the uniqueness of fixed sections, we see that η(u, x) = ω(x)(u) for all 1 ≤ p ≤ k. It follows that η is α-Hölder transversely C k . Now suppose that D is a complex disk and π : E → D × X is Lipschitz transversely complex analytic bundle. We see, from the above paragraph, that there exists a unique α-Hölder transversely C k section ηk for all k. By the uniqueness ηk is independent of k and we simply denote it by η. Then, by Formula (35), for all x ∈ X, η|D×{x} is a C k -limit of a sequence of complex analytic sections for all k, hence is complex analytic itself. It follows that η is α-Hölder transversely complex analytic.  38 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO We now notice that one may identify a neighborhood of the section σ0 in the statement of Theorem 6.5 with a ball sub-bundle of a vector bundle. Lemma 6.7. Let π : E → D × X be a transversely complex analytic (or C k ) bundle over D × X, D is a complex (or real) disk about 0, and σ is a section of E defined over {0} × X. Then there exists (1) a neighborhood U of zero in D, (2) a transversely complex analytic (or C k ) closed ball bundle B of radius R in a complex (or real) vector bundle F , (3) a transversely complex analytic (or C k ) bijective map from B to a neighborhood of the graph of σ0 so that • the graph of σ0 = σ|{0}×X is in the image of the graph of the zero section, • the fibrewise metric on B coincides along σ0 with the fiberwise metric on E. Proof. We first give the proof in the case that σ is defined over D × X. Let Z be the transversely complex analytic (or C k ) vector bundle over D × X so that the fibre over the point (u, x) is given by Tσ(u,x) (π −1 (u, x)). We equip Z with a Riemannian metric coming from E and let B(r) be the closed ball sub-bundle of radius r > 0. Using the trivializations, we can find, after restricting to an open neighborhood U of 0 in D, • a finite cover {Oi }16i6n of X, • an open neighborhood W of the graph of σ, • transversely holomorphic (or C k -diffeomorphic) bundle maps φi defined on W |U×Oi with values in Z|U×Oi so that for all (u, x) ∈ U × Oi φi (σ(u, x)) Dfσ(u,x) φi = = 0 ∈ Tσ(u,x) (π −1 (u, x)) Id. (36) Let {ψi }16i6n be a partition of unity on X subordinate to {Oi }16i6n and, for each i, let ψ̂i : W → [0, 1] be obtained by composing the projection of W to X with ψi . One may then define Φ : W → Z by letting Φ= n X ψ̂i φi . i=1 Since ψ̂i is constant in the direction of D, Φ is transversely holomorphic (or C k diffeomorphic), Φ(σ(u, x)) = 0 and Dfσ(y) Φ = Id. It then follows from the implicit function theorem, that one may further restrict U and W so that Φ is a transversely holomorphic (or C k -diffeomorphic) isomorphism of W with B(r) for some r. If σ is only defined on {0} × X, it now suffices to extend the section σ0 to a section σ defined over U × X where U is a neighborhood of 0 in D. Composing π with the projection π2 : D × X → X, we may consider the bundle π2 ◦ π : E → X. Then σ0 is a section of π2 ◦ π. We now apply the result of the previous paragraph, in the case where the disk is 0-dimensional, to identify, in a complex analytic (or C k ) way, a neighborhood of the graph of σ0 with a ball bundle B in a vector bundle F over X. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 39 Now π restricts to a bundle morphism from π ◦ π2 : B → X to π2 : D × X → X which is a fiberwise complex analytic (or C k ) submersion and whose fiberwise derivatives vary continuously. Let W be a linear sub-bundle of F , so that if Wx and Fx are the fibers over x ∈ X, then T(π −1 (0, x)) ⊕ Wx = Fx . Thus, after further restricting B, π becomes a fiberwise complex analytic (or C k ) injective local diffeomorphism from W ∩ B to D × X whose fiberwise derivatives vary continuously. Applying the Implicit Function Theorem (with parameter), we obtain a neighborhood U of 0 and a map σ : U × X → B which is fiberwise complex analytic (or C k ) and whose fiberwise derivatives vary continuously, so that π ◦ σ = Id. Thus σ is the desired section of E.  Theorem 6.5 now follows from Theorem 6.6 and Lemma 6.7. Proof of Theorem 6.5: Let V be the complex (or real) vector bundle provided by Lemma 6.7. We know that kDfσ0 (x) F k < 1 for all x in X. After further restraining U and choosing r small enough, we may assume by continuity that for all y in B(r), kDfy F k < K < 1. After further restricting U , we may assume that for all u ∈ U and x ∈ X, we have kF (σ(u, x)) − σ(u, f (x))k 6 (1 − K)r, In particular, if y ∈ B(r) is in the fiber over (u, x), kF (y) − σ(u, f (x))k 6 kF (y) − F (σ(u, x))k +kF (σ(u, x)) − σ(u, f (x))k 6 Kr + (1 − K)r = r. Thus F maps B(r) to itself and is contracting. We can therefore apply Theorem 6.6 to complete the proof of Theorem 6.5.  In the proof of Theorem 6.1, we will also need to use the fact that transverse regularity of a continuous function f : D × X → M implies regularity of the associated map of D into C α (X, M ). Let X be a compact metric space and let M be a complex analytic (or C k ) manifold. If U is an open subset of M and V is a relatively compact open subset of X, then let W(U, V ) = {g ∈ C α (X, M ) | g(V ) ⊂ U }. We will say that a map f from D to C α (X, M ) is complex analytic (or C k ) if for any U and V as above and any complex analytic function φ : U → C (or C k function φ : U → R), the function f φ defined on f −1 (W(U, V )), by f φ (x) = φ ◦ f (x)|V , with values in C α (V, C) (or C α (V, R)) is complex analytic (or C k ). Recall that the function f φ is complex analytic if and only if it has a a C-linear differential at each point, see, for example, Hubbard [31, Thm. A5.3]. The following lemma shows that an α-Hölder transversely complex analytic map from D × X to M gives rise to a complex analytic map from D to C α (X, M ). The proof is quite standard so we will omit it, see Hubbard [31, Prop. A5.9] for a very similar statement. 40 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Lemma 6.8. Suppose that D is a complex (or real) disk, M is a complex analytic (or C k ) manifold, X is a compact metric space and f : D × X → M is α-Hölder transversely complex analytic (or C k ), then the map fˆ from D to C α (X, M ) given by u → fu where fu (·) = f (u, ·) is complex analytic (or C k−1 ). 6.2. Analytic variation of the limit maps. We are now ready for the proof of Theorem 6.1 in the complex analytic case. Given a complex analytic family of representations which contains an Anosov representation, we construct an associated bundle where we can apply the results of the previous section to produce a family of limit maps. Let G be a complex Lie group and let P be a parabolic subgroup. Let {ρu }u∈D be a complex analytic family of homomorphisms of Γ into G parameterized by a complex disk D about 0 so that ρ0 is (G, P)-Anosov. We construct a G/P-bundle over D × U0 Γ. Let g à = D × U 0 Γ × G/P g which is a G/P-bundle over D × U 0 Γ. Then γ ∈ Γ acts on Ã, by γ(u, x, [g]) = (u, γ(x), [ρu (γ)g]) and we let A = Ã/Γ. g The geodesic flows on U 0 Γ and U0 Γ lift to geodesic flows {Ψ̃t }t∈R and {Ψ̃t }t∈R on à and A. (The flow {Ψ̃t }t∈R acts trivially on the D and G/P factors.) Since ρ0 is (G, P)-Anosov there exists a section σ0 of A over {0}×U0 Γ. Concretely, if ξ0 : ∂∞ Γ → G/P is the limit map, we construct an equivariant section σ̃0 of à g over {0} × U 0 Γ of the form (0, (x, y, t)) → (0, (x, y, t), ξ0 (x)). The section σ̃0 descends to the desired section σ0 of A over {0} × U0 Γ. One may identify the bundle over {0} × U0 Γ with fiber Tσ0 (x) π −1 (0, x) with Nρ− . Since the geodesic flow lifts to a flow on Nρ− whose inverse flow is contracting, the inverse flow {Φ−t }t∈R is contracting along σ0 (U0 Γ). Theorem 6.5 then implies that there exists a sub-disk D1 ⊂ D containing 0, α > 0, and an α-Hölder transversely complex analytic section η : D × U0 Γ → A that extends σ0 , is fixed by {Φt }t∈R and so that the inverse flow {Φ−t }t∈R contracts along η. (More concretely, Theorem 6.5 produces, for large enough t0 , a section fixed by Φ−t0 so that Φ−t0 contracts along η. One may then use the uniqueness portion of the statement to show that η is fixed by Φt for all t.) We may lift η to g g a section η̃ : D1 × U 0 Γ → à which we may view as a map η̄ : D1 × U0 Γ → G/P. We next observe that η̄(u, (x, y, t)) does not depend on either y or t. Since η̄ is flow-invariant, η̄(u, (x, y, t)) does not depend on t. Fix u ∈ D1 and let g η̄u : U 0 Γ → G/P be given by η̄u (·) = η̄(u, ·). Let γ be an infinite order element of Γ whose associated orbit in U0 Γ has period tγ and let d be an arbitrary metric on G/P. Since {Φ−t }t∈R is contracting along η, there exists a constant k0 > 0 such that if {pn } is a sequence in G/P with d(η̄u (γ + , γ − , 0), pn ) ≤ k0 for all n, then lim d(η̄u (γ + , γ − , 0), γ n (pn )) = lim d(η̄u (γ n (γ + , γ − , −ntγ )), γ n (pn )) = 0. (37) n→∞ n→∞ THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 41 Given z ∈ ∂∞ Γ, there exists tz ∈ R, so that, if d¯ denotes a Γ-invariant metric on g U 0 Γ, then ¯ n (γ + , γ − , 0), (γ + , z, tz + ntγ )) = 0. lim d(γ n→∞ Therefore, ¯ + , γ − , 0), γ −n (γ + , z, tz + ntγ )) = 0. lim d((γ n→∞ Applying (37) with pn = η̄(γ −n (γ + , z, tz + ntγ )), we see that lim d(η̄u (γ + , γ − , 0), γ n η̄u (γ −n (γ + , z, tz + ntγ ))) = 0. n→∞ Since η̄u is Γ-equivariant, this implies that lim d(η̄u (γ + , γ − , 0), η̄u (γ + , z, tz + ntγ )) = 0. n→∞ Since η̄u (γ + , z, t) does not depend on t, we finally obtain that η̄(u, (γ + , γ − , 0)) = η̄(u, (γ + z, t)) for any z ∈ ∂∞ Γ, u ∈ D1 and t ∈ R. Since, fixed points of infinite order elements are dense in ∂∞ Γ and η̄ is continuous, we see that η̄(u, (x, y, t)) does not depend on y or t. Therefore, we obtain a transversely complex analytic map ξ : D1 × ∂∞ Γ → G/P which extends ξ0 . The map ξ satisfies properties (2) and (3), since ξ is α-Hölder transversely complex analytic, while property (4) follows from Lemma 6.8. It remains to prove that we may restrict to a sub disk D0 of D1 so that if u ∈ D0 , then ρu is (G, P)-Anosov with limit map ξu . Let Q be a parabolic subgroup of G which is opposite to P. Then there exists a Lipschitz transversely complex analytic G/Q-bundle A′ over D × U0 Γ and we may lift the geodesic flow to a flow {Φ′t } on A′ . Since ρ0 is (G, P)-Anosov, there exists a map θ0 : ∂∞ Γ → G/Q which gives rise to a section σ0′ of A′ over {0} × U0 Γ such that the flow is contracting on a neighborhood of σ0′ ({0} × U0 Γ). We again apply Corollary 6.5 to find an α′ -Hölder (for some α′ > 0) transversely complex analytic flow invariant section η ′ : D2 × U0 Γ → A′ that extends σ0′ , for some sub-disk D2 of D which contains 0, such that the flow {Ψ′t }t∈R contracts along η ′ (D2 × U0 Γ). The section η ′ lifts to a section of η̃ ′ of Ã′ ′ g which we may reinterpret as a map η̄ ′ : D2 × U 0 Γ → G/Q so that η̄ (u, (x, y, t)) ′ depends only on u and y. So we obtain an α -Hölder transversely complex analytic map θ : D2 × ∂∞ Γ → G/Q which restricts to θ0 . Since ξ0 and θ0 are transverse, we may find a sub-disk D0 of D1 ∩ D2 so that ξu and θu are transverse if u ∈ D0 . It follows that if u ∈ D0 , then ρu is (G, P)-Anosov with limit maps ξu and θu . This completes the proof of Theorem 6.1 in the complex analytic case. Remark: Notice that the same proof applies to a C k -family {ρu }u∈D of representations of a hyperbolic group Γ into a real semi-simple Lie group G such that ρ0 is (G, P)-Anosov. It produces a sub-disk D0 and a α-Hölder transversely C k map ξ : D0 × ∂∞ Γ → G/P so that if u ∈ D0 , then ρu is (G, P)-Anosov with limit map ξu . 42 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO 6.3. Analytic variation of the reparameterization. We now turn to the proof of Proposition 6.2. Let {ρu : Γ → SLm (R)}u∈D be a real analytic family of projective Anosov representations and let DC be the complexification of D. We may extend {ρu }u∈D to a complex analytic family {ρu : Γ → SLm (C)}u∈DC of homomorphisms. Theorem 6.1 implies that, after possibly restricting DC , there exists a α-Hölder transversely complex analytic map ξ : DC × ∂∞ Γ → GC /PC = CP(m) such that if u ∈ DC , then ρu is Anosov with respect to the parabolic subgroup PC , which is the stabilizer of a complex line, with limit map ξu . (We call such representations complex projective Anosov.) We construct a Lipschitz transversely complex analytic Cm -bundle W C over C m g D × U0 Γ which is the quotient of W̃ C = DC × U associated to the family 0Γ × C {ρu }u∈DC . We can then lift the Gromov geodesic flow on U0 Γ to a Lipschitz transversely complex analytic flow {Ψt }t∈R on W C . Since the functions in the partition of unity for our trivializations of W C are constant in the the DC direction, we have: Proposition 6.9. After possibly further restricting DC , the bundle W C is equipped with a Lipschitz transversely complex analytic 2-form ω of type (1, 1) such that τ (u, v) = ω(u, v) + ω(v, u), is Hermitian. Let LC be the (complex) line sub-bundle of W C determined by ξ, i.e. LC is the g quotient of the line sub-bundle of W̃ C whose fiber over (u, (x, y, t)) ∈ DC × U 0Γ C is the complex line ξu (x). Then, L is a α-Hölder transversely complex analytic line bundle over DC × U0 Γ. Since each ρu is complex projective Anosov with limit map ξu , LC is preserved by the flow {Ψt }t∈R . We restrict ω and τ to LC (and still denote them by ω and τ ). Since LC is a line bundle, we can consider the function a : DC × U0 Γ → C such that ω(u, x)(v, v) = a(u, x)τ (u, x)(v, v). whenever v is in the fiber of LC over (u, x). Concretely. a(u, x) = ω(v, v) 2ℜ(ω(v, v)) for any non-trivial v in the fiber over (u, x). We observe that a is α-Holder transversely real analytic. If U is an open subset of U0 Γ in one of our trivializing sets, we can construct a non-zero section V : D C × U → LC which is α-Hölder transversely complex analytic. Then ω(V, V ) : DC × U → C is α-Hölder transversely complex analytic. Lemma 6.8 implies that the map from DC to C α (U, C) given by u → ω(V (u, ·), V (u, ·)) is complex analytic. Therefore, the map from DC to C α (U, R) given by u → ℜ(ω(V (u, ·), V (u, ·))) is real analytic. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 43 It follows that the map from DC to C α (U, C) given by u → a(u, ·) is real analytic since ω(V, V ) . a|DC ×U = 2ℜ(ω(V, V )) Since x was arbitrary the map from DC to C α (U0 Γ, C) given by u → a(u, ·) is real analytic. Similarly, a itself is α-Holder transversely real analytic. If we define, for all t, the map ht : DC × U0 Γ → C so that Ψ∗t ω = ht ω, then, we may argue, just as above, that ht is α-Hölder transversely complex analytic. Lemma 6.8 guarantees that the map from DC to C β (U0 Γ, C) given by u → ht (u, ·) is complex analytic. If t ∈ R, Ψ∗t τ (·) = 2ℜ(Ψ∗t ω(·)) = 2ℜ(ht (·)ω(·)) = 2ℜ (a(·)ht (·)) G(·). We define kt (·) = ℜ(aht )(·) and note that Ψ∗t τ = kt τ . Then, kt is α-Hölder transversely real analytic and the map from DC to C α (U0 Γ, R) given by u → kt (u, ·) is real analytic (since it is the real part of a product of a real analytic and a complex analytic function). We apply the construction of Lemma 4.3 to produce an α-Hölder transversely real analytic metric τ 0 on L̂ such that Ψ∗t (τ 0 ) < e−βt τ 0 . for some β > 0 and all t > 0. Concretely, Z Z t0 eβs Ψ∗s (τ )ds = τ0 = 0 t0 0  eβs ks ds G for some appropriately chosen t0 > 0. We define, for all t, Kt : DC × U0 Γ → R by R t0 +t βs e ks ds . Kt = e−βt Rt t0 βs k ds e s 0 One then checks that Ψ∗t (τ 0 ) = Kt τ 0 for all t. Then, for each u ∈ DC we define fu : U0 Γ → R, by setting fu (·) = ∂Kt eβt0 kt0 (·) − 1 . (u, ·, 0) = −β + R t0 ∂t eβs ks (·) ds 0 Then, since u → kt (u, ·) is real analytic for all t, our formula for fu guarantees that the map from DC to C β (U0 Γ, R) given by u → fu is real analytic. Therefore, the restriction of this map to the real submanifold D is also real analytic. To complete the proof of Proposition 6.2 we will show that, for each u ∈ D, the periods of the reparameterization of U0 Γ by fu and the periods of Uρu Γ agree. Livšic’s Theorem 3.3 then implies that the reparameterization of U0 Γ by fu is Hölder conjugate to Uρu Γ as desired. 44 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO For u ∈ D, let ju : U0 Γ×R be given by ju (·, t) = log Kt (u, ·). We can differentiate the equality ju (·, t + s) = ju (Ψs (·), t) + ju (·, s) with respect to t and evaluate at t = 0 to conclude that fu (·, s) = fu (Ψs (·), 0). In particular, for any t, Z t (fu (Ψs (·), 0) ds = ju (·, t). 0 Let γ ∈ Γ and let x ∈ U0 Γ be a point on the periodic orbit associated to γ (which g is simply the quotient of (γ + , γ − ) × R ⊂ U 0 Γ). If tγ is the period of the orbit of U0 Γ containing x, then e so R tγ 0 τ (x, u) = Ψ∗tγ τ 0 (u, x) = eΛ(ρu ,γ) τ 0 (u, x), fu (Ψs (u,x))ds 0 Z tγ fu (Ψs (u, x))ds = Λ(ρu , γ) 0 is the period of the reparameterization of the flow U0 Γ by fu , which agrees with the period of the orbit in Uρu Γ associated to γ (see Proposition 4.1). This completes the proof of Proposition 6.2. Remark: Notice that a simpler version of the above proof establishes that given a C k family of projective Anosov representations, one may, at least locally, choose the reparameterization functions to vary C k−1 . 7. Deformation spaces of projective Anosov representations In this section, we collect a few facts about the structure of deformation spaces of projective Anosov representations of Γ into SLm (R) and their relatives. 7.1. Irreducible projective Anosov representations. We first observe that our deformation spaces C(Γ, m) and Cg (Γ, G) are real analytic manifolds. Let e m) ⊂ Hom(Γ, SLm (R)) denote the set of regular, irreducible, projective Anosov C(Γ, representations and let e m)/SLm (R). C(Γ, m) = C(Γ, If G is a reductive subgroup of SLm (R), then we similarly let Ceg (Γ, G) ⊂ Hom(Γ, G) denote the space of G-generic, regular representations which are irreducible and projective Anosov when viewed as representations into SLm (R). Let Cg (Γ, G) = Ceg (Γ, G)/G. Proposition 7.1. Suppose that Γ is a word hyperbolic group. Then (1) The deformation spaces C(Γ, m) and Cg (Γ, SLm (R)) have the structure of a real analytic manifold compatible with the algebraic structure on Hom(Γ, SLm (R)) (2) If G is a reductive subgroup of SLm (R), then Cg (Γ, G) has the structure of a real analytic manifold compatible with the algebraic structure on Hom(Γ, G). THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 45 Proof. We may regard Hom(Γ, SLm (R)) as a subset of Hom(Γ, SLm (C)). We first notice that an irreducible homomorphism in Hom(Γ, SLm (R)) is also irreducible when regarded as a homomorphism in Hom(Γ, SLm (C)). Lubotzky and Magid ([51, Proposition 1.21 and Theorem 1.28]) proved that the set of irreducible homomorphisms form an open subset of Hom(Γ, SLm (C)), so they also form an open subset of Hom(Γ, SLm (R)). Results of Labourie [41, Prop. 2.1] and Guichard-Wienhard [26, Theorem 5.13] imply that the set of projective Anosov homomorphisms is an open e m) is an open subset of Hom(Γ, SLm (R)) (see also Proposition 6.1). Therefore, C(Γ, subset of Hom(Γ, SLm (R)). Since the former consists of regular homomorphisms, it is an analytic manifold. Lubotzky–Magid ([51, Theorem 1.27]) also proved that SLm (C) acts properly (by conjugation) on the set of irreducible representations in Hom(Γ, SLm (C)). It e m). Schur’s Lemma guarantees that the follows that SLm (R) acts properly on C(Γ, centralizer of an irreducible representation is contained in the center of SLm (R). Therefore, PSLm (R) acts freely, analytically and properly on the analytic manifold e m), so its quotient C(Γ, m) is also an analytic manifold. C(Γ, Since the set of G-generic elements of G is an open G-invariant subset of G, we may argue exactly as above to show that Ceg (Γ, G) is an open subset of Hom(Γ, G) which is an analytic manifold. The action of G/Z(G) on Ceg (Γ, G) is again free, analytic and proper, so its quotient Cg (Γ, G) is again an analytic manifold.  e m) with the space Zρ1 (Γ, slm (R)) e m), then one may identify Tρ C(Γ, If ρ ∈ C(Γ, of cocycles and one may then identify T[ρ] C(Γ, m) with the cohomology group Hρ1 (Γ, slm (R)) (see [51, 33]). In particular, the space Bρ1 (Γ, slm (R)) is identified with the tangent space of the SLm (R)-orbit of ρ. Similarly, if ρ ∈ Ceg (Γ, G), we identify Tρ Ceg (Γ, G) with Z 1 (Γ, g) and T[ρ] Cg (Γ, G) with Hρ1 (Γ, g). More generally, if ρ is an irreducible representation in Hom(Γ, G), the tangent vector to any analytic path through ρ may be identified with an element of Zρ1 (Γ, g) (see [33, Section 2]). A simple calculation in cohomology gives that irreducible projective Anosov representations of fundamental groups of 3-manifolds with non-empty boundary are regular. These include free groups and fundamental groups of closed surfaces. Proposition 7.2. If Γ is isomorphic to the fundamental group of a compact orientable 3-manifold M with non empty boundary, then C(Γ, m) is the set of conjugacy classes of irreducible projective Anosov representations. Proof. Let Γ = π1 (M ) where M is a compact orientable 3-manifold with non-empty boundary. It suffices to show that the open subset of Hom(Γ, SLm (R)) consisting of irreducible projective Anosov homomorphisms consists entirely of regular points. We recall that ρ0 ∈ Hom(Γ, SLm (R)) is regular if there exists a neighborhood U of ρ0 so that dim(Zρ1 (M, g)) is constant on U and the centralizer of any representation ρ ∈ U is trivial [51]. If ρ0 is projective Anosov and irreducible, we can take U to be any open neighborhood of ρ0 consisting of irreducible projective Anosov representations. Since ρ ∈ U is irreducible, Schur’s Lemma guarantees that the centralizer of ρ(Γ) is the center of SLm (R). Moreover, if ρ ∈ U , then dim(Hρ0 (M, g)) − dim(Hρ1 (M, g)) + dim(Hρ2 (M, g)) = χ(M ) dim(G). 46 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Since the centralizer is trivial, dim(Hρ0 (M, g)) = 0. By Poincaré duality, dim(Hρ2 (M, g)) = dim(Hρ0 (M, ∂M, g)). Since dim(Hρ0 (M, g)) = 0, the long exact sequence for relative homology implies that dim(Hρ0 (M, ∂M, g)) = 0. Thus, dim(Hρ1 (M, g)) = −χ(M ) dim(G). Therefore, dim(Zρ1 (M, g)) = (1 − χ(M )) dim(G) for all ρ ∈ U , so ρ is a regular point.  7.2. Virtually Zariski dense representations. We recall that if Γ is a word hyperbolic group, G is a semi-simple Lie group with finite center and P is a nondegenerate parabolic subgroup, then Z(Γ; G, P) is the space of (conjugacy classes of) regular virtually Zariski dense (G, P)-Anosov representations of Γ into G. We will prove that Z(Γ; G, P) is a real analytic orbifold. Proposition 7.3. Suppose that Γ is a word hyperbolic group, G is a semi-simple Lie group with finite center and P is a non-degenerate parabolic subgroup of G. Then Z(Γ; G, P) is a real analytic orbifold. Moreover, if G is connected, then Z(Γ; G, P) is a real analytic manifold. Proof. Let Hom∗ (Γ, G) be the set of regular homomorphisms. By definition, Hom∗ (Γ, G) is an open subset of Hom(Γ, G) and hence it is an analytic manifold, since it is the set of smooth points of a real algebraic variety. Results of Labourie [41, Prop. 2.1] and Guichard-Wienhard [26, Theorem 5.13] again imply that the set of (G, P)Anosov homomorphisms is open in Hom∗ (Γ, G). The main difficulty in the proof is e G, P) of virtually Zariski dense Anosov homomorphisms to show that the set Z(Γ; ∗ is open in Hom (Γ, G) and hence an analytic manifold. e G, P) is an analytic manifold, we may complete Once we have shown that Z(Γ; the proof in the same spirit as the proof of Proposition 7.1. We observe that if e G, P) then its centralizer is finite, since the Zariski closure of ρ(Γ) has ρ ∈ Z(Γ; e G, P) with finite index in G. Then, G/Z(G) acts properly and analytically on Z(Γ; finite point stabilizers, so the quotient Z(Γ; G, P) is an analytic orbifold. If G0 is the connected component of G, then the Zariski closure of any representation e G, P) contains G0 , so the intersection of the centralizer of ρ with G0 is ρ ∈ Z(Γ; e G, P)/G0 is an analytic manifold. In particular, simply Z(G) ∩ G0 . Therefore, Z(Γ; if G is connected, Z(Γ; G, P) is an analytic manifold. We complete the proof by showing that the set of virtually Zariski dense (G, P)Anosov homomorphisms is open in Hom∗ (Γ, G). If not, then there exists a sequence {ρm }m∈N of (G, P)-Anosov representations which are not virtually Zariski dense converging to a virtually Zariski dense (G, P)-Anosov representation ρ0 . 0 Since G has finitely many components, ρ−1 n (G ) has bounded finite index for all n. Since Γ is finitely generated, it contains only finitely many subgroups of a given index, so we may pass to a finite index subgroup Γ0 of Γ so that ρn (Γ0 ) is contained in the identity component G0 of G for all n. Since each ρn |Γ0 is (G, P)-Anosov and ρ0 (Γ0 ) is also virtually Zariski dense, we may assume for the remainder of the proof that G is the Zariski closure of G0 . Let Zn be the Zariski closure of Im(ρn ) and let zn be the Lie algebra of Zn . Consider the decomposition of the Lie algebra g of G p M gi , g= i=1 THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 47 where gi are simple Lie algebras. Let Gi = Aut(gi ). We consider the adjoint representation Ad : G → Aut(g). Let H be the subgroup of G consisting of all g ∈ G so that Ad(g) preserves the factors of g. Then H is a finite index, Zariski closed subgroup of G. Hence, with our assumptions, H = G. Therefore, we getLa well-defined p projection map πi : G → Gi . If p is the Lie algebra of P, then p = i=1 pi , where pi is a Lie subalgebra of gi . Let Pi be the stabilizer of pi in Gi . Then we also obtain a G-equivariant projection, also denoted πi , πi : G/P → Gi /Pi = Gpi ⊂ Grdim(pi ) (gi ) where Grdim(pi ) (gi ) is the Grassmanian space of dim(pi )-dimensional vector spaces in gi . If ξn : ∂∞ Γ → G/P is the limit map of ρn , πi ◦ ξn is a ρn -equivariant map from ∂∞ Γ to Gi /Pi . If πi ◦ ξn is constant, then ρn (Γ) would normalize a conjugate of pi . So, if πi ◦ ξn is constant for infinitely many n, then ρ0 (Γ) would normalize a conjugate of pi ,which is impossible since ρ0 (Γ) is Zariski dense and Pi is a proper parabolic subgroup of Gi . Therefore, we may assume that πi ◦ ξn is non-constant for all i and all n. Since Γ acts topologically transitively on ∂∞ Γ, we then know that the image must then be infinite. Therefore, for all i and n, dim(πi (zn )) > 0. (38) We may thus assume that {zn } converges to a proper Lie subalgebra z0 which is normalized by ρ0 (Γ) with dim(z0 ) > 0. (39) Since ρ0 is virtually Zariski dense, z0 must be a strict factor in the Lie algebra g of G. Thus, after reordering, we may assume that z0 = q M gi . (40) i=1 For n large enough, zn is thus a graph of an homomorphim fn : z0 → h = p M gi . i=q+1 Since there are only finitely many conjugacy classes (under the adjoint representation) of homomorphisms of z0 into h, we may pass to a subsequence such that fn = Ad(gn ) ◦ f0 ◦ πh1 , where f0 is a fixed isomorphism from an ideal h1 in z0 to an ideal h2 in h, πh1 is the projection from z0 to h1 and gn ∈ H2 where Hi is the subgroup of G whose Lie algebra is hi . Let Z0 be the subgroup of G whose Lie algebra is z0 and consider A1 = exp aZ0 , where aZ0 is a Cartan subspace of z0 , and let A2 = exp aH2 , where the Cartan subspace aH2 is chosen so that f0 (πh1 (A1 )) = A2 . Considering the Cartan decomposition H2 = KA2 K of H2 where K is a maximal compact subgroup, we may write gn = kn an cn with an ∈ A2 and kn , cn ∈ K. Moreover we may write Ad(cn ) = f0 (Ad(dn )), where dn lies in a fixed compact subgroup of H1 . Thus, if u ∈ aZ0 , since A2 is commutative, we have −1 fn (Ad(d−1 n )u) = Ad(gn )f0 (Ad(dn )u) = Ad(kn )f0 (u). 48 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO We may extract a subsequence so that that {kn }n∈N and {dn }n∈N converge respectively to k0 and d0 . Therefore, {(Ad(d−1 0 )u, Ad(k0 )f0 (u)) | u ∈ aZ0 } ⊂ z0 , Lq which contradicts the fact that z0 = i=1 gi . This contradiction establishes the fact that the set of Anosov, virtually Zariski dense regular homomorphisms is open, which completes the proof.  We record the following observation, established in the proof of Proposition 7.3 which will be useful in the proof of Corollary 1.9. Proposition 7.4. Suppose that Γ is a word hyperbolic group, G is a semi-simple Lie group with finite center and P is a non-degenerate parabolic subgroup of G. Then e G, P)/G0 is an analytic manifold. Z(Γ; 7.3. Kleinian groups. Let Cc (Γ, PSL2 (C)) be the set of (conjugacy classes of) convex cocompact representations of Γ into PSL2 (C)). We say that a convex cocompact representation ρ in PSL2 (C) is Fuchsian if its image is conjugate into PSL2 (R). Since every non-elementary Zariski closed, connected subgroup of PSL2 (C) is conjugate to PSL2 (R), we note that ρ ∈ Cc (Γ, PSL2 (C)) is Zariski dense unless ρ is virtually Fuchsian, i.e. there exists a finite index subgroup of ρ(Γ) which is conjugate into PSL2 (R) (see also Johnson-Millson [33, Lemma 3.2]). Notice that if ρ is virtually Fuchsian, then ρ(Γ) contains a finite index subgroup which is isomorphic to a free group or a closed surface group. Bers [8] proved that Cc (Γ, PSL2 (C)) is a complex analytic manifold. which has real dimension −6χ(Γ) if Γ is torsion-free. (See also Kapovich [34, Section 8.8] where a proof of this is given in the spirit of Proposition 7.1.) We summarize these results in the following proposition. Proposition 7.5. Let Γ be a word hyperbolic group. Then (1) Cc (Γ, PSL2 (C)) is a smooth analytic manifold. (2) ρ ∈ Cc (Γ, PSL2 (C)) is Zariski dense if and only if ρ is not virtuallyFuchsian. (3) If Γ is torsion-free, then Cc (Γ, PSL2 (C)) has dimension −6χ(Γ). 7.4. Hitchin components. Let S be a closed orientable surface of genus at least 2 and let τm : PSL2 (R) → PSLm (R) be an irreducible homomorphism. If ρ : π1 (S) → PSL2 (R) is discrete and faithful, hence uniformizes S, then τm ◦ρ is called a Fuchsian representation. A representation ρ : π1 (S) → PSLm (R) that can be deformed into a Fuchsian representation is called a Hitchin representation. Lemma 10.1 of [41] implies that all Hitchin representations are irreducible. Let Hm (S) be the space of Hitchin representations into PSLm (R) and let Hm (S) = Hm (S)/PGLm (R). Each Hm (S) is called a Hitchin component and Hitchin [30] proved that Hm (S) is 2 an analytic manifold diffeomorphic to R(m −1)|χ(S)| . One may identify the Teichmüller space T (S) with H2 (S). The irreducible representation gives rise to an analytic embedding that we also denote τm , of T (S) into the Hitchin component Hm (S) and we call its image the Fuchsian locus of the Hitchin component. Each Hitchin representation lifts to a representation into SLm (R). Labourie [41] showed that all lifts of Hitchin representations are irreducible and (SLm (R), B)Anosov where B is a minimal parabolic subgroup of SLm (R). In particular, lifts THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 49 of Hitchin representations are projective Anosov. Moreover, Labourie [41] showed that the image of every non-trivial element of π1 (S) under the lift of a Hitchin representation is diagonalizable with distinct eigenvalues. In particular, every lift of a Hitchin representation is SLm (R)-generic, so is contained in Cg (π1 (S), SLm (R)). Moreover, notice that distinct lifts of a given Hitchin representation must be contained in distinct components of Cg (π1 (S), SLm (R)). We summarize what we need from Hitchin and Labourie’s work in the following result. Theorem 7.6. Every Hitchin component lifts to a component of the analytic manifold Cg (π1 (S), SLm (R)). 8. Thermodynamic formalism on the deformation space of projective Anosov representations In Section 8.1, we show that entropy, intersection and renormalized intersection vary analytically over C(Γ, m), then in section 8.2 we construct the thermodynamic mapping of C(Γ, m) into the space of Livšic cohomology classes of pressure zero functions on U0 Γ and use it to define the pressure form on C(Γ, m) and Cg (Γ, G). 8.1. Analyticity of entropy and intersection. Let Γ be a word hyperbolic group admitting a projective Anosov representation. By Proposition 5.7, the Gromov geodesic flow on U0 Γ admits a Hölder reparametrization which turns it into a topologically transitive metric Anosov flow. Since the Gromov geodesic flow is only well defined up to reparametrization, we choose a fixed Hölder reparametrization which gives rise to a topologically transitive metric Anosov flow, and use the corresponding flow, denoted by ψ = {ψt }t∈R , as a background flow on U0 Γ. Let ρ : Γ → SLm (R) be a projective Anosov representation. By Proposition 4.1, the geodesic flow (Uρ Γ, {φt }t∈R ) of ρ is Hölder conjugate to a Hölder reparametrization of the flow {ψt }t∈R . Periodic orbits of {φt }t∈R are in one-to-one correspondence with conjugacy classes of infinite order elements of Γ. The periodic orbit associated to the conjugacy class [γ] has period Λ(ρ)(γ). If ρ : Γ → SLm (R) is projective Anosov, let fρ : U0 Γ → R be a Hölder function such that the reparameterization of U0 Γ by fρ is Hölder conjugate to Uρ Γ. Livšic’s theorem 3.3 implies that the correspondence ρ 7→ fρ is well defined modulo Livšic cohomology and invariant under conjugation of the homomorphism ρ. Therefore, we may define h(ρ1 ) = I(ρ1 , ρ2 ) = J(ρ1 , ρ2 ) = h(fρ1 ), (41) I(fρ1 , fρ2 ), and h(ρ2 ) J(fρ1 , fρ2 ) = I(ρ1 , ρ2 ), h(ρ1 ) (42) (43) for projective Anosov representations ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R). These quantities are well defined and agree with the definition given in the Introduction. Proposition 7.3.1 implies that 1 log ♯(RT (ρ1 )) T →∞ T h(fρ1 ) = lim 50 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO while equation (12) implies that  I(fρ1 , fρ2 ) = lim  T →∞ 1 ♯(RT (ρ1 )) X [γ]∈RT (ρ1 )  log(Λ(γ)(ρ2 ))  . log(Λ(γ)(ρ1 )) Proposition 6.2 implies that if {ρu }u∈D is an analytic family of of projective Anosov homomorphisms defined on a disc D, then we can choose, at least locally, the map u 7→ fρu to be analytic. Proposition 3.12 then implies that entropy, intersection and renormalized intersection all vary analytically. Proposition 8.1. Given two analytic families {ρu }u∈D and {ηv }v∈D′ of projective Anosov homomorphisms, the functions u 7→ h(ρu ), (u, v) 7→ I(ρu , ηv ) and (u, v) 7→ J(ρu , ηv ) are analytic on their domains of definition. Combining Propositions 3.8, 3.9 and 3.11 one obtains the following. Corollary 8.2. For every pair ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) of projective Anosov representations, one has J(ρ1 , ρ2 ) > 1. If J(ρ1 , ρ2 ) = 1, then there exists a constant c > 1 such that Λρ1 (γ)c = Λρ2 (γ) for every γ ∈ Γ. Moreover, if {ρt } is a smooth one parameter family of projective Anosov representations and {ft } is an associated smooth family of reparametrizations, then ∂2 ∂t2 J(ρ0 , ρt ) = 0 t=0 if and only if ∂ ∂t (hρt ft ) t=0 is Livšic cohomologous to 0. 8.2. The thermodynamic mapping and the pressure form. If ρ ∈ C(Γ, m) and fρ is a reparametrization of the Gromov geodesic flow giving rise to the geodesic flow of ρ, we define Φρ : U0 Γ → R by Φρ (x) = −h(ρ)fρ (x). Lemma 3.1 implies that Φρ ∈ P(U0 Γ). Let H(U0 Γ) be the set of Livšic cohomology classes of pressure zero function, we saw that the class of Φρ in H(U0 Γ) only depends on ρ. We define the thermodynamic mapping to be  C(Γ, m) → H(U0 Γ) T: ρ 7→ [Φρ ] By Proposition 6.2, the thermodynamic mapping is “analytic” in the following sense: for every representation ρ in the analytic manifold C(Γ, m), there exists a neighborhood U of ρ in C(Γ, m) and an analytic mapping from U to P(U0 Γ) which lifts the thermodynamic mapping. We use the thermodynamic mapping to define a 2-tensor on our deformation spaces. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 51 Definition 8.3. [Pressure Form] Let {ρu }u∈M be an analytic family of projective Anosov homomorphims parametrized by an analytic manifold M . If z ∈ M , we define Jz : M → R by letting Jz (u) = J(ρz , ρu ). The associated pressure form p on M is the 2-tensor such that if v, w ∈ Tz M , then p(v, w) = D2z Jz (v, w). Notice that, by Corollary 8.2, the pressure form is non-negative. e m) and on C(Γ, e G) when G is a In particular, we get pressure forms on C(Γ, reductive subgroup of SLm (R). Since J is invariant under the action of conjugation on each variable, these pressure forms descend to 2-tensors, again called pressure forms, on the analytic manifolds C(Γ, m) and Cg (Γ, G). 9. Degenerate vectors for the pressure metric In this section, we analyze the norm zero vectors for the pressure metric. If Γ is a word hyperbolic group, α is an infinite order element of Γ and {ρu }u∈M is an analytic family of projective Anosov homomorphisms parameterized by an analytic manifold M , one may view L(α) as an analytic function on M where we abuse notation by letting L(α)(u) = L(α)(ρu ) denote the eigenvalue of ρu (α) of maximal modulus. The following is the main result of the section. Proposition 9.1. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). Suppose that {ρu : Γ → G}u∈D is an analytic family of projective Anosov G-generic homomorphisms defined on a disc D with associated pressure form p. Suppose that z ∈ D, v ∈ Tz D and p(v, v) = 0. Then, for every element α of infinite order in Γ, Dz L(α)(v) = 0. 9.1. Log-type functions. We begin by showing that if v is a norm zero vector, then each L(α) is of log-type Kat v for some fixed K. Definition 9.2. We say that an analytic function f has log-type K at v ∈ Tu M , if f (u) 6= 0 and Du log(|f |)(v) = K log(|f (u)|), and is of log-type if it is of log-type K for some K. Lemma 9.3. Let {ρu }u∈M be an analytic family of projective Anosov homomorphims parametrized by an analytic manifold M and let p be the associated pressure form. If v ∈ Tz M and p(v, v) = 0, then there exists K ∈ R such that if α is any element of infinite order in Γ, then L(α) is of log-type K at v. Proof. Consider a smooth one parameter family {us }s∈(−1,1) in M such that u0 = z and u̇0 = v. Let ρs = ρus and let fs = fus where {fus } is a smooth family of reparametrizations obtained from Proposition 6.2. We define, for all s ∈ (−1, 1), Φs = Φρs = −h(ρs )fs , 52 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO By Corollary 8.2, Φ̇0 is Livšic cohomologous to zero. In particular, the integral of Φ̇0 is zero on any φs -invariant measure. Thus for any infinite order element α ∈ Γ one has hδα |Φ̇0 i = 0. By definition, Φs = −h(ρs )fρs and thus hδα |Φs i = −h(ρs ) log Λ(α)(us ). It then follows that dΦs (x) d(hδα |Φs i)(x) 0 = hδα | i= ds s=0 ds = s=0 d (h(ρs ) log(Λ(α)(us )) ds Applying the chain rule we get    dh(ρs ) d log(Λ(α)(us ) 0= log(Λ(α)(us )) + h(ρs ) ds s=0 ds It follows that setting K =− 1 d (h(ρs )) h(ρ0 ) ds s=0  . s=0 . , s=0 we get that for all α ∈ Γ, Dz log(Λ(α))(v) = d ds (log(Λ(α)(ρs )) = K log (Λ(α)(z)) . s=0 Since Λ(α) = |L(α)|, L(α) has log-type K at v.  9.2. Trace functions. Recall, from Proposition 2.6, that if α is an infinite order element of Γ and ρ is a projective Anosov representation in C(Γ, m), then we may write 1 ρ(α) = L(α)(ρ)p(ρ(α)) + m(ρ(α)) + q(ρ(α)), L(α−1 )(ρ) where (1) L(α)(ρ) is the eigenvalue of ρ(α) of maximum modulus and p(ρ(α)) is the projection on ξ(α+ ) parallel to θ(α− ) (2) L(α−1 )(ρ) is the eigenvalue of ρ(α−1 ) of maximal modulus and q(ρ(α)) is the projection onto the line ξ(α− ) parallel to θ(α+ ), and (3) the spectral radius of m(ρ(α)) is less than δ l(α) Λ(α)(ρ) for some δ = δ(ρ) ∈ (0, 1) which depends only on ρ. It will be useful to define 1 r(ρ(α)) = m(ρ(α)) + q(ρ(α)) −1 L(α )(ρ) which also has spectral radius less than δ l(α) Λ(α)(ρ). If {ρu }u∈D is an analytic family of projective Anosov G-generic homomorphisms defined on a disc D and α and β are infinite order elements of Γ, we consider the following analytic functions on D: T(α, β) : T(p(α), β) : T(p(α), p(β)) : T(p(α), r(β)) : T(r(α), p(β)) : T(r(α), r(β)) : u 7→ u 7→ u 7→ u 7→ u 7→ u 7→ Tr(ρu (α)ρu (β)) Tr(p(ρu (α))ρu (β)), Tr(p(ρu (α))p(ρu (β))), Tr(p(ρu (α))r(ρu (β))), Tr(r(ρu (α))p(ρu (β))), Tr(r(ρu (α))r(ρu (β))). THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 53 We say that two infinite order elements of Γ are coprime if they have distinct fixed points in ∂∞ Γ (i.e. they do not share a common power). We then have Proposition 9.4. Let {ρu }u∈D be an analytic family of projective Anosov homomorphisms defined on a disc D. If α and β are infinite order, coprime elements of Γ, then L(αn β n ) T(p(α), p(β)) = lim n→∞ L(α)n L(β)n and L(αn β) T(p(α), β) = lim . n→∞ L(α)n Moreover, if L(γ) has log-type K at v ∈ Tu D for all infinite order γ ∈ Γ, then both T(p(α), p(β)) and T(p(α), β) have log-type K at v. We say that a family {fn }n∈N of analytic functions defined on a disk D decays at v ∈ Tz D if lim fn (z) = 0 and lim Dz fn (v) = 0. n→∞ n→∞ The following observation will be useful in the proof of Proposition 9.4. Lemma 9.5. Let G be an analytic function that may be written, for all positive integers n, as G = Gn (1 + hn ), where Gn has log-type K and {hn }n∈N decays at v ∈ Tu M , then G has log-type K. Proof. Notice that Du log(G)(v) = Du log(Gn )(v) + Du log(1 + hn )(v) Du hn (v) = K log Gn (u) + . 1 + hn (u) We now simply notice that the right hand side of the equation converges to K log G(u)  Proof of Proposition 9.4: First notice that T(αn , β n ) = L(αn β n )(1 + gn ) where gn = Tr(r(αn β n )) . L(αn β n ) n n Since r(αn β n )(ρu ) has spectral radius at most δ(ρu )ℓ(α β ) |L(αn β n )|, δ(ρu ) ∈ (0, 1), and limn→∞ ℓ(αn β n ) = +∞, we see that limn→∞ gn (ρu ) = 0 for all ρu ∈ C(Γ, m). Since {gn } is a sequence of analytic functions, gn decays at v. On the other hand, ρu (αn β n ) = L(α)n L(β)n p(α)p(β)+L(α)n p(α)r(β n )+L(β)n r(αn )p(β)+r(αn )r(β n ), so T(αn , β n ) = L(α)n L(β)n T(p(α), p(β))(1 + ĝn ) 54 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO where ĝn = L(α)n T(p(α), r(β n )) + L(β)n T(r(αn ), p(β)) + T(r(αn ), r(β n )) . T(p(α), p(β))L(α)n L(β)n and again ĝn decays at v. (Notice that, since α and β are co-prime, ξρu (β + ) is not contained in θρu (α− ) for any u ∈ D, so T(p(α), p(β)) is non-zero on D.) Combining, we see that T(p(α), p(β)) = L(αn β n )(1 + gn ) , L(α)n L(β)n (1 + ĝn ) which implies, since lim gn = 0 and lim ĝn = 0, that L(αn β n ) . n→∞ L(α)n L(β)n T(p(α), p(β)) = lim Moreover, if L(γ) has log-type K at v for all infinite order γ ∈ Γ, then Gn = L(αn β n ) L(α)n L(β)n has log-type K, being the ratio of log-type K functions and we may apply Lemma 9.5 to see that T(p(α), p(β)) has log-type K. We similarly derive the claimed facts about T(p(α), β) by noting that T(αn , β) = L(αn β)(1 + hn ) where hn = Tr(r(αn β)) , L(αn β) and that T(αn , β) = L(α)n T(p(α), β)(1 + ĥn ) where ĥn = T(r(αn ), β) L(αn )T(p(α), β) and applying an argument similar to the one above.  Remark: Dreyer [23] previously established that   Λ(αn β)(ρ) Λ(α)(ρ)n has a finite limit when ρ is a Hitchin representation. 9.3. Technical lemmas. We will need a rather technical lemma, Lemma 9.7, in the proof of Lemma 9.8, which is itself the main ingredient in the proof of Proposition 9.1. We first prove a preliminary lemma, which may be viewed as a complicated version of the fact that exponential functions grow faster than polynomials. If as is a polynomial in q variables and their conjugates, we will use the notation kas k = sup{|as (z1 , . . . , zq )| | |zi | = 1}. Lemma 9.6. Let (f1 , . . . , fq ) and (θ1 , . . . , θq ) be two q-tuples of real numbers and let (g1 , . . . , gq ) be a q-tuple of complex numbers, such that 1 > f1 > · · · > fq > 0. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 55 Suppose that there exists a strictly decreasing sequence {µs }s∈N of positive real numbers so that µ1 < 1 and a sequence of complex-valued polynomials {as }s∈N in q variables and their conjugates, such that, for all n ∈ N, q X nfpn ℜ(einθp gp ) = ∞ X µns ℜ(as (einθ1 , . . . , einθq )), (44) s=1 p=1 and there exists N such that ∞ X |µs |n kas k s=1 is convergent for all n ≥ N . Then, for all p = 1, . . . , q, ℜ(gp ) = 0 gp = 0 if if θp ∈ 2πQ, θp 6∈ 2πQ. Proof. There exists r ∈ N, so that, for all i, either rθi ∈ 2πZ or rθi 6∈ 2πQ. Equation (44) remains true if we replace (θ1 , . . . , θq ) with (rθ1 , . . . , rθq ), so we may assume that either θi 6∈ 2πQ or θi ∈ Z. Let V be the set of accumulation points of {(einθ1 , . . . , einθq ) | n ∈ N}. We first show that if (z1 , . . . , zq ) ∈ V , then ℜ(g1 z1 ) = 0. This will suffice to prove our claim if p = 1, since if θi ∈ 2πZ, then z1 = 1 and ℜ(g1 ) = 0. If not, any z1 ∈ S 1 can arise in such a limit, so ℜ(z1 g1 ) = 0 for all z1 ∈ S 1 , which implies that g1 = 0. So, suppose that {nm } is an increasing sequence in N and {(einm θ1 , . . . , einm θq )} converges to (z1 , . . . , zq ). Then either (1) ℜ(as (z1 , z2 , . . . , zq )) = 0 for all s, or (2) there exists s0 ∈ N so that A0 = ℜ(as0 (z1 , z2 , . . . , zq )) 6= 0, and for all s < s0 ℜ(as (z1 , z2 , . . . , zq )) = 0. If (1) holds, then Equation (44) implies lim nm ℜ(einm θ1 g1 ) + ǫ0 (nm ) = 0. m→∞ where ǫ0 (nm ) = q X p=2 nm  fp f1 nm (45) ℜ(einm θp gp ). Since, limm→∞ ℜ(einm θ1 g1 ) = ℜ(z1 g1 ) and limm→∞ ǫ0 (nm ) = 0, we conclude that ℜ(z1 g1 ) = 0. If (2) holds, then Equation (44) implies that n  µs0 m Am (1 + ǫ1 (nm )) = 0 lim nm ℜ(z1 g1 ) + ǫ0 (nm ) − m→∞ f1 where Am Am,s inm θ1 inm θ2 , . . . , einm θq )), ,e = ℜ(as 0 (e nm
µs 1 ℜ as (einm θ1 , . . . , einm θq ) , and = Am µs0 56 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO ǫ1 (nm ) = ∞ X Am,s . (46) s=s0 +1 Observe that lim Am = A0 6= 0 If m is large enough that |Am,s | 6 µnm −N m→∞ |Am | > 12 |A0 | −N µsn0m+1 µns0m = 0 and Since limm→∞ sµ0ns+1 m 0 follows that the sequence Bs where Bs = P∞ s=1  and nm > N , then 1 nm 2 |µs |N kas k. A0 Bs is convergent, limn→∞ ǫ1 (nm ) = 0. It then  µs0 f1 nm  m∈N is bounded. Thus µs0 6 f1 and it follows that ℜ(z1 g1 ) = 0. Once we have proved that ℜ(z1 g1 ) = 0 for all (z1 , . . . , zq ) ∈ V , we may use the same argument to prove that ℜ(z2 g2 ) = 0 for all (z1 , z2 , . . . , zq ) and proceed iteratively to complete the proof for all p.  We are now read to prove the technical lemma used in the proof of Lemma 9.8 Lemma 9.7. Let {fp }qp=1 and {θp }qp=1 be 2 families of real analytic functions defined on (−1, 1) such that, for all t ∈ (−1, 1), 1 > |f1 (t)| > · · · > |fq (t)| > 0 and θ̇q (0) = 0 {gp }qp=1 Let be a family of complex valued analytic functions defined on (−1, 1) so that gq (0) ∈ R \ {0}. For all n ∈ N, let Fn = 1 + q X fpn ℜ(einθp gp ). p=1 If there exists a constant K such that for all large enough n, F˙n (0) = KFn (0) log(Fn (0)). Then, f˙q (0) = 0. Proof. We first notice that it suffices to prove the lemma in the restricted setting where fp (t) > 0 for all p and all t. In general, we can then replace each fp with fp2 and each θp with 2θp and apply the restricted form of the lemma to conclude that d 2 ˙ dt t=0 fq = 0, which implies that fq (0) = 0. For the remainder of the proof, we will assume that fp (t) > 0 for all p and all t. Let g(x) = K(1 + x)log(1 + x). Then g is analytic at 0. Consider the expansion X g(x) = am xm n>0 with radius of convergence δ > 0. Notice that there exists N such that if n ≥ N , then q X δ fp (0)n |gp (0)| < . 2 p=1 THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS If n ≥ N , then KFn (0) log(Fn (0)) = g q X n fp (0) ℜ(e inθp (0) p=1 = X am m>0 q X ! 57 gp (0)) !m fp (0)n ℜ(einθp (0) gp (0)) p=1 . If we expand this out, for each q-tuple of non-negative integers m ~ = (m1 , . . . , mq ), we get a term of the form   
m1 + · · · + mq  q q mp n Πp=1 (ℜ(gp (0)einθp (0) )mp . (47) am1 +···+mq Πp=1 fp (0) m1 m2 · · · mq Let q mp < 1. hm ~ = Πp=1 fp (0) Using the equality ℜ(z(w + w̄)) = 2ℜ(z)ℜ(w) repeatedly, we may rewrite the term in (47) in the form n inθ1 (0) hm , . . . , einθq (0) ) ~ (e ~ ℜ(Hm where P Hm ~ is a complex polynomial in q variables and their conjugates. Since the n series m ~ k is convergent for all n ≥ N , we are free to re-arrange the terms. ~ hm ~ kHm We group all terms where the coefficient hm ~ agrees (of which there are only finitely many for each value of hm ) and order the resulting terms in decreasing order of ~ co-efficient to express ∞ X KFn (0) log(Fn (0)) = hns ℜ(Hs (einθ1 , . . . , einθq )), s=0 where each Hs is a complex polynomial in q variables and their conjugates and {hs }s∈N is a strictly decreasing sequence of positive numbers less than 1. Moreover, for all n ≥ N the series ∞ X hns kHs k s=0 is convergent. On the other hand, F˙n (0) = q X p=1 nfpn ℜ e inθp gp f˙p + iθ̇p fp !! + q X fpn ℜ(einθp ġp ) p=1 where all functions on the right hand side are evaluated at 0. Since F˙n (0) = KFn (0) log(Fn (0)) we see that !! q ∞ X X f˙p n inθp gp nfp ℜ e = hns ℜ(Hs (einθ1 , . . . , einθq )). + iθ̇p f p p=1 s=1 The previous lemma then implies that for all p !! f˙p ℜ gp =0 + iθ̇p fp Since gq (0) is a non zero real number, fq (0) 6= 0 and θ̇q (0) = 0, we get that f˙q (0) = 0.  58 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO 9.4. Degenerate vectors have log-type zero. Proposition 9.1 then follows from the following lemma and Lemma 9.3. Lemma 9.8. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). If {ρu }u∈D is an analytic family of projective Anosov G-generic homomorphisms defined on a disc D and L(α) has log-type K at v ∈ Tz D for all infinite order α ∈ Γ, then Dz L(α)(v) = 0 for all infinite order α ∈ Γ. Proof. Notice that if we replace the family {ρu }u∈D by a conjugate family {ρ′u = gu ρu gu−1 }u∈D where {gu }u∈D is an analytic family of elements of SLm (R), then L(α)(ρu ) = L(α)(ρ′u ) for all u ∈ D. Therefore, we are free to conjugate our original family when proving the result. By Proposition 2.21, we may choose β ∈ Γ, so that ρu (β) is generic. After possibly restricting to a smaller disk about z, we may assume that ρu (β) is generic for all u ∈ D. We may then conjugate the family so that ρu (β) lies in the same maximal torus for all u, we can write n n ρu (β ) = L(β) p + q−1 X p=1 λnp (cos(nθp )pp + sin(nθp )pbp ) + 1 q, L(β −1 )n where L(β), L(β −1 ), λp , and θp are analytic functions of u and |L(β)(u)| > |λ1 (u)| > |λ2 (u)| > · · · > |λq−1 (u)| > 1 |L(β −1 (u)| >0 for all u ∈ D. Choose an infinite order element α ∈ Γ which is coprime to β. Proposition 9.4, implies that, for all n, n    T(p(α), β n ) Tr(p(ρ(α))q)) 1 = 1 + L(β n )T(p(α), p(β)) L(β)L(β −1 ) T(p(α), p(β))      q−1 n X λp Tr(p(ρ(α))pbp ) Tr(p(ρ(α))pp ) + . ℜ einθp +i L(β) T(p(α), p(β)) T(p(α), p(β)) p=1 has log-type K at v, since the numerator has log-type K at v and the denominator is a product of two functions which have log-type K at v. Since α and β are coprime and ρ is projective Anosov, ξ(β − ) ⊕ θ(α− ) = Rm , so Tr(p(ρ(α)), q) 6= 0 (since p(ρ(α)) is a projection onto the line ξ(α+ ) parallel to θ(α− ) and q = q(ρ(β)) is a projection onto the line ξ(β − )). Similarly, T(p(α), p(β)) 6= 0, since ξ(β + ) ⊕ θ(α− ) = Rm . Let {us }s∈(−1,1) be a smooth family in D so that u0 = z and u̇0 = v. We now apply Lemma 9.7, taking fp (s) gp (s) λp (us ) , L(β)(u s)   Tr(p(ρus (α))pbp ) Tr(p(ρus (α))pp ) , +i = T(p(α), p(β))(us ) T(p(α), p(β))(us ) = if p = 1, . . . , q − 1, and taking fq (s) = gq (s) = 1 , L(β)(us )L(β −1 )(us ) Tr(p(ρus (α))q) , and T(p(α), p(β))(us ) THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS θq (s) = 59 0. We conclude from Lemma 9.7 that f˙q = 0. Thus Dz L(β)(v) · L(β −1 )(z) = −L(β)(z) · Dz L(β −1 )(v). Since L(β) and L(β −1 (48) ) both have log-type K at v, we get that Dz L(β −1 )(v) Dz L(β)(v) = K log(|L(β)(z)|) and = K log(|L(β −1 )(z)|). L(β)(z) L(β −1 )(z) (49) Combining (48) and (49) we see that K log(|L(β)(z)|) = Dz L(β −1 )(v) Dz L(β)(v) =− = −K log(|L(β −1 )(z)|). L(β)(z) L(β −1 )(z) Since log |L(β)(z)| > 0 and log |L(β −1 )(z)| > 0, this implies that K = 0. Therefore, L(α) has log-type 0 at v for all infinite order α ∈ Γ, so Dz L(α)(v) = 0 for all infinite order α ∈ Γ.  10. Variation of length and cohomology classes The aim of this section is to prove the following proposition. Proposition 10.1. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). Suppose that η : D → Hom(Γ, G) is an analytic map such that for each u ∈ D, η(u) = ρu is irreducible, projective Anosov, and G-generic. If v ∈ Tz D and Dz L(α)(v) = 0 for all infinite order elements α ∈ Γ, then Dz η(v) defines a zero cohomology class 1 in Hη(z) (Γ, g). 1 We recall that Dz η(v) defines a zero cohomology class in Hη(z) (Γ, g) if and only r if it is tangent to the orbit Gη(z) in Hom(Γ, G) ⊂ G . Propositions 9.1 and 10.1 together imply that the pressure form is non-degenerate on Cg (Γ, G). More generally, we obtain the following corollary. Corollary 10.2. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). Suppose that η : D → Ceg (Γ, G) is an analytic map and p is the associated pressure form on D. If v ∈ Tz D and p(v, v) = 0, 1 then Dz η(v) defines a zero cohomology class in Hη(z) (Γ, g). In the course of the proof of Proposition 10.1 we also obtain the following fact which is of independent interest. Proposition 10.3. Suppose that G is a reductive subgroup of SLm (R) and ρ ∈ Cg (Γ, G). Then the set {Dρ L(α) | α infinite order in Γ} , generates the cotangent space T∗ρ Cg (Γ, G). Both propositions will be established in section 10.3. 60 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO 10.1. Invariance of the cross-ratio. We recall the definition of the cross ratio of a pair of hyperplanes and a pair of lines. First define 2 RP(m)(4) = {(ϕ, ψ, u, v) ∈ RP(m)∗ × RP(m)2 : (ϕ, v) and (ψ, u) span Rm }. We then define b : RP(m)(4) → R by b(ϕ, ψ, u, v) = hϕ|ui hψ|vi . hϕ|vi hψ|ui Notice that for this formula to make sense we must make choices of elements in ϕ, ψ, u, and v, but that the result is independent of our choices. If ρ is a projective Anosov representation with limit curves ξ : ∂∞ Γ → RP(m) and θ : ∂∞ Γ → RP(m)∗ , we define the associated cross ratio on ∂∞ Γ(4) , as in [42], to be bρ (x, y, z, w) = b(θ(x), θ(y), ξ(z), ξ(w)). (50) We first derive a formula for the cross-ratio at points associated to co-prime elements. This formula generalizes the formula in Corollary 1.6 from Benoist [4]. Proposition 10.4. If ρ : Γ → SLm (R) is a projective Anosov representation and α and β are infinite order co-prime elements of Γ, then L(αn β) . n→∞ L(α)n bρ (α− , β − , β + , α+ ) = T(p(α), p(β)) = lim Proof. Choose a+ ∈ ξ(α+ ), a− ∈ θ(α− ), b+ ∈ ξ(β + ) and b− ∈ θ(β − ). Observe that p(α)(u) = ha− |ui + a . ha− |a+ i for all u ∈ Rm . In particular, p(β)p(α)(u) = hb− |a+ i ha− |a+ i hb− |b+ i ha− |ui b+ . Therefore, T(p(α)p(β)) = ha− |b+ i hb− |a+ i = bρ (α− , β − , β + , α+ ). ha− |a+ i hb− |b+ i The last equality in the formula follows immediately from Proposition 9.4.  As a corollary, we see that if L(α) has log-type zero for all infinite order α ∈ Γ, then the cross-ratio also has log-type zero. Corollary 10.5. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). Suppose that {ρu : Γ → G}u∈D is an analytic family of projective Anosov G-generic homomorphisms parametrized by a disc D. If L(α) has log-type 0 at v ∈ Tz D for all infinite order α ∈ Γ, then for all distinct collections of points x, y, z, w ∈ ∂∞ Γ, the function u 7→ bρu (x, y, z, w), is of log-type 0 at v. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 61 Proof. Suppose that α, β ∈ Γ have infinite order. Propositions 9.4 and 10.4 imply that bρ (α− , β − , β + , α+ ) has log-type 0. Since pairs of fixed points of infinite order elements are dense in ∂∞ Γ(2) and ξu and θu vary analytically by Proposition 6.1, we see that ρ 7→ bρ (x, y, z, w), has log-type 0 for all pairwise distinct x, y, z, w ∈ ∂∞ Γ.  10.2. An useful immersion. We define a mapping from PSLm (R) into a quotient W(m) of the vector space Mm+1 of all (m + 1) × (m + 1)-matrices and use it to encode a collection of cross ratios. Consider the action of the multiplicative group (R \ {0})2(m+1) on Mm+1 given by (a0 , . . . , am , b0 , . . . , bm )(Mi,j ) = (ai bj Mi,j ). We denote the quotient by W(m) = Mm+1 /(R \ {0})2(m+1) . Given a projective frame F = (x0 , . . . , xm ) for RP(m) and a projective frame F ∗ = (X0 , . . . , Xm ) for the dual RP(m)∗ , let • x̂i be non zero vectors in xi , such that 0= m X x̂i , (51) i=0 • X̂i be non zero covectors in Xi such that m X X̂i . 0= (52) i=0 Observe that x̂i , respectively X̂i , are uniquely defined up to a common multiple. Then, the mapping µF,F ∗ : PSLm (R) → W(m) given by µF,F ∗ : A 7→ X̂i (A(x̂j )) is well defined, independent of the choice of x̂i and X̂i . Lemma 10.6. The mapping µF,F ∗ is a smooth injective immersion. Proof. Since µF,F ∗ (A) determines the projective coordinates of the image of the projective frame (x0 , . . . , xn ) by A, µF,F ∗ is injective. Let µ = µF,F ∗ . Let {At }t∈(−1,1) be a smooth one-parameter family in PSLm (R) such that Ȧ ∈ TA0 (PSLm (R)) and Dµ(Ȧ) = 0. Let {X̂it }t∈(−1,1) and {x̂tj }t∈(−1,1) be time dependent families of covectors in Xi and vectors xj respectively, and let ati,j = X̂it (At (x̂tj )). If Dµ(Ȧ) = 0, then there exists λi and µj such that ȧi,j = λi ai,j + µj ai,j . 62 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Multiplying each X̂it by e−λ0 t and each x̂ti by e−µ0 t has the effect of replacing λi and µj by λi − λ0 and µj − µ0 respectively. Thus, we may assume that λ0 = µ0 = 0. We now use the normalization (51) and (52), to see that m X λi ai,j = 0 = m X µj ai,j . j=1 i=1 On the other hand, since the collections of vectors {vi = (ai,j )16j6m } and {wj = (ai,j )16i6m } are linearly independent, this implies that λi = µj = 0 for all i and j.  The following lemma relates the immersion µ and the cross ratio. Lemma 10.7. Let {x0 , . . . , xm } and {y0 , . . . , ym } be collections of m + 1 pairwise distinct points in ∂∞ Γ. Suppose that ρ : Γ → SLm (R) is projective Anosov with limit maps ξ and θ and that F F∗ = (ξ(x0 ), . . . , ξ(xm )), = (θ(y0 ), . . . , θ(ym )). are projective frames for RP(m) and RP(m)∗ . If α ∈ Γ, then µF,F ∗ (πm (ρ(α))) = [bρ (yi , z, α(xj ), w)] where z and w are arbitrary points in ∂∞ Γ. Proof. Choose, for each i = 0, . . . , m, φi ∈ θ(yi ) and vi ∈ ξ(xi ), and choose φ ∈ θ(z) and v ∈ ξ(w). Then µF,F ∗ (πm (ρ(α))) = [hφi |α(vj )i] while   hφi |α(vj )i hφ|vi [bρ (yi , z, α(xj ), w)] = . hφi |vi hφ|α(vj )i The equivalence is given by taking ai = hφ|vi hφi |vi and bj = 1 hφ|α(vj )i .  10.3. Vectors with log type zero. Propositions 10.1 and 10.3 follow from Proposition 9.1 and the following lemma. Lemma 10.8. Let Γ be a word hyperbolic group and let G be a reductive subgroup of SLm (R). Suppose that η : D → Hom(Γ, G) is an analytic map such that for each u ∈ D, η(u) = ρu is irreducible, projective Anosov and G-generic. Suppose that v ∈ Tz D and that Dz L(α)(v) = 0 for all infinite order α ∈ Γ. Then the cohomology 1 class of Dη(v) vanishes in Hη(z) (Γ, g). Proof. Let {ut }t∈(−1,1) be a path in D so that u0 = z and u̇0 = v. Let ρt = ρut . By Corollary 10.5, d (bρt (x, y, z, w)) = 0 dt t=0 for any pairwise distinct (x, y, z, w) in ∂∞ Γ. Lemma 2.18 allows us to choose collections {x0 , . . . , xm } and {y0 , . . . , ym } of pairwise distinct points in ∂∞ Γ such that if Ft Ft∗ = = (ξt (x0 ), . . . ξt (xm )), (θt (y0 ), . . . θt (ym )). then F0 and F0∗ are both projective frames. For some ǫ > 0, Ft and Ft∗ are projective frames for all t ∈ (−ǫ, ǫ). (We will restrict to this domain for the remainder of the THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 63 argument.) We may then normalize, by conjugating ρt by an appropriate element of SLm (R), so that Ft = F0 for all t ∈ (−ǫ, ǫ). Let µt = µFt ,Ft∗ ◦ πm . Then, by Lemma 10.7, µt (ρt (α)) = [bρt (xi , z, α(yj ), w)] for all α ∈ Γ. Therefore, d µt (ρt (α)) = 0. dt t=0 for all α ∈ Γ. Notice that if χ and χ∗ are projective frames, then µχ,B ∗ χ∗ (A) = µχ,χ∗ (B −1 ◦ A), for all A, B ∈ SLm (R). If we choose Ct ∈ SLm (R) so that (Ct−1 )∗ (Ft∗ ) = F0∗ , then 0 = = d d (µt (ρt (α))) = (µ0 (Ct ρt (α))) dt t=0 dt t=0   d Dµ0 (Ct ◦ ρt (α)) . dt t=0 Lemma 10.6 implies that µ0 is an immersion, so d dt (Ct ◦ ρt (α)) = 0 t=0 Thus, C0 ◦ d dt ρt (α) + Ċ0 ◦ ρ(α) = 0. (53) t=0 Taking α = id in Equation (53), we see that Ċ0 = 0. Since C0 = I, d dt ρt (α) = 0 t=0 1 for all α ∈ Γ. Therefore the cohomology class of Dη(v) vanishes in Hη(z) (Γ, slm (R)). ⊥ 1 Since G is a reductive subgroup of SLm (R), slm R = g ⊕ g , so Hη(z) (Γ, g) injects 1 1 into Hη(z) (Γ, sln (R)). Therefore, Dη(v) vanishes in Hη(z) (Γ, g) as claimed.  11. Rigidity results In this section, we establish two rigidity results for projective Anosov representations. We first establish Theorem 1.2 which states that the signed spectral radii determine the limit map of a projective Anosov representation, up to the action of SLm (R), and that they determine the conjugacy class, in GLm (R), of an irreducible projective Anosov representation. Theorem 11.1. [Spectral rigidity] Let Γ be a word hyperbolic group and let ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) be projective Anosov representations such that L(γ)(ρ1 ) = L(γ)(ρ2 ) for all infinite order γ ∈ Γ. Then there exists g ∈ GLm (R) such that g ◦ ξ1 = ξ2 . Moreover, if ρ1 is irreducible, then ρ2 = gρ1 g −1 . 64 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO We next establish our rigidity result for renormalised intersection. If H is a Lie group, denote by Z(H) its center and by H0 the connected component of the identity. We denote by πm the projection from SLm (R) to PSLm (R). If H ⊂ SLm (R) denote by PH = πm (H) the projectivised group. Finally, if ρ : Γ → SLm (R) is a representation, denote by Gρ the Zariski closure of ρ(Γ). Theorem 11.2. [Intersection rigidity] Let Γ be a word hyperbolic group and let ρ1 : Γ → SLm1 (R) and ρ2 : Γ → SLm2 (R) be projective Anosov representations such that J(ρ1 , ρ2 ) = 1. If Gρ1 and Gρ2 are connected, then there exists an isomorphism σ : Gρ1 /Z(Gρ1 ) → Gρ2 /Z(Gρ2 ) such that σ ρ̄1 = ρ̄2 , where ρ̄i : Γ → Gρi /Z(Gρi ) is the composition of ρi and the projection of Gρi onto Gρi /Z(Gρi ). Remarks: (1) If either Gρ1 or Gρ2 is not connected, then Theorem 11.2 holds for the finite index subgroup −1 0 0 Γ0 = Γ ∩ ρ−1 1 (Gρ1 ) ∩ ρ2 (Gρ2 ). Indeed, each ρi |Γ0 is again projective Anosov (see [26, Cor. 3.4]), and Corollary 8.2 implies that J(ρ1 |Γ0 , ρ2 |Γ0 ) = 1. (2) Consequently, if G0ρ1 and G0ρ2 are not isomorphic, then Theorem 11.2 implies that J(ρ1 , ρ2 ) > 1. (3) The representations need not actually be conjugate if J(ρ1 , ρ2 ) = 1. Let ρ : π1 (S) → PSL2 (R) be a Fuchsian representation and let τk : PSL2 (R) → PSLk (R) be the irreducible representation, then J(τn ◦ ρ, τm ◦ ρ) = 1 but τn ◦ ρ and τm ◦ ρ are not conjugate if n 6= m. 11.1. Spectral rigidity. Our spectral rigidity theorem will follow from Proposition 10.4 and work of Labourie [42]. Recall, from Section 10.1, that we defined the cross ratio b of a pair of hyperplanes and a pair of lines. Then, given a projective Anosov representation ρ with limit maps ξ and θ, we defined a cross ratio bρ on ∂∞ Γ(4) by letting bρ (x, y, z, w) = b(θ(x), θ(y), ξ(z), ξ(w)). (54) Labourie [42, Theorem 5.1] showed that if ρ is a projective Anosov representation with limit map ξ, then the dimension dim hξ(∂∞ Γ)i can be read directly from the cross ratio bρ . (In [42], Labourie explicitly handles the case where Γ = π1 (S), but his proof generalizes immediately.) Consider S∗p the set of pairs (e, u) = (e0 , . . . , ep , u0 , . . . , up ) of (p + 1)-tuples in ∂∞ Γ such that ej 6= ei 6= u0 and uj 6= ui 6= e0 when j > i > 0. If (e, u) ∈ S∗p , he defines χpbρ (e, u) = det (bρ (ei , e0 , uj , u0 )). i,j>0 THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 65 Lemma 11.3. If ρ : Γ → SLm (R) is projective Anosov, then dim hξ(∂∞ Γ)i = inf{p ∈ N : χpbρ ≡ 0} − 1. Lemma 4.3 of Labourie [42] extends in our setting to give: Lemma 11.4. If ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) are projective Anosov and bρ1 = bρ2 , then there exists g ∈ GLm (R) such that g ◦ ξ1 = ξ2 . Moreover, if ρ1 is irreducible, then g (πm ◦ ρ1 ) g −1 = πm ◦ ρ2 . Proof. Lemma 11.3 implies that dim hξ1 (∂∞ Γ)i = dim hξ2 (∂∞ Γ)i = p. Choose {x0 , . . . , xp } ⊂ ∂∞ Γ so that {ξ1 (x0 ), . . . , ξ1 (xp )} and {ξ2 (x0 ), . . . , ξ2 (xp )} are projective frames for hξ1 (∂∞ Γ)i and hξ2 (∂∞ Γ)i (see Lemma 2.17). Choose u0 ∈ ξ1 (x0 ) and {ϕ1 , . . . , ϕp } ⊂ (Rm )∗ such that ϕi ∈ θ1 (xi ) and ϕi (u0 ) = 1. One may check that {ϕ1 , . . . , ϕp } is a basis for hθ1 (∂∞ Γ)i . Complete {ϕ1 , . . . , ϕp } to a basis B1 = {ϕ1 , . . . , ϕp , ϕp+1 , . . . , ϕm } m ∗ for (R ) such that ϕi (hξ1 (∂∞ Γ)i) = 0 for all i > p. For y ∈ ∂∞ Γ, the projective coordinates of ξ1 (y) with respect to the dual basis of B1 are given by [. . . : hϕi |ξ1 (y)i : . . .] = [. . . : hϕi |ξ1 (y)i hϕ1 |u0 i : . . .] hϕ1 |ξ1 (y)i hϕi |u0 i which reduces to [bρ1 (x1 , x1 , y, x0 ), . . . , bρ1 (xp , x1 , y, x0 ), 0, . . . , 0]. Now choose v0 ∈ ξ2 (x0 ) and {ψ1 , . . . , ψp } such that ψi ∈ θ2 (xi ) and ψi (v0 ) = 1. One sees that {ψ1 , . . . , ψp } is a basis of hθ2 (∂∞ Γ)i . One can then complete {ψ1 , . . . , ψp } to a basis B2 = {ψ1 , . . . , ψp , ψp+1 , . . . , ψm } m ∗ for (R ) such that ψi (hξ2 (∂∞ Γ)i) = 0 for all i > p. One checks, as above, that if y ∈ ∂∞ Γ, then the projective coordinates ξ2 (y) with respect to the dual basis of B2 are given by [bρ2 (x1 , x1 , y, x0 ), . . . , bρ2 (xp , x1 , y, x0 ), 0, . . . , 0]. We now choose g ∈ GLm (R) so that gϕi = ψi for all i. It follows from the fact that bρ1 (xi , x1 , y, x0 ) = bρ2 (xi , x1 , y, x0 ) for all i 6 p, that g ◦ ξ1 = ξ2 . Assume now that ρ1 is irreducible, so that p = m. Lemma 2.17 implies that there exists a (m+1)-tuple (x0 , . . . , xm ) of points in ∂∞ Γ such that F = (ξ1 (x0 ), . . . , ξ1 (xm )) is a projective frame for RP(m) and F ∗ = (θ1 (x0 ), . . . , θ1 (xm )) is a projective frame for RP(m)∗ . Thus, using the notation of Lemma 10.7, we have that, given arbitrary distinct points z, w ∈ ∂∞ Γ, µF,F ∗ (πm (ρ1 (γ))) = [bρ1 (xi , z, γ(xj ), w)] Similarly µF,F ∗ (g −1 πm (ρ2 (γ))g) = µgF,gF ∗ (πm (ρ2 (γ))) = [bρ2 (xi , z, γ(xj ), w)] Thus, since bρ1 = bρ2 , µF,F ∗ (ρ1 (γ)) = µF,F ∗ (g −1 ρ2 (γ)g). 66 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Since µF,F ∗ is injective, see Lemma 10.6, it follows that g (πm ◦ ρ1 ) g −1 = πm ◦ ρ2 .  We can now prove our spectral rigidity theorem: Proof of Theorem 11.1: Consider two projective Anosov representations ρ1 : Γ → SLm (R) and ρ2 : Γ → SLm (R) such that L(γ)(ρ1 ) = L(γ)(ρ2 ) for all γ ∈ Γ. Suppose that α and β are infinite order, co-prime elements of Γ. Proposition 10.4 implies that bρ1 (β − , α− , α+ , β + ) = = = L(αn β n )(ρ1 ) n→∞ L(α)(ρ1 )n L(β)(ρ1 )n L(αn β n )(ρ2 ) lim n→∞ L(α)(ρ2 )n L(β)(ρ2 )n bρ2 (β − , α− , α+ , β + ). lim Since pairs of fixed points of infinite order elements of Γ are dense in ∂∞ Γ(2) [25] and bρ1 and bρ2 are continuous, we see that bρ1 = bρ2 . Lemma 11.4 implies that there exists g ∈ GLm (R) such that g ◦ ξ1 = ξ2 . If ρ1 is irreducible, then Lemma 11.4 guarantees that g (πm ◦ ρ1 ) g −1 = πm ◦ ρ2 , so πm ◦ (gρ1 g −1 ) = πm ◦ ρ2 . Notice that if A and B are proximal matrices such that π(A) = π(B) and that the eigenvalues of A and B of maximal absolute value have the same sign, then A = B. Therefore, if α is any infinite order element of Γ, gρ2 (α)g −1 = ρ1 (α). It follows that gρ2 g −1 = ρ1 as claimed.  11.2. Renormalized intersection rigidity. Theorem 11.2 follows from Corollary 2.20, Corollary 8.2 and Corollary 11.6 below, which is a consequence of a deep result of Benoist [3]. If G is a real-algebraic semi-simple Lie group, let aG be a Cartan subspace of the + Lie algebra g of G and let a+ G be a Weyl Chamber. Let µG : G → aG be the Jordan projection. Let + ∗ ∗ (a+ G ) = {ϕ ∈ aG : ϕ|aG > 0}. + ∗ + ∗ If ϕ lies in the interior int(a+ G ) of (aG ) , then if v ∈ aG and ϕ(v) = 0, then v = 0. For a subgroup ∆ of G the limit cone L∆ of ∆ is the smallest closed cone in a+ G that contains {µ(g) : g ∈ ∆}. Benoist [3] proved that Zariski dense subgroups have limit cones with non-empty interior. Theorem 11.5. [Benoist] If ∆ is a Zariski dense subgroup of a connected realalgebraic semi-simple Lie group G, then L∆ has non empty interior. Benoist’s theorem implies the following corollary, which was explained to us by J.-F. Quint. This corollary is a stronger version of a result of Dal’Bo-Kim [21] (see also Labourie [47, Prop. 5.3.6]). THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 67 Corollary 11.6. [Quint] Suppose that ∆ is a group, Gρ and Gη are centerfree connected real-algebraic semi-simple Lie groups without compact factors, and ∗ ρ : ∆ → Gρ and η : ∆ → Gη are Zariski dense representations. If there exist ϕ1 ∈ int(a+ Gρ ) ∗ and ϕ2 ∈ int(a+ Gη ) such that for all g ∈ ∆ one has ϕ1 (µGρ (ρ(g))) = ϕ2 (µG2 (η(g))), then η ◦ ρ−1 : ∆ → ∆ extends to an isomorphism Gρ → Gη . Proof. Let H be the Zariski closure of the image of the product representation ρ × η : ∆ → Gρ × Gη , defined by g 7→ (ρg, ηg). Since the equation ϕ1 (µGρ (g1 )) = ϕ2 (µGη (g2 )) (55) holds for every pair (g1 , g2 ) ∈ ρ × η (∆), Benoist’s [3] Theorem 11.5 implies that the same relation holds for every pair (g1 , g2 ) ∈ H. The group H∩(Gρ ×{e}) is a normal subgroup of Gρ , it is hence (up to finite index) a product of simple factors. Equation (55) implies that for all (g, e) ∈ H∩(Gρ × {e}) necessarily one has ϕ1 (µGρ g) = 0. Since ϕ1 (v) > 0 for all v ∈ a+ Gρ − {0}, one has µGρ (g) = 0. This implies that H ∩ (Gρ × {e}) is a normal compact subgroup of Gρ . Since Gρ does not have compact factors and is center free one concludes that H ∩ (Gρ × e) = {e}. The same argument implies that H ∩ ({e} × Gη ) = {e} and hence H is the graph of an isomorphism extending η ◦ ρ−1 .  11.3. Rigidity for Hitchin representations. O. Guichard [27] has announced a classification of the Zariski closures of lifts of Hitchin representations. Theorem 11.7. [Guichard] If ρ : π1 (S) → SLm (R) is the lift of a Hitchin representation and H is the Zariski closure of ρ(π1 (S)), then • If m = 2n is even, H is conjugate to either τm (SL2 (R)), Sp(2n, R) or SL2n (R). • If m = 2n + 1 is odd and m 6= 7, then H is conjugate to either τm (SL2 (R)), SO(n, n + 1) or SL2n+1 (R). • If m = 7, then H is conjugate to either τ7 (SL2 (R)), G2 , SO(3, 4) or SL7 (R). where τm : SL2 (R) → SLm (R) is the irreducible representation. Notice in particular, that the Zariski closure of the lift of a Hitchin representation is always simple and connected. We can then apply our rigidity theorem for renormalized intersection to get a rigidity statement which is independent of dimension in the Hitchin setting. Corollary 11.8. [Hitchin rigidity] Let S be a closed, orientable surface and let ρ1 ∈ Hm1 (S) and ρ2 ∈ Hm2 (S) be two Hitchin representations such that J(ρ1 , ρ2 ) = 1. Then, • either m1 = m2 and ρ1 = ρ2 in Hm1 (S), • or there exists an element ρ of the Teichmà 14 ller space T (S) so that ρ1 = τm1 (ρ) and ρ2 = τm2 (ρ). Observe that the second case in the corollary only happens if both ρ1 and ρ2 are Fuchsian. 68 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO Proof. In order to apply our renormalized intersection rigidity theorem, we will need the following analysis of the outer automorphism groups of the Lie algebras of Lie groups which arise as Zariski closures of lifts of Hitchin representations. This analysis was carried about by Gündoğan [28] (see Corollary 2.15 and its proof). Theorem 11.9. [Gündoğan [28]] Let Out(g) be the group of exterior automorphism of the Lie algebra g. Then, if n > 0, 2 (1) If g = sl2n+2 (R), then Out(g) is isomorphic to (Z/2Z) and is generated by X 7→ −X t , and conjugation by an element of GL2n+2 (R). (2) If g = sl2n+1 (R), then Out(g) is isomorphic to Z/2Z and is generated by X 7→ −X t . (3) If g = so(n, n + 1, R), then Out(g) is isomorphic to Z/2Z and is generated by conjugation by an element of SL2n+1 (R). (4) If g = sp(2n + 2, R), then Out(g) is isomorphic to Z/2Z and is generated by conjugation by an element of GL2n+2 (R). (5) If g = g2 then Out(g) is trivial. (6) If g = sl2 (R), then Out(g) is isomorphic to Z/2Z and is generated by conjugation by an element of GL2 (R). (7) If g = so(n, 1, R), then Out(g) is isomorphic to Z/2Z and is generated by conjugation by an element of GLn+1 (R). Let ρ1 : π1 (S) → PSLm1 (R) and ρ2 : π1 (S) → PSLm2 (R) be two Hitchin representations such that J(ρ1 , ρ2 ) = 1. Theorem 11.7 implies that Gρ1 and Gρ2 are simple and connected and have center contained in {±I}. Theorem 11.2 implies that there exists an isomorphism σ : Gρ1 → Gρ2 such that ρ2 = σ ◦ ρ1 . If G1 is not conjugate to τm1 (SL2 (R)), then it follows from Theorem 11.7, that m1 = m2 = m, and that, after conjugation of ρ1 , Gρ1 = Gρ2 = H so that σ is an automorphism of H. We first observe that, since H is connected, there is an injective map from Out(H) to Out(h). We now analyze the situation in a case-by-case manner using Gündoğan’s Theorem 11.9. (1) If H = PG2 , then σ is an inner automorphism, so ρ1 = ρ2 in H7 (S). (2) If H = PSO(n, n + 1) or H = PSp(2n, R), σ is either the identity or the conjugation by an element of PGL2n+1 (R) or PGL2n (R), so ρ1 = ρ2 in H2n+1 (S) or H2n (S). (3) If H = SLm (R), then, after conjugation of ρ1 by an element of PGLm (R), σ is either trivial or ρ2 = η ◦ ρ1 where η(g) = transpose(g −1 ). If σ is non-trivial, then since J(ρ1 , ρ2 ) = 1 Corollary 8.2 implies that there exists c > 0 so that cµ1 (ρ1 (γ)) = µ1 ((ρ2 (γ)) = −µm (ρ1 (γ)) for all γ ∈ Γ, where (µ1 , . . . , µm ) : SLm (R) → {(a1 , . . . , am ) ∈ Rm : X ai = 0 and a1 > · · · > am } is the Jordan projection of SLm (R). Thus, the limit cone of ρ1 (Γ) has empty interior. Since ρ1 (Γ) is Zariski dense, this contradicts Benoist’s Theorem 11.5. Therefore, ρ1 = ρ2 in Hm (S) in this case as well. (4) If Gρ1 is conjugate to τm1 (SL2 (R)), then Gρ2 is conjugate to τm2 (SL2 (R)). So, after conjugation, there exist Fuchsian representations, η1 : π1 (S) → SL2 (R) THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 69 and η2 : π1 (S) → SL2 (R), such that ρ1 = τm1 ◦ η1 , ρ2 = τm1 ◦ η1 and there exists an automorphism of σ of SL2 (R) such that σ ◦ η1 = η2 . Case (6) of Gündoğan’s Theorem then implies that η1 is conjugate to η2 by an element of GL2 (R). Therefore, we are in the second case of Theorem 11.8. This completes the proof.  11.4. Benoist representations. We say that an open subset Ω of RP(m) is properly convex if its intersection with any projective line is connected and its closure Ω̄ is contained in the complement of a projective hyperplane. Moreover, a properly convex open set Ω is said to be strictly convex if its boundary ∂Ω does not contain a projective line segment. A subgroup ∆ ⊂ Aut(Ω) = {g ∈ PGLm (R) : gΩ = Ω} is said to divide the open properly convex set Ω if the quotient ∆\Ω is compact. Benoist [6, Thm. 1.1] proved that if ∆ divides the properly convex open set Ω, then Ω is strictly convex if and only if ∆ is hyperbolic. Definition 11.10. If Γ is a torsion-free hyperbolic group, a faithful representation ρ : Γ → PGLm (R) is a Benoist representation if ρ(Γ) divides an open strictly convex set Ω ⊂ RP(m). It is a consequence of Benoist’s work [6] that a Benoist representation is irreducible and projective Anosov (see Guichard-Wienhard [26, Proposition 6.1] for a detailed explanation). Benoist [7, Corollary 1.2] (see also Koszul [40]) proved that the space Bm (Γ) of Benoist representations of Γ into PSLm (R) is a collection of components of Hom(Γ, PSLm (R)). Let Bm (Γ) = Bm (Γ)/PGLm (R). We call the components of Bm (Γ) Benoist components. Benoist [5, Theorem 1.3] proved that the Zariski closure of any Benoist representation is either PSLm (R) or is conjugate to PSO(m − 1, 1). We may thus apply the technique of proof of Theorem 11.8 to prove: Corollary 11.11. [Benoist rigidity] Let ρ1 , ρ2 ∈ Bm (Γ). If J(ρ1 , ρ2 ) = 1, then ρ1 = ρ2 in Bm (Γ). The same techniques also provide the following related rigidity result for Benoist representations. Observe that if ρ is a projective Anosov representation, then so is Ad ρ : Γ → PGL(sl(m, R)) (see the discussion in Guichard-Wienhard [26, Section 10.2]) If η(g) = (g −1 )t for all g ∈ PGLm (R), and ρ ∈ Bm (Γ), then η ◦ ρ is the dual (or contragredient) representation of ρ. Corollary 11.12. If ρ1 , ρ2 ∈ Bm (Γ), then J(Ad ρ1 , Ad ρ2 ) = 1 if and only if either ρ1 = ρ2 or ρ2 = η ◦ ρ1 . As a consequence, we recover a result of Cooper-Delp [18] and Kim [38] which asserts that if ρ1 , ρ2 ∈ Bm (Γ) are the holonomies of strictly convex projective structures with the same Hilbert marked length spectrum, then ρ1 and ρ2 either agree or are dual. Recall that if ρ ∈ Bm (Γ) and γ ∈ Γ, then the length, in the Hilbert metric, of the closed geodesic on ρ(Γ)\Ωρ associated to [γ] is µ1 (ρ(γ)) − µm (ρ(γ)) 2 70 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO (see, for example, Benoist [6, Proposition 5.1]). Furthermore, if g ∈ PGLm (R) then log(Λ(Ad g)) = µ1 (g) − µm (g). Hence if ρ1 and ρ2 are the holonomies of strictly convex projective structures with the same Hilbert marked length spectrum, then Λ(Ad ρ1 (γ)) = Λ(Ad ρ2 (γ)) for all γ ∈ Γ. Hence, J(Ad ρ, Ad ρ2 ) = 1, so the result follows from Corollary 11.12. 12. Proofs of main results In this section, we assemble the proofs of the results claimed in the introduction. Several of the results have already been established. The inequality in Theorem 1.1 follows from Corollary 8.2 and rigidity follows from Theorem 11.2. Theorem 1.2 is proven in Section 11 as Theorem 11.1, while Corollary 1.5 is proven as Corollary 11.8. Theorem 1.3 follows from Proposition 8.1 and Corollary 8.2. Theorem 1.10 combines the results of Propositions 4.1 and 5.7. The proof of Theorem 1.4 is easily assembled. Proof of Theorem 1.4: Consider the pressure form defined on Cg (Γ, G) as in Definition 8.3. Recall that by Corollary 8.2 the pressure form is non-negative. Moreover, by Corollary 10.2 the pressure form is positive definite, so gives a Riemannian metric. The invariance with respect to Out(Γ) follows directly from the definition. Proof of Corollary 1.6: Corollary 7.6 implies that every Hitchin component lifts to a component of Cg (π1 (S), SLm (R)) which is an analytic manifold. Theorem 1.4 then assures that the pressure form is an analytic Riemannian metric which is invariant under the action of the mapping class group. Entropy is constant on the Fuchsian locus, so if ρ1 , ρ2 ∈ T (S), the renormalized intersection has the form X log Λ(τm ◦ ρ2 )(γ) 1 J(τm ◦ ρ1 , τm ◦ ρ2 ) = lim T →∞ #(Rτm ◦ρ1 (T )) log Λ(τm ◦ ρ1 )(γ) [γ]∈Rτm ◦ρ1 X log Λ(ρ2 )(γ) 1 = lim T →∞ #(Rρ1 (T )) log Λ(ρ1 )(γ) [γ]∈Rρ1 Wolpert [68] showed that the Hessian of the final expression, regarded as a function on T (S), is a multiple of the Weil-Petersson metric (see also Bonahon [11] and McMullen [53, Theorem 1.12]). Proof of Corollary 1.7: We may assume that Γ is the fundamental group of a compact 3-manifold with non-empty boundary, since otherwise Cc (Γ, PSL2 (C)) consists of 0 or 2 points. We recall, from Theorem 7.5, that the deformation space Cc (Γ, PSL2 (C)) is an analytic manifold. Let α : PSL2 (C) → SLm (R) be the Plücker representation given by Proposition 2.13. If we choose co-prime infinite order elements α and β of Γ, we may define a global analytic lift ω : Cc (Γ, PSL2 (C)) → Hom(Γ, PSL2 (C)) by choosing ω([ρ]) to be a representative ρ ∈ [ρ] so that ρ(α) has attracting fixed point 0 and repelling fixed point ∞ and ρ(β) has attracting fixed point 1. Then A = α ◦ ω : Cc (Γ, PSL2 (C)) → Hom(Γ, SLm (R)) THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 71 is an analytic family of projective Anosov homomorphisms. We define the associated entropy h̄ and renormalised intersection J̄ functions on Cc (Γ, PSL2 (C)) by setting h̄([ρ]) = h(A([ρ])) and J̄([ρ1 ], [ρ2 ]) = J(A([ρ1 ]), A([ρ2 )]). Since ω is analytic, both h̄ and J̄ vary analytically over Cc (Γ, PSL2 (C)) and we may again define a non-negative 2-tensor on the tangent space TCc (Γ, PSL2 (C)) which we again call the pressure form, by considering the Hessian of J̄. Let G = α(PSL2 (C)). Then G is a reductive subgroup of SLm (R). If ρ(Γ) is Zariski dense, then A(ρ)(Γ) is Zariski dense in G, so Lemma 2.21 implies that ρ(Γ) contains a G-generic element. Since α is an immersion, 1 (Γ, g) α∗ : Hρ1 (Γ, sl2 (C)) → Hα([ρ]) is injective where g is the Lie algebra of G. Corollary 10.2 then implies that the pressure form on Tρ Cc (Γ, PSL2 (C)) is Riemannian if ρ is Zariski dense. If ρ = ω([ρ]) is not Zariski dense, then its limit set is a subset of R̂ ⊂ Ĉ, and the Zariski closure of ρ(Γ) is either H1 = PSL(2, R) or H2 = PSL(2, R) ∪ (z → −z)PSL(2, R). Since each Hi is a real semi-simple Lie group, Proposition 7.2 then implies that the subset of non-Zariski dense representations in Cc (Γ, PSL2 (C)) is an analytic submanifold. We then again apply Corollary 10.2 to see that the restriction of the pressure form to the submanifold of non-Zariski dense representations is Riemannian. The pressure form determines a path pseudo-metric on the deformation space Cc (Γ, PSL2 (C)), which is a Riemannian metric off the analytic submanifold of nonZariski dense representations and restricts to a Riemannian metric on the submanifold. Lemma 13.1 then implies that the path metric is actually a metric. This establishes the main claim. Theorem 7.5 implies that if Γ is not either virtually free or virtually a surface group, then every ρ ∈ Cc (Γ, PSL2 (C)) is Zariski dense. Auxiliary claim (1) then follows from our main claim. In the case that Γ is the fundamental group of a closed orientable surface, then the restriction of the pressure metric to the Fuchsian locus is given by the Hessian of the intersection form I. It again follows from work of Wolpert [68] that the restriction to the Fuchsian locus is a multiple of the Weil–Petersson metric. This establishes auxiliary claim (2). Proof of Corollary 1.8: Let α : G → SLm (R) be the Plücker representation given by Proposition 2.13. An analytic family {ρu : Γ → G}u∈M of convex cocompact homomorphisms parameterized by an analytic manifold M , gives rise to an analytic family {α ◦ ρu }u∈M of projective Anosov homomorphisms of Γ into SLm (R). Theorem 1.3, and Corollary 2.14 then imply that topological entropy varies analytically for this family. Results of Patterson [56], Sullivan [66], Yue [69] and Corlette-Iozzi [20] imply that the topological entropy agrees with the Hausdorff dimension of the limit set, so Corollary 1.8 follows. Proof of Corollary 1.9: Given a semi-simple real Lie group G with finite center and a non-degenerate parabolic subgroup P, let α : G → SLm (R) be the Plücker representation given by Proposition 2.13. Then H = α(G) is a reductive subgroup of SLm (R). We will adapt the notation of Proposition 7.3. Let b G, P) = Z̃(Γ; G, P)/G0 Z(Γ; 72 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO b G, P) is a finite analytic where G0 is the connected component of G. Then, Z(Γ; manifold cover of the analytic orbifold Z(Γ; G, P) with covering transformations given by G/G0 , see Proposition 7.4. Since G0 acts freely on Z̃(Γ; G, P), the slice b G, P), then there exists a neighborhood U of [ρ] theorem implies that if [ρ] ∈ Z(Γ; and a lift β : U → Z̃(Γ; G, P) ⊂ Hom(Γ, G). Then ω = α◦β is an analytic family of H-generic projective Anosov homomorphisms parameterized by U . The Hessian of the pull-back of the renormalized intersection gives rise to an analytic 2-tensor, again called the pressure form, on TU . Suppose that v ∈ Tz Ũ has pressure norm zero. Then Corollary 10.2 implies that Dω(v) is 1 trivial in Hω(z) (Γ, h) where h is the Lie algebra of H. Since α is an immersion, 1 1 α∗ : Hβ(z) (Γ, g) → Hω(z) (Γ, h) 1 is an isomorphism. Since β∗ identifies Tz U with Hβ(z) (Γ, g) this implies that v = 0, so the pressure form on TU is non-degenerate. Therefore, the pressure form is b G, P). Since the pressure form is invariant an analytic Riemannian metric on Z(Γ; b G, P) This under the action of G/G0 it descends to a Riemannian metric on Z(Γ; completes the proof. 13. Appendix We used the following lemma in the proof of Corollary 1.7. Lemma 13.1. Let M be a smooth manifold and let W be a submanifold of M . Suppose that g is a smooth non negative symmetric 2-tensor g such that • g is positive definite on Tx M if x ∈ M \ W , • the restriction of g to Tx W is positive definite if x ∈ W . Then the path pseudo metric defined by g is a metric. Proof. It clearly suffices to show that if x ∈ M , then there exists an open neighborhood U of M such that the restriction of g to U gives a path metric on U . If x ∈ M \ W , then we simply choose a neighborhood U of x contained in M \ W and the restriction of g to U is Riemannian, so determines a path metric. If x ∈ W we can find a neighborhood U which is identified with a ball B in Rn so that W ∩ U is identified with B ∩ (Rk × {0n−k }). Possibly after restricting to a smaller neighborhood, we can assume that there exists r > 0 so that if v ∈ Tz B and v is tangent to Rk × {(zk+1 , . . . , zn )}, then g(v, v) ≥ r2 ||v||2 , where ||v|| is the Euclidean norm of v. If z, w ∈ B, z 6= w and one of them, say z, is contained in M \ W , then g is Riemannian in a neighborhood of z, so dU,g (z, w) > 0 where dU,g is the path pseudo-metric on U induced by g. If z, w ∈ W , then the estimate above implies that dU,g (z, w) ≥ rdB (z, w) where dB is the Euclidean metric on B. Therefore, dU,g is a metric on U and we have completed the proof.  References [1] L.M. Abramov, “On the entropy of a flow,” Dokl. Akad. Nauk. SSSR 128(1959), 873–875. [2] J.W. Anderson and A. Rocha, “Analyticity of Hausdorff dimension of limit sets of Kleinian groups,” Ann. Acad. Sci. Fenn. 22(1997), 349–364. [3] Y. Benoist, “Propriétés asymptotiques des groupes linéaires,” Geom. Funct. Anal. 7(1997), 1–47. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 73 [4] Y. Benoist, “Propriétés asymptotiques des groupes linéaires II,” Adv. Stud. Pure Math. 26(2000), 33–48. [5] Y. Benoist, “Automorphismes des cônes convexes,” Invent. Math. 141(2000) 149–193. [6] Y. Benoist, “Convexes divisibles I,” in Algebraic groups and arithmetic, Tata Inst. Fund. Res. Stud. Math. 17(2004), 339–374. [7] Y. Benoist, “Convexes divisibles III,” Ann. Sci. de l’E.N.S. 38(2005), 793–832. [8] L. Bers, “Spaces of Kleinian groups,” in Maryland conference in Several Complex Variables I, Springer-Verlag Lecture Notes in Math, No. 155(1970), 9–34. [9] M. Bridgeman, “Hausdorff dimension and the Weil-Petersson extension to quasifuchsian space,” Geom. and Top. 14(2010), 799–831. [10] M. Bridgeman and E. Taylor, “An extension of the Weil-Petersson metric to quasi-Fuchsian space,” Math. Ann. 341(2008), 927–943. [11] F. Bonahon, “The geometry of Teichmüller space via geodesic currents,” Invent. Math. 92(1988), 139–162. [12] R. Bowen, “Periodic orbits of hyperbolic flows,” Amer. J. Math. 94(1972), 1–30. [13] R. Bowen, “Symbolic dynamics for hyperbolic flows,” Amer. J. Math. 95(1973), 429–460. [14] R. Bowen, “Hausdorff dimension of quasi-circles,” Publ. Math. de l’I.H.E.S. 50(1979), 11–25. [15] R. Bowen and D. Ruelle, “The ergodic theory of axiom A flows,” Invent. Math. 29(1975), 181–202. [16] M. Burger, “Intersection, the Manhattan curve and Patterson-Sullivan theory in rank 2,” Internat. Math. Res. Notices 7(1993), 217–225. [17] C. Champetier, “Petite simplification dans les groupes hyperboliques,” Ann. Fac. Sci. Toulouse Math. 3(1994), 161–221. [18] D. Cooper and K. Delp, “The marked length spectrum of a projective manifold or orbifold,” Proc. of the A.M.S. 138(2010), 3361–3376. [19] M. Coornaert and A. Papadopoulos, “Symbolic coding for the geodesic flow associated to a word hyperbolic group,” Manu. Math. 109(2002), 465–492. [20] K.Corlette and A. Iozzi, “Limit sets of discrete groups of isometries of exotic hyperbolic spaces,” Trans. A.M.S. 351(1999), 1507–1530. [21] F. Dal’Bo and I. Kim, “A criterion of conjugacy for Zariski dense subgroups,” Comptes Rendus Math. 330 (2000), 647–650. [22] T. Delzant, O. Guichard, F. Labourie and S. Mozes, “Displacing representations and orbit maps,” in Geometry, rigidity and group actions, Univ. Chicago Press, 2011, 494–514. [23] G. Dreyer, “Length functions for Hitchin representations,” Algebraic and Geometric Topology, to appear, preprint available at: arXiv:1106.6310. [24] M. Darvishzadeh and W. Goldman, “Deformation spaces of convex real projective structures and hyperbolic structures,” J. Kor. Math. Soc. 33(1996), 625–639. [25] M. Gromov, “Hyperbolic groups,” in Essays in Group Theory, MSRI Publ. 8 (1987), 75–263. [26] O. Guichard and A. Wienhard, “Anosov representations: Domains of discontinuity and applications,” Invent. Math. 190(2012), 357–438. [27] O. Guichard, oral communication. [28] H. Gündoğan. “The component group of the automorphism group of a simple Lie algebra and the splitting of the corresponding short exact sequence,” J. Lie Theory 20(2010), 709–737. [29] M. W. Hirsch, C. C. Pugh and M. Shub, Invariant manifolds Lecture Notes in Mathematics, Vol. 583, 1977 [30] N. Hitchin, “Lie groups and Teichmüller space,” Topology 31(1992), 449–473. [31] J. Hubbard, Teichmüler theory and applications to geometry, topology and dynamics. Vol 1., Matrix Editions, Ithaca. [32] J.E. Humphreys Linear algebraic groups, Graduate Text in Mathema- tics 21, Springer Verlag , New York, 1981. [33] D. Johnson and J. Millson, “Deformation spaces associated to compact hyperbolic manifolds,” in Discrete Groups and Geometric Analysis, Progress in Math., vol. 67(1987), 48– 106. [34] M. Kapovich, Hyperbolic manifolds and discrete groups, Progr. Math. 183, Birkhäuser, 2001. [35] A. Katok, G. Knieper, M. Pollicott and H. Weiss, “Differentiability and analyticity of topological entropy for Anosov and geodesic flows,” Invent. Math. 98(1989), 581–597. 74 BRIDGEMAN, CANARY, LABOURIE, AND SAMBARINO [36] A. Katok, G. Knieper, and H. Weiss, “Formulas for the derivative and critical points of topological entropy for Anosov and geodesic flows,” Comm. Math. Phys. 138(1991), 19–31. [37] I. Kim, “Ergodic theory and rigidity on the symmetric space of non-compact type,” Erg. Thy. Dyn. Sys. 21(2001), 93–114. [38] I. Kim, “Rigidity and deformation space of strictly convex real projective structures,” J. Differential Geom. 58(2001), 189–218. [39] I. Kim and G. Zhang, “Kähler metric on Hitchin component”, preprint available at http://arxiv.org/pdf/1312.1965v1.pdf. [40] J.L. Koszul, “Déformation des connexions localement plates,” Ann. Inst. Four. 18(1968), 103–114. [41] F. Labourie, “Anosov flows, surface groups and curves in projective space,” Invent. Math. 165(2006), 51–114. [42] F. Labourie. “Cross Ratios, Surface Groups, SLn (R) and Diffeomorphisms of the Circle,” Publ. Math. de l’I.H.E.S. 106(2007), 139–213. [43] F. Labourie, “Flat projective structures on surfaces and cubic differentials,” P.A.M.Q. 3(2007), 1057–1099. [44] F. Labourie, “Cross ratios, Anosov representations and the energy functional on Teichmüller space,” Ann. Sci. E.N.S. 41(2008), 437–469. [45] F. Labourie “Cyclic surfaces and Hitchin components in rank 2”, preprint available at http://arxiv.org/pdf/1406.4637.pdf. [46] F. Labourie, R. Wentworth “The pressure metric along the fuchsian locus” in preparation. [47] F. Labourie, Lectures on representations of surface groups, Zurich Lectures in Advanced Mathematics, 2013, 145 pages. [48] Q. Li, “Teichmüller space is totally geodesic in Goldman space,” preprint, available at: http://front.math.ucdavis.edu/1301.1442 [49] A.N. Livšic, “Cohomology of dynamical systems,” Math. USSR Izvestija 6(1972). [50] J. Loftin, “Affine spheres and convex RP2 structures,” Amer. J. Math. 123(2001), 255–274. [51] A. Lubotzky and A. Magid, Varieties of representations of finitely generated groups, Mem. Amer. Math. Soc. 58(1985), no. 336. [52] R. Mañé, Teoria Ergódica, Projeto Euclides, IMPA, 1983. [53] C. McMullen, “Thermodynamics, dimension and the Weil-Petersson metric,” Invent. Math. 173(2008), 365–425. [54] I. Mineyev, “Flows and joins of metric spaces,” Geom. Top. 9(2005), 403–482. [55] W. Parry and M. Pollicott, Zeta functions and the periodic orbit structure of hyperbolic dynamics, Astérisque 187-188(1990). [56] S. Patterson, “The limit set of a Fuchsian group,” Acta Math. 136(1976), 241–273. [57] M. Pollicott, “Symbolic dynamics for Smale flows,” Am. J. of Math. 109(1987), 183–200. [58] M. Pollicott and R. Sharp, “Length asymptotics in higher Teichmüller theory,” Proc. A.M.S. 142(2014), 101–112. [59] D. Ragozin, “A normal subgroup of a semisimple Lie group is closed.” Proc. A.M.S. 32(1972), 632–633. [60] D. Ruelle, “Repellers for real analytic maps,” Ergodic Theory Dynamical Systems 2(1982), 99–107. [61] D. Ruelle, Thermodynamic Formalism, Addison-Wesley, London. [62] A. Sambarino, Quelques aspects des représentations linéaires des groupes hyperboliques, Ph.D. Thesis, Paris Nord, 2011. [63] A. Sambarino, “Quantitative properties of convex representations,” Comm. Math. Helv., 89(2014), 443–488. [64] A. Sambarino, “Hyperconvex representations and exponential growth,” Ergod. Th. Dynam. Sys. 34(2014), 986–1010. [65] M. Shub, Global Stability of Dynamical Systems, Springer-Verlag, 1987. [66] D. Sullivan, “The density at infinity of a discrete group of hyperbolic motions,” Inst. Hautes Études Sci. Publ. Math. 50(1979), 171–202. [67] S. Tapie, “A variation formula for the topological entropy of convex-cocompact manifolds,” Erg. Thy. Dynam. Sys. 31(2011), 1849–1864. [68] S. Wolpert, “Thurston’s Riemannian metric for Teichmüller space,” J. Diff. Geom. 23(1986), 143–174. THE PRESSURE METRIC FOR ANOSOV REPRESENTATIONS 75 [69] C. Yue, “The ergodic theory of discrete isometry groups on manifolds of variable negative curvature,” Trans. A.M.S. 348(1996), 4965–5005. Boston College, Chestnut Hill, MA 02467 USA University of Michigan, Ann Arbor, MI 41809 USA Univ. Nice Sophia-Antipolis, Laboratoire Jean Dieudonné, UMR 7351, Nice F-06000; FRANCE Univ. Paris-Sud, Laboratoire de Mathématiques, Orsay F-91405 Cedex; CNRS, Orsay cedex, F-91405 FRANCE