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Cusped Hitchin representations and Anosov representations of geometrically finite Fuchsian groups

We develop a theory of Anosov representation of geometrically finite Fuchsian groups in SL(d,R) and show that cusped Hitchin representations are Borel Anosov in this sense. We establish analogues of many properties of traditional Anosov representations. In particular, we show that our Anosov representations are stable under type-preserving deformations and that their limit maps vary analytically. We also observe that our Anosov representations fit into the previous frameworks of relatively Anosov and relatively dominated representations developed by Kapovich-Leeb and Zhu.


Introduction
In this paper we develop a theory of Anosov representations for geometrically finite Fuchsian groups into SL(d, K) where K is either the field of real numbers or the field of complex numbers (and more generally, into any semisimple Lie group G with finite center). This theory shares two crucial features with Anosov representations of convex cocompact Fuchsian groups. First, if Γ ⊂ PSL(2, R) is a geometrically finite Fuchsian group and ρ : Γ → SL(d, K) is Anosov, then there is a ρ-equivariant quasi-isometric embedding of an orbit of Γ in H 2 into the Riemannian symmetric space X d (K) associated to SL(d, K). Moreover, there is a ρ-equivariant map of the limit set of Γ into the appropriate (partial) flag variety. Second, small deformations of Anosov representations which preserve the conjugacy class of the images of parabolic elements in Γ remain Anosov. We further show that these limit maps vary analytically in ρ. We observe that our representations are relatively Anosov representations in the sense of Kapovich and Leeb [20] and relatively dominated representations as defined by Zhu [36]. Our concrete setting allows for simpler proofs and more explicit results. Our main motivation was to study the class of cusped Hitchin representations. A representation ρ : Γ → SL(d, R) of a geometrically finite Fuchsian group is said to be cusped Hitchin if there exists a continuous ρ-equivariant positive map from the limit set of Γ into the full flag variety F d . We show that cusped Hitchin representations are irreducible and Borel Anosov (i.e. they are P k -Anosov for all 1 ≤ k ≤ d−1). This generalizes results of Labourie [24], when Γ is cocompact, and Labourie-McShane [25] (see also Burelle-Treib [10]), when Γ is convex cocompact. (Recent results of Sambarino [32] also imply that cusped Hitchin representations are irreducible). Fock and Goncharov [16] introduced the theory of positive representations. They consider the case where Γ is not cocompact but has cofinite area, and they only require that the limit map be defined on fixed points of peripheral elements of Γ. We show that type-preserving positive representations, in their sense, are in fact cusped Hitchin representations. Other examples of cusped Anosov representations are provided by exterior powers of cusped Hitchin representations and direct products of cusped Hitchin representations with trivial representations.
In turn, the motivation for studying cusped Hitchin representations arises from an intriguing potential analogy with the augmented Teichmüller space from classical Teichmüller theory. The augmented Teichmuller space of a closed orientable surface S is obtained by appending to the Teichmüller space of S points corresponding to (marked) finite area hyperbolic structures on the complement in S of any multicurve on S. Masur [30] showed that the augmented Teichmüller space is the metric completion of Teichmüller space with the Weil-Petersson metric. Loftin and Zhang [26] construct an analytic model for an augmented Hitchin component. Bray, Canary, Kao and Martone [6] combine our work with thermodynamical results from [5] to construct pressure metrics on deformation spaces of cusped Hitchin representations, generalizing work of Bridgeman, Canary, Labourie and Sambarino [8]. The hope is that this will allow us to investigate whether the augmented Hitchin component is the metric completion of the Hitchin component with the pressure metric. For further discussion of the conjectural geometric picture see the survey paper [12].
We now turn to a more detailed discussion of our work. Let Γ ⊂ PSL(2, R) be a geometrically finite group with limit set Λ(Γ) ⊂ ∂H 2 . Suppose that ρ : Γ → SL(d, K) is a representation and that we are given a continuous ρ-equivariant map into the Grassmanians of k-planes and (d − k)-planes in K d . We require that ξ ρ is transverse, i.e.
We obtain an associated splitting and U(Γ) ⊂ T 1 H 2 is the set of tangent vectors which extend to geodesics both of whose endpoints lie in Λ(Γ). This descends to a splitting E ρ = Γ\E ρ = Θ k ρ ⊕ Ξ d−k ρ of the flat bundle associated to ρ over the non-wandering portion U(Γ) = Γ\ U(Γ) of the geodesic flow. The geodesic flow on U(Γ) lifts naturally to a flow on E ρ which preserves the splitting and is parallel to the flat connection. We say that ρ is P k -Anosov if the associated flow on Hom( Ξ d−k ρ , Θ k ρ ) is uniformly contracting with respect to the (operator) norm arising from some family of continuous norms on the fibers of E ρ (see Definition 3.1 for details). In this case, we call ξ ρ the P k -Anosov limit map of ρ. When Γ contains a parabolic element, we will sometimes refer to our P k -Anosov representations as cusped P k -Anosov representations to distinguish them from traditional Anosov representations.
In general, being P k -Anosov is not an open condition in the space of representations of a geometrically finite group. For instance, consider the case where Γ = g 1 , g 2 ⊂ PSL(2, R) is a free group, g 2 is parabolic, and ρ : Γ → SL(4, R) is P 1 -Anosov with Next define a family of representations ρ t where ρ t (g 1 ) = ρ(g 1 ) and Then ρ t is not P 1 -Anosov for any t = 0 since the sequence {ρ t (g 2 ) n } does not converge to a rank one element of P(End(R 4 )). Bowditch [3,Sec. 5] gave an example of a sequence {ρ n } of indiscrete representations of a free group on two generators into SO(4, 1) which converge to a geometrically finite representation ρ, so that ρ n (α) is parabolic if and only if ρ(α) is parabolic. However, in his example ρ n (α) is not conjugate to ρ(α) for any n. Bowditch [3, Thm. 1.5] also established a stability theorem for deformations of geometrically finite representations which preserve the structure of the Jordan decomposition of parabolic elements.
To account for these examples, we introduce the following subset of the representation variety. If ρ : Γ → SL(d, K) is a representation of a geometrically finite Fuchsian group, let Hom tp (ρ) ⊂ Hom(Γ, SL(d, K)) be the space of representations σ : Γ → SL(d, K) so that if α ∈ Γ is parabolic, then σ(α) is conjugate to ρ(α). We obtain the following stability result for type-preserving deformations. (In their preprint, Kapovich and Leeb [20] suggest that such a stability result holds more generally.) Furthermore, we show that limit maps vary analytically. Combined with work of Bray-Canary-Kao-Martone [5] this will imply that entropy and pressure intersection vary analytically over the cusped Hitchin component (which will be used crucially in the construction of the pressure metric). The proof also allows us to see that the P k -Anosov limit maps are uniformly Hölder in a neighborhood of a P k -Anosov representation.
We say that {ρ u } u∈M is a K-analytic family of representations if M is a K-analytic manifold and the map u → ρ u is a K-analytic map from M into Hom(Γ, SL(d, K)). (1) If ρ ∈ O, then ρ is P k -Anosov. (2) There exists α > 0 so that if ρ ∈ O, then its P k -Anosov limit map ξ ρ is α-Hölder. to When U(Γ) is compact, stability follows from standard arguments in hyperbolic dynamics and the Hölder regularity of the boundary maps is a consequence of standard results, e.g. [33,Cor. 5.19]. The non-compact case is more involved. Our key idea to prove stability is to observe that if ρ 2 ∈ Hom tp (ρ 1 ), then on each cusp there is a smooth conjugacy of the flows associated to ρ 1 and ρ 2 . This is made precise in Equation (21) below. This essentially means that the two flows differ by a compact perturbation and thus, essentially, reduces to the compact base case. Our key idea to prove Hölder regularity is to first introduce certain "canonical families of norms" on the flow spaces, see Section 3.1. Then we prove that if the flow is contracting with respect to any family of norms, then the flow is contracting with respect to any canonical family of norms (see Theorem 6.1 and Theorem 4.1). Finally, the canonical family of norms are well-behaved enough that we can adapt an argument from [38] to prove Hölder regularity of the boundary maps directly.
We now discuss the applications of our results to cusped Hitchin representations, which was the original goal of our work. Given an ordered basis B for R d , we say that a unipotent A ∈ SL(d, R) is totally positive with respect to B, if its matrix with respect to B is upper triangular and all its minors (which are not forced to be 0 by the fact that the matrix is upper triangular) are strictly positive. The set U >0 (B) of unipotent, totally positive matrices with respect to B is a semi-group (see Lusztig [27,Section 2.12]). Following Fock and Goncharov [16], we say that an ordered k-tuple (F 1 , F 2 , . . . , F k ) of distinct flags in F d is positive, if there exists an ordered basis for all i, and there exists u 2 , . . . , u k−1 ∈ U >0 (B) so that F i = u k−1 · · · u i F k for all i = 2, . . . , k − 1. Fock and Goncharov [16] proved that positivity of a n-tuple is invariant under cyclic permutation and reversal (also see Kim-Tan-Zhang [23, Section 3.1 -3.3]). If X is a subset of S 1 then a map ξ : X → F d is positive if whenever (x 1 , . . . , x n ) is a cyclically ordered subset of distinct points in X, then (ξ(x 1 ), . . . , ξ(x n )) is a positive n-tuple of flags.
We say that a representation ρ : Γ → SL(d, R) of a geometrically finite Fuchsian group is a Hitchin representation if there exists a continuous positive ρ-equivariant map ξ ρ : Λ(Γ) → F d . If Γ is cocompact and torsion-free then these are exactly the Hitchin representations of closed surface groups introduced by Hitchin [19] and further studied by Labourie [24], while if Γ is convex cocompact they are the same as the Hitchin representations studied by Labourie and McShane [25]. We distinguish the case where Γ contains parabolic elements by calling these cusped Hitchin representations. If d = 3 and S = Γ\H 2 is a finite area hyperbolic surface, then cusped Hitchin representations of Γ are holonomy representations of strictly convex, finite area, projective structures on S (see Marquis [29]). Further it follows from [16,Thm. 3.3] that holonomy maps of geometrically finite projective surfaces in the sense of Crampon-Marquis [13] are also cusped Hitchin. Fock and Goncharov [16] studied the a priori more general class of representations which admit equivariant positive maps from the set Λ p (Γ) of fixed points of peripheral elements of Γ into F d when Γ is not cocompact but has cofinite area. We show that all such type-preserving representations are in fact cusped Hitchin representations. Our main result here is that cusped Hitchin representations are P k -Anosov for all k. Theorem 1.4. If Γ ⊂ PSL(2, R) is geometrically finite and ρ : Γ → SL(d, R) is a Hitchin representation, then ρ is irreducible and P k -Anosov for all k. Moreover: (1) For all k the map x → ξ k ρ (x) is the P k -Anosov limit map. (2) If α ∈ Γ is parabolic, then ρ(α) = ±u for some unipotent u ∈ SL(d, R) with a single Jordan block. (3) If γ ∈ Γ is hyperbolic, then ρ(γ) is loxodromic.
If we let H d (Γ) denote the space of all Hitchin representations of Γ into SL(d, R), it is easy to see that H d (Γ) is a real analytic manifold. In fact, one may use results of Fock and Goncharov [16] to show that the space H d (Γ) of conjugacy classes of Hitchin representations is diffeomorphic to R m (for some m). (Marquis [28] gives an explicit parametrization of H 3 (Γ) as a topological cell when S = Γ\H 2 is a finite area hyperbolic surface.) Comparison to other results: As mentioned above, Kapovich-Leeb [20] and Zhu [36] have previously developed theories of Anosov representations for relatively hyperbolic groups. Their work is based on extending characterizations of Anosov representations due to Kapovich-Leeb-Porti [21] and Bochi-Potrie-Sambarino [2] respectively. Theorem 1.1 immediately implies that a P k -Anosov representation ρ is P krelatively dominated, in the sense of Zhu [36]. Theorem 9.4 in [36] then implies that ρ is P k -relatively asymptotically embedded and Remark 9.10 in [36] implies that ρ is P k -relatively uniform RCA in the sense of Kapovich-Leeb [20].
Then ρ is P k -relatively dominated, P k -relatively asymptotically embedded and P k -relatively uniform RCA.
Further, it follows from [37, Thm. C] and our Theorems 1.1 and 1.2, that a representation of a geometrically finite Fuchsian group Γ is P k -Anosov if and only if it is P k -relatively dominated with respect to P.
Kapovich and Leeb [20] mention that they can show that a cusped Hitchin representation of a geometrically finite Fuchsian is relatively Anosov in their sense.
In a subsequent preprint, Filip [14] introduces the class of log-Anosov representation of Fuchsian lattices, which also agrees with the class of cusped Anosov representations. He introduces adapted metrics, which correspond to our canonical norms. He shows that monodromy maps of certain variations of Hodge structures on finite area Riemann surfaces are log-Anosov and uses this result to study their properties. In particular many monodromy maps coming from hypergeometic differential equations can be analyzed in this manner, including those previously studied by Brav and Thomas [4] and by Filip and Fougeron [15].
Acknowledgements: The authors thank the referee for their careful reading of the original manuscript and helpful suggestions which improved the exposition. This material is based upon work supported by the National Science Foundation under Grant No. DMS-1928930 while the first author participated in a program hosted by the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2020 semester.

Preliminaries
In this section, we recall some preliminary facts and introduce notation that will be used throughout this paper.
2.1. Hyperbolic 2-space. In this paper we will identify H 2 with the Poincaré upper half plane model. For any v ∈ T 1 H 2 , let r v : R → H 2 denote the unit speed geodesic with r ′ v (0) = v and let v + := lim for all v ∈ T 1 H 2 and t ∈ R. If {g n } is a sequence in PSL(2, R), we say that g n converges to x ∈ ∂ H 2 if g n (z) → x for some (any) z ∈ H 2 . We often simply write g n → x.
Let U(Γ) := Γ\ U(Γ), and note that φ t descends to a flow on U(Γ), which we also denote by φ t . If S = Γ\H 2 , then U(Γ) is the non-wandering portion of T 1 S and its orbits are the complete geodesics which remain entirely in the convex core of S.
(2) A set C = {C 1 , . . . , C r } of disjoint embedded cusp neighborhoods in S = Γ\H 2 is full if any curve on S which is represented by a parabolic element of Γ is freely homotopic into some C i .
If C = st Γ (p)\H is an embedded cusp neighborhood, then we set If C is a full collection of cusp neighborhoods for S, then we set Notice that its complement U(Γ) c C is compact. We will sometimes informally refer to U(Γ) C as the thin part of the geodesic flow and its complement as the thick part.
2.3. The representation theory of SL(2, R): For d ≥ 1, let τ d : SL(2, R) → SL(d, K) denote the standard irreducible representation. Explicitly, ) is the representation induced by the linear SL(2, R)-action on Sym d−1 (R 2 ) given by One may also regard τ d as a representation into SL(d, C). Let ξ d : . . , f j ) and It is straightforward to verify that ξ d is a continuous, τ d -equivariant, strongly dynamics preserving map. We call ξ d the Veronese embedding associated to τ d .
One can compute that τ d 1 a 0 1 is the upper triangular matrix in SL(d, R) given by Furthermore, it is well-known that the d × d upper triangular matrix given by (1) is unipotent and totally positive with respect to the standard basis of R d when a > 0. From this, it follows easily that ξ d is a positive map.
Given an element g ∈ SL(d, K), let denote the singular values of g and let denote the absolute values of the (generalized) eigenvalues of g. Also, let g = g ss g u = g u g ss denote the Jordan-Chevalley decomposition, that is g ss is semisimple, g u is unipotent, and g ss , g u commute. We say that g is elliptic if it is semisimple and λ j (g) = 1 for all j. Notice that if g is elliptic, then the cyclic group g generated by g is relatively compact in SL(d, K). If g ∈ SL(d, K) and σ k (g) > σ k+1 (g), let U k (g) ∈ Gr k (K d ) denote the subspace spanned by the k major axes of the ellipse g S d−1 , that is U k (g) = g e 1 , . . . , e k . The following lemma relates the singular values along a sequence in SL(d, K) to the action of this sequence on the associated Grassmannian. We omit the proof as it is standard.
, then the following are equivalent: Moreover, if g n = g n for all n, then g is We will also make use of the following elementary calculations, which we recall without proof.

Anosov representations into SL(d, K)
Following Labourie [24] and Guichard-Weinhard [18] we define Anosov representations for geometrically finite Fuchsian groups into SL(d, K). In Appendix B, we will extend our definition to the setting of all semi-simple Lie groups with finite center.
If Γ ⊂ PSL(2, R) is a geometrically finite group and ρ : Γ → SL(d, K) is a representation, let where γ ∈ Γ acts on the first factor by γ and on the second factor by ρ(γ). The projection map p : E ρ → U(Γ) descends to a vector bundlep which is called the flat bundle associated to ρ. The geodesic flow on U(Γ) extends to a flow on E ρ = U(Γ) × K d whose action is trivial on the second factor. This in turn descends to a flow on E ρ which covers the geodesic flow on U(Γ). We use φ t to denote all four of these flows. We say that a continuous map Such a map induces a continuous φ t -invariant spitting which descends to a continuous φ t -invariant splitting (1) There exists a ρ-equivariant, continuous, transverse map which induces a splitting E ρ = Θ k ⊕ Ξ d−k . (2) For some family of norms · v on the fibers of E ρ there exists C > 1 and c > 0 so that We refer to ξ as a P k -Anosov limit map of ρ.
Remark 3.2. Notice that a continuous family of norms · v∈ U(Γ) on the fibers of E ρ lifts to a continuous family of norms · v∈U(Γ) on K d which is ρ-equivariant in the following sense: if v ∈ U(Γ), Y ∈ K d and γ ∈ Γ, then Conversely, any continuous family of norms satisfying Equation (2) descends to a continuous family of norms on the fibers of E ρ .
If one prefers a bundle-theoretic definition, one can consider the vector bundle Hom( Ξ d−k , Θ k ) over U(Γ). Notice that since the splitting is flow-invariant, φ t induces a flow on Hom( Ξ d−k , Θ k ) given by with some abuse of notation we use φ t to denote this flow. Moreover, any norm on the fibers of E ρ induces an operator norm on Hom( Ξ d−k , Θ k ). We say that the flow φ t on Hom( Ξ d−k , Θ k ) is uniformly contracting if there exists C, c > 0 so that One may easily check that ρ is P k -Anosov if and only if there exists a ρ-equivariant, continuous transverse map ξ = (ξ k , ξ d−k ) : Λ(Γ) → Gr k (K d ) × Gr d−k (K d ) so that the flow is uniformly contracting on Hom( Ξ d−k , Θ k ) with respect to an operator norm associated to a continuous family of norms on E ρ . (The details of this equivalence are worked out carefully in the proof of [8,Prop. 2.3].) Moreover, by duality, φ t is uniformly contracting on Hom( Ξ d−k , Θ k ) if and only if the flow, also called φ t , on Hom( Θ k , Ξ d−k ) is uniformly expanding, i.e. the inverse flow φ −t is uniformly contracting. We record these observations for future use. (1) ρ is P k -Anosov, (2) There exists a ρ-equivariant, continuous transverse map so that the flow is uniformly contracting on Hom( Ξ d−k , Θ k ) with respect to an operator norm induced by a norm on E ρ . (3) There exists a ρ-equivariant, continuous transverse map so that the flow is uniformly expanding on Hom( Θ k , Ξ d−k ) with respect to an operator norm induced by a norm on E ρ .
As an immediate corollary we obtain: Remark 3.5. If we wish to allow Γ to be a geometrically finite subgroup of PGL(2, R), then we simply consider Γ 0 = Γ ∩ PSL(2, R) and say that ρ : Γ → SL(d, R) is P k -Anosov if and only if ρ| Γ0 is P k -Anosov. With this definition, all of our results for geometrically finite Fuchsian groups remain true for geometrically finite subgroups of PGL(2, R).

3.1.
Canonical norms and cusp representations. It will be useful to construct certain "canonical" norms on the fibers of the vector bundles E ρ , and hence also on Hom( Ξ d−k , Θ k ), that are well behaved on the cusps. Later we will show that when ρ is P k -Anosov, the flow φ t on Hom( Ξ d−k , Θ k ) is uniformly contracting for any canonical norm (see Corollary 6.2).
The crucial property of our canonical norms is that they have a standard form over the thin part of the geodesic flow. In order to describe this standard form we will use the following result about representations of SL(2, R). Recall that g = g ss g u denotes the multiplicative Jordan-Chevalley decomposition of g ∈ SL(d, K). Proposition 3.6 follows easily from the Jordan normal form of a weakly unipotent matrix and we delay the proof until Appendix A.

Proposition 3.6 implies that cusp representations always exist.
Suppose that Γ is a geometrically finite Fuchsian group, ρ : Γ → SL(d, K) is type-preserving and C = α \H is an embedded cusp neighborhood where α ∈ Γ is parabolic. Further suppose that Ψ is a cusp representation for α and ρ(α), and · v is a family of norms of the fibers Such families are easy to construct: the group K := ρ(α) ss , Ψ(− id 2 ) is abelian and compact, so there exists a norm · 0 which is K-invariant, then if we fix some v 0 ∈ T 1 H 2 , the family of norms defined by Z g(v0) := Ψ(g) −1 Z 0 has the desired properties.
Also, for such a family of norms So this family descends (and restricts) to a family of norms on the fibers of E ρ over U(Γ) C which we call a canonical family of norms on the cusp neighborhood C. Observe that if C ′ is an embedded cusp neighborhood properly contained in C, then U(Γ) C − U(Γ) C ′ is precompact, and a canonical family of norms on the fibers of E ρ over U(Γ) C is determined completely by its restriction to the fibers over Definition 3.7. Let ρ : Γ → SL(d, K) be a type preserving representation of a geometrically finite Fuchsian group. A continuous family of norms · v on the fibers of E ρ is canonical if there exists a full collection C of embedded cusp neighborhoods for Γ, such that for all C ∈ C, the restriction of the family of norms to the fibers over U(Γ) C is a canonical family of norms on C.
It is straightforward to construct a canonical family of norms on the fibers of E ρ . One first chooses a full collection of embedded cusp neighborhoods C. For each C ∈ C, one chooses a canonical norm on the cusp neighborhood C. One then chooses any continuous norm on a compact neighborhood of the fibers over the thick part U(Γ) − U(Γ) C . One may then use a cut off function to interpolate between the norms on their interface and obtain a family of norms on all of E ρ which is canonical with respect to a full collection of cusp neighborhoods contained in C.
We observe that there are uniform upper and lower bounds on the growth rate of a canonical norm with respect to the (lift of the) geodesic flow φ t .
Let C be a full collection of embedded cusp neighborhoods such that for all C ∈ C, · v restricted to the fibers over each component of U(Γ) C is a canonical family of norms on C. For each C ∈ C, let C ′ be an embedded cusp neighborhood such that Then f is constant on each U(Γ) C ′ . Further, is finite. Set c 0 = log C 0 . For any t ∈ R, let n be the largest integer such that n ≤ |t|. If t > 0, then On the other hand, if t < 0, then This proves the required upper bound. The lower bound is similar.
Remark 3.9. We will not need this for our work, but with a bit more effort, one can show that any two canonical families of norms on E ρ are bilipschitz.

Basic properties of Anosov representations
In this section we prove Theorem 1.1 which we restate here.
(2) There exists B, b > 1 so that if γ ∈ Γ, then where ℓ(γ) is the translation length of γ on H 2 . (3) The P k -Anosov limit map ξ ρ is strongly dynamics-preserving and unique. In Before proving the theorem we note the following consequences which will be useful in [5].
(2) If α ∈ Γ is parabolic and j ∈ {1, . . . , d}, then there exists an integer c(j, α) and C j > 1 so that Proof. Property (1) in Corollary 4.2 is known as the P k -Cartan property and is an immediate consequence of the fact that ξ ρ is strongly dynamics-preserving and Lemma 2.2.
If α ∈ Γ is parabolic, then part (3) of Theorem 1.1 implies that ρ(α) is weakly unipotent. So the group Thus there exists a constant C 0 > 1 such that for all n ∈ N and j ∈ {1, . . . , d}.
Given j ∈ ∆ and n ∈ N, there exists i and k ∈ {1, . . . , d i }, so that Then Lemma 2.3 implies the first claim in part (2) of Corollary 4.2 , while the second claim follows from part (1) of Theorem 4.1.
Proof of Theorem 4.1. Suppose that Γ ⊂ PSL(2, R) is a geometrically finite group and ρ : Lemma 4.3 allows us to reduce much of the proof to considering hyperbolic elements γ ∈ Γ with d ∞ (γ + , γ − ) ≥ δ. Since ξ is transverse and continuous these elements have the following decomposition: there exists a compact set A ⊂ SL(d, K) so that if γ ∈ Γ is hyperbolic and There also exists a compact set K of H 2 so that any bi-infinite geodesic whose endpoints are a distance at least δ apart intersects K. Let The next two lemmas establish (1) for any hyperbolic element γ with d ∞ (γ + , γ − ) ≥ δ. The proof of the first lemma makes crucial use of the contraction properties of the flow, while the second lemma does not depend on the contraction properties. and

the ρ-equivariance of the norms and Equation (3) imply that
for all Y ∈ ξ k (γ + ) and non-zero Z ∈ ξ d−k (γ − ). So by the max-min/min-max characterization of singular values Hence So, again by the max-min/min-max characterization of singular values, In particular, Proof. Fix a ρ-equivariant family of norms · * v∈U(Γ) which descends to a canonical family of norms on E ρ . Then by Lemma 3.8 there exist C 0 , c 0 > 0 so that , the ρ-equivariance of the norms and Equation (5) imply that Finally, notice that We can now prove part (1). Since B is finite, both and n for all γ ∈ Γ. Therefore, part (2) follows immediately from part (1). (One may give a direct proof of (2) in the spirit of Lemma 4.4 by noting that there is a compact subsetK of H 2 such that every hyperbolic element of Γ is conjugate to an element of Γ whose axis passes throughK.) Part (4) is a simple consequence of part (1) and Lemma 4.5. Recall that X d (R) = SL(d, R)/SO(d) and X d (C) = SL(d, C)/SU(d). We may choose x 0 to be either [SO(d)] or [SU(d)]. Then, after possibly scaling, we have the following formula for the distance on X d (K), see for instance [9,Cor. 10.42], Therefore, applying part (1), we see that On the other hand, by Lemma 4.5 so the orbit map is a quasi-isometry, which completes the proof of part (4). We now show that ξ is strongly dynamics preserving. Notice that this immediately implies that ξ is unique. Lemma 2.2 will then imply that if α ∈ Γ is parabolic, then ρ(α) is weakly unipotent, while if γ ∈ Γ is hyperbolic, then ρ(γ) is P k -proximal.
Fix a sequence {γ n } with γ n → x ∈ Λ(Γ) and γ −1 n → y ∈ Λ(Γ). For each n there exists β n ∈ B such that γ n β n is hyperbolic and Since the set B is finite, we can divide {γ n } into finitely many subsequences and only consider the case when β n = β for all n. Then (γ n β) + → x and (γ n β) − → β −1 (y). Let be the block diagonal decomposition from Equation (4). Then by Lemma 4.4 So by the continuity of ξ, ) which completes the proof of (3).

Basic properties of cusp representations
In this section we establish some useful properties of the cusp representations associated to type preserving representations introduced in Section 3.1.
We say that a representation Ψ : is P k -proximal for some (any) hyperbolic element γ ∈ SL(2, R). We first observe that a cusp representation associated to a P k -Anosov representation is P k -proximal.
Proof. Let α ∈ SL(2, R) be the unique unipotent lift of α. Since ρ(α) is weakly unipotent, the group ρ(α) ss is compact. Then, since Ψ( α n ) = ρ(α) −n ss ρ(α n ), there exists L > 1 such that for all j ∈ {1, . . . , d} and n ∈ Z. Then, since ρ is strongly dynamics-preserving, Lemma 2.2 implies that Now, write α n = ℓ n a tn m n , where ℓ n , m n ∈ SO(2) and a t = e t 0 0 e −t . Since {Ψ(a t ) : t ∈ R} is simultaneously diagonalizable, by increasing L > 1 we may assume that for all t ∈ R. Since Ψ(SO (2)) is compact, we may increase L > 1 further and assume that 1 L ≤ σ j (Ψ(g)) ≤ L for all g ∈ SO(2) and j ∈ {1, . . . , d}. Then Next we observe that a P k -proximal representation is itself P k -Anosov and admits a P k -limit map, in the following sense.
with the following properties: (1) If · is a Ψ-invariant family of norms on the fibers of Proof. By conjugating Ψ we can assume that Let ξ di : ∂ H 2 → F (K di ) denote the τ di -equivariant boundary map described in Section 2.3.
By definition, if γ ∈ SL(2, R) is a hyperbolic element, then τ di (γ) is diagonalizable with eigenvalues having pairwise distinct absolute values. Furthermore, for all k = 1, . . . , d i − 1, ξ k di (γ + ) is the direct sum of the eigenspaces of the k largest eigenvalues of τ di (γ).
First, we construct the map η. Observe that since Ψ is P k -proximal, Ψ(γ) is P k -proximal and P d−kproximal for every hyperbolic element γ ∈ SL(2, R). Thus, for all i = 1, . . . , m, there are integers Note that this pair of maps are continuous, Ψ-equivariant, and transverse. For which completes the proof of (1). If {γ n } is a sequence in SL(2, R) with γ n → x and γ −1 n → y, one can write γ n = ℓ n a tn m n where ℓ n , m n ∈ SO(2), t n ≥ 0, and a t = e t 0 0 e −t . By assumption, t n → ∞, ℓ n → ℓ and m n → m with To prove (3), fix x ∈ ∂ H 2 and a hyperbolic element γ ∈ SL(2, R) so that The following technical result says that the image of the limit map of an Anosov representation is asymptotically homogeneous at a parabolic fixed point. In one of the arguments that follow, we need uniform control over continuous families of Anosov representations, so we introduce a parameter u.
is a P k -proximal representation with P k -limit map η, and ℓ ∈ SL(d, K) is elliptic and commutes with the elements of Ψ(SL(2, R)).
Let U be a compact metric space and {g u } u∈U be a continuous family of elements in SL(d, R). Suppose is continuous and for each u ∈ U , the map ξ u := ξ(u, ·) is transverse, ξ u (α + ) = g u η(α + ) and if lim x n = x ∈ ∂ H 2 and x n ∈ γ −n (X) for all n. Moreover, the convergence is uniform in u ∈ U .
Suppose the proposition fails for a sequence {x n } ⊂ X with lim x n = x ∈ X. Then there exist ǫ > 0, {n j } converging to infinity and a sequence {u j } in U such that For notational convenience, let So, by passing to a tail of our sequences we may assume that x nj = α + for every j.
First suppose that {y j = γ nj (x nj )} lies in a compact subset of ∂H 2 − {α + }. Then x = γ − . Since ξ uj → ξ u∞ , there exists N > 0 sufficiently large so that {g −1 uj ξ uj (y j )} n≥N lies in a compact subset of flags transverse to g −1 u∞ ξ u∞ (α + ) = η(γ + ). Hence Proposition 5.2 part (2) implies that and we have a contradiction. Now suppose that {y j } does not lie in a compact subset of ∂H 2 − {α + }. We may assume without loss of generality that γ − = 0, γ + = ∞ = α + , and α = u 1 where Then Passing to a subsequence, we can suppose that w j → w ∈ [0, 1]. Proposition 5.2 part (2) then implies that Ψ(δ j )(V ) converges to η k (x) locally uniformly for all V ∈ Gr k (K d ) which are transverse to η d−k (∞). Also, where in the first line we apply assumption (8)). We may pass to a subsequence so that ℓ zj → ℓ ∞ ∈ SL(d, K). Then ℓ ∞ is also elliptic and commutes with the elements of Ψ(SL(2, R)) and hence, by Proposition 5.2 part (3), fixes each element in the image of η. Then , and ℓ ∞ • η = η. Reversing the roles of k and d − k, we may similarly show that η d−k j (x nj ) → g u∞ η d−k (x). Hence we again have a contradiction.

A dynamical characterization of linear Anosov representations
In this section, we prove Theorem 1.2. The forward implication has already been established as part (3) of Theorem 1.1. The reverse implication follows from the following more general statement.
First, recall from Section 3 that a transverse, ρ-equivariant, continuous map ξ = (ξ k , ξ d−k ) : Λ(Γ) → Gr k (K d )×Gr d−k (K d ) induces a continuous decomposition of E ρ into a pair of sub-bundles E ρ = Θ k ⊕ Ξ d−k of rank k and d − k respectively. Further, the flow φ t induces a flow on Hom( Ξ d−k , Θ k ), which we denote by φ t , and any canonical norm on E ρ induces a canonical norm on Hom( Ξ d−k , Θ k ) which is simply given by the associated operator norm.
, then ρ is type-preserving and the flow φ t on Hom( Ξ d−k , Θ k ) is uniformly contracting with respect to any canonical norm on Hom( Ξ d−k , Θ k ). In particular, ρ is P k -Anosov and ξ is its P k -Anosov limit map.
The following is an immediate corollary of Theorem 6.1 and Theorem 4.1. As another corollary, we see that if ρ is Zariski dense, then ρ is P k -Anosov if it admits a transverse limit map, which generalizes a result of Guichard and Wienhard from the uncusped Anosov setting [18,Theorem 4.11]. Corollary 6.3. Suppose Γ ⊂ PSL(2, R) is a geometrically finite group and ρ : Γ → SL(d, K) is a representation.
, then ρ is P k -Anosov and ξ is its P k -Anosov limit map.
Proof. (1): It is enough to show that ξ is strongly dynamics preserving. Fix an escaping sequence {γ n } in Γ with γ n → x and γ −1 n → y. Let [ρ(γ n )] denote the image of ρ(γ n ) in P(End(K d )). Then it is enough to show that [ρ(γ n )] converges to the element T ∈ P(End(K d )) with ker(T ) = ξ d−1 (y) and Image(T ) = ξ 1 (x). Since P(End(R d )) is compact it is enough to show that every convergent subsequence of ρ(γ n ) converges to T . So suppose that [ρ(γ n )] → S in P(End(K d )). Then We first claim that Image(S) = ξ 1 (x). Since ρ : Since ∂Γ is perfect, we can perturb each x j and assume that we can relabel and suppose that for all 1 ≤ j ≤ m. Hence Image(S) = ξ 1 (x).
To compute the kernel, we notice that Gr d−1 (K d ) may be identified with P(K d * ) by identifying a hyperplane Q in K d with the projective class of linear functionals with kernel Q. Notice that [ t ρ(γ n )] converges to t S in P(End(K d * )). Repeating the argument above shows that Image( t S) = ξ d−1 (y), so the kernel of S is ξ d−1 (y).
Proof of Theorem 6.1. Suppose that Γ ⊂ PSL(2, R) is a geometrically finite group, ρ : Γ → SL(d, K) is a representation and that is a continuous, transverse, ρ-equivariant, strongly dynamics-preserving map. Lemma 2.2 implies that ρ is type-preserving. Let · be a canonical family of norms on E ρ and let C be a full collection of embedded cusp neighborhoods so that the restriction to the fibers over U(Γ) C is canonical for all C ∈ C. We will also use · to denote the lift of · to a continuous family of norms on the fibers of U(Γ) × K d .
The proof divides into two parts. We first use properties of the canonical family of norms to control the flow over the thin part U(Γ) C . We then use a compactness argument to control the flow on the complement. Proposition 6.4. If C ∈ C, then there exist constants b C and B C and an embedded cusp sub-neighborhood Proof. Suppose that C = α \H. Then it suffices to find a horodisc H ′ ⊂ H and constants After possibly replacing C with a subcusp, there exists a cusp representation Ψ : SL(2, R) → SL(d, K) for α and ρ(α) such that · on U(Γ) H coincides with a ρ(α) ss -invariant, Ψ-equivariant family of norms · ⋆ v∈T 1 H 2 . Let η be the P k -limit map of Ψ. Proposition 5.2 implies that there exists B, b > 0 such that We claim that there is a horodisk for all Y ∈ ξ k (v + ) and non-zero Z ∈ ξ d−k (v − ). If this is not the case, then there exists • a sequence {t n } in [T, 2T ], • a nested sequence {H n } of horodisks centered at the fixed point of α whose intersection is empty, Let u t = 1 t 0 1 and a s = e s 0 0 e −s .
As usual, by conjugating, we can assume that Ψ = ⊕ m i=1 τ di , and α = [u 1 ]. There is a sequence {s n } → ∞ of positive real numbers and a sequence {m n } of integers such that {w n := a −sn u mn (v n )} is relatively compact in T 1 H 2 . By passing to a subsequence, we may assume that w n converges to some w ∞ ∈ T 1 H 2 , and that where ξ n = Ψ(a −sn )•ξ•a sn . Proposition 5.3 (applied when U is a singleton) implies that lim n→∞ ξ n = η, so , this contradicts (11), thus proving the claim. Repeated applications of the claim implies that if v ∈ U(Γ) H ′ and t ≥ T is a number such that (notice that B C is finite by Lemma 3.8) then the proposition follows.
It remains to control the behavior of the flow on the thick part. The proof of the following proposition is inspired by arguments of Tsouvalas [35, Theorem 1.1].
Proposition 6.5. Suppose that K is a compact subset of U(Γ), {u n } is a sequence in K, Y n ∈ Θ k | un , Z n ∈ Ξ d−k | un and Y n un = Z n un = 1 for all n. If {t n } ⊂ R, lim t n = +∞ and φ tn (u n ) ∈ K for all n, then Proof. We first choose a compact set K ⊂ U(Γ) so that K ⊂ π(K) (where π : U(Γ) → U(Γ) is the quotient map). For each n, choose v n ∈ K so that π(v n ) = u n and γ n ∈ Γ so that w n = γ −1 n (φ tn (v n )) ∈ K. We may assume that v n → v ∞ and w n → w ∞ for some v ∞ , w ∞ ∈ K. Notice that γ n → v + ∞ and γ −1 Since ξ is strongly dynamics preserving, Lemma 2.2 implies that does not converge to 0 it must be the case that does not converge to 0, and hence that We may then pass to a subsequence so that the limits are equal and exist. At the same time, . This contradicts the transversality of ξ, and completes the proof.
We now combine Proposition 6.4 and Proposition 6.5 to finish the proof of Theorem 1.2. For each C ∈ C, let C ′ ⊂ C be the embedded cusp subneighborhood given by Proposition 6.4. Let C ′ = {C ′ : C ∈ C} and let K = U(Γ) − U(Γ) C ′ , which is compact. Let b = min{b C : C ∈ C} and B = max{B C : C ∈ C}. Proposition 6.5 implies that there exists T 1 > 0 so that Moreover, since K is compact, there exists R > 0 so that: Now choose T > 0 so that Once we have proven our claim, we can choose (notice that A is finite by Lemma 3.8) and observe that Hence, we will have shown that ρ is P k -Anosov. We now establish (19).
Otherwise let If s 2 − s 1 ≥ T 1 , then Proposition 6.4 and (17) give On the other hand, if s 2 − s 1 < T 1 , then by Proposition 6.4 and (18), we have

Hitchin representations are Borel Anosov
In this section, we show that Hitchin representations are irreducible and Borel Anosov, i.e. P k -Anosov for all k. Theorem 1.2 reduces the proof that Hitchin representations are Borel Anosov to the claim that their limit maps are strongly dynamics preserving.
for all V transverse to ξ d−k (y).
Proof. Our proof relies on an observation about convergence of sequences of flags.
If {F n } is a sequence in F (R d ) such that F n ∈ O(F 1,n , F 2,n , F 3,n ) for all n, then F n → F + .
We first suppose that x = y. Then by passing to the tail of the sequence {γ n }, we may assume that each γ n is hyperbolic with attractor and repellor γ + n and γ − n in Λ(Γ). Then γ + n → x, γ − n → y, and γ n (z) → x for all z ∈ Λ(Γ) − {y}.
We recall that positive tuples of flags are in general position in the following sense.
We also use the following equivalent formulation of the positivity of a quadruple of flags.
Assume for contradiction that there is some i, j ∈ {1, . . . , d} such that the i-th diagonal entry of b is 1, while the j-th diagonal entry of b is −1. Since all the upper triangular entries of v are positive, this implies that the upper triangular entries of bv along the i-th row are positive, while the upper triangular entries of bv along the j-th row are negative. But this is impossible since ua = g = bv, and for every column of ua, the upper triangular entries in that column must have the same sign. As such, b = ± id. This implies that g ss = ± id and g u = v = u.
It now suffices to show that u has a unique fixed flag in F (R d ). Observe the following linear algebra facts: (1) If w is a unipotent element that is represented in a basis (e 1 , . . . It only remains to show that ρ is irreducible. Suppose that ρ is not irreducible. Then there is a proper subspace W ⊂ R d which is invariant under ρ(Γ). By Theorem 7.1, ρ(γ) is loxodromic for any hyperbolic γ ∈ Γ, so either W contains the attracting fixed point (in P(R d )) of ρ(γ), or W lies in the repelling hyperplane (in P(R d )) of ρ(γ). Since ξ is transverse, this implies that either W lies in the repelling hyperplanes of ρ(γ) for all hyperbolic γ ∈ Γ, or W contains the attracting fixed point of ρ(γ) for all hyperbolic γ ∈ Γ. However, this contradicts Proposition 7.4 in either case. Therefore, ρ is irreducible.

Stability of Anosov representations
In this section, we prove Theorem 1.3, which we restate here.
(2) There exists α > 0 so that if ρ ∈ O, then ξ ρ is α-Hölder (with respect to any distance on Λ(Γ) induced by a Riemannian metric on ∂ H 2 and any distance on

induced by a Riemannian metric). (3) If {ρ u } u∈M is a K-analytic family of representations in O and z ∈ Λ(Γ) then the map from M
to The proof of (1) is based on the proof of stability for Anosov diffeomorphisms on compact manifolds given in Shub's book [33,Cor. 5.19]. The two key features which allow us to overcome the noncompactness of the base space are the smooth conjugacy of the flows on the cusps, see Equation (21), and uniform estimates for families of canonical norms, see Lemma 3.8.

Proof of Theorem 8.1: We define
The geodesic flow on U(Γ) extends to a flow on E(O) whose action is trivial on the first and third factor. This in turn descends to a flow on E(O). As usual, we use φ t to denote these flows. Also, notice that Let · 0 be a canonical family of norms for E ρ0 and let C be a full collection of embedded cusp neighborhoods for Γ so that · 0 is canonical with respect to C.
over ρ 0 . If C ∈ C and C = α \H, then by shrinking O if necessary, we may assume that there is a continuous With this, we may extend the splitting to a global splitting | v ) for all ρ ∈ O, C ∈ C and v ∈ U(Γ) C , and then extending this globally after perhaps shrinking O and each C.
The flow φ t does not necessarily preserve the splitting E(O) = F k ⊕ G d−k . To find the required φ t -invariant splitting, we will use the contraction mapping theorem. For that purpose, we extend · 0 to a canonical family of norms · on the fibers of E(O) over O × U(Γ) as follows. If C ∈ C, we define This gives us canonical norms over all C ∈ C. Then, perhaps after once more shrinking O and each C, we may extend this to a continuous family of norms · for the fibers of E(O) over O × U(Γ) such that the restriction to E ρ is canonical for all ρ ∈ O. Suppose that V is a subspace of E(O)| (ρ,v) , W is a subspace of E(O)| (ρ,w) and T ∈ Hom(V, W ). We define the operator norm for any subset X ⊂ U(Γ). In the case when Q = id, we may view T as an element of S (Hom(V, W )), the vector space of sections of the bundle Hom(V, W ). Note then that · U(Γ) defines a norm on S(Hom(V, W )) whose corresponding distance is Cauchy complete.
We may decompose the flow φ t as In the case when this splitting happens to be φ t -invariant, then B t ≡ 0 and C t ≡ 0. Fix ǫ ∈ (0, 1/2) so that .

First notice that for all
is a flow invariant splitting. Since ρ 0 is P k -Anosov, there exist c, C > 0 such that for all v ∈ U(Γ) and t > 0. Choose T so that Ce −cT < ǫ.
Consider the compact set On the other hand, if v ∈ U(Γ) − X, then there is some C ∈ C such that φ t (v) ∈ U(Γ) C for all t ∈ [0, 2T ]. Then by construction, for all ρ ∈ O and t ∈ [0, 2T ]. Since Φ C ρ is an isometry that preserves the splitting, it follows that on 2T ], then A t and D t are invertible, B t = 0, C t = 0, and Consider the bundle Hom( G d−k , F k ) → O × U(Γ) with its induced operator norm · . Let R r ⊂ Hom( G d−k , F k ) denote the ball bundle of radius r about the zero section. 2T ], then there is a well-defined map ψ t : R 1 → R 2ǫ given by Remark 8.4. One can verify that the map ψ t has the defining property is a well defined flow on Hom( G d−k , F k ).

Proof of Proposition 8.3 . If t ∈ [T, 2T ], then
for all ρ ∈ O, v ∈ U(Γ) and f ∈ R 1 | (ρ,v) . By Lemma 8.2, if f (ρ,v) < 1, then which implies that id +D −1 t C t f has trivial kernel for all ρ ∈ O and v ∈ U(Γ). Hence, id +D −1 t C t f is invertible. Since Lemma 8.2 also gives that D t is invertible, it follows that Thus, by Equation (22), Next, we prove our final claim. For any Thus, by Equation (22), ) . By Proposition 8.3, the map ψ S t is a contraction mapping on S(R 1 ) for each t ∈ [T, 2T ]. We may now apply the contraction mapping theorem to conclude that for each t ∈ [T, 2T ] there exists a unique ψ S t -invariant section σ (t) of the bundle R 2ǫ . We claim that σ (t) does not depend on t. If t 1 , t 2 ∈ Q ∩[T, 2T ], then there exist sequences {n j }, {m j } with n j , m j → ∞ and n j t 1 = m j t 2 for all j ≥ 1. Then by the proof of the contraction mapping theorem So σ (t) does not depend on t when t ∈ Q ∩[T, 2T ]. Then by uniqueness of invariant sections and the continuity of ψ S t , we see that σ d−k := σ (t) does not depend on t.
Applying a similar argument to the bundle Hom( F k , G d−k ) we obtain, by further shrinking O if necessary, a φ t -invariant k-dimensional subbundle Θ k of E(O).
To show that E(O) = Θ k ⊕ Ξ d−k , it now suffices to show that the fibers Ξ d−k | (ρ,v) and Θ k | (ρ,v) are transverse for every (ρ, v) ∈ O × U(Γ). Suppose for contradiction that there is some non-zero Z ∈ Ξ d−k | (ρ,v) ∩ Θ k | (ρ,v) for some ρ ∈ O and v ∈ U(Γ). We may write Z uniquely as For the same reasons, Z k = 0 and which is a contradiction. Thus, Θ k and Ξ d−k indeed give a φ t -invariant splitting of E(O).
Proof. We start by proving the following estimate: Then, by the construction of Θ k and Ξ d−k , we have and since φ t (Y ) ∈ Θ k | (ρ,φt(v))) we have Thus by Lemma 8.2 Similar reasoning shows that So by Lemma 8.2 and Equation (22), This proves the estimate in Equation (23). We then may apply Equation (23) iteratively to show that, for all n ∈ N, (notice that C 0 is finite by Lemma 3.8).
Existence of limit maps. Next, we use the φ t -invariant splitting Lift this splitting of E(O) to a splitting The flow φ t on E(O) lifts to a flow, also denoted φ t , on E(O), under which this splitting is invariant. Then the bundle Hom( Θ k , Ξ d−k ) lifts to the bundle Hom(Θ k , Ξ d−k ). Finally, we use · to denote the lifted norms on E(O) and Hom(Θ k , Ξ d−k ). By Proposition 8.5, there exists C 0 , c 0 > 0 such that and σ d−k (ρ, v) depend only on ρ, v + and v − . We now check that σ d−k depends only on ρ and v − . Let γ ∈ Γ be a hyperbolic element, let v γ ∈ U(Γ) be a vector so that v + γ = γ + and v − γ = γ − . Let ℓ(γ) be the translation distance of γ on H 2 . Then φ nℓ(γ) (v γ ) = γ n (v γ ) for all n. Since σ d−k is equivariant and φ t -invariant, it follows that Equation (24) implies that is the repelling fixed point of ρ(γ). Since every point in Λ(Γ) is a limit of repelling fixed points of hyperbolic elements of Γ, this implies that σ d−k depends only on ρ and v − . One may similarly show that σ k depends only on ρ and v + , so there exists . As such, if ρ ∈ O, then ρ is P k -Anosov. This proves (1).
The limits maps are Hölder. We now prove that, perhaps after shrinking our neighborhood O again, that the limit maps are uniformly Hölder. It is possible to establish this using Shub's C r -Section theorem [33,Thm. 5.18], however setting up bundles with the correct regularity (see [33,Cor. 5.19]) and verifying the admissibility condition is somewhat involved when U(Γ) is non-compact. Instead we provide a direct argument based on the proof of Lemma 4.4 in [38]. We will continue to work with E(O) and Hom( Notice that f ρ,φt(v),x = f ρ,v,x for all t ∈ R. If v ∈ U(Γ), we let v ⊥ ⊂ ∂H 2 denote the endpoints of the geodesic through the basepoint of v which is orthogonal to v. (One can use the orientation to canonically identify them as v ⊥,+ and v ⊥,− but this will not be needed for our purposes.) Lemma 8.6. Up to taking a subneighborhood of O, there exists C 1 > 1 so that if ρ ∈ O, v ∈ U(Γ) and x ∈ v ⊥ ∩ Λ(Γ), then In the case when Γ is convex co-compact, the lemma is a simple consequence of equivariance and compactness, but in the general geometrically finite case the proof is somewhat involved. Delaying the proof of the lemma, we first complete the proof of part (2) of Theorem 8.1.
Let O ′ ⊂ O be a subneighborhood of ρ 0 such that the closure of O ′ of O ′ is a compact subset of O. Fix a compact set K ⊂ U(Γ) such that Let d ∞ denote the distance induced by a Riemannian metric on ∂ H, and let d G denote the distance induced by a Riemannian metric on Fix ρ ∈ O ′ and v ∈ K. Then d G is bilipschitz to the norm · (ρ,v) on any compact subset of the affine chart There also exists C 3 > 0 such that: Finally, let If d ∞ (x, y) ≤ δ, then there exists t ≥ 0 and v ∈ K such that y = v − and x ∈ φ −t (v) ⊥ . Then Therefore, ξ d−k ρ is c 0 -Hölder. One may similarly prove that, perhaps after passing to another sub-neighborhood, that if ρ ∈ O ′ , then ξ k ρ is c 1 -Hölder for some c 1 > 0. So, (2) holds with α = min{c 0 , c 1 }.
By passing to a subsequence we can suppose that ρ m → ρ ∈ O. After passing to a further subsequence and translating by elements in Γ, either (1) v m → v ∈ U(Γ) and x m → x ∈ Λ(Γ), or (2) there exists an embedded cusp neighborhood C = α \H such that {v m } ⊂ U(Γ) H and {v m } projects to an escaping sequence in U(Γ).
In the first case, x ∈ v ⊥ and lim m→∞ f ρm,vm,xm (ρm,vm) = f ρ,v,x (ρ,v) = 0. Thus we must be in the second case. By conjugating, we may assume that ρ m (α) = ρ 0 (α) for all m ∈ Z + ∪{∞}, and that the restriction of the canonical norm on E(O ′ ) to each E ρm is with respect to the same cusp representation Ψ for α and ρ 0 (α). Then there exists a ρ 0 (α) ss -invariant, Ψ-equivariant family of norms · = 0 and we have a contradiction. This completes the proof of Lemma 8.6 and hence the proof of (2).
The limits maps vary analytically. It remains to prove the analytic variation of the limit maps. First suppose that K = R. The general strategy is to complexify and then exploit the fact that locally uniform limits of complex analytic functions are complex analytic.
Suppose that h : M → Hom tp (ρ 0 ) is a real analytic map and every representation in h(M ) is P k -Anosov.
If ρ : Γ → SL(d, R) is P k -Anosov, we may compose with the inclusion map ι 2 : SL(d, R) → SL(d, C) to obtain a P k -Anosov representation ρ C = ι 2 • ρ : Γ → SL(d, C). We claim, after possibly shrinking M C , that h(M C ) ⊂ Hom tp (ρ C 0 ). For any α ∈ SL(d, C), the set {g ∈ SL(d, C) : g is conjugate to α} is locally closed (i.e. open in its closure) in the Zariski topology, see for instance [11,Theorem 3.6]. This implies that Hom tp (ρ C 0 ) is itself a locally closed set in the Zariski topology on SL(d, C) N . Then since h(M ) ⊂ Hom tp (ρ C 0 ), by shrinking M C we may assume that h(M C ) ⊂ Hom tp (ρ C 0 ). Since every representation in h(M ) is P k -Anosov, by shrinking M C again if necessary, we may assume that every representation in h(M C ) is also P k -Anosov. Thus, if we can prove that for any x ∈ Λ(Γ), the map u → ξ k h(u) (x) from M C to Gr k (C d ) is complex analytic, then its restriction to M is real analytic.
If γ is a hyperbolic element and ρ ∈ h(M C ), then ρ(γ) is P k -proximal and ξ k ρ (γ + ) is the attracting k-plane of ρ(γ). Then it follows from standard results in the perturbation theory of linear operators, see, for example, [22,Chapter 6], that the function from M C to Gr k (C d ) given by u → ξ k h(u) (γ + ) is complex analytic. If x ∈ Λ(Γ), then there exists a sequence {γ n } of hyperbolic elements of Γ, so that γ + n → x. Then, since the map is a locally uniform limit of complex analytic functions, hence complex analytic. This completes the proof of (3) in the case when K = R.
The case when K = C case follows by simply repeating the argument in the previous paragraph.

Positive representations in the sense of Fock-Goncharov
In Fock and Goncharov's work [16], they define positive representations in the following way. Suppose that Γ 0 ⊂ PSL(2, R) is a discrete group and Γ 0 \ H 2 is a non-compact, finite area hyperbolic surface. Recall that γ ∈ Γ 0 is peripheral if it is represented by a curve which may be freely homotoped off of every compact subset of S = Γ\ H 2 . Let Λ p (Γ 0 ) be the set of fixed points of the peripheral elements in Γ 0 (in this case, all of which are parabolic). Notice that Λ p (Γ 0 ) inherits two natural cyclic orders as a subset of ∂H 2 . A representation ρ : Notice that, with Fock and Goncharov's definition, every Hitchin representation of a convex cocompact, but not cocompact, Fuchsian group Γ 0 is also a positive representation of a lattice Γ. (If Γ\H 2 is homeomorphic to the interior of a compact surface S, then Γ is a finite area uniformization of the interior of S.) In this case, every peripheral element is mapped to a loxodromic element and there are many different positive ρ-equivariant maps from Λ p (Γ) to F (R d ), corresponding to a choice of fixed point for the (unique) fixed point of each peripheral element of Γ.
Motivated by their definition, we define the notion of a positive type preserving representation in the following way. Let Γ ⊂ PSL(2, R) be a geometrically finite group. Then γ ∈ Γ is peripheral if it is either unipotent or it is a hyperbolic element whose fixed points both lie in the boundary of Λ(Γ). Observe that if ρ : Γ → SL(d, R) is a positive type preserving representation, then ρ•f * : Γ 0 → SL(d, R) is a positive representation for all homeomorphisms f : It is clear that every Hitchin representation from Γ to SL(d, R) is a positive type preserving representation. We show that the converse is also true. Let ρ be a positive, type preserving representation, and let ζ : Λ p (Γ) → F (R d ) denote a positive, ρ-equivariant map. We make use of the following well-known fact which is implicit in Fock-Goncharov [16], see Kim-Tan-Zhang [23,Observation 3.18] for details. It may be viewed as a generalization of the fact that every bounded monotone sequence in R is convergent. We fix for the remainder of the section, one of the two natural cyclic orders on ∂ H 2 . With this convention, it is natural to define one-sided convergence of sequences. Definition 9.4. A sequence {x n } in ∂ H 2 converges to x ∈ ∂ H 2 in the positive direction (respectively, in the negative direction) if {x n } converges to x, and there exists N > 0 such that For short, we write x n ր x (respectively, x n ց x) if {x n } converges to x in the positive direction (respectively, in the negative direction). If {x n } converges to x in either the positive direction or the negative direction, we say that {x n } converges monotonically to x. • If there are no sequences in Λ p (Γ) that converge to x in the positive direction (respectively, in the negative direction), then there necessarily are sequences in Λ p (Γ) that converge to x in the negative direction (respectively, in the positive direction). Thus, we may set Since Λ p (Γ) is dense in Λ(Γ), Proposition 9.3 implies that these limit maps are well-defined. We refer to the maps ξ + : Λ(Γ) → F (R d ) (respectively, ξ − : Λ(Γ) → F (R d )) as the plus limit map (respectively, minus limit map).
Proof. For each i = 1, . . . , k, let q i , q ′ i ∈ Λ p (Γ) be points satisfying the following conditions: • q i = x i if there are no sequences in Λ p (Γ) that converge to x i in the positive direction, • q ′ i = x i if there are no sequences in Λ p (Γ) that converge to x i in the negative direction, if there is a sequence in Λ p (Γ) that converges to x i in the negative direction, and • q ′ i−1 < q i < x i if there is a sequence in Λ p (Γ) that converges to x i in the positive direction. Here, arithmetic in the subscripts is done modulo k.
By the definition of ξ ± , for each i = 1, . . . , k, there is a sequence (y i,n ) n≥1 in Λ p (Γ) that converges monotonically to x i , and satisfies lim n→∞ ζ(y i,n ) = ξ si (x i ). By passing to the tail of the sequences (y i,n ) n≥1 , we may assume that for all n, q 1 < y 1,n < q ′ 1 < q 2 < y 2,n < q ′ 2 < · · · < q k < y k,n < q ′ k , which implies that ) is a positive tuple of flags in F (R d ). Our result then follows from repeatedly applying Proposition 9.3. We next prove that ξ ± satisfy an analogue of the strongly dynamics-preserving property of Anosov limit maps. In particular, if ξ + = ξ − , then it implies that ξ ± is strongly dynamics-preserving. In fact, the proof mimics the proof of Theorem 7.1.
The proof proceeds in three cases. In the first case, we assume that x is not the fixed point of a peripheral element. If x is the fixed point of a peripheral element, then x is the fixed point of either a parabolic element or a peripheral hyperbolic element.
Case 2: x ∈ Λ p (Γ) is the fixed point of a parabolic element α ∈ Γ. As in the proof of Theorem 1.4, Lemma 7.6 implies that ρ(α) has a unique fixed flag in F (R d ). It follows that ξ + (x) = ξ − (x) = ζ(x) is this unique fixed flag.

Appendix A. Constructing cusp representations
In this appendix, we prove Proposition 3.6. Suppose g ∈ SL(d, K) is weakly unipotent and let u = 1 1 0 1 .
Next suppose that K = R. Given two matrices A, B we will let A ⊗ B denote the Kronecker product, that is if A = [a ij ] is an m-by-n matrix, then Also given θ ∈ R let M (θ) = cos θ sin θ − sin θ cos θ .

Appendix B. Anosov representations into semisimple Lie groups
In this appendix, we develop a more general theory of Anosov representations of geometrically finite Fuchsian groups into a semisimple Lie group G with respect to a parabolic subgroup P + . We extend results of Guichard-Wienhard [18,Prop. 4.3] (see also Section 3]) to show that there exists an irreducible representation ψ : G → SL(d, R) so that a representation into G is Anosov with respect to P + if and only if its composition with ψ is P 1 -Anosov. This will allow us to immediately recover generalizations of all the results we obtained for linear Anosov representations.
For the rest of the section, we will assume that G is a semisimple Lie group of non-compact type with finite center, denote its Lie algebra by g, and let ad : g → sl(g) and Ad : G → SL(g) be the adjoint representations.
Fix a parabolic subgroup P + ⊂ G and an opposite parabolic subgroup P − ⊂ G and let F ± := G/P ± be the associated flag varieties. If ρ : Γ → G is a representation, we define the bundles ρ is a vector bundle over B ± ρ of rank dim(F ± ). The geodesic flow on U(Γ) extends to flows on U(Γ) × F ± and U(Γ) × T F ± whose action is trivial on the second factor. These in turn descends to flows on B ± ρ and V ± ρ which covers the geodesic flow on U(Γ). We use φ t to denote all of these flows.
(2) For some family of norms · on the fibers of V ± ρ → B ± ρ , the pullback of the flow φ t , also denoted φ t , is uniformly expanding/contracting on ( σ ± ξ ) * V ± ρ . We refer to any such map ξ as the P ± -Anosov limit map for ρ.
Remark B.2. With a little more work, one can show that the pullback of the flow φ t is uniformly expanding on ( σ + ξ ) * V + ρ (with respect to some norm) if an only if it is uniformly contracting on ( σ − ξ ) * V − ρ (with respect to some norm).
Remark B.3. To be precise, the flow φ t is uniformly expanding on . Remark B.4. In the case when G = SL(d, K) and (P + , P − ) = (P k , P opp k ) where P k = Stab G ( e 1 , . . . , e k ) and P opp k = Stab G ( e k+1 , . . . , e d ), we can identify F + = Gr k (K d ) and F − = Gr d−k (K d ). Then for any transverse pair (F, G) ∈ Gr k (K d ) × Gr d−k (K d ), there is a natural identification T F Gr k (K d ) ≃ Hom(F, G) and T G Gr d−k (K d ) ≃ Hom(G, F ). Thus, the pullback bundles ( σ + ξ ) * ( V + ρ ) and ( σ − ξ ) * ( V − ρ ) are canonically identified with Hom( Θ k , Ξ d−k ) and Hom( Ξ d−k , Θ k ) respectively, where Θ k and Ξ d−k are the sub-bundles of E ρ that lift to sub-bundles Θ k and Ξ d−k of E ρ with the defining property Thus, in this case, Definition B.1 agrees with Definition 3.1, see Proposition 3.3.
The following result often allows one to reduce the general study of Anosov representations to the study of P 1 -Anosov representations into SL(d, R). For Anosov representations of word hyperbolic groups the analogous result is due to Guichard-Wienhard [18,Proposition 4.3].
Theorem B.5. Let G be a semisimple Lie group with finite center and let P ± be a pair of opposite parabolic subgroups. Suppose that ψ : G → SL(V ) is a finite dimensional irreducible representation which is adapted to (P + , P − ). Then a representation ρ : Γ → G of a geometrically finite Fuchsian group Γ is P ± -Anosov if and only if ψ • ρ is P 1 -Anosov.
Adapted representations are not hard to construct. If n + is the nilpotent radical of the Lie algebra of P + , and n = dim n + , then ψ(g) = ∧ n Ad(g) and V = Span{ψ(G)(∧ n n + )} ⊂ ∧ n g is adapted to (P + , P − ), see [18,Remark 4.12]. We obtain the following immediate corollary.
Corollary B.6. Suppose that G is a semisimple Lie group with finite center and P ± is a pair of opposite parabolic subgroups. There exists a finite-dimensional irreducible representation ψ : G → SL(V ) so that a representation ρ : Γ → G of a geometrically finite Fuchsian group Γ is P ± -Anosov if and only if ψ • ρ is P 1 -Anosov.
Corollary B.6 allows us to generalize our main results about linear Anosov representations into the general setting. As in the SL(d, K) case, if ρ : Γ → G is a representation, let be the space of representations σ : Γ → G so that if α ∈ Γ is parabolic, then σ(α) is conjugate to ρ(α). Theorem 1.3 becomes: Corollary B.7. Suppose that G is a semisimple Lie group with finite center, P ± is a pair of opposite parabolic subgroups of G and Γ is a geometrically finite Fuchsian group. If ρ : Γ → G is P ± -Anosov, then there exists an open neighborhood O of ρ in Hom tp (ρ), so that (1) If ρ ∈ O, then ρ is P ± -Anosov.
(2) There exists α > 0 so that if ρ ∈ O, then its P ± -Anosov limit map ξ ρ is α-Hölder. One can also obtain analogues of parts (1) and (2) of Theorem 1.1 where the roles of singular values and eigenvalues are played by roots acting on the Cartan and Jordan projections (see [17,Section 3] for a complete discussion). Part (4) of Theorem 1.1 remains true if we replace X d (K) with the symmetric space of G.
Theorem B.5 will be a consequence of Theorem 1.2 and the following dynamical property of Anosov representations.
Delaying the proof of the lemma we prove Theorem B.5.
Next fix a maximal flat F ⊂ X 1 with W + , W − ⊂ F (∞) and fix some p ∈ F . Then let s p : X 1 → X 1 denote the involutive isometry based at p. Then F and H are transverse if and only if W + = s p (W − ). Since T ψ : X 1 → X 2 is a totally geodesic isometric embedding, there exists a maximal flatF ⊂ X 2 with T ψ (F ) ⊂F . ThenŴ + ,Ŵ − ⊂F (∞) and if sp : X 2 → X 2 is the involutive isometry based atp := T ψ (p), then ψ(F ) and ψ(H) are transverse if and only ifŴ + = sp(Ŵ − ).
Since We first prove the reverse direction of Theorem B.5.
We now prove the forward direction of Theorem B.5.
Since ψ is irreducible, ψ + (F + ) spans V and one can repeat the proof of Corollary 6.3 to show that η is strongly dynamics preserving.
It only remains to prove Lemma B.9.
Proof of Lemma B.9. Let p ± be the Lie algebra of P ± . Then there exists a Cartan decomposition g = k ⊕ p, a Cartan subspace a ⊂ p, and an element H 0 ∈ a so that is the root space decomposition associated to a. Let n ± = α(±H0)>0 g α and define T : n − → F + where T (X) = e X P + The map T has the following properties.
Proof. By definition, F is transverse to P − if and only if (F, P − ) ∈ G · (P + , P − ) if and only if F = gP + for some g ∈ P − . By the Langlands decomposition, P − = N − (P + ∩ P − ) where N − ⊂ G is the connected Lie subgroup with Lie algebra n − . Since N − is nilpotent, N − = e n − . So F is transverse to P − if and only if F = e X P + for some X ∈ n − . This proves part (1). Part (2) is an immediate consequence of the definition. Part (3) follows from the fact that g = n − ⊕ p + (as vector spaces) and p + is the Lie algebra of P + .
As a consequence of (1) and (2) in Observation B.13, we have the following.
Let K ⊂ G be the maximal compact subgroup with Lie algebra k and fix a K-invariant Riemannian metric on F + , and let | · | denote the induced family of norms on the fibers of T F + .

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