Regularity of limit sets of Anosov representations
Tengren Zhang Andrew Zimmer


 In this paper we establish necessary and sufficient conditions for the limit set of a projective Anosov representation to be a C^α-submanifold of projective space for some α∈(1,2). We also calculate the optimal value of α in terms of the eigenvalue data of the Anosov representation.


Optimising the Input Image to Improve Visual Relationship Detection
    
		Noel Mizzi

		Department of Computer Science 

		University of Malta, Malta 

		Adrian Muscat

		Department of Computer Science 

		University of Malta, Malta
	
    
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§ INTRODUCTION

Suppose that _^d is real hyperbolic d-space. Let ∂_∞_^d denote the geodesic boundary of _^d and  let Isom(_^d) denote the isometry group of _^d. Given a representation ρ: Γ→ Isom(_^d), the limit set of ρ is defined to be 

    _ρ = ρ(Γ) · x_0∩∂_∞_^d

where x_0 ∈_^d is any point. If we further assume that Γ is a hyperbolic group and ρ is a convex co-compact representation, then there is a ρ-equivariant, continuous map from ∂_∞Γ, the Gromov boundary of Γ, to the limit set _ρ. The limit set in this setting is generically very irregular, for instance when ∂_∞Γ is a topological manifold Yue <cit.> proved: unless ρ is a co-compact action on a totally geodesic subspace of _^d, its limit set is fractal like, and in particular, has Hausdorff dimension strictly greater than its topological dimension. 

The group Isom(_^d) is a semisimple Lie group. For a general semisimple Lie group G, there is a rich class of representations from a hyperbolic group Γ to G called Anosov representations, which generalize the convex co-compact representations from Γ to Isom(_^d). Anosov representations were introduced by Labourie <cit.> and extended by Guichard-Wienhard <cit.>. Since then, they have been heavily studied, <cit.>. One reason for their popularity is that they are rigid enough to retain many of the good geometric properties that convex co-compact representations have, while at the same time are flexible enough to admit many new and interesting examples. 

In this paper, we investigate the regularity of the limit sets of Anosov representations from Γ into _d().
We will give precise definitions in Section <ref> but informally: if Γ is a word hyperbolic group with Gromov boundary ∂_∞Γ, a representation ρ:Γ→_d() is said to be k-Anosov if there exist continuous ρ-equivariant  maps 

    ξ^(k) : ∂_∞Γ→_k(^d)  and ξ^(d-k) : ∂_∞Γ→_d-k(^d)

which satisfy certain dynamical properties. For a k-Anosov representation, it is reasonable to call the image of ξ^(k) in _k(^d) the “k-limit set of ρ in _k(^d).”

We will largely focus our attention on 1-Anosov representations; by a result of Guichard-Wienhard <cit.>, for any Anosov representation ρ : Γ→ G into a semisimple Lie group G, there exists d > 0 and an irreducible representation ϕ : G →_d() such that ϕ∘ρ is 1-Anosov. Thus, up to post composition with irreducible representations, the class of 1-Anosov representations contains all other types of Anosov representations. Further, the flag maps induced by ϕ are smooth. Thus, another result of Guichard-Wienhard <cit.> implies that all regularity properties of the limit set can be investigated by reducing to the case of 1-Anosov representations.

Our first main result gives a sufficient condition for the 1-limit set of a 1-Anosov representation to be a C^α-submanifold of (^d) for some α>1.


 (Theorem <ref>) Suppose Γ is a hyperbolic group, ∂_∞Γ is a topological (m-1)-manifold, and ρ: Γ→_d() is a 1-Anosov representation. If 


  (†) ρ is m-Anosov, and ξ^(1)(x) + ξ^(1)(z) +  ξ^(d-m)(y) is a direct sum for all pairwise distinct x,y,z ∈∂_∞Γ,

then 


  () M:=ξ^(1)(∂_∞Γ) is a C^α-submanifold of (^d) for some α > 1. 

Moreover, T_ξ^(1)(x) M = ξ^(m)(x) for any x ∈∂_∞Γ. 



 

  * A weaker version of this result, only deducing C^1 regularity was independently proven by Pozzetti-Sambarino-Wienhard <cit.>.

  * Property (†) and k-Anosovness in Theorem <ref> are open conditions in (Γ, _d()), see Section <ref>. 


 
 
 
Theorem <ref> is a generalization of the following theorem due to Benoist in the setting of divisible, properly convex domains in (^d). A group of projective transformations Γ⊂_d() divides a properly convex domain Ω⊂(^d) if Γ acts properly discontinuously and co-compactly on Ω. 


Let Γ⊂_d() be a hyperbolic group that divides a properly convex domain Ω⊂(^d). Then then 𝕀:Γ→_d() is a 1-Anosov representation whose 1-limit set is ∂Ω. Furthermore, ∂Ω⊂(^d) is a C^α-submanifold for some α>1.


Theorem <ref> also generalizes a result due to Labourie in the setting of Hitchin representations. Let S be a closed orientable hyperbolizable surface and fix a Fuchsian representation ρ_0: π_1(S) →_2(). Then let τ_d : _2() →_d() be the standard irreducible representation (see Section <ref>). A representation ρ : π_1(S) →_d() is Hitchin if it is conjugate to a representation in the connected component of (π_1(S), _d()) that contains τ_d ∘ρ_0. 


If ρ:π_1(S)→_d() is a Hitchin representation, then ρ is k-Anosov for every k ∈{1,…, d-1}, and the 1-limit set of ρ is a C^α-submanifold in (^d) for some α>1. 

 
Using Theorem <ref>, we can find more examples of representations that preserve C^α-submanifolds in (^d). 

(See Section <ref>) Suppose τ: (m,1) →_d() is a irreducible representation, Γ≤(m,1) is a co-compact lattice, and ρ := τ|_Γ : Γ→_d(). If ρ is 1-Anosov, then there exists a neighborhood  of ρ in (Γ, _d()) such that every representation in  is a 1-Anosov representation whose 1-limit set is a C^α submanifold of (^d) for some α > 1. 


 (See Section <ref>) If ρ : π_1(S) →_d() is a Hitchin representation, then for all k=1,…,d-1, there is an open set  of ⋀^kρ in (Γ,(⋀^k^d)) so that every representation in 
is a 1-Anosov representation whose 1-limit set is a C^α-submanifold of (⋀^k^d) for some α>1. See Section <ref> for the definition of ⋀^kρ. In particular, by applying <cit.>, the k-limit set of ρ is a C^α-submanifold of _k(^d) for some α>1. 


 Example <ref> was independently observed by Pozzetti-Sambarino-Wienhard <cit.>.









 
In fact, Theorem <ref> is a consequence of a more general theorem, see Theorem <ref>, that is stated using ρ-controlled subsets M⊂(^d), of which the 1-limit set of ρ is an example, see Definition <ref>. In the main body of our paper, all our results will be stated for ρ-controlled subsets. These statements are stronger than the results we mention in this introduction, but are more technical to state.
 
We also investigate the extent to which the converse of Theorem <ref> holds. In general, there are 1-Anosov representations ρ whose 1-limit set are C^∞-submanifolds of (^d), but for which (†) in Theorem <ref> does not hold, see Example <ref>. However, we prove that when Γ is a surface group and ρ is irreducible, the conditions in Theorem <ref> are both necessary and sufficient. 

(Theorem <ref>) Suppose Γ is a hyperbolic group, ρ: Γ→_d() is an irreducible 1-Anosov representation, and ∂_∞Γ is homeomorphic to a circle. Then the following are equivalent: 


  (†) ρ is a 2-Anosov representation and ξ^(1)(x) + ξ^(1)(y) + ξ^(d-2)(z) is a direct sum for all x,y,z ∈∂Γ distinct,

  () ξ^(1)(∂_∞Γ) is a C^α-submanifold of (^d) for some α > 1.














From Theorem <ref> and (3) of Remark <ref>, we have the following corollary. 

 Suppose Γ is a hyperbolic group with ∂_∞Γ homeomorphic to a circle. Let ⊂(Γ, _d()) denote the set of representations that are irreducible, 1-Anosov, and whose 1-limit set is a C^α-submanifold of (^d) for some α> 1 (which may depend on ρ).  Then  is an open set in (Γ, _d()).












For non-surface groups the situation is more complicated; there exist irreducible 1-Anosov representations ρ: Γ→_d() whose 1-limit set is a C^∞-submanifold of (^d), but ρ does not satisfy the condition (†) in Theorem <ref>, see Example <ref>. However, if one assumes a stronger irreducibility condition on ρ, then the conditions in Theorem <ref> are both necessary and sufficient. 


 (Theorem <ref>) Suppose Γ is a hyperbolic group, ∂_∞Γ is a (m-1)-dimensional topological manifold, and ρ: Γ→_d() is an irreducible 1-Anosov representation such that ⋀^m ρ: Γ→(⋀^m ^d) is also irreducible. Then the following are equivalent: 


  (†) ρ is a m-Anosov representation and ξ^(1)(x) + ξ^(1)(y) + ξ^(d-m)(x)
is a direct sum for all pairwise distinct x,y,z ∈∂_∞Γ,

  () ξ^(1)(∂_∞Γ) is a C^α-submanifold of (^d) for some α > 1.












Recall that if ρ:Γ→ H is a Zariski-dense representation and τ: H→_d() is an irreducible representation, then τ∘ρ:Γ→_d() is irreducible. Thus, (3) of Remark <ref> and Theorem <ref> give the following corollary. 

 Suppose Γ is a hyperbolic group. Let ⊂(Γ, _d()) denote the set of representations ρ: Γ→_d() where ρ is 1-Anosov, has Zariski dense image, and whose 1-limit set is a C^α-submanifold of (^d) for some α> 1 (which may depend on ρ). Then  is an open set in (Γ, _d()).


Finally, for representations satisfying certain irreducibility conditions, we also determine the optimal regularity of the 1-limit set in terms of the spectral data of ρ(Γ). More precisely, given g∈_d(), let g∈_d() be a lift of g, and let λ_1(g) ≥λ_2(g) ≥…≥λ_d(g) denote the absolute values of the eigenvalues of g. Note that for all i,j, the ratio 

    λ_i/λ_j(g):=λ_i(g)/λ_j(g)

does not depend on the choice of lift g of g. Then given a representation ρ : Γ→_d() and 2 ≤ m ≤ d-1 define

    α_m(ρ) = inf_γ∈Γ{logλ_1/λ_m+1(ρ(γ))/logλ_1/λ_m(ρ(γ)) : λ_1/λ_m(ρ(γ)) ≠ 1 }.

If ρ is (1,m)-Anosov, it follows from definition that α_m(ρ) > 1 (see Section <ref>). 

 (Theorem <ref>) Suppose Γ is a hyperbolic group and ρ: Γ→_d() is an irreducible (1,m)-Anosov representation so that ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-m)(z) is a direct sum for all x,y,z ∈∂_∞Γ distinct. Then

    α_m(ρ) ≤sup{α∈ (1,2) : ξ^(1)(∂_∞Γ)  is a C^α-submanifold}

with equality if 

    ξ^(1)(∂_∞Γ) ∩(ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-m)(z))

spans ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-m)(z) for all x,y,z ∈∂_∞Γ distinct.


   

  * In Theorem <ref>, when ξ^(1)(∂_∞Γ) has either dimension one or co-dimension one, the extra hypothesis for equality is automatically satisfied. If the dimension is one (i.e. m=2), then 

    ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-m)(z)= ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-2)(z) = ^d.

So the extra hypothesis follows from the irreducibility of ρ. If the co-dimension is one (i.e. m=d-1), then 

    ξ^(1)(x) + ξ^(1)(y) +  ξ^(d-m)(z)= ξ^(1)(x) + ξ^(1)(y) +  ξ^(1)(z)

is spanned by ξ^(1)(x), ξ^(1)(y), ξ^(1)(z). Hence the extra hypothesis always holds in this case. 

  * In general, the extra hypothesis for equality is an open condition, see Section <ref>.

  * The irreducibility of ρ is necessary in Theorem <ref>. For instance, if τ_d : _2() →_d() is the standard irreducible representation, see Section <ref>, and Γ≤_2() is a co-compact lattice, then ρ = (τ_5 ⊕τ_2) |_Γ is 1-Anosov and ξ^(1)(∂_∞Γ) is a 1-dimensional C^∞-submanifold of (^7). At the same time, for any infinite order γ∈Γ,

    logλ_1/λ_3(ρ(γ))/logλ_1/λ_2(ρ(γ))= 3/2.
 

  * Notice that the quantity α_m(ρ) is invariant under passing to finite index subgroups. In particular, if Γ_0 ≤Γ is a finite index subgroup and γ∈Γ, then there exists some k ∈ such that γ^k ∈Γ_0. Further, 

    logλ_1/λ_m+1(ρ(γ^k))/logλ_1/λ_m(ρ(γ^k)) = logλ_1/λ_m+1(ρ(γ))/logλ_1/λ_m(ρ(γ)).

Hence α_m(ρ|_Γ_0) = α_m(ρ). 
 




In Section <ref>, we establish a generalization of Theorem <ref> which holds for ρ-controlled subsets. One example of such a subset is the boundary of a properly convex domain Ω⊂(^d) that admits a Γ-action induced by a 1-Anosov representation ρ. In this case, Theorem <ref> implies the following. 


 Suppose Γ is a hyperbolic group and ρ: Γ→_d() is an irreducible 1-Anosov representation. Also, suppose Ω⊂(^d) is a ρ(Γ)-invariant properly convex domain so that ξ^(d-1)(x)∩∂Ω=ξ^(1)(x) for all x∈∂Γ. If 


  (⋆)p_1+p_2+ξ^(1)(y) is a direct sum for all pairwise distinct p_1,p_2,ξ^(1)(y)∈∂Ω,

then 


  (⋆⋆) ∂Ω is C^α along ξ^(1)(∂_∞Γ) for some α>1.

Moreover, T_ξ^(1)(x)∂Ω = ξ^(d-1)(x) for any x ∈∂_∞Γ, and

    α_d-1(ρ) = sup{α∈ (1,2) : ∂Ω is  C^α along ξ^(1)(∂_∞Γ) }.



In the case when the Γ-action on Ω is co-compact, Theorem <ref> was previously proven by Guichard <cit.> using different techniques. Also, a weaker version of Theorem 1.13 (without the optimal bound for α) was previously proven independently by Danciger-Gueritaud-Kassel <cit.> and the second author <cit.>. 



 §.§ Terminology

Through out the paper we will use the following terminology: 


  * ·_2 will always denote the standard ℓ^2-norm on ^d,

  * a (m-1)-dimensional topological manifold M ⊂(^d) is C^α for some α∈ (1,2) if for every p ∈ M there exists local coordinates around p and a differentiable map f:^m-1→^d-m such that M coincides with the graph of f near p and 

    f(u+h)=f(u)+df_u(h)+ O(h_2^α)
 
for all u,h∈^m-1. 

  * a (m-1)-dimensional topological manifold M ⊂(^d) is C^α along a subset N⊂ M for some α∈ (1,2) if for every p ∈ N there exists local coordinates around p and a continuous map f:^m-1→^d-m such that M coincides with the graph of f near p and if (u,f(u)) ∈ N, then f is differentiable at u and satisfies 

    f(u+h)=f(u)+df_u(h)+ O(h_2^α)
 
for all h∈^m-1.








§ ANOSOV REPRESENTATIONS



For the rest of this article, Γ will denote a hyperbolic group, and ∂_∞Γ will be its Gromov boundary. In this section, we define Anosov representations from Γ to _d(), and mention some of their properties. 




 §.§ A definition of Anosov representations


Since they were introduced, several other characterizations of Anosov representations have been given by Kapovich et al. <cit.>, Guéritaud et al. <cit.>, and Bochi et al. <cit.>. The definition we give below comes from <cit.>. 
 
First, let S be a finite symmetric generating set of Γ, and d_S the induced word metric on Γ. For γ∈Γ, let ℓ_S(γ) denote the minimal translation distance of γ acting on Γ, that is

    ℓ_S(γ) := inf_x ∈Γ d_S(γ· x, x).

Also, recall that for any g∈_d() and any i,j∈{1,…,d}, we have defined

    λ_i/λ_j(g):=λ_i(g)/λ_j(g),

where λ_1(g)≥…≥λ_d(g) are the absolute values of the (generalized) eigenvalues of a representative g∈_d() of g. 

A third ingredient we need to define Anosov representations are appropriate definitions of “well-behaved " flag maps. More precisely, we have the following.

 Let ρ: Γ→_d() be a representation.  If 1 ≤ k ≤ d-1, then a pair of maps ξ^(k): ∂_∞Γ→_k(^d) and ξ^(d-k): ∂_∞Γ→_d-k(^d) are called:


  * ρ-equivariant if ξ^(k)(γ· x) = ρ(γ)·ξ^(k)(x) and ξ^(d-k)(γ· x) = ρ(γ)·ξ^(d-k)(x) for all x ∈∂_∞Γ and γ∈Γ,

  * dynamics-preserving if for every γ∈Γ of infinite order with attracting fixed point γ^+ ∈∂_∞Γ, the points ξ^(k)(γ^+) ∈_k(^d) and ξ^(d-k)(γ^+) ∈_d-k(^d) are attracting fixed points of the action of ρ(γ) on _k(^d) and _d-k(^d), and

  * transverse if for every distinct pair x, y ∈∂_∞Γ we have 

    ξ^(k)(x) + ξ^(d-k)(y) = ^d.




With these definitions, we can now define Anosov representations.
 
 
A representation ρ: Γ→_d() is k-Anosov if 


  * there exist continuous, ρ-equivariant, dynamics preserving, and transverse maps ξ^(k): ∂Γ→_k(^d), ξ^(d-k): ∂Γ→_d-k(^d), and

  * for any sequence {γ_i}_i=1^∞⊂Γ so that lim_i→∞ℓ_S(γ_i)=∞, we have

    lim_i→∞logλ_k/λ_k+1(ρ(γ_i))=∞.


If ρ is k-Anosov for all k ∈{ k_1,…, k_j} we say that ρ is (k_1,…, k_j)-Anosov. 

 
If S' is another finite symmetric generating set of Γ, then 𝕀:(Γ,d_S)→(Γ,d_S') is a quasi-isometry. In particular, the notion of an Anosov representation does not depend on the choice of S. Also, it follows from the definition that a representation ρ:Γ→_d() is k-Anosov if and only if it is (d-k)-Anosov. We refer to ξ^(k) as the k-flag map of ρ, and ξ^(k)(∂Γ)⊂_k(^d) as the k-limit set of ρ.
 
 Given a subspace V ⊂^N define 

    [V]={ [v] ∈(^N) : v ∈ V}.
 
Often, we will view ξ^(k)(x) as the projective subspace [ξ^(k)(x)]⊂(^d). However, to simplify notation, we will denote [ξ^(k)(x)] simply by ξ^(k)(x) in those settings.
 

In many other places in the literature, what we call a k-Anosov representation is usually known as a P_k-Anosov representation, where P_k is the stabilizer in _d() of a point in _k(^d). This notation is an artifact of a more general definition of Anosov representations to an arbitrary non-compact semisimple Lie group. Since we do not use that generality here, we will use k in place of P_k to simplify the notation.
 




 §.§ Singular values and Anosov representations


We will now briefly discuss singular values, which we use to give an alternate description of Anosov representations. This description was initially due to Kapovich et al. <cit.>, but was also later proven by Bochi et al. <cit.> using different techniques.


 Let |·| and · be norms on ^d, and let L:(^d,|·|)→(^d,·) be a linear map.


  * For any X∈(^d,|·|), the stretch factor of X under L is the quantity 

    σ_X(L):=L(X)/|X|.


  * For i=1,…,n, the i-th singular value of L is the quantity

    σ_i(L)  :=max_W⊂^d, W=i   min_X∈ Wσ_X(L) = min_W ⊂^d,  W=d-i+1max_X ∈ Wσ_X(L).




Observe that for all i=1,…,d-1, σ_i(L)≥σ_i+1(L), and if L is invertible, then σ_i(L)=1/σ_d-i+1(L^-1). 

When L=g∈_d() and ·=|·| is the standard norm ·_2 on ^d, we denote σ_i(L) by μ_i(g). In that case, if A is a d× d real-valued matrix representing g in an orthonormal basis for the standard inner product on ^d, then the singular values μ_1(g)≥…≥μ_d(g)>0 are the square roots of the eigenvalues of A^TA. Using this, we may define, for any g∈_d() and all i,j∈{1,…,d}, the quantity

    μ_i/μ_j(g):=μ_i(g)/μ_j(g),

where g∈_d() is a lift of g. 

We can now state the following theorem due to Kapovich et al. <cit.>, (see Bochi et al. <cit.>).


Suppose Λ is a finitely generated group and S is a finite symmetric generating set. A representation ρ:Λ→_d() is k-Anosov if and only if there are constants C,c>0 such that

    logμ_k/μ_k+1(ρ(γ)) ≥ C d_S(γ,𝕀)-c

for all γ∈Λ.



In Theorem <ref>, it is implied, not assumed, that Λ is a hyperbolic group.



 

 §.§ Properties of Anosov representations
 
 
Next, we recall some important properties of Anosov representations. 

Define respectively the Cartan and Jordan projection μ,λ:_d()→^d by 

    μ(g):=  ( logμ_1(g), …, logμ_d(g) )    and    λ(g) =  ( logλ_1(g), …, logλ_d(g) ).
 
Observe that while the Jordan projection is invariant under conjugation in _d(), the Cartan projection is not. These two projections can be interpreted geometrically in the following way. 

Associated to the Lie group _d() is the Riemannian symmetric space X, on which _d() acts transitively  and by isometries. As a _d()-space, X=_d()/(d). Furthermore, the distance d_X on X induced by its Riemannian metric can be computed from the Cartan projection by the formula

    d_X(g_1·(d),g_2·(d))=μ(g_1^-1g_2)_2,

where g_1^-1g_2∈^±_d():={g ∈_d() : (g) = ± 1 } is a lift of g_1^-1g_2, and ·_2 is the l^2-norm. On the other hand, if g∈_d() and g∈^±_d() is a representative of g, then 

    inf_p∈ Xd_X(p,g· p)=λ(g)_2.

As such, if g∈_d() and g∈^±_d() is a lift of g, then μ(g) is a refinement of the distance by which g translates the identity coset in X, and λ(g) is a refinement of the minimal translation distance of g in X.

As an immediate consequence of Theorem <ref> an Anosov representations coarsely preserve the metric d_S on Γ.

 
Let ρ:Γ→_d() be k-Anosov for any k. Then the map Γ→ X defined by γ↦ρ(γ)·(d) is a quasi-isometric embedding. In other words, there are constants C≥ 1 and c≥ 0 such that for all γ_1,γ_2∈Γ,

    1/Cμ(ρ(γ_1^-1γ_2))_2-c≤ d_S(γ_1,γ_2)≤ Cμ(ρ(γ_1^-1γ_2))_2+c,

where ρ(γ_1^-1γ_2)∈^±_d() is a lift of ρ(γ_1^-1γ_2).
  
 
We also have the following proposition due to Quint (see <cit.> for a proof), which restricts the possible Zariski closures of Anosov representations to _d().
 
 Let ρ: Γ→_d() be a 1-Anosov representation. If ρ is irreducible, then the Zariski closure of ρ(Γ) is a semisimple Lie group without compact factors. 


We will also use the following observation of Guichard-Wienhard.

<cit.> Let ρ: Γ→_d() be an irreducible 1-Anosov representation. If Γ_0 ≤Γ is a finite index subgroup, then ρ|_Γ_0 is also irreducible. 

 
In many places in our argument, it will be more convenient to work with representations into _d() instead of _d(). The next observation allows us to make this reduction. Let π:_d()→_d() denote the obvious projection.

 For any representation ρ: Γ→_d(), there exists a subgroup Λ_ρ≤_d() so that π|_Λ_ρ:Λ_ρ→_d() is a representation whose kernel is a subgroup of _2, and whose image is a subgroup of ρ(Γ) with index at most two. 


 Define Λ_0 := { g ∈^±_d() : [g] ∈ρ(Γ) }, and let Λ_ρ := Λ_0 ∩_d(). Then π(Λ_0)⊂_d() coincides with ρ(Γ), and Λ_ρ has index at most two in Λ_0.


In particular, if Γ is a hyperbolic group, then so is Λ_ρ, and there are canonical identifications ∂_∞Γ=∂_∞ρ(Γ)=∂_∞π(Λ_ρ)=∂_∞Λ_ρ. Furthermore, the following proposition is an immediate consequence of <cit.> .


Let ρ:Γ→_d() be a representation. The representation 

    ρ':=π|_Λ_ρ:Λ_ρ→_d()
 
is k-Anosov if and only if ρ is k-Anosov. If so, the k-flag maps of ρ and ρ' agree.



To prove any properties about the k-limit sets of ρ, it is now sufficient to show those properties hold for the k-limit sets of ρ'. The advantage of working with ρ' in place of ρ is that ρ':Λ_ρ→(d,) admits a lift to a representation from Λ_ρ to _d(). With this, we can henceforth assume that ρ:Γ→(d,) admits a lift to a representation ρ:Γ→(d,).





 §.§ Gromov geodesic flow space
 

In their proof of Theorem <ref>, Bochi et al. <cit.> gave a description of Anosov representations using dominated splittings. Our next goal is to give this description. To do so, we recall the definition of the flow space of a hyperbolic group, and state some of their well-known properties. For more details, see for instance <cit.>, <cit.>, or <cit.>. 

As a topological space, the flow space for Γ, denoted UΓ, is homeomorphic to ∂_∞Γ^(2)×, where ∂_∞Γ^(2):={(x,y)∈∂_∞Γ^2:x≠ y}. This flow space admits a natural -action by translation in the -factor called the geodesic flow on UΓ. We will use the notation v=(v^+,v^-,v_0)∈UΓ, and denote the geodesic flow on UΓ by ϕ_t, i.e. 

    ϕ_t(v)=(v^+,v^-,v_0+t)=(ϕ_t(v)^+,ϕ_t(v)^-,ϕ_t(v)_0).


There is a proper, co-compact  Γ-action on UΓ that commutes with ϕ_t, and satisfies γ·(v^+,v^-,)=(γ· v^+,γ· v^-,). There is also a natural ℤ/2ℤ action on UΓ which satisfies 

    (1+2) · (x,y,) = (y,x,).

This action commutes with the Γ action, but not the ϕ_t action. Instead:

    αϕ_t α = ϕ_-t

where α = (1+2). So the actions of Γ, ϕ_t, and ℤ/2ℤ combine to yield an action of Γ× (⋊_ψℤ/2ℤ) on UΓ where ψ : ℤ/2ℤ→() is given by ψ(α)(t) = -t. 

Since the Γ action commutes with ϕ_t, the geodesic flow on UΓ descends to a flow on the compact space UΓ:=UΓ/Γ, which we refer to as the geodesic flow on UΓ, and denote by ϕ_t. This also implies that if v^+=γ^+ and v^-=γ^- are the attracting and repelling fixed points of some infinite order γ∈Γ, then the orbit (γ^+,γ^-,)⊂UΓ of ϕ_t descends to a closed orbit of ϕ_t in UΓ. We will denote the period of this closed orbit by T_γ∈, and refer to T_γ as the period of γ. In other words, for all v_0∈, γ· (γ^+,γ^-,v_0)=(γ^+,γ^-,v_0+T_γ). 

Furthermore, UΓ admits a Γ×ℤ/2ℤ-invariant metric so that every orbit (v^+,v^-,) of ϕ_t is a continuous quasi-geodesic. Since the Γ-action on UΓ is also co-compact, any Γ-orbit is a quasi-isometry. As a consequence, there is a canonical Γ-invariant homeomorphism ∂_∞UΓ≃∂_∞Γ between the Gromov boundaries of UΓ and Γ, and v^+ and v^- in ∂_∞UΓ are the forward and backward endpoints of (v^+,v^-,)⊂UΓ respectively.







In the case when Γ is the fundamental group of a compact Riemannian manifold X with negative sectional curvature,  this geodesic flow space is what one would expect. In particular, let T^1 X denote the unit tangent bundle of X, let X denote the universal cover of X, and let T^1X denote the unit tangent bundle of X. Then we may take UΓ to be T^1X and UΓ to be T^1X. The geodesic flow on both T^1X and T^1X is the usual geodesic flow associated to the Riemannian metrics on X and X, and the Γ-invariant metric d_UΓ is the lift of the Riemannian metric on T^1X that is locally given by the product of the Riemannian metric on X and the spherical metric on the fibers. Further, the ℤ/2ℤ action is given by v → -v. 


Gromov proved that the geodesic flow space UΓ is unique up to homeomorphism. 



<cit.> Suppose that  is a proper Gromov hyperbolic metric space such that


  * Γ× (⋊_ψℤ/2ℤ) acts on ,

  * the actions of Γ and ℤ/2ℤ are isometric, 

  * for every v ∈, the map γ∈Γ→γ· v ∈ is a quasi-isometry. 

  * the  action is free and every -orbit is a quasi-geodesic in . Further, the induced map / →∂_∞^(2) is a homeomorphism.

Then there exists a Γ×ℤ/2ℤ-equivariant homeomorphism T : →UΓ that maps -orbits to -orbits. 

 












 §.§ Dominated Splittings
 
Next, we describe an alternate characterization of Anosov representations in _d() using dominated splittings due to Bochi et al. <cit.>.

Let ρ:Γ→_d() be a representation. Let E:=UΓ×^d be the trivial bundle over UΓ, and define the vector bundle E_ρ:=E/Γ over UΓ, where the Γ action on E is given by γ·(v,X)=(γ· v,ρ(γ)· X). Since E_ρ is naturally a flat vector bundle over UΓ, it admits a continuous norm, and the compactness of UΓ ensures that any two such norms are bi-Lipschitz. For any continuous norm on E_ρ, choose a lift of this norm to a Γ-invariant, continuous norm · on E. With this, we can state the following theorem due to Bochi et al (see Theorem 2.2, Proposition 4.5 and Proposition 4.9 in <cit.>). 



 A representation ρ:Γ→_d() is k-Anosov if and only if there exist


  * continuous, ϕ_t-invariant, ρ-equivariant maps 

    F_1:UΓ→_k(^d)    and    F_2:UΓ→_d-k(^d)

so that F_1(v)+F_2(v)=^d for all v∈UΓ, and

  * constants C > 0, β > 0 such that 

    X_1_ϕ_tv/X_2_ϕ_tv≤ C e^-β tX_1_v/X_2_v

for all v ∈UΓ, X_i ∈ F_i(v) non-zero, and t ≥ 0.  



Here, we may think of F_1 and F_2 as Γ-invariant sub-bundles of E. The maps F_1 and F_2 are related to the flag maps ξ^(k) and ξ^(d-k) by

    F_1(v) = ξ^(k)(v^+)    and    F_2(v) = ξ^(d-k)(v^-)

for all v = (v^+,v^-,v_0)∈UΓ. 





§ Ρ-CONTROLLED SETS


In this section we introduce ρ-controlled sets and construct a useful family of projections. 

Suppose that ρ : Γ→_d() is a 1-Anosov representation. A closed ρ(Γ)-invariant subset M⊂(^d) is ρ-controlled if 


  (i) ξ^(1)(∂_∞Γ)⊂ M and

  (ii) M∩ξ^(d-1)(x)=ξ^(1)(x) for every x∈∂_∞Γ.

If ρ also happens to be m-Anosov for some m=2,…,d-1, then a ρ-controlled subset M⊂(^d) is m-hyperconvex if 

    p_1+p_2+ξ^(d-m)(y)

is a direct sum for all p_1,p_2 ∈ M and y ∈∂_∞Γ with p_1, p_2, ξ^(1)(y) pairwise distinct. 


 We will typically consider the case when M is a topological (m-1)-dimensional manifold and then require that M is m-hyperconvex. 
 

The three main examples of ρ-controlled subsets M⊂^d that we will be concerned with are the following. 

 When ρ : Γ→_d() is a 1-Anosov representation, then the 1-limit set ξ^(1)(∂_∞Γ) for ρ is obviously ρ-controlled. Furthermore, if ρ is m-Anosov for some m=1,…,d-1, then ξ^(1)(∂_∞Γ) is m-hyperconvex if and only if 

    ξ^(1)(x)+ξ^(1)(z)+ξ^(d-m)(y)

 is a direct sum for all pairwise distinct x,y,z∈∂_∞Γ.










 Suppose that ρ:Γ→_d() is a 1-Anosov representation and ρ(Γ) preserves a properly convex domain Ω⊂(^d) so that ξ^(d-1)(x)∩∂Ω=ξ^(1)(x) for all x∈∂_∞Γ, see Section <ref>. Then M:=∂Ω is obviously ρ-controlled. Notice that in this case, the requirement that M is (d-1)-hyperconvex is simply that 

    p_1+p_2+ξ^(1)(y)

is a direct sum for all pairwise distinct p_1,p_2, ξ^(1)(y)∈∂Ω. This is satisfied if and only if ξ^(1)(∂_∞Γ) does not intersect any proper line segments in ∂Ω. 


 Suppose ρ : Γ→_d() is a 1-Anosov representation and Γ_1 ≤Γ is a quasi-convex subgroup. Then ρ_1:=ρ|_Γ_1: Γ_1 →_d() is also 1-Anosov, and its 1-limit set ξ^(1)(∂_∞Γ) for ρ is obviously ρ_1-controlled. Furthermore, if ρ_1 is m-Anosov for some m=1,…,d-1, then ξ^(1)(∂_∞Γ) is m-hyperconvex if and only if 

    ξ^(1)(x)+ξ^(1)(z)+ξ^(d-m)(y)

 is a direct sum for all pairwise distinct x,y,z ∈∂_∞Γ with y ∈∂_∞Γ_1. 



Recall that ∂_∞Γ^(2) is the set of all pairs (x,y) ∈∂_∞Γ^2 with x ≠ y. Then for any (x,y)∈∂_∞Γ^(2), let L_x,y denote the orbit (x,y,)⊂UΓ of ϕ_t. The following proposition is one of the key tools we use to investigate regularity properties of ρ-controlled subsets.

 Suppose that ρ : Γ→_d() is a 1-Anosov representation and M ⊂(^d) is ρ-controlled. Then there exists a continuous family of continuous maps 

    π_x,y : M ∖{ξ^(1)(x), ξ^(1)(y) }→ L_x,y

indexed by (x,y) ∈∂_∞Γ^(2) such that 

    π_x,y    =ρ(γ)^-1∘π_γ· x,γ· y∘ρ(γ), 
    
    x     = lim_p→ξ^(1)(x)π_x,y(p),  and
    
    y     = lim_p→ξ^(1)(y)π_x,y(p)

for all (x,y) ∈∂_∞Γ^(2) and γ∈Γ.


Delaying the proof of Proposition <ref> until Section <ref>, we describe the main application. Suppose that ρ : Γ→_d() is a 1-Anosov representation and M ⊂(^d) is ρ-controlled. Let {π_x,y : (x,y)∈∂_∞Γ^(2)} be a family of maps satisfying Proposition <ref>. Then define the following space

    P(M) : = { (v,p) ∈U Γ× M : p ≠ξ^(1)(v^±)  and  v = π_v^+,v^-(p) }.

Notice that there is a natural Γ action on P(M) given by 

    γ· (v,p) = (γ· v, ρ(γ)p).

This space has the following properties.

 With the notation above, 


  * Γ acts co-compactly on P(M), 

  * for any v ∈UΓ and z ∈ M ∖{ξ^(1)(v^+), ξ^(1)(v^-)}, there exists t ∈ such that (ϕ_t(v), z) ∈ P(M),

  * for any compact set K ⊂UΓ there exists δ > 0 such that: If v ∈ K and p ∈ M∖{ξ^(1)(v^+)} satisfies d_(ξ^(1)(v^+), p) ≤δ, then (ϕ_t (v), p) ∈ P(M) for some t > 0. 




(1): Since the Γ-action on UΓ is co-compact, there exists a compact set K ⊂UΓ such that Γ· K = UΓ. Since  {π_x,y : (x,y)∈∂_∞Γ^(2)} is a family of maps satisfying Proposition <ref>, the set 

    K : = {(v,z) ∈ K × M:v ≠ξ^(1)(v^±)  and  v=π_v^+,v^-(z)  }

is compact. Further, by definition, Γ·K = P(M).

(2): Follows directly from the definition. 

(3): Fix a compact set K ⊂UΓ. If such a δ > 0 does not exist, then there exists v_n ∈ K, p_n ∈ M∖{ξ^(1)(v_n^+)}, and t_n ≤ 0 such that  

    d_(ξ^(1)(v_n^+), p_n) ≤ 1/n

and ϕ_t_n(v_n)=π_v_n^-,v_n^+(p_n). By passing to a subsequence we can suppose that v_n → v ∈ K. But then p_n →ξ^(1)(v^+) as n→∞, so

    v^+ = lim_n →∞π_v_n^-,v_n^+(p_n) = lim_n →∞ϕ_t_n(v_n) ∈ L_v^+, v^-∪{ v^-}

which is a contradiction. 


 The set P(M) is designed to be a generalization of the following construction: Suppose Γ is the fundamental group of X a compact negatively curved Riemannian manifold, X is the universal cover of X,  T^1 X is the unit tangent bundle of X, and ϕ_t is the geodesic flow on T^1 X. Then we define 

    Perp⊂ T^1 X×∂_∞Γ

to be the set of pairs (v,z) such that there exists w ∈ T^1_π(v)X with w  v and lim_t →∞π(ϕ_tw) = z. 
 




 §.§ Properly convex domains

We now describe properly convex domains and some of their relevant properties. These will be used to prove Proposition <ref>.

  

  * An open set Ω⊂(^d) is a properly convex domain if its closure lies in an affine chart in (^d), and it is convex, i.e. for every pair of distinct points x,y∈Ω, there is a projective line segment in Ω whose endpoints are x and y.

  * Given a subset X ⊂(^d) the projective automorphism group of X is defined to be 

    (X) = { g ∈_d() : g X = X}.




Given a properly convex domain Ω⊂(^d), there is a canonical distance on Ω which is defined as follows. For any pair of points x,y∈Ω, let l be a projective line through x and y, and let a and b be the two points of intersection of l with ∂Ω, ordered so that a<x≤ y<b lie along l. Then define

    H_Ω(x,y):=log C(a,x,y,b).

Here, C is the cross ratio along the projective line l, i.e.

    C(a,x,y,b):=a-y_2b-x_2/a-x_2b-y_2,

where ·_2 is the standard norm on some (equiv.) any affine chart of (^d) containing the closure of Ω. One can verify from properties of the cross ratio that the map H_Ω:Ω×Ω→^+∪{0} is a well-defined, continuous, distance function. This is commonly known as the Hilbert metric on Ω. 

Let TΩ denote the tangent bundle of Ω and π : TΩ→Ω the natural projection. Also, for any v∈ TΩ, let l_v denote the oriented projective line segment in Ω through π(v) in the direction given by v, and with endpoints in ∂Ω. Then let v^+ and v^- be the forward and backward endpoints of l_v respectively. The Hilbert metric H_Ω is infinitesimally given by the norm 

    h_Ω:   TΩ   →∪{0},
       v   ↦v_2(1/π(v)-v^+_2+1/π(v)-v^-_2),

where ·_2 is the standard norm on any affine chart of (^d) containing the closure of Ω. With this, define the unit tangent bundle of Ω to be

    T^1 Ω = { v ∈ TΩ : h_Ω(v) = 1}.


We recall the definition of a convex co-compact action on Ω.

 A discrete subgroup Λ≤_d() acts convex co-compactly on a properly convex domain Ω if Λ≤(Ω) and there exists a closed non-empty convex subset ⊂Ω such that Λ≤() and the quotient Λ\ is compact.
 
 
  This is not the definition of convex co-compactness used in <cit.>, instead they say groups satisfying Definition <ref> act naive convex co-compactly. 
 
We now use work of Danciger-Gueritaud-Kassel <cit.> and the second author <cit.> to construct a convex co-compact action. 

 
 <cit.>, <cit.> Suppose that ρ : Γ→_d() is a 1-Anosov representation and there exists a properly convex domain Ω_0 ⊂(^d) with ρ(Γ) ≤(Ω_0). Then ρ(Γ) acts convex co-compactly on a properly convex domain Ω⊂(^d). Moreover, 
 
 
  * ξ^(1)(∂_∞Γ) ⊂∂Ω,
 
  * for every x,y ∈∂_∞Γ distinct, Ω_0 and Ω are contained in the same connected component of 
 
    (^d) ∖( ξ^(d-1)(x) ∪ξ^(d-1)(y) ),

 
  * we can assume
 
    = Ω∩ ConvHull{ξ^(1)(∂_∞Γ) }

 and 
  
    ∩∂Ω = ξ^(1)(∂_∞Γ).

 
 
 
  In <cit.> it is assumed that ρ is irreducible. 
 




 §.§ The proof of Proposition <ref>


We will prove Proposition <ref> by constructing a projective model of the geodesic flow space U Γ. This construction has several steps: first we post compose to obtain a new 1-Anosov representation that preserves a properly convex domain. Theorem <ref> then gives us a convex co-compact action, which we then use to construct a projective model of the geodesic flow space. Finally we use this projective model to construct the maps π_x,y. 



  §.§.§ Constructing an invariant properly convex domain


In general, a 1-Anosov representation will not preserve a properly convex domain. 

<cit.> If d is even and ρ : π_1(S) →_d() is Hitchin (see Definition <ref>), then ρ(π_1(S)) does not preserve a properly convex domain. 


However, we will show that after post composing with another representation we can always find an invariant properly convex domain.  In Section <ref> we will study the regularity of these sets. 


Denote the vector space of symmetric 2-tensors by _2(^d) and let D := _2(^d). Then let S : _d() →(_2(^d)) be the representation 

    S(g)(v ⊗ v) = gv ⊗ gv.

Given a representation ρ : Γ→_d(), let S(ρ) : Γ→_d() be the representation S(ρ) = S ∘ρ. 

Associated to S are smooth embeddings Φ : (^d) →(_2(^d)) and Φ^* : _d-1(^d) →_D-1(_2(^d)) defined by

    Φ(v) = [v ⊗ v]

and

    Φ^*(W) = { v ⊗ w + w ⊗ v : w ∈ W, v ∈^d }.

Notice that Φ and Φ^* are both S-equivariant. 

 If ρ : Γ→_d() is 1-Anosov with boundary maps ξ^(1) and ξ^(d-1), then S(ρ) is 1-Anosov with boundary maps Φ∘ξ^(1) and Φ^* ∘ξ^(d-1). 


 The maps Φ∘ξ^(1) and Φ^* ∘ξ^(d-1) are clearly S(ρ)-equivariant, dynamics-preserving, and transverse. 

Suppose that γ∈Γ and g is a lift of ρ(γ). If λ_1(g) ≥…≥λ_d(g) are the absolute values of the eigenvalues of g and λ_1(S(g)) ≥λ_2(S(g)) ≥… are the absolute values of the eigenvalues of S(g), then 

    λ_1(S(g)) = λ_1(g)^2

and 

    λ_2(S(g)) = λ_1(g) λ_2(g).

So 

    λ_1/λ_2(S(ρ)(γ)) =λ_1(g)/λ_2(g)= λ_1/λ_2(ρ(γ)).

Then since ρ is 1-Anosov, we see that S(ρ) is also 1-Anosov. 


Now we construct a properly convex domain in (_2(^d)) which is invariant under the action of S(_d()). Given X ∈_2(^d) we say that X is positive definite, and write X > 0, if (f⊗ f)(X) > 0 for every f ∈^d*∖{0}. Also, we say that X is positive semidefinite, and write X ≥ 0, if (f⊗ f)(X) ≥ 0 for every f ∈^d*. Then define 

    ^+ := { [X] : X ∈_2(^d), X > 0}.


  

  * ^+ is a properly convex domain in (_2(^d)),

  * S(_d()) ≤(^+),

  * Φ((^d)) ⊂^+. 



 
(1): Clearly C := { X : X∈_2(^d), X > 0} is a convex open cone in _2(^d). Since 

    (f⊗ f)(X+tY) = (f⊗ f)(X)+t(f⊗ f)(Y)

it is clear that C does not contain any real affine lines. Thus C is properly convex. Since C projects to ^+ we see that ^+ is a properly convex domain. 

(2): Notice that

    (f⊗ f)(S(g)X) = ((f ∘ g) ⊗ (f ∘ g) )(X)

when f ∈^d*, g ∈_d(), and X ∈_2(^d). So S(_d()) ≤(^+).

(3): Suppose that [v] ∈(^d). Then Φ([v]) = [v ⊗ v] and 

    (f⊗ f)(v ⊗ v) = f(v)f(v) ≥ 0

when f ∈^d*. Thus Φ([v]) ⊂^+. 

  
 

  §.§.§ Constructing a projective geodesic flow

 

In this step we construct a “projective” geodesic flow for any 1-Anosov representation that acts convex co-compactly on a properly convex domain. For the rest of this subsection, suppose that ρ : Γ→_d() is a 1-Anosov representation and ρ(Γ) acts convex co-compactly on a properly convex domain Ω. We can further assume that 

    = Ω∩ ConvHull{ξ^(1)(∂_∞Γ) }

 and 
  
    ∩∂Ω = ξ^(1)(∂_∞Γ).

 
 




Every projective line segment in Ω can be parametrized to be a geodesic in H_Ω and thus T^1 Ω has a natural geodesic flow, which we denote by ψ_t, obtained by flowing along the projective line segments. Using this flow we can construct a model of the flow space UΓ. 

For x,y ∈∂_∞Γ distinct, let ℓ_x,y⊂ T^1 Ω be the unit tangent vectors whose based points are contained in the line segment joining ξ^(1)(x) to ξ^(1)(y) and who point in the direction of ξ^(1)(y). Then the set

    : = ⋃_(x,y) ∈∂_∞Γ^(2)ℓ_x,y

is invariant under the action of ρ(Γ), the flow ψ_t, and the natural ℤ/2ℤ action on T^1 Ω given by v → -v. Using Theorem <ref> we will construct a homeomorphism →UΓ. 


 With the notation above, there exists a homeomorphism T : →UΓ with the following properties:


  * T is equivariant relative to the Γ and ℤ/2ℤ actions

  * for every (x,y) ∈∂_∞Γ^(2), T maps the flow line ℓ_x,y to the flow line L_x,y.



 By construction Γ× (⋊_ψℤ/2ℤ) acts on  and  the  action is free. Further,  is homeomorphic to ∂_∞Γ^(2)× and so we have a homeomorphism 

    / →∂_∞Γ^(2).

Hence to apply Gromov's theorem we just have to verify that  has a complete metric d with the following properties: 


  * the actions of Γ and ℤ/2ℤ are isometric, 

  * for every v ∈, the map γ∈Γ→γ· v ∈ is a quasi-isometry. 

  * every -orbit is a quasi-geodesic in . 

To do this, define d:𝒢×𝒢→ by

    d(v,w) =1/√(π)∫_ H_Ω(γ_v(t), γ_w(t)) e^-t^2 dt

where γ_v and γ_w are the unit speed geodesics with γ_v^'(0)=v and γ_w^'(0)=w. Then conditions (1) and (3) are easy to check. To verify condition (2), notice that 

    H_Ω(γ_v(t), γ_w(t))-H_Ω(π(v), π(w))≤ 2t

and 

    1/√(π)∫_2 te^-t^2 dt = 2/√(π).

Hence 

    H_Ω(π(v), π(w)) -2/√(π)≤ d(v,w) ≤ H_Ω(π(v), π(w)) +2/√(π).

And so π : (, d) → (, H_Ω) is a quasi-isometry. Since (, H_Ω) is a geodesic metric space and Γ acts co-compactly on , the fundamental lemma of geometric group theory states that for every c ∈, the map γ∈Γ→γ· c ∈ is a quasi-isometry. So the Γ orbits in  are also quasi-isometries. 








  §.§.§ Finishing the proof of Proposition <ref>


Suppose that ρ : Γ→_d() is a 1-Anosov representation and M ⊂(^d) is ρ-controlled. 

By Proposition <ref>, the representation S(ρ) : Γ→(_2(^d)) is 1-Anosov with boundary maps ξ^(1)_S : = Φ∘ξ^(1) and ξ^(d-1)_S : = Φ^* ∘ξ^(d-1). By Observation <ref>, 

    S(ρ)(Γ) ≤(^+).

Then by Theorem <ref> there exists a properly convex domain Ω⊂(_2(^d)) where S(ρ)(Γ) acts convex co-compactly on Ω. We can assume that 
 
    = Ω∩ ConvHull{ξ^(1)(∂_∞Γ) }

 and 
  
    ∩∂Ω = ξ^(1)(∂_∞Γ).

 Now let ⊂ T^1 Ω be the projective model of the geodesic flow constructed in Section <ref> and let T : →UΓ denote the homeomorphism in Corollary <ref>.
 
 Next, for x,y ∈∂_∞Γ distinct we define a projection

    p_x,y : (_2(^d)) ∖( ξ_S^(d-1)(x) ∩ξ_S^(d-1)(y) ) →ξ_S^(1)(x) + ξ^(1)_S(y)

by 

    { p_x,y(v) } = ( ξ_S^(1)(x) + ξ^(1)_S(y)) ∩( v + ξ_S^(d-1)(x) ∩ξ_S^(d-1)(y) ).


 If m ∈ M ∖{ξ^(1)(x),ξ^(1)(y) }, then p_x,y(Φ(m)) is contained in the line segment joining ξ^(1)_S(x) to ξ^(1)_S(y) in Ω. 

 Since M is ρ-controlled, 

    m ∉ξ^(d-1)(x) ∪ξ^(d-1)(y).

Hence 

    Φ(m) ∉ξ^(d-1)_S(x) ∪ξ^(d-1)_S(y).

Observation <ref> implies that Φ(m) ∈^+ and (2) of Theorem <ref> says that ^+ and Ω are in the same connected component of 

    (_2(^d)) ∖( ξ_S^(d-1)(x) ∪ξ_S^(d-1)(y) ).

Hence p_x,y(Φ(m)) is contained in the line segment joining ξ^(1)_S(x) to ξ^(1)_S(y) in Ω.


 Next, for x,y ∈∂_∞Γ distinct we define a map

    p_x,y : M ∖{ξ^(1)(x),ξ^(1)(y) }→

by letting p_x,y(m) be the unit vector above π(p_x,y(Φ(m))) pointing towards y. 

Finally, we define

    π_x,y : M ∖{ξ^(1)(x), ξ^(1)(y) }→ L_x,y

by π_x,y = T ∘p_x,y. Recall that T is defined in Corollary <ref>.

By construction we have

    π_x,y=ρ(γ)^-1∘π_γ· x,γ· y∘ρ(γ)

for all (x,y) ∈∂_∞Γ^(2) and γ∈Γ. Further, by (2) of Corollary <ref> 

    lim_p→ξ^(1)(x)π_x,y(p)=x  and lim_p→ξ^(1)(y)π_x,y(p)=y

for all (x,y) ∈∂_∞Γ^(2).

 


  §.§.§ The construction for non-surface groups:

 
It is worth noting that for many word hyperbolic groups, post composing with the representation S : _d() →(_2(^d)) is not necessary to construct a  convex co-compact action. 

<cit.>  Suppose Γ is a non-elementary word hyperbolic group which is not commensurable to a non-trivial free product or the fundamental group of a closed hyperbolic surface. Then any irreducible 1-Anosov representation ρ: Γ→_d() acts convex co-compactly on a properly convex domain Ω⊂(^d). 







§ SUFFICIENT CONDITIONS FOR DIFFERENTIABILITY OF Ρ-CONTROLLED SUBSETS





The goal of this section is to prove Theorem <ref>, which is a generalization of Theorem <ref> in terms of ρ-controlled subsets of (^d) instead of the 1-limit set. 




 §.§ The quantity α^m(ρ)


Suppose that ρ is (1,m)-Anosov for some m=2,…,d-1. To state Theorem <ref>, we first define a quantity α^m(ρ) as follows. Recall that E:=UΓ×^d. Let ρ:Γ→_d() be a lift of ρ (see Remark <ref>), and let · be a Γ-invariant norm on E, i.e. v↦·_v is a continuous family of norms on ^d parameterized by UΓ, so that ρ(γ)· X_γ· v=X_v for all γ∈Γ, v∈UΓ and X∈^d. For any v=(v^+,v^-,v_0) ∈UΓ, let 

    E_1(v)     = ξ^(1)(v^+),
    
    E_2(v)     = ξ^(d-1)(v^-) ∩ξ^(m)(v^+),
    
    E_3(v)     = ξ^(d-m)(v^-),

and define f:UΓ×→ by

    f(v,t):=inf_X_i∈ S_i(v){logX_3_ϕ_t(v)/X_1_ϕ_t(v)/logX_2_ϕ_t(v)/X_1_ϕ_t(v)},

where S_i(v):={X∈ E_i(v):||X||_v=1} for i=1,2,3. Then define

    α^m(ρ):=lim inf_t→∞inf_v∈UΓf(v,t).


To see that α^m(ρ) is well-defined and strictly larger than 1, we need the following observation.

 
There exists C_1 ≥1 and β_1 ≥0 such that 

    1/C_1 e^-β_1 tX_v≤X_ϕ_t(v)≤ C_1 e^β_1 tX_v

for all v ∈UΓ, t > 0, and X∈^d.


 Since Γ acts co-compactly on UΓ there exists β_1 ≥ 0 such that 

    e^-β_1X_v≤X_ϕ_t(v)≤ e^β_1X_v

for all v ∈UΓ, t ∈ [0,1], and X∈^d. Then for any t>0, let k∈^+ so that t∈[k-1,k), and note that 

    e^-kβ_1X_v≤X_ϕ_t(v)≤ e^kβ_1X_v.

Thus, if we let C_1:=e^β_1, then

    1/C_1e^-tβ_1X_v≤1/C_1e^-(k-1)β_1X_v≤X_ϕ_t(v)≤ C_1e^(k-1)β_1X_v≤ C_1e^tβ_1X_v.




By Theorem <ref>, the assumption that ρ is (1,m)-Anosov ensures that there are constants C_2,C_3≥ 1 and β_2,β_3≥0 so that for all v ∈UΓ, X_i ∈ F_i(v) non-zero, and t ≥ 0, we have

    X_2_ϕ_t(v)/X_1_ϕ_t(v)≥1/C_2 e^β_2 tX_2_v/X_1_v    and    X_3_ϕ_t(v)/X_2_ϕ_t(v)≥1/C_3 e^β_3 tX_3_v/X_2_v.

This, together with Observation <ref>, then implies that

    logX_3_ϕ_t(v)/X_1_ϕ_t(v)/logX_2_ϕ_t(v)/X_1_ϕ_t(v)   =1+logX_3_ϕ_t(v)/X_2_ϕ_t(v)/logX_2_ϕ_t(v)/X_1_ϕ_t(v)
       ≥1+β_3 t-log C_3+logX_3_v/X_2_v/2β_1 t+2log C_1+logX_2_v/X_1_v

and

    logX_3_ϕ_t(v)/X_1_ϕ_t(v)/logX_2_ϕ_t(v)/X_1_ϕ_t(v)   ≤1+2β_1 t+2log C_1+logX_3_v/X_2_v/β_2 t-log C_2+logX_2_v/X_1_v

In particular, α^m(ρ) is a well-defined real number that is strictly larger than 1. Also, observe that α^m(ρ) does not depend on the choice of ·, nor on the choice of lift ρ of ρ. 

With this, we can state the main theorem of this section.

 Let ρ:Γ→_d() be (1,m)-Anosov for some m=2,…, d-1. Suppose that M⊂(^d) is a ρ-controlled subset that is also a topological (m-1)-dimensional manifold. If



  (†) ρ is m-Anosov and M is m-hyperconvex, 

then 


  () M is C^α along ξ^(1)(∂_∞Γ) for all α so that 1<α<α^m(ρ).

Moreover, for all x∈∂_∞Γ, the tangent space to M at ξ^(1)(x) is ξ^(m)(x). 


   
 
  * See Section <ref>, for the definition of “C^α along.”
 
  * As mentioned in the introduction, in the special case when M = ξ^(1)(∂_∞Γ) this theorem was independently proven by Pozzetti-Sambarino-Wienhard <cit.> without the estimate on α. 
  
 

It is clear from Example <ref> and <ref> that Theorem <ref> and the first part of Theorem <ref> follow immediately from Theorem <ref>.




 §.§ The key inequality 


Suppose that ρ is (1,m)-Anosov for some m=2,…,d-1. Fix a distance d_ on (^d) induced by a Riemannian metric. The following lemma is the key inequality needed to prove Theorem <ref>.

 
Suppose that M⊂(^d) be ρ-controlled and m-hyperconvex. Then for all α satisfying 0<α<α^m(ρ), there exists D ≥ 1 with the following property: for every x ∈∂_∞Γ and p∈ M, we have

    d_(p, ξ^(m)(x) ) ≤ D d_(p, ξ^(1)(x) )^α.



We prove Lemma <ref> via a series of small observations. First, from the definition of α^m(ρ), one observes the following.


If 0<α<α^m(ρ), then there is a constant B≥ 1 so that 

    X_1_ϕ_t(v)/X_3_ϕ_t(v)≤ B ( X_1_ϕ_t(v)/X_2_ϕ_t(v))^α

for all v∈UΓ, t≥ 0, and X_i∈ S_i(v)



Since 0<α <α^m(ρ), there exists T > 0 such that α < f(v,t) for all t ≥ T and v ∈UΓ, so

    X_1_ϕ_t(v)/X_3_ϕ_t(v) < ( X_1_ϕ_t(v)/X_2_ϕ_t(v))^α

for all t ≥ T, v ∈UΓ, and X_i ∈ S_i(v). On the other hand, if 0≤ t≤ T, then the Γ-invariance of · implies that both sides of the inequality (<ref>) are continuous positive functions on [0,T]× S_ρ, where S_ρ⊂ E_ρ is a compact fiber bundle over UΓ whose fiber over [v]∈ UΓ is S_1(v)× S_2(v)× S_3(v). Thus, there exists some B ≥ 1 so that 
(<ref>) holds. 


Next, for i=1,2,3 and v ∈UΓ define P_i,v : ^d → E_i(v) to be the projection with kernel E_i-1(v)+E_i+1(v), where the arithmetic in the subscripts are done modulo 3.

The following observation is an immediate consequence of the fact that M is ρ-controlled and m-hyperconvex.

 
If v ∈UΓ and p∈ M ∖{ξ^(1)(v^+), ξ^(1)(v^-)} then P_i,v(X) ≠ 0
for all non-zero X∈ p and i=1,2,3. 











Choose a compact set K⊂UΓ so that Γ· K=UΓ. By enlarging K if necessary, we can ensure that {v^+:v∈ K}=∂_∞Γ. 

Next let

    π_x,y : M ∖{ξ^(1)(x), ξ^(1)(y) }→ L_x,y

be a family of maps  which satisfy Proposition <ref>. Then, as in Section <ref>, define

    P(M):={ (v,z) ∈U Γ× M : p ≠ξ^(1)(v^±)  and  v = π_v^+,v^-(p) }.


Using the fact that Γ\ P(M) is compact (see (1) of Observation <ref>) and Observation <ref>, we deduce the next three observations, which we use to prove Theorem <ref>.


There is a constant C≥ 1 so that
 
    1/C≤ P_i,v( X ) _v/P_j,v( X )_v≤ C

for all (v,p) ∈ P(M), all non-zero X ∈ p, and all i,j ∈{1,2,3}, and

    1/C≤X_v /X_2≤ C

for all v ∈ K and non-zero X∈^d.




Since · is Γ-invariant, Observation <ref> implies that the map P(M)/Γ→ defined by

    [v,p]   ↦ P_i,v( X ) _v/P_j,v( X )_v

where X∈ p is a non-zero vector, is a well-defined, continuous, positive function on P(M)/Γ. Hence, (1) of Observation <ref> implies that there exists C≥ 1, so that (<ref>) holds. Also, since the function K×(^d)→ defined by 

    (v,[X])↦X_v/X_2
 
is also well-defined, continuous, and positive, by further enlarging C if necessary, we may assume that (<ref>) holds.


Using the fact that d_ is induced by a Riemannian metric we have the following estimates. 

 
For any sufficiently small δ>0, there exists A ≥ 1 such that: for all v ∈ K, p ∈ M so that d_(ξ^(1)(v^+), p) ≤δ, and X ∈ p non-zero, we have

    1/AP_3,v(X)_2/P_1,v(X)_2≤ d_( p, ξ^(m)(v^+) ) ≤  A P_3,v(X)_2/P_1,v(X)_2

and 

    1/AP_2,v(X)_2/P_1,v(X)_2≤ d_(p, ξ^(1)(v^+)) ≤ A P_2,v(X)_2+P_3,v(X)_2/P_1,v(X)_2.


















Using Observations <ref> and <ref>, we will now prove Lemma <ref>.

 
Let δ>0 be sufficiently small so that Observation <ref> holds. Using (3) of Observation <ref> and possibly decreasing δ > 0 we may also assume that for all v ∈ K and p ∈ M ∖{ξ^(1)(x^+)} satisfying d_(ξ^(1)(v^+), p) ≤δ, there is some t>0 so that (ϕ_t (v), p) ∈ P(M).

Elementary considerations imply that it is sufficient to prove Lemma <ref> for all x ∈∂_∞Γ and p∈ M ∖{ξ^(1)(x)} so that d_(ξ^(1)(x), p) ≤δ. By the assumptions on K, there exist some v ∈ K such that v^+ = x. Further, by our choice of δ, there exists t > 0 such that (ϕ_t(v),p) ∈ P(M).

For any non-zero X ∈ p and for i=1,2,3, let

    X_i := P_i,v(X)/P_i,v(X)_v∈ S_i(v).
 
By (<ref>), (<ref>), and (<ref>),

    d_( p, ξ^(m)(x) ) 
       ≤  A P_3,v(X)_2/P_1,v(X)_2
       ≤ AC^3 P_3,v(X)_v/P_1,v(X)_vP_1,v(X)_ϕ_t(v)/P_3,v(X)_ϕ_t(v)
       = AC^3 X_1_ϕ_t(v)/X_3_ϕ_t(v)





Repeating a similar argument, but with (<ref>) in place of (<ref>), proves

 



    d_( p, ξ^(1)(x))  ≥1/AC^3X_1_ϕ_t(v)/X_2_ϕ_t(v).


Finally, since 0<α<α^m(ρ), Observation <ref> and (<ref>) gives

    d_( p, ξ^(m)(x) ) ≤ A BC^3 ( X_1_ϕ_t(v)/X_2_ϕ_t(v))^α.

Combining this with (<ref>) yields

    d_( p, ξ^(m)(x) )  ≤ D d_( p, ξ^(1)(x))^α

where D := A^1+αB C^3+3α. 





















 §.§ Proof of Theorem <ref>


We now use Lemma <ref> to prove Theorem <ref>. Again, suppose that ρ is (1,m)-Anosov for some m=2,…,d-1. 

We begin by making the following simple observation. Fix a hyperplane ⊂^d and a (d-m)-dimensional subspace ⊂. Then consider the affine chart _:=(^d)∖[] of (^d). Recall that [] denotes the projectivization of , see (<ref>). For any m-dimensional subspace ⊂^d that is transverse to , let 

    Π_,:_→ []∩_
 
be the projection given by [X]↦ [U_X], where X=U_X+V_X with U_X∖{0}∈ and V_X∈. Observe that the fibers of Π_, are of the form [· X+]∩_ for some X∈∖. In particular, the fibers of Π_, do not depend on , i.e. if '⊂^d is another m-dimensional subspace of ^d that is transverse to , then the fibers of Π_, and the fibers of Π_', agree.

Now, fix y∈∂_∞Γ. We will specialize the observation in the previous paragraph to the case where =ξ^(d-1)(y) and =ξ^(d-m)(y). This yields the following statement, which we record as an observation.

 Let _y :=_ξ^(d-1)(y). If x∈∂_∞Γ∖{y}, then

    Π_x,y:=Π_ξ^(m)(x),ξ^(d-m)(y) : _y→ξ^(m)(z)∩_y

is a projection whose fibers do not depend on x.


Since M is ρ-controlled, M∖{ξ^(1)(y)}⊂_y, so we may define 

    F_x,y:=Π_x,y|_M∖{ξ^(1)(y)}.


 If x∈∂_∞Γ∖{y}, then the map

    F_x,y:M∖{ξ^(1)(y)}→ξ^(m)(x) ∩_y.

is a homeomorphism.


 Lemma <ref> implies that M∖{ξ^(1)(y)} can be viewed as the graph of a map from 

    ξ^(m)(x) ∩_y    to    Π_x,y^-1(ξ^(1)(x))=(ξ^(1)(x)+ξ^(d-m)(y)) ∩_y.
 
In particular, M∖{ξ^(1)(y)} is diffeomorphic to ^m-1. 

The proof of Lemma <ref> requires a basic result from topology.

 If U ⊂^d is open and f: U →^d is continuous injective map, then f(U) is open and f induces a homeomorphism U → f(U). 




We first observe that the map F_x,y is injective. If p_1, p_2 ∈ M ∖{ξ^(1)(y)} and F_x,y(p_1) = F_x,y(p_2), then 

    p_1 + ξ^(d-m)(y) = p_2 + ξ^(d-m)(y),

so p_1 + p_2 + ξ^(d-m)(y) is not direct. The assumption that M is m-hypercovex implies that p_1=p_2. 


Since F_x,y is continuous and injective, we can now apply the invariance of domain theorem to deduce that F_x,y is a homeomorphism onto an open set in I(x,y) in ξ^(m)(x)∩_y. To finish the proof, we now need to show that 

    I(x,y) = ξ^(m)(x) ∩_y.







Suppose γ∈Γ has infinite order, and denote its attracting and repelling fixed points in ∂_∞Γ by γ^+ and  γ^- respectively. Note that ρ(γ)· I(γ^+, γ^-) = I(γ^+,γ^-) and ξ^(1)(γ^+)∈ I(γ^+,γ^-). Since ρ is 1-Anosov, 

    ξ^(m)(γ^+)∩_γ^- = ⋃_n ∈ρ(γ)^-n·

for any open set ⊂ξ^(m)(γ^+) ∩_γ^- containing ξ^(1)(γ^+). Hence

    I(γ^+, γ^-) = ⋃_n ∈ρ(γ)^-n· I(γ^+, γ^-) = ξ^(m)(γ^+) ∩_γ^-.

The density of { (γ^+, γ^-) : γ∈Γ has infinite order} in ∂_∞Γ×∂_∞Γ proves (<ref>).


With Lemma <ref>, we can now proceed to the proof of Theorem <ref>.


Fix y∈∂_∞Γ, and as before, consider the affine chart 

    _y:=(^d) ∖ξ^(d-1)(y).

By working in some particular affine coordinates in the affine chart _y, we will show that Theorem <ref> holds for all x∈∂_∞Γ∖{y}. Since y was chosen arbitrarily, this suffices to prove the theorem.

Let x ∈∂_∞Γ∖{y} and choose affine coordinates _y≃^d-1 so that in these coordinates,


  * ξ^(1)(x) = 0,

  * ξ^(m)(x)∩_y = ^m-1×{0},

  * (ξ^(1)(x)+ ξ^(d-m)(y) )∩_y  = {0}×^d-m.

For any z∈∂_∞Γ sufficiently close to x, there exists a unique affine map 

    A_z : ^m-1×{0}→{0}×^d-m

whose graph is H_z:=ξ^(m)(z)∩_y, i.e.

    H_z = { u+A_z(u) : u ∈^m-1×{0}},

see Figure <ref>. Let L_z: ^m-1×{0}→{0}×^d-m denote the linear part of A_z (in our choice of affine coordinates). Note that the maps z ↦ A_z and z ↦ L_z are continuous.



For any z∈∂_∞Γ∖{y}, Observation <ref> implies that Π_z,y^-1(ξ^(1)(z)) is parallel to Π_x,y^-1(ξ^(1)(x))={0}×^d-m in _y. Thus, as a consequence of Lemma <ref>, there exists a map 

    f_z : H_z →{0}×^d-m

whose graph is ξ^(1)(∂_∞Γ∖{y} ), i.e.

    ξ^(1)(∂_∞Γ∖{y} )= { u + f_z(u) : u ∈ H_z}.

Further, Theorem <ref> implies that for all α satisfying 1≤α <α^m(ρ) and all ξ^(1)(z)+ h ∈ H_z, we have

    f_z(ξ^(1)(z)+h) =  o(h^α).


Now, for any u ∈^m-1×{0} and z∈∂_∞Γ∖{y},

    u + f_x(u) = ( u+ A_z(u) ) + f_z( u+ A_z(u) ).

Also, if u_z:= Π_x, y(ξ^(1)(z)) then u_z+ A_z(u_z) = ξ^(1)(z), which means that f_x(u_z) = A_z(u_z). Thus, for all h∈^m-1×{0},

    f_x(u_z+h) 
       = A_z(u_z+h) + f_z( u_z+h+ A_z(u_z+h) ) 
        = A_z(u_z)+L_z(h) + f_z( u_z+h+ A_z(u_z)+L_z(h) ) 
       = f_x(u_z) + L_z(h) + f_z(ξ^(1)(z)+h+L_z(h) ) 
        = f_x(u_z) +L_z(h)+  o(h+L_z(h)^α). 
        = f_x(u_z) +L_z(h)+  o(h^α).

This proves the theorem.














§ EIGENVALUE DESCRIPTION OF Α^M(Ρ)



For the rest of this section, let ρ:Γ→(d,) be a (1,m)-Anosov representation. Recall that in the introduction, we defined 

    α_m(ρ) := inf_γ∈Γ{logλ_1/λ_m+1(ρ(γ))/logλ_1/λ_m(ρ(γ)): λ_1/λ_m(ρ(γ)) ≠ 1 }.


The main result of this section is the following theorem.

 If ρ is irreducible, then

    α_m(ρ) = α^m(ρ),

where α^m(ρ) is the quantity defined by (<ref>).


The proof of Theorem <ref> will be given in the following two subsections. In the first, we will use general properties of singular values to relate the quantity f(v,t) (the function f was defined by (<ref>)) to the ratios of eigenvalues of ρ(γ) when v^±=γ^±. In the second, we will use a deep result due to Benoist to finish the proof.


Before starting the proof we make several reductions. First, α_m(ρ) and α^m(ρ) are invariant under passing to a finite index subgroup (see Remark <ref>). So by Remark <ref>, we can assume that ρ admits a lift ρ:Γ→_d(). Then, by passing to another finite index subgroup, we may also assume that the Zariski closure of ρ(Γ) is connected. By Proposition <ref> this representation is still irreducible. 





 §.§ Singular values along closed orbits


Let E:=UΓ×^d, and for i=1,2,3, let E_i be the Γ-invariant sub-bundle of E defined by (<ref>). (Recall that the Γ-action on E is given by γ· (v,X)=(γ· v,ρ(γ)· X).) Also, choose a Γ-invariant inner product ⟨·,·⟩ on E so that E=E_1⊕ E_2⊕ E_3 is an orthogonal splitting. We may assume that the norm · used in the definition of α^m(ρ) is given by ·_v=√(⟨·,·⟩_v) for all v∈UΓ. 

For any v,w∈UΓ, let σ_i(v,w) denote the i-th singular value of 

    𝕀=𝕀_v,w:(^d,·_v)→(^d,·_w),
 
and for (v,t)∈UΓ×, denote σ_i(v,t):=σ_i(v,ϕ_t(v)). 








Using this, define the fuction h:UΓ×→ by 

    h(v,t):=logσ_d-m(v,t)/σ_d(v,t)/logσ_d-m+1(v,t)/σ_d(v,t).

The functions h and f (recall that f is defined by (<ref>)) are related by the following lemma.

  For all v∈UΓ and for sufficiently large t, we have

    f(v,t)=h(v,t).

In particular, α^m(ρ)=lim inf_t→∞inf_v∈UΓh(v,t).



Since E=E_1⊕ E_2⊕ E_3 is an orthogonal splitting, Theorem <ref> implies that for all v∈UΓ and for sufficiently large t, 


  * σ_d(v,t)=X_ϕ_t(v) for all X∈ S_1(v),

  * σ_d-m+1(v,t)=sup_X∈ S_2(v)X_ϕ_t(v),

  * σ_d-m(v,t)=inf_X∈ S_3(v)X_ϕ_t(v).

Thus,

    h(v,t)   =   logσ_d-m(v,t)/σ_d(v,t)/logσ_d-m+1(v,t)/σ_d(v,t)
       =   loginf_X∈ S_3(v)X_ϕ_t(v)/sup_X∈ S_1(v)X_ϕ_t(v)/logsup_X∈ S_2(v)X_ϕ_t(v)/inf_X∈ S_1(v)X_ϕ_t(v)
       =   inf_X_i∈ S_i(v){logX_3_ϕ_t(v)/X_1_ϕ_t(v)/logX_2_ϕ_t(v)/X_1_ϕ_t(v)}
       =   f(v,t)

Notice that in the third equality we used the fact that E_1(v) = 1. 


The following observation gives a simple but important bound for ratios of singular values. The proof is a straightforward calculation which we omit.


Suppose that for i=1,…,4, ·_(i) are norms on ^d so that for all X∈^d, 1/A≤X_(1)/X_(2)≤ A and 1/A'≤X_(3)/X_(4)≤ A' for some A,A'>1. Let L:(^d,·_(1))→(^d,·_(3)) and L':(^d,·_(2))→(^d,·_(4)) denote the identity maps. Then

    1/AA'≤σ_i(L)/σ_i(L')≤ AA'.



The next lemma relates the function h to the eigenvalues of ρ(γ).


Let γ∈Γ∖{𝕀} be an infinite order element, and let v=(v^+,v^-,v_0)∈UΓ so that v^±=γ^±. Then

    lim_t→∞h(v,t)=logλ_1/λ_m+1(ρ(γ))/logλ_1/λ_m(ρ(γ)).




Let T denote the period of γ (see Section <ref>). For all k ∈^+ and X ∈^d

    X_ϕ_kT(v)=X_γ^k· v=ρ(γ^-k)· X_v.

Hence, the singular values of the two linear maps 

    𝕀:(^d,·_v)→(^d,·_ϕ_kT(v))    and ρ(γ^-k):(^d,·_v)→(^d,·_v)
 
agree.









It is a straightforward calculation to show that for any inner product ⟨·,·⟩ on ^d and any invertible linear map g:(^d,⟨·,·⟩)→(^d,⟨·,·⟩),

    lim_k→∞1/klogσ_i(g^k)=logλ_i(g).

Thus, we can deduce that

    lim_k→∞σ_i(v,kT)^1/k=lim_k→∞σ_i(ρ(γ^-k))^1/k=|λ_i(ρ(γ^-1))|=1/|λ_d+1-i(ρ(γ))|,

which implies that

    lim_k→∞h(v,kT)   =   lim_k→∞(logσ_d-m(v,kT)/σ_d(v,kT)/logσ_d-m+1(v,kT)/σ_d(v,kT))
       =   logλ_1/λ_m+1(ρ(γ))/logλ_1/λ_m(ρ(γ)).


For any t>0, let k∈^+ so that t∈[kT,(k+1)T). Then Lemma <ref> implies that there are constants C≥ 1 and β≥ 0 so that 

    1/Ce^-β T≤X_ϕ_tv/X_ϕ_kTv≤ Ce^β T

for all t ∈ and X∈^d. This, together with Observation <ref>, implies that for all i=1,…,d,

    1/Ce^-β T≤σ_i(v,kT)/σ_i(v,t)≤ Ce^β T.
 
Also, since ρ is (1,m)-Anosov, we know that

    lim_k→∞logσ_d-m(v,kT)/σ_d(v,kT)=∞=lim_k→∞logσ_d-m+1(v,kT)/σ_d(v,kT).

Hence,

    lim sup_t→∞h(v,t)   =   lim sup_t→∞logσ_d-m(v,t)/σ_d(v,t)/logσ_d-m+1(v,t)/σ_d(v,t)
       ≤   lim sup_k→∞2log C+2β T+logσ_d-m(v,kT)/σ_d(v,kT)/-2log C-2β T+logσ_d-m+1(v,kT)/σ_d(v,kT)
       =   lim_k→∞h(v,kT).

By a similar argument, lim inf_t→∞h(v,t)≥lim_k→∞h(v,kT), so lim_t→∞h(v,t)=lim_k→∞h(v,kT). This, together with (<ref>) implies the lemma.





 §.§ Asymptotic cones and eigenvalues


Recall that λ,μ:_d()→^d respectively denote the Jordan and Cartan projections defined in Section <ref>. For any subgroup G ≤_d(), let _λ(G) ⊂^d denote the smallest closed cone containing λ(G), that is

    _λ(G) := ⋃_g∈ G_>0·λ(g).

Also, let _μ(G) denote the asymptotic cone of μ(G), that is

    _μ(G) := { x ∈^d : ∃g_n∈ G, ∃ t_n ↘ 0,  with lim_n →∞ t_n μ(g_n) =x}.


A deep result of Benoist <cit.> implies the following. 


If G ≤_d() is a connected semisimple real algebraic subgroup which acts irreducibly on ^d and Λ≤ G is a Zariski dense subgroup, then 

    _μ(Λ) = _λ(Λ).



 Notice that for any subgroup Λ⊂_d(), the fact that _λ(Λ) ⊂_μ(Λ) is a consequence of (<ref>).


A proof of Theorem <ref> is given in the appendix. Theorem <ref> can be used to prove the following lemma.

 
For any ϵ > 0 there exists R > 0 such that 

    α_m(ρ)-ϵ < logμ_1/μ_m+1(ρ(γ))/logμ_1/μ_m(ρ(γ))

for all γ∈Γ with μ(ρ(γ))_2≥ R. 


 
By Proposition <ref> and Theorem <ref>, _μ(ρ(Γ)) = _λ(ρ(Γ)). Fix ϵ > 0 and suppose for contradiction that there exists a sequence {γ_n}_n=1^∞⊂Γ such that for all n, μ(ρ(γ_n))_2≥ n and

    α_m(ρ)-ϵ≥logμ_1/μ_m+1(ρ(γ_n))/logμ_1/μ_m(ρ(γ_n)).

By passing to a subsequence we can suppose that 

    1/μ(ρ(γ_n))_2μ(ρ(γ_n)) → x =(x_1,…,x_d)∈_μ(ρ(Γ)) = _λ(ρ(Γ)).

It follows that α_m(ρ)-ϵ≥x_1-x_m+1/x_1-x_m. On the other hand, the definition of α_m(ρ) and _λ(ρ(Γ)), implies that

    α_m(ρ) ≤x_1-x_m+1/x_1-x_m,
 
which is a contradiction. 
 



It is clear from Lemma <ref> and Lemma <ref> that α^m(ρ)≤α_m(ρ). We will now prove α^m(ρ)≥α_m(ρ). Let K⊂UΓ be a compact fundamental domain for the Γ-action on UΓ. Since h(v,t)=h(γ· v,t) for all γ∈Γ and all v∈UΓ, by Lemma <ref>, it is enough to show that 

    α_m(ρ) ≤lim inf_t→∞inf_v∈ Kh(v,t)


Fix C > 1 such that 1/CX_2≤X_v≤ C X_2 for all v∈ K and X ∈^d. By Lemma <ref>, there exists, for every ϵ>0, a positive number R' > 0 such that

    α_m(ρ)-ϵ < logμ_1/μ_m+1(ρ(γ))/logμ_1/μ_m(ρ(γ))

for all γ∈Γ with μ(ρ(γ))_2≥ R'. Since ρ is 1-Anosov and

    logμ_1/μ_2(ρ(γ)) ≤logμ_1/μ_k(ρ(γ))

for k > 1, Theorem <ref> and Corollary <ref> together imply that 

    logμ_1/μ_m+1(ρ(γ)) , logμ_1/μ_m(ρ(γ))≥1/A”μ(ρ(γ))_2-B”

for some A”≥ 1 and B”≥ 0. Hence, there exists R ≥ R' such that 
 
    α_m(ρ)-2ϵ < (logμ_1/μ_m+1(ρ(γ))-4log C)/(logμ_1/μ_m(ρ(γ))+4log C)

for all γ∈Γ with μ(ρ(γ))_2≥ R.

Let d=d_UΓ denote the Γ-invariant metric on UΓ specified in Section <ref>, and let D be the diameter of K. By Corollary <ref> and the fact that any Γ-orbit in UΓ is a quasi-isometry, there exists A ≥ 1 and B ≥ 0 such that 

    1/Aμ(ρ(γ))_2 - B ≤ d(v, γ· v) ≤ A μ(ρ(γ))_2 + B

for all v ∈UΓ. Also, since every ϕ_t-orbit in UΓ is a quasi-isometric embedding, there exists A'≥1 and B'≥0 so that ,

    1/A' |t|- B' ≤ d(v,ϕ_t(v))≤ A' |t| + B'

for all t∈ and v ∈UΓ. 

Fix  t > A'(B'+D + AR + B) and v ∈ K. Let γ∈Γ such that γ^-1·ϕ_t (v) ∈ K. By the definition of C, we see that for any X∈^d,

    1/C≤X_v/X_2, ρ(γ)^-1· X_γ^-1·ϕ_t(v)/ρ(γ)^-1· X_2≤ C.



Since ρ(γ)^-1· X_γ^-1·ϕ_t (v)=X_ϕ_t (v) and  X↦ρ(γ)^-1· X_2 are both norms on ^d, it follows from Proposition <ref> that 

    1/C^21/μ_d+1-i(ρ(γ))=1/C^2μ_i(ρ(γ)^-1)≤σ_i(v,t)≤ C^2μ_i(ρ(γ)^-1)= C^21/μ_d+1-i(ρ(γ)).

Also, d(γ· v,v)≥ d(v,ϕ_t(v))-d( ϕ_t (v), γ· v) ≥1/A't-B'-D, which means

    μ(ρ(γ))_2 ≥1/A(d(γ· v, v) - B) ≥1/A't-B'-D-B/A≥ R.

Hence, 

    h(v,t)    = logσ_d-m(v,t)/σ_d(v,t)/logσ_d-m+1(v,t)/σ_d(v,t)
       ≥(logμ_1/μ_m+1(ρ(γ))-4log C)/(logμ_1/μ_m(ρ(γ))+4log C)
        > α_m(ρ)-2ϵ.

Since v ∈ K and t > A'(B'+D + AR+ B) was arbitrary, 

    α_m(ρ)-2ϵ≤lim inf_t→∞inf_v∈ Kh(v,t).

Then since ϵ > 0 was also arbitrary we see that 

    α_m(ρ) ≤lim inf_t→∞inf_v∈ Kh(v,t).



 
 
  
 

§ OPTIMAL REGULARITY

  
   
   
 In this section we prove Theorem <ref> and the second part of Theorem <ref>. By Example <ref> and <ref>, it is sufficient to prove the following theorem.
 
  Suppose that ρ: Γ→_d() is an irreducible, (1,m)-Anosov representation for some m=2,…,d-1, and suppose that M⊂(^d) is a ρ-controlled, m-hyperconvex, topological (m-1)-dimensional submanifold. Then 

    α_m(ρ)≤sup{α∈ (1,2) : M  is  C^α along ξ^(1)(∂_∞Γ) }
 
with equality if 


  (∗) M ∩(p_1 + p_2 +  ξ^(d-m)(y)) spans p_1 + p_2 +  ξ^(d-m)(y) for all pairwise distinct p_1,p_2,ξ^(1)(y)∈ M.



As mentioned in the introduction (see (2) of Remark <ref>), the condition (∗) is trivial when m=2 and m=d-1. In Section <ref>, we show that when M=ξ^(1)(∂_∞Γ), (∗) is an open condition in (Γ,_d()). Then, in Section <ref>, we prove Theorem <ref>.




 §.§ Stability of hypotheses



To show that (∗) is an open condition when M=ξ^(1)(∂_∞Γ), we  use the following two statements. The first is a standard fact about hyperbolic groups.












  The Γ-action on ∂_∞Γ^(3): = { (x,y,z)  ∈∂_∞Γ^3 : x,y,z  distinct} is  co-compact.


The second is a well-known result about Anosov representations due to Guichard-Wienhard. In the case when Γ is the fundamental group of a negatively curved Riemannian manifold, this result was established by Labourie <cit.>. Before stating the result we need some notation: If ρ: Γ→_d() is a k-Anosov representation, let ξ_ρ^(k):∂_∞Γ→_k(^d) denote the k-flag map of ρ.


<cit.> Let

    _k : = {ρ∈(Γ, _d()) : ρ is k-Anosov}.

Then _k is open, and the map 

    ρ∈_k →ξ^(k)_ρ∈ C( ∂_∞Γ, _k(^d))

is continuous. 
















 Suppose ∂_∞Γ is a topological (m-1)-manifold, and ρ_0: Γ→_d() is a (1,m)-Anosov representation. If ξ^(1)_ρ_0(x) + ξ^(1)_ρ_0(z) +  ξ^(d-m)_ρ_0(y) is a direct sum and

    ξ_ρ_0^(1)(∂_∞Γ) ∩(ξ_ρ_0^(1)(x) + ξ_ρ_0^(1)(z) +  ξ_ρ_0^(d-m)(y))

spans ξ_ρ_0^(1)(x) + ξ_ρ_0^(1)(z) +  ξ_ρ_0^(d-m)(y) for all x,y,z ∈∂_∞Γ distinct, then any sufficiently small deformation of ρ_0 also has these properties. 


 It follows easily from Theorem <ref> and Proposition <ref> that there exists a neighborhood ⊂(Γ, _d()) of ρ_0 with the following property: if ρ∈, then ρ is a (1,m)-Anosov representation and ξ^(1)_ρ(x) + ξ^(1)_ρ(z) +  ξ^(d-m)_ρ(y) is a direct sum for all x,y,z ∈∂_∞Γ distinct.

By Proposition <ref>, it is enough to fix (x_0,y_0,z_0) ∈∂_∞Γ^(3) and prove that there exists a neighborhood U of (x_0,y_0,z_0) in ∂_∞Γ^(3) such that 

    ξ_ρ^(1)(∂_∞Γ) ∩(ξ_ρ^(1)(x) + ξ_ρ^(1)(z) +  ξ_ρ^(d-m)(y))

spans ξ_ρ^(1)(x) + ξ_ρ^(1)(z) +  ξ_ρ^(d-m)(y) for all (x,y,z) ∈ U and any ρ that is a sufficiently small deformation of ρ_0. 

Let e_1,…, e_d be the standard basis of ^d. By changing coordinates we can assume that 

    ξ^(1)_ρ_0(x_0)     = · e_1, 
    ξ^(m)_ρ_0(x_0)     = {e_1,…,e_m},
    ξ^(d-m)_ρ_0(y_0)    = { e_m+1,…, e_d},
    ξ^(d-1)_ρ_0(y_0)    = {e_2,…, e_d},  and
    ξ^(1)_ρ_0(z_0)    = ·(e_1+e_2+e_d).
 
 Using Theorem <ref> and possibly shrinking , we can find a neighborhood U_0 of (x_0,y_0,z_0) such that there exists a continuous map 
 
    (ρ, (x,y,z)) ∈× U_0 → g_ρ,(x,y,z)∈_d()

 such that  g_ρ_0,(x_0,y_0,z_0) =𝕀, 
 
    g_ρ,(x,y,z)·ξ^(1)_ρ(x)     = · e_1, 
    
    g_ρ,(x,y,z)·ξ^(m)_ρ(x)     = {e_1,…,e_m},
    
    g_ρ,(x,y,z)·ξ^(d-m)_ρ(y)    = { e_m+1,…, e_d},
    
    g_ρ,(x,y,z)·ξ^(d-1)_ρ(y)    = {e_2,…, e_d},  and
     
     g_ρ,(x,y,z)·ξ^(1)_ρ(z)    = ·(e_1+e_2+e_d).
 
 
By Theorem <ref>, for each (ρ, (x,y,z)) ∈× U_0, there exists a unique C^1 function f_ρ, (x,y,z) : ^m-1→^d-m such that 
 
    g_ρ,(x,y,z)·ξ^(1)_ρ( ∂_∞Γ∖{y}) = { [1:v:f_ρ, (x,y,z)(v)] : v ∈^m-1}.

Then by Theorem <ref>, the map × U_0→ C(^m-1, ^d-m) given by
 
    (ρ, (x,y,z)) ↦ f_ρ,(x,y,z)

is continuous. Notice that
 
    ξ^(1)_ρ( ∂_∞   Γ∖{y}) ∩(ξ_ρ^(1)(x) + ξ_ρ^(1)(z) +  ξ_ρ^(d-m)(y)) 
        =   g_ρ,(x,y,z)^-1·{ [1:te_2:f_ρ, (x,y,z)(te_2)] : t ∈}.

So if 

    [1:t_1e_2:f_ρ_0, (x_0,y_0,z_0)(t_1e_2)], …, [1:t_d-m+1e_2:f_ρ_0, (x_0,y_0,z_0)(t_d-m+1e_2)]

spans ξ_ρ_0^(1)(x_0) + ξ_ρ_0^(1)(y_0) +  ξ_ρ_0^(d-m)(z_0), then 

    g_ρ,(x,y,z)^-1[1:t_1e_2:f_ρ, (x,y,z)(t_1e_2)], …, g_ρ,(x,y,z)^-1[1:t_d-m+2e_2:f_ρ, (x,y,z)(t_d-m+2e_2)]

spans ξ_ρ^(1)(x) + ξ_ρ^(1)(y) +  ξ_ρ^(d-m)(z) when (ρ,(x,y,z)) is sufficiently close to (ρ_0, (x_0,y_0,z_0)). 



 
 

 §.§ Proof of Theorem <ref>

  
We begin with the following observation. Let e_1,…, e_d denote the standard basis of ^d, and let g∈_d() be a proximal element so that 


  * e_1 spans the eigenspace corresponding to λ_1(g), 

  * e_m lies in the generalized eigenspace corresponding to λ_m(g), and

  * e_m+1 lies in the generalized eigenspace corresponding to λ_m+1(g). 
 
Then observe that

    logλ_1(g)   =lim_n →∞1/nlogg^n· e_1,
    logλ_m (g)   = lim_n →∞1/nlogg^n· e_m, and 
    logλ_m+1(g)    = lim_n →∞1/nlogg^n·∑_j=m+1^d v_j e_j when v_m+1≠ 0.


 From Theorem <ref> and Theorem <ref>, we see that 
 
    α_m(ρ) ≤sup{α∈ (1,2): M  is  C^α along ξ^(1)(∂_∞Γ) }.


To prove the equality case, fix some γ∈Γ with infinite order and let γ^±∈∂_∞Γ denote the attracting and repelling fixed points of γ. We can make a change of basis and assume that ξ^(1)(γ^+) = · e_1, ξ^(m)(γ^+) = { e_1,…, e_m}, ξ^(d-m)(γ^-) = { e_m+1, …, e_d}, and ξ^(d-1)(γ^-) = {e_2,…, e_d}. Now fix a lift g∈_d() of ρ(γ)∈_d(). Then 
 
    g = [ λ    ;   U  ;     V ]

where λ∈, U ∈_m-1(), and V ∈_d-m(). By a further change of basis, we can assume that e_m lies in the generalized eigenspace corresponding to λ_m(g), and e_m+1 lies in the generalized eigenspace corresponding to λ_m+1(g). 
 
By Theorem <ref>, M is C^1 along ξ^(1)(∂_∞Γ), and the tangent space to M at ξ^(1)(γ^+) is ξ^(m)(γ^+). Thus, for any ϵ > 0 sufficiently small there exists some p ∈ M such that 

    p = [e_1 + ϵ e_m + ∑_j=m+1^d y_j e_j ].

Then 

    ξ^(1)(γ^+) + p +  ξ^(d-m)(γ^-) = {e_1,e_m, e_m+1, …, e_d}

and by hypothesis there exists some q ∈ M such that 

    q= [z_1e_1 + z_m e_m + ∑_j=m+1^d z_j e_j ]

and z_m+1≠ 0. The sums q+ξ^(d-1)(γ^-) and ξ^(1)(γ^+) + q +  ξ^(d-m)(γ^-) are both direct, so z_1 ≠ 0≠ z_m. 

Next fix a distance d_ on (^d) induced by a Riemannian metric. Since 

    lim_n→∞ρ(γ^n)· q= ξ^(1)(γ^+),
  
Observation <ref> implies that if 

    X_n:=ρ(γ^n)·(z_1e_1 + z_m e_m + ∑_j=m+1^d z_j e_j),
 
then there is some A≥ 1 so that for sufficiently large n,

    1/AP_3,v(X_n)_2/P_1,v(X_n)_2≤ d_( ρ(γ^n)· q, ξ^(m)(v^+) ) ≤  A P_3,v(X_n)_2/P_1,v(X_n)_2

and 

    1/AP_2,v(X_n)_2/P_1,v(X_n)_2≤ d_( ρ(γ^n)· q, ξ^(1)(v^+) ) ≤  A P_2,v(X_n)_2+P_3,v(X_n)_2/P_1,v(X_n)_2.

It then follows from (<ref>) that

    lim_n →∞1/nlog d_(ρ(γ^n)· q, ξ^(m)(γ^+) )=logλ_m+1/λ_1.

and 

    lim_n →∞1/nlog d_(ρ(γ^n)· q, ξ^(1)(γ^+) )= logλ_m/λ_1


Finally, if M is C^α along ξ^(1)(∂_∞Γ), then there exists C > 0 such that

    d_( ξ^(m)(γ^+), ρ(γ^n)· q ) ≤ C d_( ξ^(1)(γ^+), ρ(γ^n)· q )^α

for all sufficiently large n. By taking the logarithm to both sides, dividing by n, and then taking the limit, we see that

    α≤logλ_1/λ_m+1/logλ_1/λ_m.

Since γ∈Γ was arbitrary, we see that α≤α_m(ρ). 





 




§ NECESSARY CONDITIONS FOR DIFFERENTIABILITY OF Ρ-CONTROLLED SUBSETS



In this section, we establish Theorem <ref>. By Example <ref>, it is sufficient to prove the following theorem.

 Suppose Γ is a hyperbolic group and ρ: Γ→_d() is an irreducible 1-Anosov representation such that ⋀^m ρ: Γ→(⋀^m ^d) is also irreducible. Also, suppose that M is a ρ-controlled, (m-1) dimensional topological manifold. If


  () M is C^α along ξ^(1)(∂_∞Γ) for some α>1,

then


  (†') ρ is m-Anosov and ξ^(1)(x) + p + ξ^(d-m)(y)
is a direct sum for all pairwise distinct ξ^(1)(x),p,ξ^(1)(y) ∈ M.




Note that (†') in Theorem <ref> is a weaker condition then (†) in Theorem <ref>. However, when M=ξ^(1)(∂_∞Γ), then the two conditions are identical.




First, in Section <ref>, we define, for any 1≤ m≤ d and any representation ρ:Γ→_d(), the representation 

    ⋀^mρ:Γ→(⋀^m^d),
 
whose irreducibility appears as a hypothesis in the statements of Theorem <ref>. Then, in Section <ref>, we give an example to demonstrate the necessity of the irreducibility of ⋀^mρ as a hypothesis of Theorem <ref> (and also in Theorem <ref>). Next, we prove Theorem <ref>, whose proof can be broken down into two main steps. In Section <ref>, we use the fact that M is an (m-1)-dimensional topological manifold that is C^α along the 1-limit set of ρ for some α>1, to deduce that logλ_m/λ_m+1(ρ(γ)) grows linearly with respect to the word length of γ. Then, in Section <ref>, we use this to deduce that ρ is m-Anosov, and obtain the required transversality condition. 




 §.§ The wedge representation



Observe that for any m≤ d-1, there is a natural linear _d()-action on ⋀^m^d given by 

    g·(u_1∧…∧ u_m):=(g· u_1)∧…∧ (g· u_m),
 
where u_i∈^d for all i. This defines a representation 

    ι_d,m:_d()→(⋀^m^d),

which in turn defines a representation

    ι_d,m:_d()→(⋀^m^d).

Using this, we may define the m-wedge representation of ρ:Γ→_d() (resp. ρ:Γ→_d()) to be

    ⋀^mρ:=ι_d,m∘ρ:Γ→(⋀^m^d)   (resp. ⋀^mρ:=ι_d,m∘ρ:Γ→(⋀^m^d)).


Also, if ⟨·,·⟩_^d denotes the standard inner product on ^d with orthonormal basis e_1,…,e_d, then we may define a bilinear pairing ⟨·,·⟩_⋀^m^d on ⋀^m^d by first defining 

    ⟨ u_i_1∧…∧ u_i_m,v_j_1∧…∧ v_j_m⟩_⋀^m^d:=∏_σ∈ S_m∏_k=1^m(σ)⟨ u_i_k,v_j_σ(k)⟩

for all u_i_k,v_j_k∈^d, and then extending it linearly to all of ⋀^m^d. Observe that 
    {e_i_1∧…∧ e_i_m:1≤ i_1<…<i_k≤ d}

is an orthonormal basis of ⋀^d, so ⟨·,·⟩_⋀^m^d is an inner product. Using this, we may define the norm ·_⋀^m^d on ⋀^m^d associated to ⟨·,·⟩_⋀^m^d.

Next, let g∈_d(). For all i, let μ_i(ι_d,m(g)) denote the i-th singular value of ι_d,m(g) with respect to the norm ·_⋀^m^d on ⋀^m^d. One can verify from the definition of the _d()-action on ⋀^m^d that for all i, there exists i_1 < i_2 < … < i_m such that

    λ_i(ι_d,m(g))=λ_i_1(g)…λ_i_m(g)    and    μ_i(ι_d,m(g))=μ_i_1(g)…μ_i_m(g).
 
This implies that

    λ_1(ι_d,m(g))=∏_i=1^mλ_i(g)    and    λ_2(ι_d,m(g))=λ_m+1(g)∏_i=1^m-1λ_i(g)

and

    μ_1(ι_d,m(g))=∏_i=1^mμ_i(g)    and    μ_2(ι_d,m(g))=μ_m+1(g)∏_i=1^m-1μ_i(g).

Hence, for any γ∈Γ,

    λ_1/λ_2(⋀^mρ(γ))=λ_m/λ_m+1(ρ(γ))

and

    μ_1/μ_2(⋀^mρ(γ))=μ_m/μ_m+1(ρ(γ)).



























 §.§ Irreducibility of ⋀^mρ


Now, we will discuss an example to demonstrate that the irreducibility of ⋀^mρ is a necessary hypothesis for Theorem <ref> to hold.

The identification of ^3 with ^6 given by 

    (z_1,z_2,z_3)↦((z_1),(z_1),(z_2),(z_2),(z_3),(z_3))

defines an inclusion j:_3()→_6(). The image of j can be characterized as the subgroup of _6() that commutes with the linear endomorphism J on ^6 defined by J(x_1,y_1,x_2,y_2,x_3,y_3):=(-y_1,x_1,-y_2,x_2,-y_3,x_3). Let (2,1)⊂_3() be the subgroup that leaves invariant the bilinear pairing that is represented in the standard basis of ^3 by the matrix

    ([ 0 0 1; 0 1 0; 1 0 0 ]),

and define 

    τ_0:=(ι_6,2∘ j)|_(2,1) : (2,1) →(⋀^2 ^6).
 

Recall that ι_d,m was defined in Section <ref>. Let ⋀^2J be the linear endomorphism on ⋀^2^6 given by 

    (⋀^2J)(u_1∧ u_2)=J(u_1)∧ J(u_2).

Consider the τ_0-invariant subspace 

    E = { v ∈⋀^2 ^6: (⋀^2J)(v) = v},

and let τ : (2,1) →(E) be the representation defined by the τ_0 action on E. Observe that if e_1,…,e_6 is the standard basis for ^6, then 

    f_1:=e_1∧ e_2,      f_4:=e_3∧ e_4,      f_7:=e_3∧ e_5+e_4∧ e_6,
       f_2:=e_2∧ e_3-e_1∧ e_4,      f_5:=e_2∧ e_5-e_1∧ e_6,      f_8:=e_4∧ e_5-e_3∧ e_6,
       f_3:=e_1∧ e_3+e_2∧ e_4,      f_6:=e_1∧ e_5+e_2∧ e_6,      f_9:=e_5∧ e_6

is a basis of E. One can then explicitly verify that τ is irreducible. 

If g ∈(2,1), then there exists some λ≥ 1 such that the (complex) eigenvalues of g ∈(2,1) have absolute values λ, 1, λ^-1. By conjugating g by an appropriate element in h ∈(2,1), we may also assume that the generalized eigenvectors of hgh^-1 corresponding to λ,1,λ^-1 are (1,0,0)^T, (0,1,0)^T and (0,0,1)^T respectively. This implies that the eigenvalues of j(hgh^-1) have absolute values λ,1,λ^-1 (each with multiplicity 2), and the corresponding invariant subspaces are _{e_1,e_2}, _{e_3,e_4} and _{e_5,e_6} respectively. Using the basis of E described above, one can then compute that the eigenvalues of τ(hgh^-1), and hence τ(g), have absolute values

    λ^2, λ, λ, 1,1,1, λ^-1, λ^-1, λ^-2.

In particular, the image of τ lies in (E).

With this set up, we can now give our example. It describes a 1-Anosov, irreducible representation of a co-compact lattice Γ⊂(2,1) to (E), where E is a 9-dimensional vector space. We show that the 1-limit set of this representation is a 3-dimensional, C^∞-submanifold of (E), but ρ is not 4-Anosov.

 Fix a co-compact lattice Γ≤(2,1). Since (2,1) is a rank-one Lie group, it acts transitively and by isometries on a negatively curved Riemannian symmetric space _^2 (the 2-dimensional complex hyperbolic space), whose visual boundary ∂_∞_^2 has the structure of a 3-dimensional smooth sphere. Thus, the inclusion of Γ into (2,1) specifies an identification of ∂_∞Γ=∂_∞_^2.

As (2,1)-spaces, _^2≃(2,1)/B, where B⊂(2,1) is the subgroup of upper triangular matrices. It is straightforward to check that τ(B)⊂(E) lies in P∩ Q, where P⊂(E) is the subgroup that preserves the line spanned by f_1, and Q⊂(E) is the subgroup that preserves _(f_1,…,f_8). In particular, there are smooth, τ-equivariant maps

    ξ^(1):∂_∞Γ=(2,1)/B→(E)/P=(E)

and

    ξ^(8):∂_∞Γ=(2,1)/B→(E)/Q=^*(E)=_8(E).

Furthermore, a result of Guichard-Wienhard <cit.> imply that τ_Γ:Γ→(E) is a 1-Anosov representation whose 1-flag map and 8-flag map are ξ^(1) and ξ^(8) respectively. The eigenvalue calculation above implies that ρ is not 4-Anosov. However, the 1-limit set ξ^(1)(∂_∞Γ) is a 3-dimensional C^∞-submanifold of (E). 







 §.§ Eigenvalue gaps from the C^α property along the 1-limit set
 

Our goal now will be to prove the following proposition. 


Suppose that ρ:Γ→_d() is a 1-Anosov representation. Also, suppose that ⋀^m ρ: Γ→(⋀^m ^d) is irreducible and M is ρ-controlled, (m-1)-dimensional topological manifold that is C^α along the 1-limit set of ρ for some α>1. If γ∈Γ, then 

    λ_m+1/λ_m(ρ(γ)) ≤(λ_2/λ_1(ρ(γ)) )^α-1.

In particular, logλ_m/λ_m+1(ρ(γ)) grows linearly with the word-length of γ.


The proof of Proposition <ref> requires two observations and a lemma.

 Let g ∈_d() be proximal, let g∈_d() be a lift of g, and let g^+ ∈(^d) and g^-∈_d-1(^d) be the attracting fixed point and repelling fixed hyperplane of g respectively. Also, let d_ be a distance on (^d) induced by a Riemannian metric. If p∈(^d) satisfies p≠ g^+ and p∉ g^-, then

    logλ_2/λ_1(g) ≥lim sup_n →∞1/nlog d_(g^n· p, g^+ ).

Moreover, there is a proper subspace V⊂^d so that if p∉ [V], then the above inequality holds as equality.



In the above observation, we identify g^-∈_d-1(^d) with a hyperplane of (^d), which we also denote by g^-.


 
Note that the affine chart _g^- contains both p and g^+. Equip _g^- with an Euclidean metric d_, and let  be the unit ball in _g^- centered at g^+. Since p∉ g^-, g^n· p∈ for sufficiently large n. On , d_ and d_ are bi-Lipschitz, so there is a constant A so that for sufficiently large n,

    1/AP_2(X)_2/P_1(X)_2≤ d_(g^n· p,g^+)≤ AP_2(X)_2/P_1(X)_2,

where X∈^d is a non-zero vector in g^n· p, P_1:^d→ g^+ is the projection with kernel g^-, and P_2:^d→ g^- is the projection with kernel g^+. On the other hand, it is straightforward that 

    logλ_2/λ_1(g) ≥lim sup_n →∞1/nlogP_2(g^n· X)_2/P_1(g^n· X)_2,

thus giving the desired inequality.

To determine V, choose a basis {e_1,…,e_d} for ^d so that g is in real Jordan normal form in this basis. We may assume that e_1 is an eigenvector of g corresponding to λ_1, and there is some l so that e_2,…,e_l spans the invariant corresponding to λ_2. Let V be the span of e_1,e_l+1,…,e_d, and it is easy to see that the inequality (<ref>) holds with equality when v∉ [V]. This proves the observation.
 
 
 

Let g∈_d() be so that λ_1/λ_d(g)>1. Then let ^d=V_1+V_2 be the g-invariant decomposition so that every eigenvalue of g|_V_1 has absolute value λ_1 and every eignevalue of g|_V_2 has absolute value strictly less than λ_1. Suppose that (V_1)>1 and g has an invariant line l∈[V_1]. Then for all p∈(^d)∖[l+V_2], 

    0  = lim_n →∞1/nlog d_(g^n· p, l),

where d_ is a distance on (^d) induced by a Riemannian metric.



First, note that since d_ has bounded diameter, 

    lim sup_n →∞1/nlog d_(g^n· p, l) ≤ 0.

Now assume for a contradiction that Equation (<ref>) does not hold for some p∈(^d)∖[l+V_2]. Then by taking a subsequence, we may assume that 

    lim_k →∞1/n_klog d_(g^n_k· p, l) < 0.

Notice that this implies that g^n_k· p → l as k→∞. 

Using the real Jordan normal form of g, we can decompose V_1 = ⊕_j=1^r V_1,j where


  * V_1,1 = l,
 
  * for 2 ≤ j ≤ r 
 
 
  * V_1,j is either one or two dimensional,
 
  * there exists a linear transformation L_j : V_1,j→ V_1,j such that

    g · Y ∈ L_j· Y + V_1,j-1

for all Y∈ V_1,j,

  * L_j · Y_2 = λ_1(g)Y_2 for all Y ∈ V_1,j. 
 









Also, let P_1,j : ^d → V_1,j and P_2 : ^d → V_2 be the projections relative to the decomposition ^d = V_1,1⊕…⊕ V_1,r⊕ V_2. 

Since g^n_k· p converges to l, (<ref>) in the first part of the proof of Observation <ref> implies that there exists A ≥ 1 such that 

    1/A( ∑_j=1^ℓP_1,j(g^n_k· X)_2 + P_2(g^n_k· X)_2/P_1,1(g^n_k· X)_2) ≤ d_(g^n_k· p, Φ(ξ^(1)(γ^+)))

for all non-zero X∈ p and all sufficiently large k. Since X ∉ l+V_2, there exists 2 ≤ j_0 ≤ r such that P_1,j_0(X) ≠ 0. By increasing j_0 if necessary, we can also assume that P_1,j(X) = 0 for j_0 < j ≤ r. This implies that

    P_1,j_0(g^n· X)_2 = λ_1(g)^n P_1,j_0(X)_2.

Further, by increasing A ≥ 1 if necessary, we can assume that

    P_1,1(g^n· X)_2≤ Ag^n· X_2≤ Ag^n_opX_2

for all n ≥ 0. 

Then by Equations (<ref>), (<ref>), and (<ref>),

    lim_k →∞1/n_klog d_(g^n_k· p, l)    ≥lim sup_k →∞1/n_klogP_1,j_0(g^n_k· X)_2/AP_1,1(g^n_k· X)_2
       ≥log(λ_1(g))+lim sup_k →∞1/n_klogP_1,j_0(X)_2/A^2g^n_k_opX_2
       ≥log(λ_1(g))-lim inf_k →∞1/n_klogg^n_k_op.

But Gelfand's formula states that 

    lim_n →∞1/nlogg^n_op = log(λ_1(g)),

so by (<ref>),

    0     >  lim_k →∞1/n_klog d_(g^n_k· p, Φ(ξ^(1)(γ^+))) ≥ 0.

and we have a contradiction. 



Next, suppose ρ:Γ→_d() and M⊂(^d) satisfy the hypothesis of Proposition <ref>. Define the map

    F_d,m:_m(^d)→(⋀^m^d)    by    F_d,m:V↦[⋀_i=1^mv_i],

where v_1,…,v_m is a basis of V. 




Note that F_d,m is well defined, smooth, and ι_d,m-equivariant. Since M is differentiable along the 1-limit set ξ^(1)(∂_∞Γ) of ρ, we can define Φ:ξ^(1)(∂_∞Γ)→_m(^d) to be the map that associates to every point in ξ^(1)(∂_∞Γ) its tangent space. Then define 

    Φ:= F_d,m∘Φ: ξ^(1)(∂_∞Γ) →(⋀^m^d).



Fix distances d_1 on (^d) and d_2 on (⋀^m^d) which are induced by Riemannian metrics. Since M is C^α along ξ^(1)(∂_∞Γ) and F_d,m is smooth, a calculation shows that there is some C≥ 1 so that 

    d_2(Φ(q_1), Φ(q_2)) ≤ C d_1(q_1, q_2)^α-1

for all q_1, q_2 ∈ξ^(1)(∂_∞Γ).


 
Suppose that ρ:Γ→_d() is a 1-Anosov representation. Also, suppose that ⋀^m ρ: Γ→(⋀^m ^d) is irreducible and M is ρ-controlled, (m-1)-dimensional topological manifold that is C^α along the 1-limit set of ρ for some α>1. If γ∈Γ has infinite order, then g:=(⋀^mρ)(γ) is proximal and Φ(ξ^(1)(γ^+))∈(⋀^m^d) is the attracting fixed point of g. 



 Let h∈_d() be a lift of ρ(γ), g:=⋀^mh, and λ_i = λ_i(h) for i=1,…,d. Then by (<ref>), λ_1(g)=λ_1⋯λ_m. Thus it is equivalent to prove that g is proximal and Φ(ξ^(1)(γ^+)) is the eigenline of g whose eigenvalue has absolute value λ_1 ⋯λ_m. 


We first show that Φ(ξ^(1)(γ^+)) is an eigenline of g whose eigenvalue has absolute value λ_1 ⋯λ_m. Let {n_k}_k=1^∞ be an increasing sequence of integers such that 

    1/g^n_kg^n_k

converges to some T ∈(⋀^m^d). Also, let ⋀^m ^d = V_1 ⊕ V_2 be a g-invariant decomposition of ⋀^m ^d, where every eigenvalue of g|_V_1 has absolute value λ_1 ⋯λ_m and every eignevalue of g|_V_2 has absolute value strictly less than λ_1 ⋯λ_m. Observe that the image of T is contained in V_1. Since 

    g·Φ(ξ^(1)(γ^+))=Φ(ξ^(1)(γ·γ^+))=Φ(ξ^(1)(γ^+)),
 
Φ(ξ^(1)(γ^+)) is an eigenline of g. Thus, we only need to show that Φ(ξ^(1)(γ^+)) is contained in the image of T. 

We claim that the image of T is exactly Φ(ξ^(1)(γ^+)). Notice that if p=[v] ∈(⋀^m^d) and v ∉ T then 

    [T(v)] = lim_k →∞ g^n_k· p

(recall that [v] denotes the projective line containing v). Further, since ⋀^mρ : Γ→(⋀^m^d) is irreducible, the set {Φ(x) : x ∈ξ^(1)(∂_∞Γ)} spans ⋀^m^d. Thus there exists x_1, …, x_N ∈∂_∞Γ such that 

    Φ(ξ^(1)(x_1)), …, Φ(ξ^(1)(x_N))

span ⋀^m^d. By perturbing and relabelling the x_i (if necessary) we can also assume that γ^- ∉{x_1, …, x_N}, and that there exists 1 ≤ℓ≤ N such that

    Φ(ξ^(1)(x_1)) + … + Φ(ξ^(1)(x_ℓ)) +  T = ⋀^m^d

is a direct sum. For 1 ≤ i ≤ℓ,

    T ( Φ(ξ^(1)(x_i)) ) = lim_k →∞ g^n_kΦ(ξ^(1)(x_i)) =  lim_k →∞Φ( ξ( γ^n_k· x_i)) = Φ(ξ^(1)(γ^+)),

so the image of T is Φ(ξ^(1)(γ^+)). Thus, Φ(ξ^(1)(γ^+)) is an eigenline of g whose eigenvalue has absolute value λ_1 ⋯λ_m.

We next argue that g is proximal, or equivalently that V_1 = 1. Let d_1 and d_2 be as defined in Remark <ref>. Let

    W := V_2 + Φ(ξ^(1)(γ^+)),

and suppose for contradiction that V_1 > 1.  This implies that W⊂^d is a proper subspace. By Observation <ref>,

    0  = lim_n →∞1/nlog d_2(g^n· p, Φ(ξ^(1)(γ^+)))

when p ∈(⋀^m^d) ∖ [W].

Since {Φ(x) : x ∈ξ^(1)(∂_∞Γ)} spans ⋀^m^d, there exists x ∈∂_∞Γ such that Φ(ξ^(1)(x)) ∉ [W]. By perturbing x (if necessary) we can assume that x ≠γ^-. Then 

    lim_n→∞ρ(γ)^n·ξ^(1)(x)=ξ^(1)(γ^+)  and lim_n→∞g^n·Φ(ξ^(1)(x))=Φ(ξ^(1)(γ^+)).

So, by Observation <ref>,

    0 > logλ_2/λ_1   ≥lim sup_n →∞1/nlog d_1(ρ(γ)^n·ξ^(1)(x), ξ^(1)(γ^+)) 
    
       ≥lim sup_n →∞1/(α-1) nlog d_2( Φ(ξ^(1)(γ^n· x)),Φ( ξ^(1)(γ^+))) 
        = lim sup_n →∞1/(α-1) nlog d_2( g^n·Φ(ξ^(1)(x)),Φ( ξ^(1)(γ^+))) =0,

where the last inequality is Remark <ref>. This is a contradiction, so g is proximal.



 




 






 
 
 
 





 



















 
 


























  Fix some γ∈Γ. If γ has finite order, then 

    λ_i/λ_j(ρ(γ)) = 1

for all 1 ≤ i,j ≤ d and there is nothing to prove. So suppose that γ has infinite order and let γ^+ ∈∂_∞Γ be the attracting fixed point of γ. By Lemma <ref>, g:=⋀^m ρ(γ) is proximal and Φ(ξ^(1)(γ^+)) = g^+.

By (<ref>) and Observation <ref>, there exists a proper subspace V ⊂⋀^m ^d such that:
if p∈(⋀^m ^d) ∖ [V] and p is not in the repelling hyperplane of g, then

    logλ_m+1/λ_m(ρ(γ)) =  lim_n →∞1/nlog d_2( g^n· p, Φ(ξ^(1)(γ^+)) ).

Since {Φ(q) : q ∈ξ^(1)(∂_∞Γ)} spans ⋀^m ^d we can find x ∈∂_∞Γ such that Φ(ξ^(1)(x)) ∉ [V]. By perturbing x if necessary, we can also assume that x ≠γ^-. Then 

    lim_n →∞ g^n·Φ(ξ^(1)(x)) = Φ(ξ^(1)(γ^+)),

so Φ(ξ^(1)(x)) does not lie in the repelling hyperplane of g. Thus, by Observation <ref> and Remark <ref>

    logλ_m+1/λ_m(ρ(γ))     ≤ (α-1) lim_n →∞1/nlog d_1( ρ(γ)^n·ξ^(1)(x), ξ^(1)(γ^+) )
       ≤ (α-1) logλ_2/λ_1(ρ(γ)).






 §.§ Anosovness from eigenvalue gaps


To prove Theorem <ref>, we will use the following proposition.


Suppose that ρ:Γ→_d() is an irreducible 1-Anosov representation and M is a ρ-controlled subset which is C^1 along ξ^(1)(∂_∞Γ). Suppose also that for all γ∈Γ with infinite order,

    λ_m+1/λ_m(ρ(γ)) ≤(λ_2/λ_1(ρ(γ)) )^α-1,

g:=(⋀^mρ)(γ) is proximal, and Φ(ξ^(1)(γ^+))∈(⋀^m^d) is the attracting fixed point of g. Then ρ is m-Anosov, and ξ^(1)(x) + p +  ξ^(d-m)(y) is a direct sum for all pairwise distinct ξ^(1)(x), p,  ξ^(1)(y) ∈ M.


Assuming Proposition <ref>, we can prove Theorem <ref>.


If Condition () holds, then Proposition <ref>, Lemma <ref> and Proposition <ref> imply Condition (†').


We start the proof of Proposition <ref> by making some initial reductions. First, notice that the reduction made in Remark <ref> does not impact the hypothesis or conclusion of the Proposition. So we may assume that there exists a lift ρ:Γ→_d() of ρ. Second, notice that passing to a finite index subgroup also does not impact the hypotheses or conclusion of the Proposition (see Proposition <ref>). Hence we may also assume that the Zariski closure of ρ(Γ) is connected. 

The proof of Proposition <ref> requires the following lemma.

 





If 1<β < α, then there exists C > 0 such that 

    logμ_m+1/μ_m(ρ(γ)) ≤ (β-1) logμ_2/μ_1(ρ(γ)) +C

for all γ∈Γ. 


 
Let _μ=_μ(ρ(Γ)) and _λ=_λ(ρ(Γ)) be the cones defined in Section <ref>. Then _μ = _λ by Proposition <ref> and Theorem <ref>. By hypothesis, if x=(x_1, …, x_d) ∈_λ, then 

    x_m+1-x_m ≤ (α-1) (x_2 - x_1).

Further, since ρ is 1-Anosov, x_2 - x_1 <0 for all x=(x_1, …, x_d) ∈_λ.

Next, we will prove that there exists R > 0 with the following property: if μ(ρ(γ))_2 ≥ R, then 
 
    logμ_m+1/μ_m(ρ(γ)) ≤ (β-1) logμ_2/μ_1(ρ(γ)).

Suppose for contradiction that there exists {γ_n}_n=1^∞⊂Γ with μ(ρ(γ_n))_2 →∞ and
 
    logμ_m+1/μ_m(ρ(γ_n)) > (β-1) logμ_2/μ_1(ρ(γ_n)).

By passing to a subsequence, we can assume that 

    1/μ(ρ(γ_n))_2μ(ρ(γ_n)) → x=(x_1, …, x_d).

Then x ∈_μ  = _λ and

    x_m+1-x_m ≥ (β-1)( x_2 -x_1) > (α-1) (x_2 - x_1)

so we have a contradiction. 

The lemma then follows from the observation that since ρ is 1-Anosov, the set {γ∈Γ : μ(ρ(γ))_2 < R} is finite.









 
Since ρ is 1-Anosov, Theorem <ref> implies that there exists C_0, c_0  > 0 such that 

    μ_2/μ_1(ρ(γ)) ≤ C_0 e^-c_0 d_S(γ, 𝕀)

for all γ∈Γ. Then by Lemma <ref>, there exists C,c > 0 such that 

    μ_m+1/μ_m(ρ(γ)) ≤ C e^-c d_S(γ, 𝕀)

for all γ∈Γ. Thus, Theorem <ref>  implies that ρ is m-Anosov. 

To finish the proof, we will now show that ξ^(1)(x) + p + ξ^(d-m)(y)
is a direct sum for all pairwise distinct ξ^(1)(x),p,ξ^(1)(y) ∈ M. Let 

    M: = { (ξ^(1)(x),p,ξ^(1)(y)) ∈ M^3 : ξ^(1)(x),p,ξ^(1)(y)  are pairwise distinct}
 

and let 

    :={(ξ^(1)(x),p,ξ^(1)(y))∈M:ξ^(1)(x) + p+ ξ^(d-m)(y) is a direct sum}.

Notice that  is open and Γ-invariant. Also, recall that Γ acts co-compactly on UΓ, the flow space associated to Γ described in Section <ref>. Hence, there exists a compact set K ⊂UΓ such that Γ· K = UΓ. Then define

    0 < ϵ := min{ d_(ξ^(1)(v^+), ξ^(1)(v^-)) : v ∈ K},

where d_ is a distance on (^d) induced by a Riemannian metric. 

Given two proper subspaces V, W ⊂^d define 

    d(V,W):= min{v-w_2 : v ∈ V, w ∈ W, v_2=w_2=1}.


Note that {(x,y)∈∂_∞Γ^2:d_(ξ^(1)(x), ξ^(1)(y)) ≥ϵ} is compact. Since ξ^(m)(x) + ξ^(d-m)(y)=^d when x ≠ y, this implies that there exists θ_0 > 0 with the following property: if x,y ∈∂_∞Γ and d_(ξ^(1)(x), ξ^(1)(y)) ≥ϵ, then 

    d(ξ^(m)(x),ξ^(d-m)(y))≥θ_0.

 
Also, by hypothesis, if γ∈Γ has infinite order and γ^+∈∂_∞Γ is the attracting fixed point of γ, then 

    ξ^(m)(γ^+) = T_ξ^(1)(x) M.

 So by the continuity of ξ^(m) and the density of {γ^+: γ∈Γ has infinite order} in ∂_∞Γ we see that ξ^(m)(x) = T_ξ^(1)(x) M for all x ∈∂_∞Γ. Thus, the compactness of M implies that there exists δ > 0 with the following property: if ξ^(1)(x),p ∈ M and d_(ξ^(1)(x), p) ≤δ, then

    d( ξ^(1)(x)+p,ξ^(m)(x)  ) < θ_0/2.

Using this, define

    :={(ξ^(1)(x),p,ξ^(1)(y))∈M:d_(ξ^(1)(x),ξ^(1)(y)) ≥ϵ and  d_(ξ^(1)(x), p) ≤δ}.
  
We claim that ⊂. Indeed, by the definition of θ_0 and δ,  if (ξ^(1)(x),p,ξ^(1)(y)) ∈ then 

    d( ξ^(1)(x)+p,ξ^(d-m)(x)  )  > θ_0/2.

This implies that ξ^(1)(x) +p+ ξ^(d-m)(y) is direct, so (ξ^(1)(x),p,ξ^(1)(y)) ∈. 

















Next, let P(M) ⊂UΓ× M be the set defined by (<ref>), and recall that ϕ_t denotes the geodesic flow on UΓ. Note that there exists T ≥ 0 such that if v ∈ K, t ≥ T, and (ϕ_t(v),p) ∈ P(M), then (ξ^(1)(v^+),p,ξ^(1)(v^-)) ∈⊂. Now, choose any (ξ^(1)(x),p,ξ^(1)(y)) ∈M. From the definition of P(M), there exists v ∈UΓ such that v^+ = x, v^- = y, and (v,p) ∈ P(M). Further, there exists γ∈Γ such that w := γ·ϕ_-T(v) ∈ K. Since the Γ-action on UΓ commutes with the geodesic flow,

    (ϕ_T(w),ρ(γ)· p) = γ· ( v, p) ∈ P(M)

and so γ· (ξ^(1)(x),p,ξ^(1)(y))=(ξ^(1)(w^+),ρ(γ)· p,ξ^(1)(w^-))∈, which means (ξ^(1)(x),p,ξ^(1)(y)) ∈. Thus, =M. 







§ NECESSARY CONDITIONS FOR DIFFERENTIABILITY OF 1-DIMENSIONAL Ρ-CONTROLLED SUBSETS




In this section we prove Theorem <ref>. Again by Example <ref>, it is sufficient to prove the following theorem.

  Suppose Γ is a hyperbolic group and ρ: Γ→_d() is an irreducible 1-Anosov representation. Also, suppose that M is a ρ-controlled, topological circle. If


  () M is C^α along ξ^(1)(∂_∞Γ) for some α>1,

then


  (†') ρ is m-Anosov and ξ^(1)(x) + p + ξ^(d-m)(y)
is a direct sum for all pairwise distinct ξ^(1)(x),p,ξ^(1)(y) ∈ M.



Before proving Theorem <ref>, we give an example to demonstrate that the irreducibility of ρ is a necessary hypothesis in Theorem <ref> (and also in Theorem <ref>) to hold.




 §.§ Irreducibility of ρ


For d ∈, let τ_d : _2() →_d() be the standard irreducible representation, which is constructed as follows. First, identify ^d with the space of homogeneous degree d-1 polynomials in two variables with real coefficients by 

    (a_1,…,a_d)↦∑_i=1^da_iX^d-iY^i-1.
 
Using this, we may define an _2()-action on ^d by 

    ([ a b; c d ]) · P(X,Y) = P( ([ a b; c d ])^-1·(X,Y)).

It is easy to check that this _2()-action is linear. Thus, it has an associated linear representation τ_d : _2() →_d(), which descends to a representation τ_d:_2()→_d().

One can verify that if λ, λ^-1 are the absolute value of the eigenvalues of g∈_2^±(), then 

    λ^d-1, λ^d-3, …, λ^-(d-1)

are the absolute values of the eigenvalues of τ_d(g). Further, if B_k⊂_k() denotes the subgroup of upper triangular matrices, then τ_d(B_2)⊂ B_d. In particular, τ_d induces a smooth map 

    Ψ_d:(^2)≃_2()/B_2→_d()/B_d.
 
Since _d()/B_d is the space of complete flags in ^d, there is an obvious smooth projection p_m:_d()/B_d→_m(^d) for each m=1,…,d-1. Using this, define Ψ_d,m:=p_m∘Ψ_d:(^2)→_m(^d). It is clear that Ψ_d,m is τ_d-equivariant.

Next, observe that the subgroup (_d()×_d+2())⊂_2d+2() preserves both the subspaces (^d) and (^d+2) of (^2d+2) induced respectively by the obvious inclusions of ^d≃^d⊕{0} and ^d+2≃{0}⊕^d+2 into ^d⊕^d+2≃^2d+2. Similarly, the subspaces _d-1(^d) and _d+1(^d+2) of _2d+1(^2d+2) that are respectively defined by the inclusions V↦ V⊕^d+2 and U↦^d ⊕ U are (_d()×_d+2())-invariant.
In particular, the representation

    τ_d⊕τ_d+2:_2()→_d()×_d+2()⊂_2d+2()

defines the representation τ_d⊕τ_d+2:_2()→_2d+2() by projectivizing, and the maps

    Ψ_d,1:(^2)→(^d)⊂(^2d+2),
       Ψ_d+2,1:(^2)→(^d+2)⊂(^2d+2),
       Ψ_d,d-1:(^2)→_d-1(^d)⊂_2d+1(^2d+2),
       Ψ_d+2,d+1:(^2)→_d+1(^d+2)⊂_2d+1(^2d+2).

are smooth and τ_d⊕τ_d+2-equivariant.

One can check that (Ψ_d,1,Ψ_d,d-1) and (Ψ_d+2,1,Ψ_d+2,d+1) are transverse pairs of maps. It also follows from (<ref>) that for any g∈_2(), τ_d⊕τ_d+2(g) is proximal. However, the attracting eigenline and repelling hyperplane 
of τ_d⊕τ_d+2(g) lies in the image of Ψ_d+2,1 and Ψ_d+2,d+1 respectively, so only the pair of maps (Ψ_d+2,1,Ψ_d+2,d+1) is dynamics preserving.
 
Fix a co-compact lattice Γ≤_2(). The inclusion of Γ into _2() induces an identification ∂_∞Γ≃(^2), and thus equips ∂_∞Γ with the structure of a smooth manifold. Consider the representation 

    ρ :=τ_d ⊕τ_d+2|_Γ:Γ→(^d⊕^d+2).
 
By the discussion above, 

    Ψ_d+2,1:∂_∞Γ→(^2d+2)    and    Ψ_d+2,d+1:∂_∞Γ→_2d+1(^2d+2)

is a pair of smooth, dynamics preserving, ρ-equivariant, transverse maps. Thus, one deduces from (<ref>) that ρ is 1-Anosov, but it is not 2-Anosov because 
    λ_2/λ_3(ρ(γ)) =1

for any γ∈Γ.  
However, since Ψ_d+2,1 is a smooth map, the 1-limit set of ρ is a 1-dimensional, C^∞-submanifold of (^2d+2). This shows that the conclusion of Theorem <ref> does not hold if we do not assume the irreducibility hypothesis of Theorem <ref>.
 




 §.§ Proof of Theorem <ref>


Lemma <ref> and Lemma <ref> stated below are respectively the analogs of Lemma <ref> and Proposition <ref> in the case when M is a 1-dimensional topological manifold. With these two lemmas, we can replicate the proof of Theorem <ref> to prove Theorem <ref>


 In Lemma <ref> and Proposition <ref>, we assumed that ⋀^2ρ is irreducible, but in Lemma <ref> and Lemma <ref> we assume that ρ is irreducible.


 
Suppose that ρ:Γ→_d() is a 1-Anosov representation. Also, suppose that M is ρ-controlled, topological circle that is C^α along the 1-limit set of ρ for some α>1, and let Φ:M→(⋀^2^d) be as defined in (<ref>). If γ∈Γ has infinite order, then ⋀^2ρ(γ) is proximal and Φ(ξ^(1)(γ^+))∈(⋀^2^d) is the attracting fixed point of ⋀^2ρ(γ). 


 Define Ψ: ξ^(1)(∂_∞Γ)×ξ^(1)(∂_∞Γ)→_2(^d) by letting Ψ(p,q) be the projective line containing p,q when p ≠ q and letting Ψ(p,p) be the projective line tangent to M at p. Then define 

    Ψ:=F_d,2∘Ψ: ξ^(1)(∂_∞Γ)×ξ^(1)(∂_∞Γ)→(⋀^2^d),
 
where F_d,2 is defined by (<ref>). Observe that Ψ is continuous and Φ(p)=Ψ(p,p) for all p∈ξ^(1)(∂_∞Γ).

Fix distances d_1 on (^d) and d_2 on (⋀^2^d) that are induced by Riemannian metrics. Since M is C^α along the 1-limit set of ρ for some α>1, there exists C > 0 such that 

    d_2( Ψ(p,p), Ψ(p, q) ) ≤ Cd_1(p, q)^α-1

for all p,q ∈ξ^(1)(∂_∞Γ). Also, since ρ is irreducible, the elements of ξ^(1)(∂_∞Γ) span ^d, so 

    Ψ( ξ^(1)(∂_∞Γ) ×ξ^(1)(∂_∞Γ))
 
spans ⋀^2 ^d. Now the rest of the proof closely follows the proof of Lemma <ref>, but we use Ψ(ξ^(1)(γ^+),ξ^(1)(γ^+)) in place of Φ(ξ^(1)(γ^+)) and Ψ(ξ^(1)(x),ξ^(1)(γ^+)) in place of Φ(ξ^(1)(x)).



In the case when M is a topological (m-1)-dimensional manifold with m>2, it is not true that ξ^(1)(x_1)+…+ξ^(1)(x_m) converges to ξ^(m)(x) as x_i→ x, so the direct analog of (<ref>) cannot hold. As such, we need the additional assumption that ⋀^mρ(γ) is irreducible in Theorem <ref>.



Suppose that ρ:Γ→_d() is a 1-Anosov representation. Also, suppose that M is ρ-controlled, topological circle that is C^α along the 1-limit set of ρ for some α>1. If γ∈Γ, then 

    λ_m+1/λ_m(ρ(γ)) ≤(λ_2/λ_1(ρ(γ)) )^α-1.



 Use the same argument as we did in the proof of Proposition <ref>, but with Ψ(ξ^(1)(γ^+),ξ^(1)(γ^+)) and Ψ(ξ^(1)(x),ξ^(1)(γ^+)) in place of Φ(ξ^(1)(γ^+)) and Φ(ξ^(1)(x)) respectively, and Lemma <ref> in place of  Lemma <ref>.



Use the same proof as Theorem <ref>, but replace Lemma <ref> and Proposition <ref> by Lemma <ref> and Lemma <ref> respectively.





§ _D()-HITCHIN REPRESENTATIONS



 In this section, let Γ:=π_1(Σ), where Σ is a closed, orientable, connected hyperbolic surface of genus at least 2. 
 
 
 A _d()-Hitchin representation is a continuous deformation (in (Γ,_d())) of τ_d∘ j, where j:Γ→_2() is a Fuchsian representation, and τ_d:_2()→_d() is the representation defined in Section <ref>.
 


The goal of this section is to show that if ρ is a _d()-Hitchin representation, then for all k=1,…,d-1, ⋀^kρ:Γ→(⋀^k^d) satisfies the hypothesis of Theorem <ref> (see Example <ref>). The following proposition is a straightforward consequence of Labourie's deep work on the Hitchin component <cit.> and has also been observed by Pozzetti-Sambarino-Wienhard <cit.>.

 Let ρ be a _d()-Hitchin representation and D := (⋀^k ^d).  If k ∈{1,…, d-1}, then ⋀^k ρ : Γ→(⋀^k ^d) is (1,2)-Anosov, and its 1-flag map ζ^(1) and (D-2)-flag map ζ^(D-2) satisfy the property that

    ζ^(1)(x)+ζ^(1)(y) + ζ^(D-2)(z),

is a direct sum for all x,y,z ∈∂_∞Γ distinct. 


For the rest of the section fix some _d()-Hitchin representation ρ and some finite generating set S of Γ.




 §.§ Preliminaries


Before proving the proposition, we recall some results of Labourie. By Theorem 4.1 and Proposition 3.2 in <cit.>,
 
 
  *  ρ is k-Anosov for every 1 ≤ k ≤ d. Denote the k-flag map of ρ by ξ^(k).
 
  *  If x,y,z ∈∂_∞Γ are distinct, k_1,k_2,k_3 ≥ 0, and k_1+k_2+k_3 =d, then 
 
    ξ^(k_1)(x) + ξ^(k_2)(y) + ξ^(k_3)(z) = ^d

 is a direct sum.
  
  *  If x,y,z ∈∂_∞Γ are distinct and 0≤ k < d-2, then
 
    ξ^(k+1)(y) + ξ^(d-k-2)(x) + (ξ^(k+1)(z) ∩ξ^(d-k)(x) )= ^d

 is a direct sum.
   
  * ρ admits a lift ρ:Γ→_d() whose image lies in _d().
 
  *  If γ∈Γ∖{1}, then the absolute values of the eigenvalues of ρ(γ) satisfy
 
    λ_1(ρ(γ)) > … > λ_d(ρ(γ)).

 
  *  If γ∈Γ∖{1}, then ξ^(k)(γ^+) is the span of the eigenspaces of ρ(γ) corresponding to the eigenvalues 
 
    λ_1(ρ(γ)), …, λ_k(ρ(γ)).
 





 §.§ Proof of Proposition <ref>



Since ρ is k-Anosov, Theorem <ref> implies that there exists C,c>0 such that 

    logμ_k/μ_k+1(ρ(γ)) ≥ C d_S(1,γ) -c

for all γ∈Γ and 1 ≤ k ≤ d. 

 ⋀^k ρ is (1,2)-Anosov. 


By Theorem <ref> it is enough to prove that there exists A,a>0 such that 

    logμ_1/μ_2(⋀^kρ(γ)) ≥ A d_S(1,γ) -a

 and

    logμ_2/μ_3(⋀^k ρ(γ)) ≥ A d_S(1,γ) -a

for all γ∈Γ.

Fix γ∈Γ and let g∈_d() be a lift of ρ(γ). Then let

    σ_1 ≥…≥σ_d

denote the singular values of g (in the Euclidean norm on ^d), and let 

    χ_1 ≥…≥χ_D

denote the singular values of ⋀^k g (in the induced norm on ⋀^k^d).

Recall, that Equation (<ref>) says that 

    χ_1 = σ_1 ⋯σ_k  and χ_2 = σ_1 ⋯σ_k-1σ_k+1.

Hence 

    logμ_1/μ_2(⋀^kρ(γ)) = logχ_1/χ_2 = logσ_k/σ_k+1≥ C d_S(1,γ) -c.


To verify the other inequality, pick 1 ≤ i_1 < … <i_k ≤ d such that

    χ_3 = σ_i_1⋯σ_i_k.

We consider two cases based on the value of i_k-1. 



Case 1: Suppose i_k-1 = k-1. Then i_j = j for j ≤ k-1 and i_k ≥ k. Since 

    (i_1,…, i_k) ∉{ (1,…, k), (1,…,k-1,k+1)}

we must have i_k ≥ k+2. So

    logχ_2/χ_3 = log(σ_1/σ_i_1⋯σ_k-1/σ_i_k-1σ_k+1/σ_i_k) = logσ_k+1/σ_i_k≥logσ_k+1/σ_k+2≥ C ℓ_S(γ)-c.


Case 2: Suppose i_k-1≥ k. Then i_k ≥ k+1 and i_j ≥ j for all j so
 
    logχ_2/χ_3 = log(σ_1/σ_i_1⋯σ_k-2/σ_i_k-2σ_k-1/σ_i_k-1σ_k+1/σ_i_k) ≥logσ_k-1/σ_i_k-1≥logσ_k-1/σ_k≥ C ℓ_S(γ)-c.

In either case 

    logμ_2/μ_3(⋀^k ρ(γ))=logχ_2/χ_3≥ C ℓ_S(γ) -c.

Then since γ∈Γ was arbitrary, we see that ⋀^k ρ is (1,2)-Anosov. 


Given subspaces V_1,…,V_k⊂^d, we will let V_1∧…∧ V_k denote the subspace of ⋀^k^d that is spanned by {X_1∧…∧ X_k:X_i∈ V_i}. For ℓ∈{1,2,D-2, D-1} define maps 

    ζ^(ℓ) : ∂_∞Γ→_ℓ(⋀^k ^d)
  
by

    ζ^(1)(x) = ⋀^k ξ^(k)(x),


    ζ^(2)(x) =( ⋀^k-1ξ^(k-1)(x) ) ∧ξ^(k+1)(x),


    ζ^(D-2)(x) =ξ^(d-k-1)(x) ∧( ⋀^k-1^d )  + ξ^(d-k)(x) ∧ξ^(d-k+1)(x) ∧( ⋀^k-2^d ),


    ζ^(D-1)(x) = ξ^(d-k)(x) ∧(⋀^k-1^d ).

These maps are clearly continuous and ⋀^k ρ-equivariant.


ζ^(1), ζ^(2), ζ^(D-2), ζ^(D-1) are the flag maps of ⋀^k ρ. 


 By the density of attracting fixed points in ∂_∞Γ and the continuity of the maps, it is enough to verify that ζ^(j)(γ^+) is the attracting fixed point of ⋀^k ρ(γ) in _j(⋀^k ^d) when γ^+ ∈∂_∞Γ is the attracting fixed point of γ∈Γ. 

By Property (<ref>) in Section <ref>, there exists a basis v_1, …, v_d of ^d of eigenvectors of ρ(γ) such that 

    ξ^(j)(γ^+) = { v_1,…, v_j} for  j=1,…, d.

Let I_1 = { d-k+1, d-k+2, …, d} and I_2 = { d-k, d-k+2, d-k+3,…, d}. Then a calculation shows that 

    ζ^(1)(γ^+) = [ v_1 ∧…∧ v_k],


    ζ^(2)(γ^+) = { v_1 ∧…∧ v_k-1∧ (av_k+bv_k+1) : a,b ∈},


    ζ^(D-2)(γ^+) = { v_i_1∧…∧ v_i_k : { i_1, …, i_k}∉{ I_1, I_2}},


    ζ^(D-1)(γ^+) = { v_i_1∧…∧ v_i_k : { i_1, …, i_k}≠ I_1 }.

So ζ^(j)(γ^+) is the attracting fixed point of ⋀^k ρ(γ) in _j(⋀^k ^d). 


 
ζ^(1)(x) + ζ^(1)(y) + ζ^(D-2)(z) is a direct sum for all x,y,z ∈∂_∞Γ distinct.  


 
Fix x,y,z ∈∂_∞Γ distinct, and choose a basis v_1,…, v_d ∈^d such that 

    [v_ℓ] = ξ^(ℓ)(x) ∩ξ^(d-ℓ+1)(y)

for 1 ≤ℓ≤ d. Next pick u_1, …, u_k ∈^d such that 

    ξ^(k)(z) = { u_1,…, u_k}.

Then ζ^(1)(z) = [ u_1 ∧…∧ u_k]. 


If I = { 1,…, k-1, k+1}, then a computation shows that

    ζ^(1)(x) + ζ^(D-2)(y) = { v_i_1∧…∧ v_i_k : { i_1, …, i_k}≠ I }.

Since 

    ξ^(k)(z) + ( ξ^(k)(x) ∩ξ^(d-k+1)(y) ) + ξ^(d-k-1)(y) = ^d

is a direct sum and 

    ( ξ^(k)(x) ∩ξ^(d-k+1)(y) ) + ξ^(d-k-1)(y) = { v_k, v_k+2, …, v_d}

we see that 

    (u_1 ∧…∧ u_k) ∧ (v_k ∧ v_k+2∧…∧ v_d) ≠ 0.

This implies that  

    ζ^(1)(x) +ζ^(1)(y) + ζ^(D-2)(z) = ⋀^k ^d.





 



§ REAL HYPERBOLIC LATTICES



The goal of this section is to justify Example <ref>. More precisely, we need to prove the following proposition.

 Suppose τ: (m,1) →_d() is a representation, Γ≤(m,1) is a co-compact lattice, and ρ = τ|_Γ : Γ→_d() is the representation obtained by restricting τ to Γ. If ρ is irreducible and 1-Anosov, then ρ is also m-Anosov and 

    ξ^(1)(x) + ξ^(1)(y) + ξ^(d-m)(z)

is a direct sum for all x,y,z ∈∂_∞Γ distinct. Thus, the same is true for any small deformation of ρ.
 
 
 Let (m,1)⊂_m+1() be the subgroup that leaves invariant the bilinear pairing that is represented in the standard basis of ^m+1 by the matrix
 
    (
     [  1  0  …  0  0;  0  1  …  0  0;  ⋮  ⋮  ⋱  ⋮  ⋮;  0  0  …  1  0;  0  0  …  0 -1 ]).

 
 

 §.§ Preliminaries
 







Consider the unit ball _m ⊂^m endowed with the metric

    d(x,y) = 1/2logy-a_2x-b_2/x-a_2y-b_2

where a,b ∈∂_m ∩ ( x+(y-x) ) ordered a,x,y,b, and ·_2 is the standard Euclidean norm on ^m. The metric space (_m, d) is usually known as the Klein-Beltrami model of real hyperbolic m-space. Further, (m,1) acts transitively and by isometries on (_m, d) via fractional linear transformations, that is 

    [   A   u; ^tv   a ]· x =  Ax + u/^tvx + a.


Using the formula for the distance, one can compute that d(e^sH· 0, 0) = s, where

    H:= [      0    e_1; ^t e_1      0 ].
 
In fact, one can verify that the map γ_0:→_m given by 

    γ_0:s↦tanh(s)e_1=e^sH· 0
 
is a unit-speed geodesic in _m with -e_1 and e_1 as its backward and forward endpoints respectively. 

A computation also verifies that K:={g∈(m,1):g· 0=0} is given by

    K =  {[ A 0; 0 σ ]: σ∈{-1,1},  A ∈ O(m)}.

In particular, K acts transitively on the set of unit vectors in T_0_m. Since (m,1) acts transitively on _m, this implies that (m,1) acts transitively on the unit tangent bundle of _m. Also, if p∈_m, then d(0,p)=d(0,k· p) for all k∈ K. This, together with the KAK-decomposition theorem <cit.>, implies the following observation.

 If g ∈(m,1), then there exists k_1, k_2 ∈ K such that 

    g = k_1 e^d(g · 0, 0)H k_2.














Recall that an element g ∈(m,1) is called hyperbolic if there exists some geodesic γ: →_m and some ℓ(g) > 0 such that 

    g(γ(t)) = γ(t+ℓ(g))

for all t ∈. The number ℓ(g) is called the translation length of g. For co-compact lattices in (m,1), we have the following proposition.


 If Γ≤(m,1) is a co-compact lattice and γ∈Γ has infinite order, then γ is a hyperbolic element. 


 See for instance <cit.>. 

Let γ_0 be the geodesic as defined above, and let M be the subgroup of (m,1) that fixes the image of the geodesic γ_0 pointwise, i.e.
 
    M := { k  ∈ K:  k · e_1 = e_1}.


 If h ∈(m,1) is hyperbolic, then h=ge^ℓ(h)Hk g^-1 for some k ∈ M that commutes with e^ℓ(h)H. 



Since h is hyperbolic, there exists some geodesic γ: →_m such that h γ(t) = γ(t+ℓ(h)) for all t ∈. Also, (m,1) acts transitively on the unit tangent bundle of _m,  there exists g ∈(m,1) so that g∘γ=γ_0. Since h translates along γ by ℓ(h) and e^-ℓ(h)H translates along γ_0 by -ℓ(h), we see that e^-ℓ(h)Hghg^-1=ghg^-1e^-ℓ(h)H fixes the image of γ_0 pointwise, and therefore lies in M. Hence, there is some k∈ M so that 
 
    ghg^-1=e^ℓ(h)Hk=ke^ℓ(h)H
 
 for some k ∈ M. 





 §.§ Proof of Proposition <ref>

Let τ, ρ, and Γ satisfy the hypothesis of Proposition <ref>. To prove Proposition <ref>, we use the following two lemmas.

Let τ_0 = τ|_e^· H and let τ_0 : e^· H→_d() be the lift of τ_0 (since  is simply connected, such a lift exists). 
 
  
 τ_0(e^H) is proximal and the eigenvalue with maximal modulus is a positive real number.  
 
 The group τ(M) is a compact subgroup of _d(), so every element in τ(M) is elliptic. Now suppose that γ∈Γ has infinite order.  Since ρ is 1-Anosov, τ(γ) has a representative in _d^±() whose eigenvalue of maximal absolute value has multiplicity 1. On the other hand, by Proposition <ref>, γ is conjugate to k e^s H for some s > 0 and k ∈ M. Then since τ(k) is elliptic and commutes with τ(e^sH), the eigenvalues of τ(e^sH) and τ(k)τ(e^sH) = τ(k e^sH) have the same absolute values. So τ(e^sH) also has a unique eigenvalue with maximal absolute value. This implies that τ(e^tH) is proximal for every t ≥ 0. 
 

Since τ_0(𝕀) = 𝕀 has all positive eigenvalues, we see that  the eigenvalue with maximal modulus of τ_0(e^tH) is positive for all t ≥ 0.
 




Let e^λ denote the eigenvalue of τ_0(e^H) with maximal modulus. There is some integer k so that the set of eigenvalues of τ_0(e^H) is 

    {e^λ-n:0≤ n≤ k}.
 
Furthermore, the eigenspace corresponding to e^λ-1 has dimension m-1.
 

The proof of Lemma <ref> is a standard argument from the theory of weight spaces. We give this argument in Appendix <ref>. With Lemma <ref> and <ref>, we can prove Proposition <ref>.


By Lemma <ref> and <ref>, the eigenvalues of τ_0(e^sH) are

    e^λ s, e^(λ-1)s, …, e^(λ-1)s, e^(λ-2)s, …,

and the multiplicity of e^(λ-1)s is m-1. In particular,

    μ_m/μ_m+1(τ(e^sH)) = e^s.

Also, the group τ(K)⊂_d() is compact, so it lifts to a compact subgroup K⊂_d^±(). Hence, there exists some C > 1 such that 

    1/Cμ_i(T) ≤μ_i(k_1 T  k_2) ≤ C μ_i(T)

for all 1 ≤ i ≤ n, all k_1, k_2 ∈K, and all T ∈(^n). By Observation <ref>,

    logμ_m/μ_m+1(ρ(γ)) ≥logμ_m/μ_m+1(τ(e^d(γ· 0,0)H))-log (C^2) =λ d(γ· 0, 0)-log (C^2),

which implies that ρ is m-Anosov. 

Since ρ is the restriction of τ to Γ, the ρ-equivariant flag maps 

    ξ^(i) : ∂Γ≃∂_m →_i(^d)
 
are τ-equivariant for i=1,d-1,m,d-m. Further, by the description of K given by (<ref>), we see that  (m,1) acts transitively on triples of distinct points x,y,z ∈∂_m. Thus it is enough to show that 

    ξ^(1)(x) + ξ^(1)(y) + ξ^(d-m)(z)

is direct for some x,y,z ∈∂_m distinct. Fix y,z ∈∂_m distinct. Then since τ is irreducible we must have

    ^d = {ξ^(1)(x) : x ∈∂_m}

and so there exists some x ∈∂_m such that 

    ξ^(1)(x) + ξ^(1)(y) + ξ^(d-m)(z)

is direct.








§ THEOREM <REF>


 First notice that _λ(Λ) is invariant under conjugation in _d(), i.e. _λ(Λ)=_λ(gΛ g^-1) for all g∈_d(). Further, if h ∈_d(), then from the geometric description of the Cartan projection given in <ref>, there exists some C > 0 such that 

    μ(g) - μ(hgh^-1) _2 ≤ C

for all g ∈_d(). Hence _μ(Λ) is also invariant under conjugation in _d(). 

Let _d() =  + denote the standard Cartan decomposition of _d(), that is 

    = { X ∈_d() : ^tX = -X}    and     = { X ∈_d() : ^tX = X}.

Let  denote the Lie algebra of G. Using Theorem 7 in <cit.> and conjugating G we may assume that 

    = ∩ + ∩

is a Cartan decomposition of . Fix a maximal abelian subspace ⊂∩. By <cit.>, there exists some k ∈(d) such that (k) is a subspace of the diagonal matrices in _d(). Since (k)  =, by replacing G with kGk^-1 we can assume that  is itself a subspace of the diagonal matrices. Finally fix a Weyl chamber ^+ of . 

Next let K ⊂ G denote the subgroup corresponding to ∩, let A = exp(), and let A^+ = exp(^+). By <cit.>, each g ∈ G can be written as 

    g = k_1 exp( μ_G(g) ) k_2

where k_1, k_2 ∈ K and μ_G(g) ∈^+ is unique. The map μ_G : G →^+ is called the Cartan projection of G relative to the decomposition G = K A^+ K. Since K ⊂(d) and  is a subspace of the diagonal matrices,  the diagonal entries of μ_G(g) coincide with the entries of μ(g) up to permuting indices.

Every g ∈ G can be written as a product g=g_e g_h g_u of commuting elements, where g_e is elliptic, g_h is hyperbolic, and g_u is unipotent. This is called the Jordan decomposition of g in G. The element g_h is conjugate to a unique element exp(λ_G(g)) ∈A^+ and the map λ_G : G →^+ is called the Jordan projection. Since G is an irreducible real algebraic subgroup of _d(), the Jordan decomposition in G coincides with the standard Jordan decomposition in _d(). Then, since  is a subspace of the diagonal matrices,  the diagonal entries of λ_G(g) coincide with the entries of λ(g) up to permuting indices. 


Next define cones _1, _2 ⊂^+ as follows: 

    _1 := ⋃_γ∈Γ_>0·λ_G(γ)

and 

    _2 := { x ∈^d : ∃γ_n ∈Γ, ∃ t_n ↘ 0,  with lim_n →∞ t_n μ_G(γ_n) =x}.

Then the main result in <cit.> says that _1 = _2. Since μ_G(g) and μ(g) (respectively λ_G(g) and λ(g)) coincide up to permuting indices, this implies that _μ(Λ) = _λ(Λ). 




§ PROOF OF LEMMA <REF> 


 Let 𝔰𝔬(m,1) denote the Lie algebra of (m,1), and let e_1,…, e_m+1 be the standard basis of  ^m+1. By fixing the signature (m,1)-form on ^m+1 that is represented in this basis by the matrix 

    [  1  0  ⋯  0  0;  0  1        0;  ⋮     ⋱     ⋮;  0        1  0;  0  0  ⋯  0 -1 ],

one can compute that
 
    𝔰𝔬(m,1) = {[   A   u; ^tu   0 ] : ^tA=-A }.

Define vector following subspaces of 𝔰𝔬(m,1):
 
    =  {[      0  λ e_1; λ^te_1      0 ] : λ∈}, 
    _0     =   {[      A  λ e_1; λ^te_1      0 ] : ^tA=-A,    Ae_1 =0,  and λ∈}, 
    _-1    =  {[ -u^te_1 +e_1 ^tu                u;              ^tu                0 ] :  u,e_1 =0},  and
    _1    =  {[ u^te_1 - e_1 ^tu                u;              ^tu                0 ] : u,e_1 =0}.

Then 𝔞⊂𝔤_0 is a maximal abelian subalgebra, and the decomposition 

    𝔰𝔬(1,m) = _0 + _-1 + _1
 
is the associated (restricted) root space decomposition of 𝔰𝔬(1,m). 

Recall that 

    H:= [      0    e_1; ^t e_1      0 ]∈(m,1).

The following lemma states some basic properties of the root space decomposition <cit.>, and can be verified explicitly in this special case.

 

  * Let  σ∈{0,1,-1}, and Y ∈_σ. Then 

    [H,Y]=σ Y    and    ( e^s H) Y= e^σ s Y.


  * Let α, β∈{0,-1,1}. Then [_α, _β] ⊂_α+β, where _-2:={0}=:_2. 



Next, suppose that τ:(m,1)→_d() is an irreducible representation so that τ(e^H) is proximal and τ_0 : e^· H→_d() is the lift of τ_0:=τ|_e^· H. 

Let  𝔰𝔩_d() denote the Lie algebra of _d() and let dτ:𝔰𝔬(m,1)→𝔰𝔩_d() be the derivative at the identity of the homomorphism τ:(m,1)→_d(). The next lemma gives a description of the eigenvalues and eigenspaces of τ_0(e^H).

 Let e^λ denote the largest eigenvalue of τ_H(e^H) and let V_0 ⊂^d denote the eigenspace of τ_0(e^H) corresponding to e^λ. For n ∈, define

    V_n+1 := dτ(_-1) V_n,



  * If v ∈ V_n, then τ_0(e^H)v = e^λ -nv.

  * If Z ∈_0, then dτ(Z)V_n ⊂ V_n. 

  * If Z ∈_1, then dτ(Z)V_0 = {0} and dτ(Z) V_n ⊂ V_n-1 when m>0.

  * ∑_n ≥ 0 V_n = ^d.



 (1): By definition v = dτ(Y)w for some Y ∈_-1 and w ∈ V_n-1. Then by induction

    τ_0(e^H)dτ(Y)w 
       = dτ( (e^H)Y )τ_0(e^H)w 
       = dτ( e^-1Y )(e^λ-(n-1)w) 
        = e^λ-n dτ( Y )w,

where the second equality is a consequence of (1) of Lemma <ref>.

 (2): Fix some v ∈ V_n. Then by definition v = dτ(Y)w for some Y ∈_-1 and w ∈ V_n-1. Then [Z,Y] ∈_-1 by (2) of Lemma <ref>, so 

    dτ(Z)dτ(Y)w = dτ([Z,Y])w - dτ(Y)dτ(Z)w ∈ V_n

by induction. 

(3): If v_0 ∈ V_0, then 

    τ_0(e^H)dτ(Z)v_0 
       = dτ( (e^H)Z )τ_0(e^H)v_0 
        = e^λ+1 dτ( Z )v_0.

Since e^λ is the largest eigenvalue of τ_0(e^H) we must have dτ(Z)v_0=0. Since v_0 ∈ V_0 was arbitrary, we then have dτ(Z)V_0 = {0}. 

Next fix some v ∈ V_n. Then by definition v = dτ(Y)w for some Y ∈_-1 and w ∈ V_n-1. Then [Z,Y] ∈_0 by (2) of Lemma <ref>, so

    dτ(Z)dτ(Y)w = dτ([Z,Y])w - dτ(Y)dτ(Z)w ∈ V_n-1

by (2) and induction. 
 
(4): The previous parts show that ∑_n ≥ 0 V_n is an dτ and hence τ invariant subspace. Since τ is irreducible, we then have ∑_n ≥ 0 V_n=^d. 












The first statement of the lemma is an immediate consequence of Lemma <ref> and <ref>. To prove the second statement, fix some non-zero v_0 ∈ V_0, and consider the linear map T: _-1→ V_1 given by 

    T(Y) = dτ(Y)v_0.

Since T is onto and __-1 = m-1, we see that _ V_1 ≤ m-1. It is now sufficient to prove that T = {0}. To see this, again let

    M := { k ∈ K: k · e_1 = e_1}.

Then a calculation shows that (M) preserves and acts irreducibly on _-1. Notice that τ(M)v_0 ⊂ V_0 is a compact connected set and so τ(M)v_0 = v_0. Further, if Y ∈_-1 and k ∈ M, then

    T((k)Y) = dτ((k)Y)v_0 = τ(k)dτ(Y) τ(k^-1) v_0 = τ(k)T(Y).

So T is an (M)-invariant subspace. So either T = {0} or T = _-1. If T = _-1, then V_0= ^d and since d > 1 this is impossible. So T ={0} and hence V_1 ≥ m-1.







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