Anosov representations with Lipschitz limit set

. We study Anosov representation for which the image of the boundary map is the graph of a Lipschitz function, and show that the orbit growth rate with respect to an explicit linear function, the unstable Jacobian, is integral. Several applications to the orbit growth rate in the symmetric space are provided. We further study regularity of the limit curves of θ -positive representations as introduced by Guichard-W. [27]


Introduction
Let Γ Ă PGL d pRq be a discrete subgroup.Following Guivarc'h, Benoist [4] has shown that if Γ contains a proximal element and acts irreducibly on R d then its action on projective space PpR d q has a smallest closed invariant set.This is usually called Benoist's limit set or simply the limit set of Γ on PpR d q and denoted by L Γ .
In contrast with the negatively curved situation, the limit set of a subgroup Γ whose Zariski closure has rank ě 2 needs not be a fractal object.Examples of infinite co-volume Zariski-dense groups whose limit set is a proper C 1 submanifold A.S. was partially financed by ANR DynGeo ANR-16-CE40-0025.B.P. and A.W acknowledge funding by the Deutsche Forschungsgemeinschaft within the Priority Program SPP 2026 "Geometry at Infinity", and from the National Science Foundation under Grant No. 1440140, while they were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the semester of fall 2019.A.W. acknowledges funding by the European Research Council under ERC-Consolidator grant 614733, and by the Klaus-Tschira-Foundation.This work is supported by the Deutsche Forschungsgemeinschaft under Germany's Excellence Strategy EXC-2181/1 -390900948 (the Heidelberg STRUCTURES Cluster of Excellence).
This paper provides the first systematic investigation of this intermediate phenomenon -its main object are discrete groups whose limit set is a Lipschitz manifold.We will restrict our investigation to the class of Anosov subgroups, a robust and rich class of strongly undistorted subgroups of semisimple Lie groups (see Section 2.2 for the precise definition).
For discrete subgroups Γ of SOp1, nq Sullivan [48] established a beautiful relation between a geometric invariant of the limit set L Γ , its Hausdorff dimension, and a dynamical invariant for the action of Γ on the symmetric space H n , the orbit growth rate.This was further used by Bowen [7] to prove a strong rigidity result: for fundamental groups of surfaces acting on H 3 , the Hausdorff dimension of the limit set is minimal if and only if the limit set is C 1 and Γ preserves a totally geodesic copy of H 2 on which it acts cocompactly.When G has higher rank, the situation is more complicated as one can additionally consider orbit growth rates with respect to different linear functionals ϕ (as in, for example, Quint [44]).It is a challenging problem to understand what are the functionals ϕ whose orbit growth rate carries geometric information on the group Γ or on its limit set L Γ .
The main contribution of the paper is to single out an explicit linear functional, the unstable Jacobian, whose critical exponent is integral on Anosov subgroups whose limit set is a Lipschitz submanifold.In order to prove such a result, we import ideas from non-conformal dynamics, such as the study of the affinity exponent, to the setting of Anosov groups, and use the Anosov property, together with ideas from geometric group theory, to establish a strengthening of the theory of Patterson-Sullivan denisities developed by Quint; these two results are of independent interest.We then showcase the strength of our main result by applying it to several well studied classes of representations: maximal representations, H p,q -convex cocompact subgroups and Θ-positive representations.
The unstable Jacobian and the affinity exponent.We now introduce some notation useful to explain more precisely our results.We denote by E " ta " pa 1 , . . ., a d q P R d : the Cartan subspace of the Lie group PGL d pRq, by a i paq " a i ´ai`1 the i-th simple root and by E `Ă E the Weyl chamber whose associated set of simple roots is Π " ta i : i P 1, d ´1 u.Let a : PGL d pRq Ñ E `be the Cartan projection with respect to the choice of a scalar product τ .Concretely apgq " plog σ 1 pgq, . . ., log σ d pgqq , where σ i pgq denote the singular values of the matrix g, the square roots of the eigenvalues of the matrix gg ˚, where g ˚is the adjoint operator of g with respect to τ .
Given a discrete subgroup Γ ă PGL d pRq, the critical exponent of a linear form ϕ P E ˚, denoted by h Γ pϕq, is defined as h Γ pϕq :" lim T Ñ8 log # γ P Γ| ϕ `apγq ˘ă T ( T .
In this paper we introduce the p-th unstable Jacobian J u p P E ˚defined by J u p " pp `1qω a1 ´ωap`1 , where ω ap paq " ř p 1 a i is the fundamental weight relative to the p-th simple root a p .Our main result is Theorem A. Let Γ ă PSL d pRq be a strongly irreducible, projective Anosov subgroup whose limit set L Γ ă PpR d q is a Lipschitz submanifold of dimension p. Then h Γ pJ u p q " 1.If p " 1 the same holds replacing strong irreducibility with weak irreducibility 1 .
A similar result was proven, in the context of fundametal groups of compact strictly convex projective manifolds by Potrie-S.[40,Theorem B]; our approach is entirely different and, since we require less regularity, its scope of application is considerable broader.Note that up to postcomposing with a suitable linear representation, any Anosov representation can be turned in a projective Anosov representation.
We prove the two inequalities in Theorem A as corollary of two different results that are applicable in more general settings.We focus first on the lower bound on the critical exponent (Corollary 1.1) that follows from a general result on the Hausdorff dimension of limit sets (of projective Anosov representations).
An important step in the proof is the study, in the context of Anosov representations, of the affinity exponent : a key notion from non-conformal dynamics that first appeared in Kaplan-Yorke [33] and Douady-Oesterlé [19], and played a prominent role in Falconer's work [21].More specifically, for every discrete subgroup Γ ă PSL d pRq, we consider the piecewise Dirichlet series defined, for p P N and s P rp ´1, ps, by Φ Aff Γ psq " ÿ γPΓ ˆσ2 σ 1 pγq ¨¨¨σ p σ 1 pγq ˙ˆσ p`1 σ 1 pγq ˙s´pp´1q : s P rp ´1, ps.
The affinity exponent is the critical exponent of this series: h Aff Γ : " inf s : Φ Aff Γ psq ă 8 ( " sup s : Φ Aff Γ psq " 8 ( P p0, 8s. The second main result of this paper is the following (see §3 for a statement for arbitrary local fields): Theorem B. Let Γ ă PGL d pRq be projective Anosov, then dim Hff `LΓ ˘ď h Aff ρ .It is easy to deduce from Theorem B relations between the Hausdorff dimension of the limit set of a projective Anosov subgroup and the orbit growth rate with respect to explicit linear functionals on the Weyl chamber.Since the quantity h Γ pJ u p´1 q appearing in Theorem A is also the critical exponent of the Dirichlet series s Þ Ñ ÿ γPΓ ˆσ1 ¨¨¨σ p σ p 1 pγq ˙s , we get: Corollary 1.1.Let Γ ă PGL d pRq be projective Anosov and assume furthermore that dim Hff pL Γ qq ě p. Then dim Hff pL Γ q ď ph ρ pJ u p q. Observe that J u 1 " a 1 , and thus, whenever dim Hff pL Γ q ě 1 we obtain as a consequence the results of Glorieux-Monclair-Tholozan [25,Theorem 4.1] and P.-S.-W.[41,Proposition 4.1].

Existence of Patterson Sullivan measures. The second inequality in Theorem
A follows from an improvement on a result by Quint [45,Théorème 8.1] concerning the relation between critical exponents and the existence of pΓ, ϕq-Patterson-Sullivan measures.
Given a set Θ Ă Π of simple roots, we denote by F Θ the associated partial flag manifold, this consists of the space of flags of subspaces of dimension indexed by Θ.We denote by E Θ the Levi subspace of E defined by The restrictions of the fundamental weights tω ap | EΘ : p P Θu span its dual pE Θ q ˚.
Using the Iwasawa decomposition of PGL d pRq, Quint introduced an Iwasawa cocycle b Θ : PGL d pRq ˆFΘ Ñ E Θ that is the higher rank analog of the more studied Busemann cocycle in negative curvature (see Quint [45,Lemma 6.6] and Section 5.3 for the precise definition).
With this notation at hand we can recall the definition of a pΓ, ϕq-Patterson-Sullivan measure from [45].

˘.
Inspired by a classical result by Sullivan [48], Quint shows [45, Théorème 8.1] that the existence of a pΓ, ϕq-Patterson-Sullivan measure on F Θ gives an upper bound on a related critical exponent h Γ pϕ `ρθ c q ď 1. (1.1) Here ρ θ c is an explicit linear functional which is positive on the interior of the Weyl chamber and accounts for the possible growth along the fibers of the projection Lemme 8.3].In general h Γ pϕ `ρθ c q ă h Γ pϕq and thus Quint's result is not sharp enough for our purposes.Using ideas from geometric group theory we show that, provided the group Γ is Anosov with respect to one of the roots in Θ, there is no contribution from the fibers: Theorem C. Let Γ ă PGL d pRq be projective Anosov and consider Θ Ă Π such that a 1 P Θ.Let ϕ P pE Θ q ˚.If there exists a pΓ, ϕq-Patterson-Sullivan measure on F Θ whose support is not contained on a complementary subspace 2 , then We refer the reader to §5 and Theorem 5.14 for a version of Theorem C where the target group is an arbitrary semi-simple group over a local field.
We provide the link between Theorem C and Theorem A in Section 6, where we establish that, if Γ ă PSL d pRq is a projective Anosov subgroup whose limit set L Γ is a Lipschitz submanifold of dimension p then there exists a pΓ, J u p q Patterson-Sullivan measure on F ta1,apu .In fact we explicitly construct such a measure using Rademacher's Theorem and an explicit volume form on the almost everywhere defined tangent space to L Γ (Proposition 6.4).
Example 1.3.If ρ : π 1 S Ñ PSpp4, Rq is a maximal representation (see §9 for the definition), the combination of Theorems B and C gives h ρpπ1Sq pa 2 q " 1 while Quint's result (equation (1.1)) becomes h ρpπ1Sq pω a2 q ď 1.This latter inequality is implied by the former equality, and often far from being sharp: one can find representations ρ for which h ρpπ1Sq pω a2 q is arbitrarily small.Theorem C is complementary and independent from the Patterson-Sullivan theory for Anosov representations developed by Dey-Kapovich [18].They only consider Patterson-Sullivan densities with respect to functionals ϕ that, as opposed to the unstable Jacobian, belong to pE θ q ˚where the representation is assumed to be Anosov with respect to all elements of θ, and induce Finsler distances on the symmetric space (see also Ledrappier [38] for a different approach yielding similar results); a drawback of their approach is that they can only relate the critical exponent with a pre-metric induced from a Finsler distance on the symmetric space that is hard to compute.On the opposite we begin with a natural measure supported on the limit set, which belongs to the Lebesgue measure class, find a suitable functional, the unstable Jacobian, turning such measure into a Patterson-Sullivan measure, and deduce from this geometric properties of the action of Γ on the symmetric space.
As another contribution of the paper of independent interest we show that also Θpositive representations of fundamental groups of surfaces in SOpp, qq (Guichard-W.[30]) yield subgroups with this property.We refer the reader to §10 for the precise definition of Θ-positive representations.We will only 3 consider here the Θ-positive representations that are furthermore Θ-Anosov for Θ " ta 1 , . . ., a p´1 u, as a result, 2 Given Θ Ă Π denote by i Θ " td ´p : p P Θu.Two points px, yq P F Θ ˆFi Θ , are transverse if for every p P Θ one has that x p X y d´p " t0u.A complementary subspace of F Θ is a subset of F Θ of the form tx P F Θ : x is not transverse to y 0 u for a given y 0 P F i Θ . 3Guichard-Labourie-W.announced that all Θ-positive representations are Θ-Anosov, so this should not pose any restriction.
for each k P Θ, they admit a boundary map ξ k : BΓ Ñ Is k pR p,q q parametrizing the limit set in the Grassmannian of k-dimensional isotropic subspaces.In Section 10 we prove: Then the images of the boundary maps ξ k : BΓ Ñ Is k pR p,q q are C 1 submanifolds for each 1 ď k ă p ´1, and the image ξ p´1 pBΓq is Lipschitz.
We will prove the two parts of Theorem D separately, respectively in Corollary 10.4 and Proposition 10.5.
At least for representations of fundamental groups of surfaces, the regularity of the limit set on a given (maximal) flag space seems to be related to the position of the associated root among the Anosov roots.By definition, a simple root is an Anosov root (for a subgroup Γ) if its kernel intersects trivially the limit cone L Γ of Γ.Among such roots one can consider the internal ones, i.e. such that every neighboring root in the Dynkin diagram is also an Anosov root, or the ones in the boundary, i.e. connected to a root that non-trivially intersects L Γ .For example, for a Θ-positive representation in SOpp, qq, the roots ta 1 , . . ., a p´2 u are internal, while a p´1 is the only boundary root.
The intermediate regularity (Lipschitz but not C 1 ) of limit sets for surface groups seems only to occur for boundary roots.For internal roots, we can prove a C 1dichotomy ruling out intermediate regularity in several interesting cases.More specifically we consider fundamental groups Γ of compact surfaces and study small deformations of representations of the form that are ta 1 , a 2 u-Anosov (this latter assumption can be rephrased as a proximality assumption on the linear representation R).For any such representation we have an explicit dichotomy: the associated limit set is either C 1 or not even Lipschitz (Corollary 7.8).We refer the reader to Section 7 for the precise statement of the dichotomy.
Entropy rigidity results.We conclude the introduction by discussing three well studied classes of representations to which Theorem A applies.Interestingly, in all these cases, the mere information on the critical exponent of the unstable Jacobian provided by Theorem A, allows to obtain a sharp upper bound on the critical exponent for the action on the symmetric space endowed with the Riemannian (!) distance function.In the case of Θ-positive representations this is even sufficient to prove that the bound is rigid: it is attained only on the specific Fuchsian locus, the generalization, in our setting, of Bowen's aforementioned result.
Maximal representations.Maximal representations are well studied representations of fundamental groups of surfaces in Hermitian Lie groups G R that were introduced by Burger-Iozzi-W.[11] through a cohomological invariant, the Toledo invariant.For these representations Theoem A applies and gives Theorem 1.4.Let G R be a classical simple Hermitian Lie group of tube type.Let ρ : Γ Ñ G R be a maximal representation, and let ǎ denote the root associated to the stabilizer of a point in the Shilov boundary of G R .Then h ρ pǎq " 1.
Concretely, in the case G R P tSpp2p, Rq, SUpp, pq, SO ˚p4pqu the root ǎ computes the logarithm of the square of the middle eigenvalue, while for G " SO 0 p2, pq the root ǎ is the first root, computing the logarithm of the first eigenvalue gap.
Theorem 1.4 also holds for the exceptional Hermitian Lie group of tube type if the representation is Zariski-dense, and we expect it to hold unconditionally.We refer the reader to §9 for a slightly more general statement, further explanations and consequences, in particular concerning a sharp upper bound on the exponential orbit growth rate for the action on the symmetric space (see Proposition 9.9).
H p,q -convex-cocompact representations.Generalizing work of Mess [39] and Barbot-Mérigot [2], Danciger-Guéritaud-Kassel [17] introduced a class of representations called H p,q -convex cocompact.Here H p,q is the pseudo-Riemannian hyperbolic space, consisting of negative lines in PpR d q for a fixed non-degenerate form Q of signature pp, q `1q.It follows then from [17,Theorem 1.11], that a projective Anosov subgroup Γ ă POpQq " POpp, q `1q is H p,q -convex cocompact if for every pairwise distinct triple of points x, y, z P L Γ , the restriction Q| xx,y,zy has signature p2, 1q.
Consider a representation Λ : POpp, 1q Ñ POpp, q `1q whose image stabilizes a pp `1q-dimensional subspace V of R d where Q| V has signature pp, 1q.Endow the symmetric space X p,q`1 with the POpp, q `1q-invariant Riemannian metric normalized so that the totally geodesic copy of H p in X p,q`1 stabilized be Λ has constant curvature ´1.Definition 1.5.For a subgroup Γ ă SOpp, q`1q and x 0 P X p,q`1 denote by h Xp,q`1 ρ the critical exponent of the Dirichlet series s Þ Ñ ÿ γPΓ e ´sdpx0,ρpγqx0q .
We have the following upper bound.
We expect this upper bound to be rigid, namely the upper bound should only be attained at an inclusion of a co-compact lattice in POpp, 1q preserving a totally geodesic copy of H p of the type induced by Λ.However, only the case p " 2 is known due to Collier-Tholozan-Toulisse [15].
Section 8 contains more information on H p,q -convex cocompact representations, in particular the relation with recent work by Glorieux-Monclair [23].
Θ-positive representations.Thanks to Theorem D, Theorem A also applies to Θ-positive representations of fundamental groups of surfaces in SOpp, qq and gives: Corollary 1.7.Let ρ : Γ Ñ SOpp, qq be a Θ-Anosov representation that is Θpositive and weakly irreducible, then h ρ pa k q " 1 for every k ď p ´1.
Inspired from Potrie-S.[40], we deduce from Corollary 1.7 a rigid upper bound for the critical exponent of the action of a positive representation on the Riemannian symmetric space X p,q (see Theorem 10.7).More precisely, we now normalize the SOpp, qq-invariant Riemannian metric on X p,q so that the totally geodesic copy of H 2 induced by the representation Λ : SL 2 pRq Ñ SOpp, qq that stabilizes a subspace of R d of signature pp, p ´1q, has constant curvature -1.We consider the critical exponent in Definition 1.5 with this normalization of distance.
Theorem 1.8.Let Γ be the fundamental group of a surface and let ρ : Γ Ñ SOpp, qq be Θ-positive.Then the critical exponent with respect to the Riemannian metric satisfies h Xp,q ρ ď 1.
Furthermore if equality is achieved at a totally reducible representation η, then η splits as W ' V , W has signature pp, p ´1q, η| W has Zariski closure the irreducible POp2, 1q in POpp, p ´1q, and η| V lies in a compact group.
New arguments are needed with respect to [40] since the Anosov-Levi space of a Θ-positive representation has codimension one (instead of 0, which is the case treated in [40]), see §10.
1.1.Plan of the paper.In §2 we introduce some required preliminaries, and recall some needed results from Bochi-Potrie-S.[6] and P.-S.-W.[41].Section 3 deals with the affinity exponent and Hausdorff dimension for Anosov representations, in it we prove Theorem B for any local field.Section 4 is a reminder on (more or less) standard definitions on semi-simple algebraic groups over a local field.In §5 we recall objets from higher rank Patterson-Sullivan Theory and in subsection 5.3 we prove Theorem 5.14 (a broader version of Theorem C).Section §6 completes the proof of Theorem A. The remaining sections deal with the applications of this result discussed in the introduction.

Preliminaries
We recall in this section the notions we will need concerning Anosov representations and cone types.We refer the reader to [41] and the references therein for more details.
Throughout the paper K will denote a local field with absolute value If K is non-Archimedean, we require that |ω| " 1 q where ω denotes the uniformizing element, namely a generator of the maximal ideal of the valuation ring O, and q is the cardinality of the residue field O{ωO (this is finite because K is, by assumption, local).This guarantees that the Hausdorff dimension of P 1 pKq equals 1.
2.1.Singular values and Anosov representations into PGL d pV K q.A K-norm } } on a K vector space V K induces a norm on every exterior power of V ; the angle between two vectors >pv, wq is the unique number in r0, πs such that sin >pv, wq :" }v ^w} }v}}w} Given two points rvs, rws P PpV q, we define their distance as dprvs, rwsq :" sin >pv, wq, and given any two subspaces P, Q ă V we define their minimal angle as >pP, Qq " min vPP zt0u min wPQzt0u >pv, wq.
An element a P GLpV K q is a semi-homothecy (for a norm } ¨}) if there exists a a-invariant K-orthogonal 4 decomposition V " V 1 ' ¨¨¨' V k and σ 1 , ¨¨¨, σ k P R such that for every i P 1, k and every v i P V i one has The numbers σ i are called the ratios of the semi-homothecy a.
Following Quint [43,Théorème 6.1], we fix a maximal abelian subgroup of diagonalizable matrices A Ă GLpV K q, a compact subgroup K Ă GLpV K q such that, if N is the normalizer of A in GLpV K q, then N " pN X KqA, and a K-norm } } on V preserved by K, and such that A acts on V by semi-homothecies.Let e 1 ' ¨¨¨' e d be the eigenlines of A (here d " dim V ) and choose the Weyl chamber A `consistsing of those elements a P A whose corresponding semi-homothecy ratios verify σ 1 paq ě ¨¨¨ě σ d paq.
For every g P GLpV K q we choose a Cartan decomposition g " k g a g l g with a g in A `, k g , l g P K, and denote by σ 1 pgq ě σ 2 pgq ě ¨¨¨ě σ d pgq the semi-homotecy ratios of the Cartan projection a g P A `(these do not depend on the choice of the Cartan decomposition once K and } ¨} are fixed).In order to simplify notation we will often write σi σj pgq " σipgq σj pgq .We define, for p P 1, d ´1 , u p pgq " k g ¨ep P V.
The set tu p pgq : p P 1, d´1 u is an arbitrary orthogonal choice of the axes (ordered in decreasing length) of the ellipsoid tAv : }v} " 1u, and, by construction, for every v P g ´1u p pgq one has }gv} " σ p pgq}v}.Let U p pgq " u 1 pgq ' ¨¨¨' u p pgq " k g ¨pe 1 ' ¨¨¨' e p q.
If g is such that σ p pgq ą σ p`1 pgq, then we say that g has a gap of index p.In that case the decomposition U d´p pg ´1q ' g ´1pU p pgqq is orthogonal (cfr.[41,Remark 2.4]) and, if K is Archimedean, the p-dimensional space U p pgq is independent of the Cartan decomposition of g.
We will denote by Π " ta 1 , . . ., a d´1 u the root system of PGLpV K q, and, given a subset θ Ă Π, by F θ the associated partial flag manifold.Given θ Ă Π we also denote by U θ pgq the partial flag U θ pgq " tU p pgq : a p P θu.The θ-basin of attraction of g B θ,α pgq " tx θ P F θ pK d q : min apPθ is the complement of the α-neighborhood of U θ c pg ´1q.When θ consists of a single root a we will write B a,α pgq instead of B tau,α pgq Remark 2.1.If g has a gap of index p, then U d´p pg ´1q is well defined if K is Archimedean, and any two possible choices have distance at most σp`1 σp pgq if K is non-Archimedean.It follows that, also in the non-Archimedean case, B θ,α pgq only depends on K provided α is bigger than the minimal singular value gap. 4 Recall that for K non-Archimedean a decomposition We recall for later use the following lemma, which explains the choice of the term basin of attraction: An a 1 -Anosov representation will be called projective Anosov.

Hausdorff dimension of the limit set and the affinitiy exponent
Generalizing the definition given in the introduction, we define the affinity exponent h Aff ρ of a projective Anosov representation ρ : Γ Ñ PGLpV K q as the critical exponent of the broken Dirichlet series where the dimension Recall furthermore that for a metric space pΛ, dq and for s ą 0 one defines its s-capacity as and that the Hausdorff dimension of Λ is defined by dim Hff pΛq " inf ts : H s pΛq " 0u " sup ts : H s pΛq " 8u . (3.1) The goal of the section is to prove the following result: The proof of Theorem 3.1 is elementary and based on the construction of a good cover of the image of the limit map (explicitely constructed in § 3.1) which we show, in § 3.2 to be contained in ellipses of controlled axis.
3.1.Coarse Cone types.In P.-S.-W.[41, Section 2.3.1]we used cone types at infinity to construct well behaved coverings of the boundary of the group.For the purposes of this paper a coarse version of these sets will be more useful, which we now introduce.
Recall that a sequence pα j q 8 0 is a pc 0 , c 1 q-quasigeodesic if for every pair j, l it holds 1 c 0 |j ´l| ´c1 ď d Γ pα j , α l q ď c 0 |j ´l| `c1 .
We associate to every element γ a coarse cone type at infinity, consisting of endpoints at infinity of quasi geodesic rays based at γ ´1 passing through the identity: ! rpα j q 8 0 s P BΓ| pα i q 8 0 is a pc 0 , c 1 q-quasigeodesic with α 0 " γ ´1, e P tα j uq ) .
Hyperbolicity of Γ lets us understand the overlaps of coarse cone types; this will be crucial in Section 5.3 to guarantee bounded overlap of suitable covers of the limit set.
Proposition 3.2.Let Γ be word-hyperbolic.For every c 0 , c 1 there exists pηq.Since Γ is hyperbolic, by the Morse Lemma, there exists K ą 0 (only depending on c 0 , c 1 and the hyperbolicity constant of Γ) such that γ is at distance at most K from a geodesic ray from e to x.The same holds then for η, and, using the hyperbolicity of Γ again, we can assume, up to making the constant K worse (but still depending on c 0 , c 1 only), that the two rays agree.This implies that there exist g 0 and g 1 on a geodesic ray from e to x such that dpγ, g 0 q ď K and dpη, g 1 q ď K. Since g 0 and g 1 lie in a geodesic we have dpg 0 , g 1 q ď ˇˇ|g 0 | ´|g 1 | ˇˇand thus dpγ, ηq ď 4K `ˇ| γ| ´|η| ˇˇ.
Our next goal is to show that, for an Anosov representation, the intersections of Cartan's basins of attraction B θ,α pρpγqq with the image of the boundary map are contained in the image of a suitably big coarse cone type of γ.Let now θ Ă Π be a subset containing the first root a 1 .We will denote by π θ,1 : F θ pK d q Ñ PpK d q the canonical projection.Recall from (2.1) that, for every α, we asociate to each g P PGLpV K q a basin of attraction B θ,α pgq Ă F θ .We will now use Lemma 2.6 to show that, for every α, there exist c 0 , c 1 such that the intersection of a θ-basin of attraction B θ,α pρpγqq with the image of the boundary map is contained in a pc 0 , c 1 q-coarse cone type.Proposition 3.3.Let ρ : Γ Ñ PGLpV K q be projective Anosov and consider α ą 0. There exist c 0 , c 1 only depending on α and ρ such that for every θ Ă Π containing a 1 , and every γ P Γ Proof.It is enough to show that if ξ 1 pxq P π θ,1 pB θ,α pρpγqqq and |γ| is big enough, then there is a quasi-geodesic ray from γ ´1 to x that passes through the identity whose constants only depend on α and ρ.Consider a quasigeodesic ray tα j u converging to x, and fix 1 ą α 1 ą α.Since, by assumption, ξ 1 pxq P B a1,α pρpγqq, we can find a constant L depending on ρ only, such that for every j ą L it holds U 1 pρpα i qq P B a1,α 1 pρpγqq (the uniformity of L follows from the last statement in Proposition 2.5).By definition we have >pU 1 pρpα j qq, U d´1 pρpγ ´1qqq ą α 1 , and thus, in particular, dpU 1 pρpα j qq, U 1 pρpγ ´1qq ą α 1 .Let now pα j q ´|γ|S i"0 be a geodesic segment with α 0 " e, α ´|γ|S " γ.Up to further enlarging α 1 and L (depending on the representation only) we have also that dpU 1 pρpα ´Lqq, U 1 pρpα L qq ą α 1 .Lemma 2.6 implies that the sequence pα j q 8 i"´|γ|S obtained as concatenation of the geodesic between γ ´1 and the identity and the ray from the identity to x is a quasi geodesic ray, thus the result.
Corollary 3.4.Let ρ : Γ Ñ PGLpV K q be projective Anosov and consider α ą 0. There exists C only depending on α and ρ such that for every θ Ă Π containing a 1 , if Proof.This follows immediately combining Proposition 3.3 and Proposition 3.2.
In particular we can use basins of attraction to construct coverings of the image of the boundary map with bounded overlap: Proposition 3.5 (Cfr.P.-S.-W.[41,Lemma 2.21]).Let ρ : Γ Ñ PGLpV K q be projective Anosov.There exists α small enough so that, for every T ą 0, the family of open sets U T :" tρpγq ¨Ba1,α pρpγqq : |γ| " T u defines an open covering of ξ 1 pBΓq.Furthermore there exists a constant C depending on α (and ρ) such that for every x P BΓ and every T , ξpxq is contained in at most C elements of U T .
Proof.Let x P BΓ, let tγ j u be a geodesic ray based at the identity representing x.Propositions 2.4 and 2.5 guarantee that there exists α " α ρ such that >pρpγ ´1 T qξ 1 pxq, U d´1 pρpγ ´1 T qq ą α, therefore ξ 1 pxq P ρpγ T qB a1,α pρpγ T qq.The second statement is a direct consequence of Corollary 3.4.

3.2.
Ellipses.The purpose of this section is to prove that for a projective Anosov representation, the set ρpγq ¨Ba1,α pρpγqq is coarsely contained in an ellipsoid with axes of size

*
The vector spaces u 1 ' u j are the axes of the ellipsoid and the size of the axis u 1 ' u j is 1{a j .We need the following covering lemma.Lemma 3.7.Let E be an ellipsoid with axis of size 1 ě β 2 ě . . .ě β d .For every p P 2, d , E can be covered by Proof.We consider the affine chart of PpV q corresponding to u 1 " 1.The ellipsoid E is contained in the product of the balls Since the product of d balls of radius β p is contained in a ball of radius ?dβ p we obtain that E can be covered by ] balls of radius β p , and hence E can be covered with balls of radius β p .Proposition 3.8.Consider α ą 0. For every g P PGLpV K q one has that the image of the corresponding Cartan's basin of attraction g ¨Ba1,α pgq is contained in the ellipsoid about U 1 pgq with axes u 1 pgq ' u j pgq of size 1 sin α σ j σ 1 pgq.
Proof.Assume first that K is Archimedean.By definition of B a1,α pgq, for every v P K d with K ¨v P B a1,α pgq one has where pv 1 , ¨¨¨, v d q are the coefficients in the decomposition of v with respect to the orthogonal splitting V " À g ´1u j pgq.Since the coefficients w j of gv in the decomposition induced by the orthogonal decomposition V " À u j pgq satisfy |w j | " σ j pgq|v j |, one has One concludes that gv lies on the corresponding ellipsoid.The non-Archimedean case follows analogously.

3.3.
The lower bound on the affinity exponent.We now have all the ingredients needed to prove Theorem 3.1: Proof.For each T ą 0 denote by U T the covering of ξ 1 pBΓq given by Proposition 3.5.By definition, U " U γ P U T is of the form ρpγq ¨Ba1,α pρpγqq for some γ satisfying |γ| " T. Proposition 3.8 applied to ρpγq implies that ρpγq ¨Ba1,α pρpγqq is contained in an ellipsoid about Ku 1 pρpγqq of axes with sizes Furthermore, since ρ is Anosov, we deduce from Lemma 2.2 that sup UPUT diam U is arbitrarily small as T goes to infinity.Recall that the s-capacity H s was defined by Equation (3.1).Applying the covering Lemma 3.7 to these ellipses and any p P 2, d , we obtain By definition, the affinity exponent h Aff ρ is such that for all s ą h Aff ρ the broken Dirichlet series is convergent and thus for all s ą h Aff ρ we have As a result we conclude that for all s ą h Aff ρ the s-capacity H s `ξpBΓq ˘vanishes, hence h Aff ρ ě dim Hff `ξpBΓq ˘.
This completes the proof.
The following generalization of Corollary 1.1 is also immediate: Corollary 3.9.Let ρ : Γ Ñ PGLpV K q be projective Anosov.If dim Hff `ξpBΓq ˘ě pd K , then dim Hff `ξpBΓq ˘ď ph ρ pJ u p q. Proof.Observe that for every s P rd K p, d K pp `1qs, the value of the broken Dirichlet series defining the affinity exponent is smaller than or equal to the value of the series associated to the p-th unstable Jacobian divided by p: The result follows as, for all k P 1, d ´1 and v P E `one has that which implies kh ρ pJ u k q ď pk ´1qh ρ pJ u k´1 q.

Semi-simple algebraic groups
Let G be a connected semi-simple K-group, G K the group of its K-points, A a maximal K-split torus and XpAq the group of its K ˚-characters.Consider the real vector space E ˚" XpAq b Z R and E its dual.For every χ P XpAq, we denote by χ ω the corresponding linear form on E.

4.1.
Restricted roots and parabolic groups.Let Σ be the set of restricted roots of A in g, the set Σ ω is a root system of E ˚.Let Σ `be a system of positive roots and Π the associated subset of simple roots.Let E `be the Weyl chamber determined by the positive roots pΣ ω q `.
Let W be the Weyl group of Σ, it is isomorphic to the quotient of the normalizer N G K pA K q of A K in G K by its centralizer Z G K pA K q.Let i : E Ñ E be the opposition involution: if u : E Ñ E is the unique element in the Weyl group with upE `q " ´Et hen i " ´u.
A subset Θ Ă Π determines a pair of opposite parabolic subgroups P Θ and PΘ whose Lie algebras are defined by and pΘ " à The group PΘ is conjugated to the parabolic group be the Lie algebra of the associated Levi group.The K-flag space associated to Θ is the closed submanifold of flags in F Θ pG K q that are not transverse to y.
Denote by p¨, ¨q a W -invariant inner product on E, p¨, ¨q the induced inner product on E ˚and define xχ, ψy " 2pχ, ψq pψ, ψq and let tω a u aPΠ be the dual basis of Π, i.e. xω a , by " δ ab .The linear form ω a is the fundamental weight associated to a.
4.2.Cartan decomposition.Let ν : A K Ñ E be defined, for z P A K , as the unique vector in E such that for every χ P XpAq one has χ ω pνpzqq " log |χpzq|.

Denote by
Let K Ă G K be a compact group that contains a representative for every element of the Weyl group W.This is to say, such that the normaliser N G K pA K q verifies N G K pA K q " pN G K pA K q X KqA K .One has G K " KA K K and if z, w P A K are such that z P KwK then νpzq " νpwq.There exists thus a function a : G K Ñ E such that for every g 1 , g 2 P G K one has that g 1 P Kg 2 K if and only if apg 1 q " apg 2 q.It is called the Cartan projection of G K .
In the case of G K " PGLpV K q this is nothing but the ordered list of semihomotecy ratioes defined in Section 2.1.

4.3.
Representations of G K .Let Λ : G Ñ PGLpV q be a finite dimensional rational 6 irreducible representation and denote by φ Λ : g Ñ slpV q the Lie algebra homomorphism associated to Λ. Then the weight space associated to χ P XpAq is the vector space V χ " tv P V : φ Λ paqv " χpaqv @a P A K u and if V χ ‰ 0 then we say that χ ω P E ˚is a restricted weight of Λ. Theorem 7.2 of Tits [49] states that the set of weights has a unique maximal element with respect to the order χ ě ψ if χ ´ψ is positive on E `.This is called the highest weight of Λ and denoted by χ Λ .Definition 4.1.Let Θ Λ be the set of simple roots a P Π such that χ Λ ´a is still a weight of Λ.
Remark 4.2.The subset Θ Λ is the subset of simple roots such that the following holds: Consider a P Σ `, n P g ´a and v P χ Λ , then φ Λ pnqv " 0 if and only if a P xΠ ´ΘΛ y.Definition 4.3.We denote by } } Λ a good norm on V invariant under ΛK and such that ΛA K consists on semi-homotecies, if K is Archimedean the existence of such a norm is classical, if K is non-Archimedean then this is the content of Quint [43, Théorème 6.1].
For every g P G K one has log }Λg} Λ " χ Λ papgqq. (4.2) If g " k g z g l g with k, l P K and z g P A K then for all v P Λpl ´1 g qV χΛ one has }Λgpvq} Λ " }Λg} Λ }v} Λ .
Denote by W χΛ the ΛA K -invariant complement of V χΛ .Note that the stabilizer in G K of W χΛ is PΘ,K , and thus one has a map of flag spaces This is a proper embedding which is an homeomorphism onto its image.Here G p2q dim Vχ Λ pV q is the open PGLpV K q-orbit in the product of the Grassmannian of pdim V χΛ q-dimensional subspaces and the Grassmannian of pdim V ´dim V χΛ qdimensional subspaces.
One has the following proposition by Tits (see also Humphreys [34, Chapter XI]).
Proposition 4.4 (Tits [49]).For each a P Π there exists a finite dimensional rational irreducible representation Λ a : G Ñ PSLpV a q, such that χ Λa is an integer multiple of the fundamental weight ω a and dim V χΛ a " 1.All other weights of Λ a are of the form where n b P N.
We will fix from now on such a set of representations and call them, for each a P Π, the Tits representation associated to a.

4.4.
The center of the Levi group P Θ,K X PΘ,K .We now consider the vector subspace together with the unique projection π Θ : E Ñ E Θ , that is invariant under the subgroup W Θ of the Weyl group spanned by reflections associated to roots in Π´Θ: The dual space pE Θ q ˚is canonically the subspace of E ˚of π Θ -invariant linear forms and it is spanned by the fundamental weights of roots in Θ pE Θ q ˚" tϕ P E ˚: ϕ ˝πΘ " ϕu " xω a : a P Θy.

Since π 2
Θ " π Θ , the pre-composition with π Θ induces a projection The following examples will be relevant in Section 7 and 8 respectively: Example 4.5.Let G K " PGLpV K q, consider p P 2, d´2 and let Θ " ta 1 , a p , a d´1 u, so that E Θ " tpa 1 , . . ., a d q P E : a 2 " ¨¨¨" a p and a p`1 " ¨¨¨" a d´1 u is three dimensional.Using the fact that the fundamental weights ω i (for i " 1, p, d ´1) belong to pE Θ q ˚one checks that the projection is pRq preserving a signature pp, qq bilinear form with p ă q.One has that E " tpa 1 , . . ., a p q : a i P Ru equipped with the root system A Weyl chamber can be chosen as E `" ta P E : a i ě a i`1 @i P 1, p ´1 and a p ě 0u and the associated set of simple roots Π " ta i : i P 1, p ´1 u Y tε p u. Consider then Θ " ta i : i P 1, p ´1 u, so that E Θ " ker ε p and thus a i P pE Θ q ˚for i P 1, p ´2 .Moreover, a Θ p´1 " ε p´1 and one has that a Θ p´1 | E `´t0u ě a p´1 | E `´t0u .4.5.Gromov product.Recall from S. [47] that the Gromov product7 based at K is the map p¨|¨q K : F p2q Θ pG K q Ñ E Θ defined to be the unique vector px|yq K P E Θ such that χ a ppx|yq K q " ´log sin > } }Λ a pξ Λa x, ξ Λa yq for all a P Θ, where χ a is the fundamental weight associated to the Tits representation Λ a of a.Note that max aPΘ χ a ppx|yq K q " max aPΘ |χ a ppx|yq K q| " ´log min aPΘ sin > } }Λ a pξ Λa x, ξ Λa yq. (4.4) On has the following remark from Bochi-Potrie-S.[6].
Remark 4.7 ([6, Remark 8.11]).Let Λ : G Ñ PGLpV q be a finite dimensional rational irreducible representation, if px, yq P F where } } Λ denotes the (stabilizer of the) inner product on V such that ΛK is orthogonal (see Definition 4.3).4.6.Iwasawa cocycle and its relation to representations of G. Another important decomposition of Lie groups that will play a role in our work is the Iwasawa decomposition: where P Π,K is the minimal parabolic subgroup, and U Π,K is its unipotent radical.For general local field K the decomposition of an element is not necessarily unique, but if z 1 , z 2 P A K are such that z 1 P Kz 2 U Π,K , then νpz 1 q " νpz 2 q.
Quint used the Iwasawa decomposition to define the Iwasawa cocycle b Π pg, xq " νpzq where x " krP Θ,K s P F Θ pG K q with k P K, g P G K and gk has Iwasawa decomposition gk " lzu.
Quint [45] proves the following lemma.One also has the following behavior of b Θ under the representations of G.
Lemma 4.9 (Quint [45,Lemma 6.4]).Let Λ : G Ñ PGLpV q be a proximal irreducible representation, then for every x P F ΘΛ pG K q and g P G K one has where v P ξ Λ pxq ´t0u.

4.7.
Cartan attractors and Cartan's attracting basins.Consider g P G K and let g " k g z g l g be a Cartan decomposition.Given Θ Ă Π, the Cartan attractor of g in F Θ pG K q is defined by and the Cartan basin of g is defined, for α ą 0, by Notice that the flag U Θ pgq is an arbitrary choice of a "most expanding" flag of type Θ for g, however, it is clear from the definition that given α ą 0 there exists a constant K α such that if y P F Θ pG K q belongs to B Θ,α pgq then for all a P Θ one has |χ a `apgq ´bΘ pg, yq ˘| ď K α .
(4.5) where tv 1 , . . ., v p u is any basis of the p-dimensional space x p of x and } } is the norm on ^pK d induced by τ .Notice that, by definition, the number ω p pbpg, xqq only depends on x p , so in order to simplify notation we will also denote it by ω p pbpg, x p qq.

Patterson-Sullivan measures in non-Anosov directions
An interesting quantity associated to a discrete subgroup Γ ă G K is its critical exponent h X Γ which measures the exponential orbit growth rate of orbit points in balls (in the symmetric space of G K ) as the radius grows.The theory of Quint's growth indicator function, which we briefly recall in Section 5.1 allows to deduce information on h X Γ from information on the critical exponent of linear forms φ on the Weyl chamber E, that are often easier to handle with the aid of Patterson-Sullivan measures.When the discrete group Γ ă G K is the image of an Anosov representation ρ : Γ Ñ G K , and the form φ belongs to the dual of the Levi-Anosov subspace E θρ , then the thermodynamical formalism applies (see the Theorem 5.12).
In this section we will, instead, be interested in studying forms φ that do not belong to pE θρ q ˚.Our main result is Theorem 5.14 in which we show that, provided a representation ρ is Anosov with respect to some root, the existence of Patterson-Sullivan measure in any flag manifold, and thus also in non-Anosov directions φ, have strong implications on the critical exponent of φ.
Let Γ Ă G K be a discrete subgroup, its Quint growth indicator function where the infimum is taken over all open cones containing v. One can easily check that Ψ Γ does not depend on the chosen norm } } and is 1-positively-homogenous.
Dually one considers the growth on linear forms.The limit (or Benoist [4]) cone L Γ of Γ is defined as the limit points of sequences t n apg n q where pt n q nPN Ă R converges to 0 and pg n q nPN Ă Γ. Denote its dual cone by pL Γ q ˚" tϕ P E ˚: ϕ|L Γ ´t0u ě 0u, and, for ϕ P pL Γ q ˚let h Γ pϕq be the critical exponent of the Dirichlet series One has the following.
Lemma 5.1.It holds Proof.One inequality is clear.For the other one, one has One can then define the subset D Γ " tϕ P pL Γ q ˚: h Γ pϕq P p0, 1su.
The next lemma is clear from the definitions, but is very useful in applications: Lemma 5.2.If φ belongs to D Γ , then φ `ψ P D Γ for every ψ P pL Γ q ˚.
The following result from Quint [44] allows to deduce information on te critical exponent of various norms in terms of growth of linear functions, that are often easier to compute: Proposition 5.3 (Quint [44]).One has that D Γ " tϕ P E ˚: @v P E `ϕpvq ě Ψ Γ pvqu, and thus it is a convex set.Moreover, for any 1-positively-homogenous function Θ : E `Ñ R the critical exponent h Γ pΘq of the Dirichlet series The importance of the set D Γ is provided by the following theorem: it is possible to compute the orbit growth rate with respect to various norms studying properties of the set D Γ : Theorem 5.4 (Quint [44]).If the Zariski closure of Γ is semi-simple then Ψ Γ is concave, consequently for every norm } } on E one has h } } Γ " inft}ϕ} ˚: ϕ P D Γ u where } } ˚is the induced operator norm on E ˚. Remark 5.5.Recall that, if we endow the symmmetric space (or the affine building) X associated to G K with a G K -invaraint Riemannian metric, there exists an Euclidean norm } } X on E such that for every g P G K one has d X prKs, grKsq " }apgq} X .
Theorem 5.4 provides then the following formula for the critical exponent of a discrete group with reductive Zariski-closure in the symmetric space X: h X Γ " inft}φ} X : φ P D Γ u.The topological boundary Q Γ of D Γ will be called Quint's indicator set of Γ.We will also denote by Let us record here a useful direct consequence of the convexity of D Γ .
Lemma 5.6.Let φ, ϕ P pL Γ q ˚, then We end this subsection with the following definition from Quint [45].
Definition 5.7.Given Θ Ă Π and ϕ P pE Θ q ˚a pΓ, ϕq-Patterson-Sullivan measure on F Θ pG K q is a finite Radon measure µ such that for every g P Γ one has

5.2.
Anosov representations with values in G K .Let Γ be a discrete group and fix Θ Ă Π.
We will sometime use the notation introduced in [41] and, if x P BΓ is a point, denote by x Θ ρ :" ξ Θ ρ pxq P F Θ pG K q the image of x via the boundary map.If θ " ta k u consists of a single root we will also write ξ k ρ and x k ρ instead of ξ ta k u ρ and x ta k u ρ .If Θ Ă Π contains the root a, we denote by π a : F θ pG K q Ñ F a pG K q the natural projection.It is easy to deduce from Corollary 3.4 the following more general statement: Corollary 5.10.Let ρ : Γ Ñ G K be a-Anosov and consider α ą 0. There exists C only depending on α and ρ such that for every θ Ă Π containing a, if ξ a ρ pBΓq X π a ´ρpγq ¨Bθ,α pρpγqq X ρpηq ¨Bθ,α pρpηqq ¯‰ H then dpγ, ηq ď ˇˇ|γ| ´|η| ˇˇ`C.
It is spanned by the fundamental weights tω a : a P Θ ρ u.
A more precise description of the indicator set of ρ can be given on its Anosov-Levi space.The following is a combination of Bridgeman-Canary-Labourie-S.[9, Theorem 1.3], Potrie-S.[40, Proposition 4.11] and S. [46].Theorem 5.12.Let ρ : Γ Ñ G K be a representation, then Q ρpΓq,Θρ is an analytic co-dimension 1 embedded sub-manifold of pE Θρ q ˚that varies analytically with ρ; moreover its restriction to the dual of the vector space spanned by the periods is strictly convex.

5.3.
When some wall is not attained.The purpose of this subsection is to explore Q ρpΓq in directions that are not controlled by the roots with respect to which ρ is Anosov.Definition 5.13.Let ρ : Γ Ñ G K be an a-Anosov representation.Consider Θ Ă Π with a P Θ and let µ ϕ be a `ρpΓq, ϕ ˘-Patterson-Sullivan measure on F Θ pG K q for some ϕ P pE Θ q ˚.We say that ρ is µ ϕ -irreducible if for every y P F i Θ pG K q one has µ ϕ `Annpyq ˘ă µ ϕ `FΘ pG K q ˘.
It is clear that if ρpΓq is Zariski dense in G K then it is µ ϕ -irreducible for any Patterson-Sullivan measure.Even assuming Zariski-density, the following result is a refinement of Quint [45, Théorème 8.1] when Θ contains a root with respect to which ρ is Anosov.Indeed, in the general case treated by Quint, one needs to control the mass of shadows on the flag space associated to Π ´Θ, and, as a result, the existence of a pρpΓq, ϕq-Patterson Sullivan measure only ensures that ϕ `ρθ c is in D ρpΓq , where ρ θ c is a suitable form that is non-negative on the Weyl chamber.In our case, the Anosov condition with respect to one root in Θ permits to control ϕ directly.
The rest of the section is devoted to the proof of this result.We begin with the following lemma from Quint [45].Quint assumes that the representation is Zariski dense, an hypothesis that is too strong for the appliations we have in mind.We observe however that for the proof to work only µ ϕ -irreducibility is needed.We sketch the proof for completeness.).Let ρ : Γ Ñ G K be a representation, µ ϕ be a `ρpΓq, ϕ ˘-Patterson-Sullivan measure on F Θ pG K q.Assume ρ is µ ϕ -irreducible, then there exists α 0 ą 0 such that for every given 0 ă α ă α 0 there exist k ą 0 only depending on α, such that for every γ P Γ one has k ´1e ´ϕ`a pρpγqq ˘ď µ ϕ ´ρpγqB Θ,α pρpγqq ¯ď ke ´ϕ`a pρpγqq ˘.
Proof.Observe that µ ϕ -irreducibility guarantees that there exist α, k ą 0 such that for every γ P Γ, µ ϕ pB θ,α pρpγqqq ě k: indeed otherwise there would be a sequence of reals α n Ñ 0 and elements γ n P Γ with µ ϕ pB θ,αn pρpγ n qqq ď 1{n.We can assume, up to extracting a subsequence, that the complement of B θ,αn pρpγqq converges to Annpyq for some y P F i θ , and this contraddicts µ ϕ -irreducibility.
The result then follows from the definition of pρpΓq, φq-Patterson-Sullivan measure using Equation (4.5).
The rest of the proof of Theorem 5.14 is similar to the argument showing that if there exists a Patterson-Sullivan density of a given exponent, then this exponent must be greater than the critical exponent (see for example Sullivan [48] where C is independent of t.This is to say, there exists K ą 0 independent of t P R `such that ÿ γPΓt e ´ϕ`a pρpγqq ˘ă K.
One concludes that for every s ą 0 one has e ´δsn ă 8, as desired.

Anosov representations with Lipschitz limit set
In this section we will prove Theorem A. We will hence fix some notation throughout this section.Assumption 6.1.The group Γ will be a word-hyperbolic group whose boundary BΓ is homeomorphic to a sphere of dimension d Γ .We will also fix a projective Anosov representation ρ : Γ Ñ PSL d pRq such that the sphere ξ 1 ρ pBΓq is a Lipschitz submanifold of PpR d q, i.e. it is locally the graph of a Lipschitz map.Note that we have restricted ourselves to K " R.
6.1.The p-th Jacobian.Given a line ℓ contained in a p `1-dimensional subspace V of R d , the space of infinitesimal deformations of ℓ inside V T ℓ PpV q Ă T ℓ PpR d q carries a natural volume form induced by the choice of a scalar product τ on R d .Namely, if one considers the τ -orthogonal decomposition V " ℓ ' ℓ K V , then one canonically identifies T ℓ PpV q " hompℓ, ℓ K V q and thus one can define Ω ℓ,V P ^ppT l PpV qq by Ω ℓ,V pϕ 1 , . . ., ϕ p q " v ^ϕ1 pvq ^¨¨¨^ϕ p pvq }v} p`1 for any v P ℓ ´t0u.
Definition 6.2.The linear form J u p P pE ta1,ap`1u q ˚defined by J u p " pp `1qω 1 ´ωp`1 is called the p-th unstable Jacobian.
Lemma 6.3.Given g P PSL d pRq and a partial flag pℓ, V q P F ta1,ap`1u pR d q, one has g ˚Ωgℓ,gV " exp ´´J u p `bta1,ap`1u pg, pℓ, V qq ˘¯Ω ℓ,V .

6.2.
Existence of a J u d Γ -Patterson-Sullivan measure.Let us prove the following proposition.Proposition 6.4.Under assumption 6.1, there exists a pρpΓq, J u d Γ q-Patterson-Sullivan measure on F ta1,a d Γ u , which we will denote by ν ρ .
Proof.It follows from Rademacher's theorem [20,Theorem 3.2] that ξ 1 ρ pBΓq has a well defined Lebesgue measure class (cfr.[22,Section 3.2]), and that Lebesgue almost every point ξ 1 ρ pxq P ξ 1 ρ pBΓq has a well defined tangent space, this defines a d Γ `1 dimensional vector subspace Consider the ρ-equivariant measurable map We can then define a volume form on ξ 1 ρ pBΓq via ξ 1 ρ pxq Þ Ñ Ω ζρpξ 1 ρ pxqq .This form is defined Lebesgue almost everywhere and thus defines a Lebesgue measure on ξ 1 ρ pBΓq, which we will denote by ν ρ .Lemma 6.3 implies directly that the push-forward pζ ρ q ˚νρ is the desired measure.

6.3.
When BΓ is a circle.Recall from the introduction that we say that ρ is weakly irreducible if the vector space span `ξ1 ρ pBΓq ˘is the whole space.Lemma 6.5.Under assumption 6.1 together with weakly irreducibility of ρ and d Γ " 1, one has that ρ is µ ϕ -irreducible for any pρpΓq, ϕq-Patterson-Sullivan measure on F ta1,a2u pR d q whose projection is absolutely continuous with the measure ν ρ constructed in Proposition 6.4.
Proof.If this were not the case, there would exist pW 0 , P 0 q P F ta d´2 ,a d´1 u pR d q such that AnnpW 0 , P 0 q would have full µ ϕ -mass; as ρ is projective Anosov we can furthermore assume that P 0 " ξ d´1 ρ pxq for some x P BΓ and thus the condition ξ 1 ρ pyq Ă P 0 only occurs for y " x.
Hence, since the projection of µ ϕ is absolutely continuous with respect to ν ρ one has that for µ ϕ -almost every ξ 1 ρ pxq P ξ 1 ρ pBΓq the vector space x 2 ρ from subsection 6.2 intersects W 0 .
Since Lipschitz maps are absolutely continuous, and in particular satisfy the fundamental theorem of calculus, we deduce that π|ξ 1 ρ pBΓq is constant.This implies that ξ 1 ρ pBΓq Ă W 0 ' ξ 1 ρ pxq, for any x P BΓ, which contradicts the weak irreducibility assumption.
We can now prove Theorem A when d Γ " 1: Corollary 6.6.Let Γ be a word-hyperbolic group such that BΓ is homeomorphic to a circle.Let ρ : Γ Ñ PGL d pRq be a weakly irreducible a 1 -Anosov representation such that ξ 1 ρ pBΓq is a Lipschitz curve.Then Proof.Note that a 1 " J u 1 is the first unstable Jacobian.Since ξ 1 ρ pBΓq is a Lipschitz circle, it has Hausdorff dimension 1 and thus Corollary 1.1 implies that h a1 ρ ě 1.On the other hand, Proposition 6.4 provides a pρpΓq, J u 1 q-Patterson-Sullivan measure µ J u 1 on F ta1,a2u pV R q that projects to the Lebesgue measure on ξ 1 ρ pBΓq.Since ρ is weakly irreducible, Lemma 6.5 implies that it is µ J u 1 -irreducible, thus Theorem 5.14 applies to give a 1 " J u 1 P D ρpΓq , this is to say h ρ pa 1 q ď 1 which concludes the proof.
Before proceeding to arbitrary d Γ let us record a direct consequence of Corollary 6.6.Let us say that ρ is coherent if the first root arising in span `ξ1 ρ pBΓq ˘is a 1 .Corollary 6.7.Let Γ be a word-hyperbolic group such that BΓ is homeomorphic to a circle.Let ρ : Γ Ñ G K be an a-Anosov representation and assume there exists proximal, real representation Λ : G K Ñ PGLpV R q with first root a, such that Λ ˝ρ is coherent, then a P Q ρpΓq .

6.4.
When BΓ has arbitrary dimension.Recall that a subgroup Γ Ă PGLpV K q is strongly irreducible if any finite index subgroup acts irreducibly.It is well known that this is equivalent to the fact that the connected component of the identity of the Zariski closure of Γ acts irreducibly on K d .We will need the following lemma (that does not require assumption 6.1).
Proof.Otherwise we would be able to find a subspace W 0 P F ta d´p u pR d q such that for almost every8 ξ 1 ρ pxq P ξ 1 ρ pBΓq one has ζpxq p X W 0 ‰ t0u.Since ζ is η-equivariant, we would find a p-dimensional subspace V such that for every γ P Γ, This implies that for every g in the Zariski closure of ηpΓq it holds that dim gV X W 0 ě 1.The contradiction comes from Labourie [37,Proposition 10.3] stating that the identity component of such a Lie group cannot act irreducibly.
We can now prove Theorem A for arbitrary d Γ .Corollary 6.9.Under assumption 6.1 together with strong irreducibility of ρ one has ρ pBΓq is a Lipschitz sphere, it has Hausdorff dimension d Γ and thus Corollary 1.1 implies that h ρ pJ u d Γ q ě 1. Proposition 6.4 guarantees the existence of a pρpΓq, J u d Γ q-Patterson-Sullivan measure.Moreover, the equivariant map from equation (6.2) allows us to apply Lemma 6.8 and thus we are in the hypothesis of Theorem 5.14, consequently h ρ pJ u d Γ q ď 1, which concludes the proof.
7. p1, 1, pq-hyperconvex representations and a C 1 -dichotomy for surface groups In this section we will consider projective Anosov representations whose image of the boundary map is a C 1 submanifold.In the second part of the section we will prove Corollary 7.8 providing a a C 1 -dichotomy for surface groups.Hyperconvex representations were introduced by Labourie [37] for surface groups and further studied by Zhang-Zimmer [50] when the boundary of Γ is a topologically a sphere and by P.-S.-W.[41] for arbitrary hyperbolic groups.In both [41,Proposition 7.4] and [50, Theorem 1.1] one finds the following result.Theorem 7.3 (P.-S.-W.and Zhang-Zimmer).Assume that BΓ is topologically a sphere of dimension p ´1 and let ρ : Γ Ñ PGL d pRq be a p1, 1, pq-hyperconvex representation.Then ξ 1 ρ pBΓq is a C 1 -sphere.Theorem A then gives: Corollary 7.4.Assume that BΓ is topologically a sphere of dimension p ´1 and let ρ : Γ Ñ PSL d pRq be strongly irreducible and p1, 1, pq-hyperconvex.Then h ρ pJ u p q " 1. Remark 7.5.This generalizes Potrie-S.[40,Corollary 7.1].Observe however that, since the limit set ξ 1 pBΓq is a C 1 -submanifold of PpR d q, the arguments of [40] adapt directly to give a version of Corollary 7.4 without requiring strong irreducibility.
is non-zero, moreover observe that, for every v P E `one has Since ρ is a p -Anosov, the last computation implies that ker φ X L ρpΓq " t0u this is to say that φ P pL ρpΓq q ˚, in particular φ has a well defined entropy h ρ pφq P p0, 8q.Moreover, where the equality comes from the equality between the corresponding linear forms and the inequality follows from Lemma 5.6 together with Corollary 7.4 stating that h ρ pJ u p´1 q " 1.Finally, observe that where i : E Ñ E is the opposition involution.Together with equation 7.1 and Lemma 5.6, this yields since entropy is i-invariant.
To conclude the proof we observe that the functional φ belongs to the Anosov-Levi space of every ta 1 , a p u-Anosov representation, its entropy thus varies continuously (Theorem 5.12) and hence is bounded away from 1 on compact subsets of X ta1,apu `Γ, PGL d pRq ˘.
C 1 -dichotomy.Now we prove the C 1 -dichotomy announced in the introduction.As we will later see (Section 9 and Section 10) there are many projective Anosov representations of surface groups where the image of the boundary map is Lipschitz.However, when we embed the surface group into PSL 2 pRq and look small deformations of representations where R satisfies additional proximality assumptions ensuring that the representation is ta 1 , a 2 u-Anosov, then the image of the boundary map is never Lipschitz.
Recall that an element g P PGL d pRq is proximal if the generalized eigenspace associated to its greatest eigenvalue (in modulus) has dimension 1.A representation R : G Ñ PGL d pRq of a given group G is proximal if its image contains a proximal element.
Corollary 7.8.Let R : PSL 2 pRq Ñ PSL d pRq be a (possibly reducible) proximal representation such that ^2R is also proximal.Let S be a closed connected surface of genus ě 2 and let ρ 0 : π 1 S Ñ PSL 2 pRq be discrete and faithful.Then we have the following dichotomy: i) If the top two weights spaces of R belong to the same irreducible factor, then for every small deformation ρ : π 1 S Ñ PSL d pRq of Rρ 0 the curve ξ 1 ρ pBπ 1 Sq is C 1 .ii) Otherwise, for every weakly irreducible small deformation ρ : π 1 S Ñ PSL d pRq of Rρ 0 the curve ξ 1 ρ pBπ 1 Sq is not Lipschitz.Proof.By the proximality assumptions on R, the representation Furthermore, if the first two weights of R belong to the same irreducible factor, the representation ρ is also p1, 1, 2q-hyperconvex, this is an open property in X `π1 S, PSL d pRq ˘(P.-S.-W [41]) and thus Theorem 7.3 implies that every small deformation of ρ has C 1 limit set.
If, instead, the two top weights of R were belonging to different irreducible factors, then it follows from the representation theory of SL 2 pRq that h ρ pa 1 q " h ρ pJ u 1 q " 2. Note that the entropy if J u 1 is continuous on X ta1,a2u `π1 S, PSL d pRq ˘(Theorem 5.12), in particular there exists a neighborhood U of ρ such that h η pJ u 1 q ą 1 for every η P U. Theorem A implies that no weakly irreducible representation in U can have Lipschitz limit set.The regular case, item i) in Corollary 7.8, is inspired by Labourie [37], who treated the case (of arbitrary deformations) of the irreducible representations, and was proven in P.-S.-W [41,Proposition 9.4].The novelty of this paper is item ii), inspired by Barbot [1] who proved it for d " 3. We believe both items placed together give a clearer picture.
It is easy to obtain similar results for other group G by considering suitable linear representations.On the other hand the double proximality assumption is necessary: the composition of a maximal representation not in the Hitchin component and the irreducible linear representation of Spp2n, Rq of highest weight w n is proximal but its second exterior power is not proximal; it is possible to check that no small Zariski dense deformation satisfies either (i) or (ii).
Along the same lines we can deduce that some natural Anosov representations of hyperbolic lattices do not have Lipschitz boundary maps: Corollary 7.9.Let Γ ă POp1, nq be a lattice, n ě 3 and ρ 1 : Γ Ñ POp1, mq strictly dominated by the lattice embedding ρ 0 .Then for any Zariski dense small deformation of ρ 0 ' ρ n´1 8. H p,q convex-cocompact representations Generalizing work of Mess [39] and Barbot-Mérigot [2], Danciger-Guéritaud-Kassel [17] introduced a class of representations called H p,q -convex cocompact.These form another interesting class of representations with Lipschitz boundary map where Theorem A apply.
Let d " p `q with p, q ě 1 and let Q be a symmetric bilinear form on R d of signature pp, qq.The subspace of PpR d q consisting on negative definite lines is called the pseudo-Riemannian hyperbolic space and denoted by H p,q´1 " tℓ P PpR d q : Q| ℓ´t0u ă 0u.
The cone of isotropic lines is usually denoted by BH p,q´1 .
Instead of the original definition of convex-cocompactness, we recall the characterization given by [17,Theorem 1.11].Definition 8.1.An a 1 -Anosov representation ρ : Γ Ñ POpp, qq is H p,q´1 -convex cocompact if for every pairwise distinct triple of points x, y, z P BΓ, the restriction ρ pzq has signature p2, 1q.When Γ 0 is a cocompact lattice in SOpp, 1q, H p,1 -convex cocompact representations of Γ 0 are usually referred to as AdS-quasi-Fuchsian groups.Barbot [3] proved that these groups form connected components of the character variety X `Γ0 , SOpp, 2q ˘only consisting of Anosov representations.In [24] Glorieux-Monclair prove that the limit set of an AdS-quasi-Fuchsian group is never a C 1 -submanifold, except for Fuchsian groups.
Proof.The space BH p,q´1 admits a twofold cover that splits as the product S p´1 Ŝq´1 . It is furthermore immediate to verify that, since for every pairwise distinct triple px, y, zq P BΓ, Q| ξ 1 ρ pxq'ξ 1 ρ pyq'ξ 1 ρ pzq has signature p2, 1q, each one of the two lifts of ξ 1 ρ pBΓq to S p´1 ˆSq´1 is the graph of a 1-Lipschitz function f : S p´1 Ñ S q´1 , and, as such, is a Lipschitz submanifold of BH p,q´1 .
Theorem A then yields: Corollary 8.3.Assume that BΓ is homeomorphic to a p ´1-dimensional sphere and let ρ : Γ Ñ POpp, qq be H p,q´1 -convex cocompact, then -if p " 2 and ρ is weakly irreducible then h ρ pJ u 1 q " 1, -if p ě 3 and ρ is strongly irreducible then h ρ pJ u p´1 q " 1.One concludes the following upper bound for the entropy of the spectral radius inspired by Glorieux-Monclair [23].
Proof.Assume first p ď q and note that for every g P POpp, qq one has pω p ´ω1 qpλpgqq " λ 2 pgq `¨¨¨`λ p pgq ě 0.
By definition, J u p´1 " pω 1 ´ωp and thus h ρ pω 1 q p ´1 " h ρ `pp ´1qω 1 ˘ď h ρ pJ u p´1 q " 1, by Corollary 8.3.The only difference in the case q ă p is that J u p´1 " pω 1 ´ωq , but the same argument applies verbatim.
The entropy for the first fundamental weight has a particular meaning for projective Anosov representations into POpp, qq, notably for q ě 2. Fix o P H p,q´1 and consider S o " tW ă R d : o Ă W, dim W " q and Q|W is negative definiteu.This is a totally geodesic embedding of the symmetric space X p,q´1 of POpp, q ´1q in the symmetric space X p,q .
We finish the section with a direct application of Theorem 5.4 and Corollary 8.3 allowing us to get a bound for the Riemannian critical exponent.We use freely the notation from Remark 5.5.
Consider a representation Λ : POpp, 1q Ñ POpp, qq such that its image stabilizes a p `1-dimensional subspace V of R d where Q|V has signature pp, 1q.Endow the symmetric space X p,q with a POpp, qq-invariant Riemannian metric such that the totally geodesic copy of H p in X p,q induced be Λ has constant curvature ´1.In particular, if ι : Γ Ñ POpp, 1q is the lattice embedding, h X Λ˝ι " p ´1.We show that this is an upper bound for any strongly irreducible, H p,q´1 -convex-cocompact representation: Proposition 8.5.Assume that BΓ is homeomorphic to a p ´1-dimensional sphere and let ρ : Γ Ñ POpp, qq be strongly irreducible and H p,q´1 -convex-cocompact and, then h X ρ ď p ´1. Proof.In view of Theorem 5.4 (or more precisely Remark 5.5), it suffices to recall that D ρpΓq is convex (Lemma 5.3) and that, by Corollary 8.3, J u p´1 P Q ρpΓq .See Potrie-S.[40, Section 1.1] for more details.

Maximal Representations
An important class of representations that are in general only Anosov with respect to one maximal parabolic subgroup, but admit boundary maps with Lipschitz image are maximal representations into Hermitian Lie groups.In this case the Lipschitz property for the image of the boundary map is a consequence of a positivity/causality property of the boundary map.We first describe the causal structure on the Shilov boundary of a Hermitian symmetric space of tube type, introduce the notion of a positive curve and show that the image of any positive curve (that is not necessarily equivariant with respect to a representation) is a Lipschitz submanifold.We then show how this applies to maximal representations and allows us to prove Theorem 9.8, the main result of this section.We also deduce consequences for the orbit growth rate on the symmetric space.9.1.Causal structure and positive curves.Let G R be a simple Hermitian Lie group of tube type.Examples to keep in mind are the symplectic group G R " Spp2n, Rq or the orthogonal group G R " SO 0 p2, nq.The Shilov boundary Š of the bounded domain realization of the symmetric space associated to G R is a flag variety G R { P , where P is a maximal parabolic subgroup determined by a specific simple root tǎu.In the two cases that serve as our main examples, G R " Spp2n, Rq and G R " SO 0 p2, nq, the parabolic subgroup P in question is, respectively, the stabilizer of a Lagrangian subspace L P L pR 2n q and the stabilizer of an isotropic line l P Is 1 pR 2,n q, so that ǎ " a n , resp.ǎ " a 1 .
In general, for a simple Hermitian Lie group of rank n, there is a special set of n strongly orthogonal roots b 1 , ¨¨¨b n of the complexification g C , see [32, p.582-583].The set of strongly orthogonal roots give rise to a (holomorphic) embedding of a maximal polydisk.If the symmetric space is of tube type, the simple root ǎ is the smallest strongly orthogonal root ǎ " b n .All the other strongly orthogonal roots are of the form b i " b n `ϕ, where ϕ P E ˚is non-negative on the Weyl-chamber.We record the following for later use: For Hermitian groups of tube type, the Shilov boundary carries a natural causal structure: for every p P Š there is an open convex acute cone C p Ă T p Š which we now define.
Recall that G R { P can be identified as the space of parabolic subgroups of G R that are conjugate to P .Let us fix a point p " P P Š, which one should think of as a point at infinity.Then at any point p " P P Š that is transverse to p, i.e. such that the parabolic groups P and P are opposite, the tangent space T p Š is identified with the Lie algebra ň of the unipotent radical of P , and the cone C p is an open convex acute cone Č Ă ň invariant under the action of the connected component of P X P .
In the case of Spp2n, Rq this is the cone of positive definite symmetric matrices, and in the case of SO 0 p2, nq it is the cone of vectors with positive first entry, that are positive for the induced conformal class of Lorentzian inner products on T P Is 1 pR 2,n q.
This invariant cone Č Ă ň in fact also gives rise to the notion of maximal triples in Š via the exponential map.A triple pP, Q, P q is said to be maximal if there exists an s P Č such that Q " exp s ¨P .Extending this by the action of G leads to a notion of maximal triples in Š, which actually coincides exactly with those triples which have maximal (generalized) Maslov index as introduced by Clerc-Ørsted [14].Definition 9.2.Let Š be the Shilov boundary of a Hermitian symmetric space of tube type.A curve ξ : S 1 Ñ Š is positive if the image of any positively oriented triple is a maximal triple.Proposition 9.3.Let ξ : S 1 Ñ Š be a positive curve.Then ξpS 1 q is a Lipschitz submanifold of Š.
Proof.Note that whenever we pick two points p 1 " P 1 , p 2 " P 2 on the image of ξ, the image ξpS 1 q can be covered by the two charts consisting of parabolic subgroups that are transverse to p 1 respectively p 2 .
In any of these charts the inverse image of ξ, under the exponential map gives a map ξ : R Ñ n i such that for every t 1 ă t 2 we have ξpt 2 q ´ξpt 1 q is contained in the open convex acute cone Č, it then follows (see for example Burger-Iozzi-Labourie-W. [10,Lemma 8.10]) that the restriction of ξ to any bounded interval has finite length.As a result ξpS 1 q Ă Š is rectifiable.It is thus possible to reparametrize S 1 so that ξ is a Lipschitz map.
Remark 9.4.Note that we did not assume that the positive map is equivariant with respect to a representation.This will be important in Section 10, where we will apply Proposition 9.3 in this generality.9.2.Maximal representations.Let now G denote an Hermitian semisimple Lie group and let Γ denote the fundamental group of a closed hyperbolic surface S. We consider representations ρ : Γ Ñ G that are maximal, i.e. they maximize the Toledo invariant, whose definition was recalled in the introduction.Important for us is that they can be characterized in terms of boundary maps by the following theorem.
In order to apply Corollary 6.7 we need to verify some weak irreducibility assumption.Let us first treat the case when the Zariski closure of ρpΓq is simple.
Corollary 9.6.Let G be a simple Hermitian Lie group of tube type and let ǎ be the root associated the Shilov boundary of G.If ρ : Γ Ñ G is a Zariski-dense maximal representation then ǎ P Q ρpΓq , this is to say, h ρ pǎq " 1.
Proof.Follows from Corollary 6.7 and Proposition 9.3 by considering the representation Λ ǎ from Proposition 4.4.
In the remainder of this section we show how the case of maximal representations with semi-simple target group that are not necessarily Zariski-dense, can be reduced to Corollary 9.6.To this aim, we will use a result from Burger-Iozzi-W.[12] describing the Zariski closure H of a maximal representation: H splits as H 1 ˆ¨¨¨ˆH n , each factor is Hermitian, and the inclusion in H Ñ G is tight.In the following we will not need the definition of a tight homomorphism, and therefore refer the interested reader to [12, Definition 1] for it.
The following lemma will then be useful: As ι is tight, and G is classical, the classification of Hamlet-Pozzetti [31] applies and gives that we have an orthogonal decomposition E G " B 1 ' ¨¨¨' B k so that π| À E i is a direct sum of maps π i : E i Ñ B i ; furthermore, there are only few possibilities for the linear map π i : if H i has rank greater than one, then B i " E mi i for some m i and π i is a diagonal inclusion; instead, if E i is one dimensional, or equivalently H i -PSL 2 pRq, then π i is induced from a direct sum of non-trivial irreducible representations (of varying degrees).It is easy to check that the subspace B i is then the span of the real vectors in p associated to the strongly orthogonal roots that do not vanish on πpE i q.Setting b i " min j,bj | E i ‰0 b j , we have b i | πpEiq " ǎHi .And hence, with Lemma 9.1, we have ǎG " min i pǎ Hi q.
Since every maximal representation factors through a representation into the normalizer of a maximal tube type subgroup H ă G (Burger-Iozzi-W.[11, Theorem 5 (3)]), which is simple, has the same rank as G, and is such that ǎG " ǎH , we can restrict to the tube type case as the limit set in ŠG is contained in ŠH and coincides with the limit set in ŠH .The maximal tube type domains are always classical Hermitian symmetric spaces, except for the one exceptional Hermitian symmetric space of tube type.
If now ρ is not Zariski dense, then the Zariski closure is reductive and of tube type, so it is of the form H 1 ˆ¨¨¨ˆH n and the representations into H i are Zariski dense and maximal.Therefore we have h ρ pǎ Hi q " 1 for all i.As the inclusion H 1 ˆ¨¨¨ˆH n Ñ G is tight, the result follows from Lemma 9.7 and Lemma 5.1.9.3.Application to the Riemannian critical exponent.Any simple Hermitian Lie group G admits a diagonal embedding ι ∆ : SL 2 pRq Ñ G, which is equivariant with the inclusion of a diagonal disk in a maximal polydisk.We say that a representation ρ : Γ Ñ G is diagonal-Fuchsian if it has the form ρ " ι ∆ ˝ρ0 where ρ 0 : Γ Ñ SL 2 pRq is the lift of the holonomy of a hyperbolization.
Let K ∆ ă G be the centralizer of the image of ι ∆ , which is compact.Then a diagonal Fuchsian representation ρ can be twisted by a representation χ : Γ Ñ K ∆ .We call the corresponding representation ρ χ : Γ Ñ G a twisted diagonal representation.Observe that the Riemannian critical exponent h X is constant on twisted diagonal representations (the exact value h X diag depends on the choice of the normalization of the Riemannian metric) Proposition 9.9.Let Γ be the fundamental group of a closed surface and let ρ : Γ Ñ G be a maximal representation, then h X ρ ď h X diag .
Proof.Let b 1 , ¨¨¨b n be the set of strongly orthogonal roots for G C .It is immediate to verify that the limit cone L ρ0pΓq of a representation ρ 0 in the Fuchsian locus is concentrated in the span of the vertex of the Weyl chamber is ř n i"1 b i , where b ˚is the basis of E dual to tb 1 ¨¨¨b n u.We know from Corollary 9.6 that, for every ρ, the growth rate h ρ pǎq " 1.Thus, if we denote by pE `q˚t he cone of functionals that are non-negative on the Weyl chamber, we get that ǎ`pE `q˚Ă D ρpΓq , and in particular all the strongly orthogonal roots are in D ρpΓq .A simple computation shows that the affine simplex determined by the strongly orthogonal roots meets the ray R ř n i"1 b i orthogonally in a point (it is just the diagonal in a positive quadrant meeting the span of the basis vectors), whose norm has to compute the Riemannian orbit growth rate of any representation ρ 0 in the Fuchsian locus: Q ρ0pΓq is the affine hyperplane orthogonal to R ř n i"1 b i that contains ǎ.Remark 5.5 concludes the proof.
Remark 9.10.Note that when G is Spp4, Rq, or more generally SO ˝p2, nq, it follows from Collier-Tholozan-Toulisse [15] that the bound is furthermore rigid: the equality is strict unless ρ is equal to ρ 0 up to a character in the compact centralizer of its image.Note that for maximal representations into Spp2n, Rq n ě 3, every connected component of the space of maximal representations contains a twisted diagonal representation.However for Spp4, Rq there are exceptional components, discovered by Gothen, where every representation is Zariski dense (see Bradlow-Garcia-Prada-Gothen [8] and Guichard-W.[28]).In these components it is easy to verify that the bound we provide is sharp, despite not being achieved.
In the special case of the Hitchin component of Spp2n, Rq, the bound of Proposition 9.9 is never attained, as the irreducible representations provide a better bound that is furthermore rigid (Potrie-S.[40]).

Θ-positive representations
Throughout this section we will write G " SOpp, qq with p ă q.We consider the subset Θ " ta 1 , . . ., a p´1 u of the simple roots discussed in Example 4.6 and denote by P Θ the corresponding parabolic group, by L Θ its Levi factor and by U Θ its unipotent radical.
The group G admits a Θ-positive structure as defined by Guichard-W.[30].This means that for every b P Θ there exists an L 0 Θ -invariant sharp convex cone c b in Here Σ Θ " Σ `z SpanpΠ´Θq.For b P ta 1 , . . ., a p´2 u, the space u b is one dimensional and the sharp convex cone c b " R `Ă R consists of the positive elements, while u ap´1 " R q´p`2 endowed with a form q J of signature p1, q ´p `1q preserved by the action of L 0 Θ " R p´2 ˆSO 0 p1, q ´p `1q.The cone c ap´1 consists precisely of the positive vectors for q J whose first entry is positive.
Following [30, §4.3] we denote by W pΘq the subgroup of the Weyl group W generated by the reflections tσ i u p´2 i"1 together with the longest element σ p´1 of the Weyl group W ap´1,ap of the subroot system generated by the last two simple roots.W pΘq is, in our case, a Weyl group of type B p´1 .We denote by w 0 Θ the longest element of W pΘq, and choose a reduced expression w 0 Θ " σ i1 . . .σ i l .Of course every reflection σ i appears at least once among the σ i k .We consider the map The Θ-positive semigroup U Θ is defined as the image of the map F σi 1 ...σi l , and doesn't depend on the choice of the reduced expression [30,Theorem 4.5].
A θ-positive structure on G gives rise to the notion of a positive triple in G{P Θ .
Definition 10.1.A pairwise transverse triple in pG{P Θ q 3 is Θ-positive if it lies in the G-orbit of a triple of the form pF 1 , u ¨F1 , F 3 q, where StabpF 3 q " P Θ , F 1 is transverse to F 3 and i u P U Θ [30, Definition 4.6].
Remark 10.2.The stabilizer in SO 0 p1, q ´p `1q of a vector v P c ap´1 is compact.As a result one readily checks that the stabilizer in G of a Θ-positive triple is compact.
Let now Γ g be the fundamental group of a hyperbolic surface.A representation ρ : Γ g Ñ G is Θ-positive if there exists a ρ-equivariant map BΓ g Ñ G{P Θ sending positive triples to Θ-positive triples [30,Definition 5.3].Guichard-Labourie-W. show that every Θ-positive representation is necessarily Θ-Anosov [30,Conjecture 5.4], but since the proof did not yet appear in print, in this section we will freely add this last assumption, and only discuss Θ-positive Anosov representations.
Proof.We denote by ξ : BΓ g Ñ G{P Θ the Θ-positive continuous equivariant boundary map, and by ξ i : BΓ g Ñ Is i pR p,q q the induced maps.By assumption, ξpyq " s ¨ξpxq for some element s in the positive semigroup of the unipotent radical of the stabilizer of ξpzq.In turn s " exppv 1 q . . .exppv l q with v t P c 0 ai t (recall that i t P t1, . . ., p ´1u).
We set d " p `q.It follows from [41,Proposition 8.11] that, in order to check that ^kρ is p1, 1, 2q-hyperconvex, it is enough to verify that the sum

‚
We can furthermore assume that ξ l pzq " xe 1 , . . ., e l y and ξ l pxq " xe d , . . ., e d´l`1 y, so that ξ k pxq X ξ d´k`1 pzq " e d´k`1 .In order to check that the representation is p1, 1, 2q-hyperconvex, we only have to verify that, given s as above, writing s ëd´k`1 " ř α i e i , the coefficient α d´k never vanishes.But we claim that such coefficient is just ř it"k v t ą 0. Indeed, by construction, if v t P c 0 am with m P t1, . . ., p ´2u, then exppv t q P SOpp, qq differs from the identity only in the positions pt, t`1q and pd´t, d´t`1q where it is equal to v t (cfr.[30, §4.5]), while if v t P c 0 ap´1 we have exppv t q " ¨Id p´2 0 0 0 0 0 1 v t q J pvq 0 0 0 Id q´p`2 Jv 0 0 0 0 The result is then immediate.
In particular we deduce from [41,Proposition 7.4] the following Corollary 10.4.Let ρ : Γ Ñ SOpp, qq be Θ-positive Anosov.For every 1 ď k ď p ´2 the image of ξ k ρ pBΓq is a C 1 submanifold of Is k pR p,q q.We now turn to the proof of the last statement in Theorem D. Instead of directly verifying that the map ξ p´1 ρ has Lipschitz image, we will study properties of the map ξ Θ0 ρ : BΓ g Ñ G{P Θ0 where Θ 0 " ta p´2 , a p´1 u.
The flag manifold G{P Θ0 consists of nested pairs of isotropic subspaces of dimension p ´2 and p ´1.
Proof.We fix a point z P BΓ and we assume without loss of generality that ξ k ρ pzq " xe 1 , . . ., e k y.We denote by A Ă G{P Θ0 the set of points transverse to ξ p´2,p´1 ρ pzq.We will show that the image of ξ Θ0 ρ | BΓztzu is a Lipschitz submanifold of A. Denote by A p´2 Ă G{P ap´2 the set of isotropic subspaces of dimension p ´2 transverse to ξ p´2 ρ pzq " xe 1 , . . ., e p´2 y, by Z p´1 the pp ´1q-isotropic subspace Z p´1 :" ξ p´1 ρ pzq " xe 1 , . . ., e p´1 y, and by Z K p´1 its orthogonal with respect to the form Q defining SOpp, qq.Observe that we have a smooth map A Þ Ñ A p´2 ˆIs 1 pZ K p´2 {Z p´2 q pY p´2 , Y p´1 q Þ Ñ pY p´2 , rY p´1 X Z K p´2 sq whose image is the product of A p´2 with the set I Z of isotropic lines transverse to the image of Z K p´1 .Indeed for every pair pY p´2 , vq P A p´2 ˆIZ , the subspace v `Zp´2 has dimension p ´1 and, dim `pv `Zp´2 q X Y K p´2 ˘" 1 as Y K p´2 and Z p´2 are transverse.We then have Y p´1 " Y p´2 `ppv `Zp´2 q X Y p´2 q.
We denote by ξ Z : BΓztzu Ñ I Z the composition of the map ξ p´2,p´1 and the projection to the second factor in the product decomposition.The form Q induces a form of signature p2, q ´p `2q on Z K p´2 {Z p´2 , which gives rise to the notion of positive curves (as introduced in Section 9).We claim that ξ Z is a positive curve.This amounts to showing that, if px, y, zq P BΓ is positively oriented, then ξ Z pyq " s Z ξ Z pxq for some positive element s Z in the unipotent radical of the stabilizer of rZ p´1 s P Is 1 pZ K p´2 {Z p´2 q.Since the representation ρ is Θ-positive, we know that ξpyq " s ¨ξpxq for some element in the positive semigroup U Θ , and, as in the proof of Proposition 10.3 we can write s " exppv 1 q . . .exppv l q with v t P c 0 ai t .Observe that, for every v t P c 0 βi t , exppv t q induces an element exppv t q Z in the unipotent radical of the stabilizer of rZ p´1 s P Is 1 pZ K p´2 {Z p´2 q, and the element exppv t q Z is trivial unless β it " a p´1 , in which case exppv t q Z belongs to the positive semigroup of the unipotent radical of the stabilizer of rZ p´1 s.As at least one of the v t in the decomposition of s belongs to such subgroup, we deduce that ξ Z is positive, as we claimed.It follows from Proposition 9.3 that ξ Z pBΓztzuq is a Lipschitz submanifold of Is 1 pZ K p´2 {Z p´2 q.As we know from Proposition 10.3 that ξ p´2 is a C 1 -curve, we deduce that the curve ξ p´2,p´1 is Lipschitz, being the image of a monotone map between a C 1submanifold and a Lipschitz submanifold.This concludes the proof.10.1.The critical exponent on the symmetric space is rigid.Let ι 2p´1 : POp1, 2q Ñ POpp, p ´1q Ñ POpp, qq be the composition of the the irreducible representation of dimension 2p ´1 with the standard embedding of POpp, p ´1q Ñ POpp, qq.We call any representation ρ : Γ Ñ POpp, qq, which is the composition of a Fuchsion representation with ι 2p´1 , a pp, p ´1q-Fuchsian representation.
Proof.Recall that, in the case of Θ-positive representations in POpp, qq, the Levi-Anosov subspace is E Θ :" kerpa p q.In particular, for every k ď p ´2 we have that a k belongs to the dual of E Θ , and belongs to the boundary of D ρpΓq,Θ by Corollary 10.4.Furthermore ε p´1 " a p´1 `ap belongs to D ρpΓq,Θ being the sum of a linear form with entropy one (the form a p´1 has entropy one by Proposition 10.5) and a linear form positive on the Weyl chamber (the root a p ).In particular the form corresponding to the barycenter of the affine simplex they determine in E Θ belongs to D ρpΓq,Θ .
If equality is achieved at a totally reducible representation η then η splits as W ' V where (i) W has signature pp, p ´1q and η|W has Zariski closure the irreducible POp2, 1q in POpp, p ´1q (ii) η|V lies in a compact group.
Proof.The inequality follows from Lemma 10.6, together with convexity of D ρpΓq,Θ established by Theorem 5.12.
Assume now that η is a totally reducible representation such that equality holds.We can assume that p ě 3, as the result for p " 2 was proven by Collier-Tholozan-Toulisse [15,Theorem 4].
Let G " ηpΓq Z be its Zariski closure.By definition, G is a real reductive group.
We consider G as an abstract group, denote by Λ : G Ñ SOpp, qq the inclusion representation, and by φ : g Ñ sopp, qq the associated Lie algebra morphism.Denote by a G a Cartan subspace of g.Since h X η attains it maximal value, Theorem 5.12 forces the Quint indicator set Q ηpΓq,Θ to be the affine hyperplane of pE Θ q ˚spanned by ∆.The strict convexity guaranteed by Theorem 5.12 implies that G has real rank at most 2. Moreover we have that φpa G q " xp2pp ´1q, 2pp ´2q, . . ., 2, 0q, p0, . . ., 0, 1qy.
Denote by T " xξ 1 η pBΓqy the vector space spanned by the projective limit curve of η.Since η is totally reducible, the action of ηpΓq, and hence that of G, on T is irreducible.
Fix then a Weyl chamber a G and let χ P a G be the highest weight of φpgq|T.Since η is a 1 -Anosov, the attracting eigenvector of every element in ηpΓq, and hence of every purely loxodromic element of G, belongs to V .We therefore conclude that for every a P a G χpaq " λ 1 `φpaq ˘.

Lemma 4 . 8 (
Quint [45, Lemmas 6.1 and 6.2]).The map p Θ ˝bΠ factors trough a map b Θ : G K ˆFΘ pG K q Ñ E Θ .The map b Θ verifies the cocycle relation: for every g, h P G K and x P F Θ,K pG K q one has b Θ pgh, xq " b Θ pg, hxq `bΘ ph, xq.

Theorem 9 . 8 .
Let G be a Hermitian semi-simple Lie group such that all factors of G that are of tube type are classical.Let θ Ă ∆ be the subset of simple roots associated to the Shilov boundary of G. Then for every maximal representation ρ : Γ Ñ G one has Θ Ă Q ρpΓq .
or, equivalently that the sumξ k ρ pxq `s ¨`ξ k ρ pxq X ξ d´k`1 ρ pzq ˘`ξ d´k´1 ρ pzqis direct (recall that s belongs to the stabilizer of ξ ρ pzq).Without loss of generality we can assume that the form Q defining the group SOpp, qq is represented by Q " Anosov representations.Let Γ be a word-hyperbolic group with identity element e, fix a finite symmetric generating set S Γ .For γ P Γ ´teu denote by |γ| the least number of elements of S Γ needed to write γ as a word on S, and define the induced distance d Γ pγ, ηq " |γ ´1η|.A geodesic segment on Γ is a sequence tα i u k 0 of elements in Γ such that d Γ pα i , α j q " |i ´j|.Definition 2.3.A representation ρ : Γ Ñ PGL d pKq is a p -Anosov 5 if there exist positive constants c, µ, the a p -Anosov constants of ρ, such that for all γ P Γ one has P p `1, d ´1 , ε d pπ Θ paqq " a d .E `´t0u ě a p | E `´t0u .Example 4.6.Consider the group SOpp, qq of transformations in PSL p`q 4.8.The PSL d pKq case.Given a good norm τ on K d , and considering the exterior power representations of PSL d pKq, one sees that Lemma 4.9 provides the following computation for the Iwasawa cocycle b : PSL d pKq ˆFpK d q Ñ E associated to a maximal compact group stabilizing τ.For p P 1, d and given g P PSL d pKq, x P FpK d q one has ω p pbpg, xqq " log }gv 1 ^¨¨¨^gv p } }v 1 ^¨¨¨^v p } (4.6) Proof of Theorem 5.14.We have to show that for every s ą 0 one has ÿ γPΓ e ´p1`sqϕ `apρpγqq ˘ă 8. Corollary 5.10 implies that given α ą 0 there exists N P N, such that if t ą 0 and Γ t " tγ P Γ : t ď |γ| ď t `1u, then for every x P BΓ one has # !γ P Γ t : π ´1 a pξ a ρ pxqq X ρpγqB Θ,α pρpγqq ‰ H ) ď N, Lemma 5.15 now yields for every t ě 0 8 ą µ ϕ ´π´1 a `ξa ρ pBΓq ˘¯ě C ÿ γPΓt e ´ϕ`a pρpγqq ˘, Proof.Denote by π : h 1 ' ¨¨¨' h n Ñ g the associated Lie algebra homomorphism.Let E i be a Cartan subspace of H i and E G a Cartan subspace of G such that πpE Lemma 9.7.Let G be a classical simple Hermitian Lie group of tube type and consider a tight embedding ι : H " H 1 ˆ¨¨¨ˆH n Ñ G.If we denote by ι ˚: E H Ñ E G the induced map, then ǎG ˝ι˚" min i ǎHi .