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Degeneration of Hitchin representations along internal sequences

Using the work of Bonahon-Dreyer and Fock-Goncharov, one can construct a real-analytic parameterization for the PSL(n,R) Hitchin component of a surface S, that is explicitly analogous to the Fenchel-Nielsen coordinates on the Teichmuller space of S. Given a Hitchin representation, we give a lower bound on the"length"of any closed curve on S in terms of the parameters describing this representation. We then show that this lower bound is good enough to produce large families of sequences in the Hitchin component along which the topological entropy converges to 0.


Introduction
Let S be a closed, oriented topological surface of genus at least 2, and denote its fundamental group by Γ. The Teichmüller space of S, denoted T (S), is the space of marked hyperbolic metrics on S. From a representation theoretic point of view, one can think of T (S) as a component of the space of conjugacy classes of discrete, faithful representations from Γ to P SL(2, R). An advantage of taking this point of view is that it allows one to define a higher rank generalization of T (S), which was first studied by Hitchin [15]. Presently, this "higher Teichmüller space" is known as the Hitchin component, and can be defined as follows. Let ι : P SL(2, R) → P SL(n, R) be the unique (up to conjugation) irreducible representation. This induces, via post-composition, an embedding i from T (S) into the character variety X n (S) := Hom(Γ, P SL(n, R))/P SL(n, R).
The n-th Hitchin component of S, denoted Hit n (S), can then be defined to be the connected component of X n (S) that contains the image of i, which is also known as the Fuchsian locus.
It is well-known that T (S) = Hit 2 (S). Also, by the work of Choi-Goldman [6] and Guichard-Wienhard [13], we know respectively that Hit 3 (S) is the space of marked convex RP 2 structures on S and that Hit 4 (S) is the space of marked convex foliated RP 3 structures on T 1 S. These realizations of the lower rank Hitchin components as deformations spaces of geometric structures associated to S provide a strong motivation for studying Hitchin representations. Guichard-Wienhard [14] also constructed domains of discontinuities for the image of Hitchin representations in Hit n (S) for all n.
Interestingly, the Hitchin components have many of the desirable properties that T (S) possess. Hitchin [15] proved using Higgs bundle techniques that Hit n (S) is a cell of real dimension (n 2 − 1)(2g − 2), where g is the genus of S. By understanding the dynamics of the Γ-action induced by Hitchin representations on the space of complete flags in R n , Labourie [16] proved that they are discrete, faithful, and their images consist only of diagonalizable elements with eigenvalues that have pairwise distinct norms. Using the work of Fock-Goncharov [10], Bonahon-Dreyer [1] gave a real-analytic parameterization of Hit n (S) that is a generalization of Thurston's shear coordinates in T (S). There is also a parameterization of Hit 3 (S) by Goldman [11] which generalizes the Fenchel-Nielsen coordinates on T (S).
By taking a special case of the parameterization by Bonahon-Dreyer [1] and performing a linear reparameterization, one can obtain another parameterization that is explicitly analogous to the Fenchel-Nielsen coordinates on T (S). More specifically, if we choose an oriented pants decomposition P of S, then Hit n (S) can be parameterized by the following parameters: • n − 1 boundary invariants for each simple closed curve in P.
• n − 1 gluing parameters for each simple closed curve in P.
• (n − 1)(n − 2) internal parameters for each pair of pants given by P. The boundary invariants take values in R + while the gluing and internal parameters take values in R. Here, one should think of the boundary invariants and gluing parameters as analogs of the Fenchel-Nielsen length and twist coordinates respectively. We call this parameterization of Hit n (S) the modified shear-triangle parameterization.
In view of this parameterization, one can ask if there is any geometric meaning behind deforming the internal parameters. One way to approach this question is to study sequences {ρ i } ∞ i=1 in Hit n (S), along which the boundary invariants are held bounded away from 0 and ∞, while the (n − 1)(n − 2) internal parameters for each pair of pants escape every compact set in the cell where they take values. Such sequences are called internal sequences. More informally, the internal sequences are those where we do not change the boundary invariants by much while deforming the internal parameters as much as possible.
One can also study Hitchin representations by considering the induced flows on T 1 S. For any Hitchin representation ρ, define the length function l ρ : Γ → R by l ρ (X) = log ω n (ρ(X)) ω 1 (ρ(X)) , where ω n (ρ(X)) and ω 1 (ρ(X)) are the eigenvalues of ρ(X) with largest and smallest norm respectively. These are related to the length functions studied by Dreyer [8].
When n = 2, l ρ (X) is the length of the closed geodesic in S corresponding to X in Γ, measured in the hyperbolic metric on S corresponding to the representation ρ. Sambarino [23] constructed, for each ρ, a unique (up to Livšic cohomology) Hölder reparameterization (φ ρ ) t of the geodesic flow on T 1 S, so that the closed orbit of (φ ρ ) t corresponding to the conjugacy class [X] in [Γ] has period l ρ (X). In the case when ρ is in T (S), (φ ρ ) t is Livšic equivalent to the geodesic flow of the hyperbolic metric corresponding to ρ. This allows us to define the topological entropy of ρ, h top (ρ), to be the topological entropy of the flow (φ ρ ) t . It then follows from the work of Bowen [2] and Pollicot [21] that In this paper, we study how the dynamics of these induced flows degenerate along internal sequences. The main goal is to prove the following theorem.
Theorem (Theorem 4.5). Let {ρ i } ∞ i=1 be an internal sequence. If X in Γ does not correspond to a curve homotopic to a multiple of a curve in P, then  This theorem is a generalization of the results in [26], where the author proved the same statement for Hit 3 (S) using the Goldman parameterization. Nie [19] also has some related results. In addition to the consequences mentioned in [26], the main theorem has several other interesting geometric corollaries. Let This corollary has further implications on the minimal immersions that arise from Hitchin representations. For any ρ in Hit n (S) and any conformal structure Σ on S, a special case of the work of Corlette [5] or Eells-Sampson [9] implies the existence of a unique (up to P SL(n, R) action) harmonic map f : Σ → ρ(Γ)\M.
Labourie [17] then proved that for every ρ in Hit n (S), there are conformal structures Σ on S so that f is a branched minimal immersion. Recently, Sanders [25] showed that these harmonic maps are always immersions. Furthermore, using the work of Sanders [25], we can deduce the following.
Corollary (Corollary 4.9). Let {ρ i } ∞ i=1 be an internal sequence in Hit n (S), and let Σ i be a conformal structure on S for which the harmonic immersion f i : Σ i → ρ i (Γ)\M is minimal. Then Here, B fi is the second fundamental form of f i , m i is the Riemannian metric on ρ i (Γ)\M , Sec i is the sectional curvature in ρ i (Γ)\M , and the integral is taken using the volume measure of f * i m i . More informally, this corollary says that the minimal immersions corresponding to the Hitchin representations along any internal sequence are on average becoming flatter and more totally geodesic as we move along the sequence. Collier-Li [4] also have results that are similar in flavor to this corollary.
We will now give a sketch of the proof of the main theorem in three main steps. Choose a hyperbolic metric on S and consider the ideal triangulation on S that is obtained by further subdividing each pair of pants given by P into two ideal triangles. Also, fix ρ in Hit n (S). For the first step, we obtain a combinatorial description of every oriented closed geodesic γ on S using the intersection pattern of γ with the ideal triangulation. Roughly, this combinatorial description keeps track of how many times γ "winds around" a collar neighborhood of a simple closed curve in P, and how many times γ "crosses between" these collar neighborhoods.
By design, two oriented geodesics on S have the same combinatorial description if and only if they are the same oriented geodesic.
In the second step, we find, for any Hitchin representation ρ and any X in Γ, a lower bound for l ρ (X) of the form (1.1) l ρ (X) ≥ r(X) · K(ρ) + s(X) · L(ρ).
Here, K, L : Hit n (S) → R + are continuous functions, s(X) is the number of times the oriented closed geodesic γ corresponding to X "winds around" collar neighborhoods of the simple closed curves in P, and r(X) is the number of times γ "crosses between" these collar neighborhoods. Finally, in the third step, we show that lim i→∞ K(ρ i ) = ∞ for any internal sequence {ρ i } ∞ i=1 in Hit n (S). This fact, combined with an elementary counting argument demonstrated in [26], proves the main theorem.
The second and third steps of the proof rely heavily on the work of Labourie [16] and Guichard [12]. Together, they proved that a representation ρ in X n (S) is a Hitchin representation if and only if there exists a ρ-equivariant Frenet curve ξ : ∂ ∞ Γ → F (R n ). Understanding the way the cross ratio interacts with ξ is of central importance to the arguments used in the second and third steps.
The rest of the paper is structured as follows. In Section 2, we investigate how the cross ratio interacts with the ρ-equivariant Frenet curve and develop the tools required to perform the second and third steps. Then, in Section 3, we give a detailed explanation of the modified shear-triangle parameterization, and how to derive it from the work of Bonahon-Dreyer and Fock-Goncharov. Next, we present a careful restatement of the main theorem and prove the corollaries mentioned above in Section 4. Section 5 contains the first step, i.e. the combinatorial description of the oriented closed curves on S, and also the proof of the lower bound (1.1). Finally, we combine all the proof ingredients and execute the second and third step of the proof of the main theorem in Section 6. (1) Let x 1 , . . . , x k be pairwise distinct points in S 1 and let n 1 , . . . , n k be positive integers so that (2) Let x 1 , . . . , x k be pairwise distinct points in S 1 and let n 1 , . . . , n k be positive integers so that m := k i=1 n i ≤ n. Then for any x ∈ S 1 , One should think of Frenet curves as having the property that points along the curve are "maximally transverse". Often, we will also refer to the image of the Frenet curve ξ by ξ.
For the rest of the paper, let S = S g be a closed oriented smooth surface of genus g > 1 and let Γ := π 1 (S). It is well-known that Γ is Gromov hyperbolic, so the Cayley graph of Γ has a natural boundary, denoted by ∂ ∞ Γ. The action of Γ on its Cayley graph also extends to an action on ∂ ∞ Γ. Moreover, the fact that S can be equipped with a complete hyperbolic metric allows us to identify ∂ ∞ Γ with the boundary of the Poincaré disc, so ∂ ∞ Γ is topologically a circle.
By the work of Labourie [16] and Guichard [12], we have the following useful characterization of the representations in Hit n (S).

Theorem 2.3 (Guichard, Labourie). A representation ρ in the character variety
Hom(Γ, P SL(n, R))/P SL(n, R) lies in Hit n (S) if and only if there exists a ρ-equivariant Frenet curve ξ : ∂ ∞ Γ → F(R n ). If ξ exists, then it is uniquely determined (up to P SL(n, R)) by ρ.
This theorem thus allows us to understand any Hitchin representation ρ via its corresponding Frenet curve ξ.
(2) Let s be a closed subsegment of ξ with endpoints A and B. Then there exists a closed subsegment t of P(L (1) is a homeomorphism with f n k+1 (A) = L Proof. Proof of (1). The continuity and well-definedness of f 1 is clear by the definition of the Frenet curve. Suppose for contradiction that there exist E = E such In particular, which contradicts the fact that ξ is a Frenet curve. This proves that f 1 is an injective continuous map between two spaces homeomorphic to S 1 , so f 1 has to be a homeomorphism. It is easy to verify that f 1 (A) = L (1) Proof of (2). As before, the continuity of f n k+1 is clear. We will prove that f n k+1 is a homeomorphism by induction. The base case when n k+1 = 1 follows from (1). For the inductive step, consider the case when n k+1 = m + 1. Pick any pair of distinct points E 0 and E 1 in the interior of s, and assume without loss of generality that E 1 lies between E 0 and B on s. Since ξ is a Frenet curve, B that contains f m+1 (E 0 ). Let this subsegment be t. By the inductive hypothesis, the map f m : s → t is a homeomorphism. Hence, the point lies on t, strictly between the points By the base case, the map f 1 : s → t is a homeomorphism. Thus, we can conclude that the point lies on t, strictly between the points B . Since A and f m+1 (B) = L B , so the continuity of f m+1 implies that it is surjective. This finishes the inductive step.
The homeomorphisms f n k+1 should be thought of as projections of subsegments of ξ (or all of ξ in the case when n k+1 = 1) onto the projective line P(L (1) 2.3. Cross ratio. We will now describe the main tool used to prove the main theorem (Theorem 4.5) and establish some of its properties. Here is its definition. Definition 2.6. Let L 1 = [l 1 ], . . . , L 4 = [l 4 ] be four lines in R n through the origin, and let M = Span{m 1 , . . . , m n−2 } be a (n − 2)-dimensional subspace of R n not containing L i for any i = 1, . . . , 4, so that no three of the four (n − 1)-dimensional subspaces M + L i agree. Define the cross ratio of the lines L 1 , L 2 , L 3 , L 4 based at M by Here, we choose a linear identification of n (R n ) with R to evaluate the expression on the right as a number in the one point compactification R ∪ {∞} of R, and then observe that the cross ratio does not depend on the identification we chose. Also, it is easy to verify that the cross ratio depends neither on the choice of basis {m 1 , . . . , m n−2 } for M , nor the choice of representatives l i for L i .
If we let c i be a linear function on R n with kernel M + L i for i = 1, . . . , 4, then this cross ratio takes a more familiar guise which is a well-studied object in projective geometry. The next proposition summarizes some basic properties of this cross ratio.
Proposition 2.7. Let L 1 , . . . , L 5 be pairwise distinct lines in R n through the origin. Let M , M be (n − 2)-dimensional subspaces of R n not containing L i for any i = 1, . . . , 5, so that no three of the five (n − 1)-dimensional subspaces M + L i agree and no three of the five (n − 1)-dimensional subspaces M + L i agree.
(2) For all i, let L i be a line in R n such that L i ⊂ M + L i and L i ⊂ M . Then (3) Suppose L 1 , L 2 , L 3 , L 4 lie in a plane. Then Proof. It is clear that (1) holds because X preserves the volume form.
(2) follows from the observation that replacing any of the l i in Definition 2.6 with a linear combination of m 1 , . . . , m n−2 , l i so that the coefficient of l i is non-zero does not change the cross ratio. To prove (3), note that we can choose X in P SL(n, R) so that X · M = M and X fixes L 1 , L 2 , L 3 . Since L 1 , L 2 , L 3 and L 4 lie in a plane, X also fixes L 4 . The P SL(n, R)-invariance of the cross ratio stated in (1) then proves (3). Parts (4), (5), (6) and (8) are immediate from the formula in Definition 2.6, and (7) can be checked via choosing a normalization and performing a simple computation.
Part (2) of Proposition 2.7 allows one to think of the cross ratio as a projective invariant associated to the four (n − 1)-dimensional subspaces of R n that intersect along a (n − 2)-dimensional subspace. This is thus a natural generalization of the classical cross ratio of four lines intersecting at a point in RP 2 .
There will be two main ways we use the cross ratio. The first is to capture some eigenvalue data of X in P SL(n, R). This is described in the next proposition. Proposition 2.9. Let X in P SL(n, R) be an element with a representative in SL(n, R) that is diagonalizable with n real, positive, pairwise distinct eigenvalues ω 1 < · · · < ω n . Let X − and X + be the eigenspaces corresponding to ω 1 and ω n respectively, and let M be the sum of the eigenspaces corresponding to ω 2 , . . . , ω n−1 . Then for any line L in R n through the origin such that L ⊂ M + X − and L ⊂ M + X + , we have (X − , L, X · L, X + ) M = ω n ω 1 .
Given three pairwise distinct (n − 1)-dimensional subspaces in R n that intersect along a common (n − 2)-dimensional subspace M , we can also use this cross ratio to parameterize the set of (n − 1)-dimensional subspaces in R n that contain M . More precisely, we have the following proposition. Proof. First, note that by (2) of Proposition 2.7, f is independent of the choice of L 1 , L 2 , L 3 and L N . For convenience, we choose L 2 = N 2 ∩ (L 1 + L 3 ) and In the case when a = ∞, set l N = l 1 − l 3 .
One can then easily compute that where in the case when a = ∞, we set The proposition follows immediately.
Next, we discuss how the cross ratio interacts with a Frenet curve. For that purpose, we introduce the following notation.  (1) D . Using this notation, we can state the following proposition, which is a collection of useful inequalities involving the cross ratio and the Frenet curve. Proof. Part (1) follows from (1) of Lemma 2.5 and Proposition 2.10. Since the proofs for (2) to (5) are very similar, we will only show the proof for (2).
Proof of (2). Consider the lines

Shear-triangle parameterization
In the first parts of this section, we will briefly describe a particular case of what we call the shear-triangle parameterization of Hit n (S) given by Bonahon-Dreyer [1]. A version of this parameterization can also be found in the monumental work of Fock-Goncharov [10], though in a much less explicit form. We will also give a geometric interpretation of the parameters in terms of flags. After that, we slightly modify this parameterization to obtain a parameterization of Hit n (S) that is more explicitly analogous to the Fenchel-Nielsen coordinates on Hit 2 (S) and the Goldman parameterization [11] on Hit 3 (S).
3.1. Ideal triangulations. One main ingredient needed to describe the sheartriangle parameterization is an ideal triangulation of the surface S. Here, we will give a description of ideal triangulations of S in terms of ∂ ∞ Γ.
We say {a, b} and {c, d} in ∂ ∞ Γ [2] intersect if neither of the closed subsegments of ∂ ∞ Γ with endpoints a, b contain both c and d.
Definition 3.2. An (undirected) ideal triangulation of the universal cover S of S is a maximal Γ-invariant subset T of ∂ ∞ Γ [2] such that the following hold: (1) Any two pairs {a, b}, {c, d} in T do not intersect.
(2) For any {a, b} in T , either one of the following must hold: • There is some c in ∂ ∞ Γ such that {b, c} and {c, a} both lie in T .
• There is some X in Γ such that {a, b} is the set of repelling and attracting fixed points of X.
An ideal triangulation of S is the quotient of an ideal triangulation of S by Γ. If T is an ideal triangulation of S, then we denote by T its quotient by Γ.
If {a, b} in T is the set of repelling and attracting fixed points of some X in Γ, then we call {a, b} a closed leaf. Also, [a, b] in T is called a closed leaf if some (or equivalently, all) of its representatives in T are closed leaves. By a triangle in T , we mean a subset of T that is of the form {{a, b}, {b, c}, {c, a}}, where a, b, c are points in ∂ ∞ Γ. Each of the three pairs in any triangle is called an edge of that triangle, and a point in any edge is called a vertex of that edge. Also, we say that two triangles in T are adjacent if they share a common edge. We will denote by ∆ = ∆ T the set of triangles in T . There is an obvious Γ-action on ∆, so we can consider ∆ = ∆ T := ∆/Γ and call any element in ∆ a triangle in T . An edge of a triangle T in ∆ is an element e in T so that T has a representative (in ∆) which has a representative (in T ) of e as an edge. As before, we say two triangles in ∆ are adjacent if they share an edge, or equivalently, if they have adjacent representatives in ∆.
If we choose a marked complete hyperbolic structure on S, then there is a canonical Γ-equivariant identification of ∂ ∞ Γ with the boundary of the Poincaré disc, ∂D. The ideal triangulation T then gives us an ideal triangulation of D (in the classical sense) by assigning to each pair {a, b} in T ⊂ (∂D × ∂D)/Z 2 the unique geodesic in D between a and b. Moreover, this ideal triangulation is Γ-invariant, so T can be thought of as an ideal triangulation (in the classical sense) of S equipped with the marked hyperbolic structure.

3.2.
A special ideal triangulation. In [1], Bonahon-Dreyer gave an explicit description of the shear-triangle parameterization of Hit n (S) for any closed surface S using the Frenet curve ξ guaranteed by Theorem 2.3 and the notion of positive configurations of flags developed by Fock-Goncharov [10]. To specify this parameterization, one needs to first choose an ideal triangulation of the surface S. For our purposes, we will only be considering this parameterization for a particular ideal triangulation, which we will now describe.
For the rest of this paper, fix a pants decomposition P for S, i.e. a maximal, pairwise disjoint, pairwise non-homotopic collection of simple closed curves in S. This pants decomposition cuts S into finitely many pairs of pants, P 1 , . . . , P 2g−2 . For each of these pairs of pants P j , let A j , B j and C j := A −1 j B −1 j be three elements in π 1 (P j ) that correspond to the three boundary components of P j , oriented so that P j lies on the left of each of these boundary components. Let f j : P j → S be the obvious inclusion, and let (f j ) * : π 1 (P j ) → Γ be the induced injection on fundamental groups. This is well-defined up to conjugation by elements in Γ. To simplify notation, we will also denote the images of A j , B j and C j under (f j ) * by A j , B j and C j respectively.
The action of any non-identity element X in Γ on ∂ ∞ Γ has a repelling and attracting fixed point. Let a j , b j , c j be the repelling fixed points and a + j , b + j , c + j be the attracting fixed points of A j , B j , C j respectively. Let Q j and P j be the subsets of ∂ ∞ Γ [2] defined by and let One can check that Q and P are disjoint, and Q ∪ P is an ideal triangulation of S. For the rest of this paper, we denote this particular ideal triangulation by T and let T be the quotient of T by Γ. Since Q and P are Γ-invariant, we can define P := P/Γ and Q := Q/Γ. It is easy to see that P is the pants decomposition we chose for S, T = P ∪ Q and P is exactly the set of closed leaves in T . (See Figure 7 for a picture of T restricted to a pair of pants given by P.) We will now state some easily verified properties of T . Since we can realize T as an ideal triangulation (in the classical sense) of S by choosing a hyperbolic metric on S, the Gauss-Bonnet theorem tells us that the cardinalities of ∆, T , P and Q are 4g − 4, 9g − 9, 3g − 3 and 6g − 6 respectively. Define Q j := Q j /Γ and observe that for any j = 1, . . . , 2g − 2, Q j has exactly three elements, which are the Γ orbits of {b j , a j }, {a j , c j } and {c j , b j }. Moreover, a triangle in ∆ has an edge in Q j if and only if all its edges lie in Q j . Furthermore, there are exactly two triangles in ∆ with edges in Q j , and we can describe them explicitly. One of them, denoted T j , is the Γ-orbit of the trian- Figure 1.) The triangles T j and T j share all their edges, and any adjacent pair of triangles in ∆ is the pair T j , T j for some j.

Triangle invariants.
In their description of the shear-triangle parameterization, Bonahon-Dreyer used two different kinds of parameters which they call the triangle invariants and the shear invariants. We will spend the next two subsections studying these invariants in detail. In this subsection, we describe the triangle invariants and what they mean geometrically. Before we do so, we need the following definition.
First, we define a set of numbers one can associate to each generic triple (F, G, H) in F (R n ) 3 , and describe how these numbers determine (F, G, H) up to the action of P SL(n, R). These numbers are called the triple ratios associated to (F, G, H).
Definition 3.5. Let (F, G, H) be a generic triple of flags in F (R n ) 3 . Then for any (x, y, z) in A, define the triple ratio to be the quantity . We will now explain the notation used in the formula above. For any triple of Then for any triple (x, y, z) in (Z ≥0 ) 3 such that x + y + z = n, define By choosing a linear identification between n (R n ) and R, we can treat F (x) ∧ G (y) ∧ H (z) as a real number. Of course, the real number F (x) ∧ G (y) ∧ H (z) is not actually well-defined because it depends on the choice of the three bases, and the choice of the identification between n (R n ) and R. However, one can check that T x,y,z (F, G, H) is independent of these choices. Also, since P SL(n, R) preserves the volume form, T x,y,z (F, G, H) = T x,y,z (X · F, X · G, X · H) for all X in P SL(n, R).
From the definition of a triple ratio, one can immediately observe the next lemma.
Geometrically, the collection of triple ratios determine the generic triple of flags (F, G, H) up to the action of P SL(n, R). In fact, we have the following lemma. Proof. Let {f 1 , . . . , f n }, {g 1 , . . . , g n }, {h 1 , . . . , h n } be bases for R such that for k = 1, . . . , n, is a basis for R n for any triple (x, y, z) in A. Hence, we can write By an easy computation and using the identification Combining the definition of T x,y,z (F, G, H) with the computation above yields The assumption that (F, G, H) is a generic triple implies that T x,y,z (F, G, H) is a non-zero real number. We know what α i and γ i are for i = 1, . . . , n because we are given the flags F , H and G (i) for i = 1, . . . , y. Hence, T x,y,z (F, G, H) determines the ratio βx βn . Since thus determine n − y − 1 hyperplanes in R n that all contain G (y+1) . Moreover, the genericity of (F, G, H) ensures that the intersection of these n − y − 1 subspaces has dimension y + 1, and so it has to be G (y+1) .
Given the P SL(n, R) orbit of the triple of flags (F, G, H), we can choose a normalization so that where {e 1 , . . . , e n } is the standard basis of R n . Lemma 3.7 then implies that we can recover the rest of the flag G from the triple ratios of (F, G, H), so these triple ratios determine the generic triple (F, G, H) up to the action of P SL(n, R).
Moreover, we can make the following pair of observations, which we record as Proposition 3.8 and Proposition 3.9 respectively. We will only give the proof for Proposition 3.8; the proof for Proposition 3.9 is similar.
for any positive integer i. By Lemma 3.7, the fact that is a non-zero real number for all (x, y, z) in A with y < y 0 , ensures that is well-defined for y = 1, . . . , y 0 . Also, this fact, together with Equation (3.1), imply that F (x) + G (y) + H (n−x−y) = R n for y = 1, . . . , y 0 and x = 1, . . . , n − y − 1.
As in the proof of Lemma 3.7, choose a basis Also, let g i,1 , . . . , g i,y0 be vectors in R n so that G (k) i = Span{g i,1 , . . . , g i,k } for all k = 1, . . . , y 0 and lim i→∞ g i,j = g j for j = 1, . . . , y 0 . Observe that for all i, is also a basis for R n .
Now, we will apply the above to Hitchin representations. Choose any ρ in Hit n (S) and let ξ be the Frenet curve for ρ. For each j = 1, . . . , 2g − 2, assign to each (x, y, z) in A the real numbers τ (x,y,z),j (ρ) and τ (x,y,z),j (ρ) given by the formulas The positivity condition on these flags described in Section 7.1 and 7.2 of [10] (alternatively, see Lemma 8.4.2 of [18]) ensures that are positive, so τ (x,y,z),j (ρ) and τ (x,y,z),j (ρ) are well-defined real numbers.
We can do this for every ρ in Hit n (S), so τ (x,y,z),j and τ (x,y,z),j can be viewed as functions τ (x,y,z),j , τ (x,y,z),j : Hit n (S) → R.
Henceforth, we will call A j := {τ (x,y,z),j : (x, y, z) ∈ A} and A j := {τ (x,y,z),j : (x, y, z) ∈ A} the set of triangle invariants for T j and T j respectively. Also, we will call A j ∪ A j the set of triangle invariants for P j . By Lemma 3.7, A j determines the triple of flags (ξ(a j ), ξ(c j ), ξ(b j )) and A j determines the triple of flags (ξ(a j ), ξ(b j ), ξ(A · c j )) up to the action of P SL(n, R).

Shear invariants.
The goal of this subsection is to describe the shear invariants, which are the second kind of invariants used in the shear-triangle parameterization. As before, let ρ be a representation in Hit n (S) and let ξ be the corresponding Frenet curve. We will use the set x + y + z = n and exactly one of x, y, z is 0} to label the shear invariants. For each j = 1, . . . , 2g − 2, assign to each vertex (x, y, z) in C the number σ (x,y,z),j (ρ) given by the formulas The cross ratio notation used in these formulas are as described in Notation 2.11. These invariants are called the shear invariants.
are positive, so the shear invariants are well-defined. By allowing ρ to vary over Hit n (S), we can view each shear invariant as a real valued function on Hit n (S).
By Proposition 2.10, the shear invariant σ x,y,0 determines the (n−1) (1) . The Frenet property of ξ ensures that the intersection of these n − 1 subspaces is 1-dimensional, so it has to be ξ(A · c j ) (1) . Similarly, C y,j determines the flags ξ(c j ), ξ(a j ) and the lines ξ(

Parameterizing the Hitchin component.
To obtain the shear-triangle parameterization, Bonahon-Dreyer found linear relations between these shear and triangle invariants, which we will now describe. As before, let ρ be a representation in Hit n (S) and ξ the corresponding Frenet curve. By a theorem of Labourie (Theorem 1.5 of [16]), we know that for every non-identity element X in Γ, ρ(X) is diagonalizable with real eigenvalues that have pairwise distinct norms. In fact, there is a lift of ρ(X) (which lies in P SL(n, R)) to SL(n, R) so that all the eigenvalues of the lift are positive. To see this, observe that this property holds for the Fuchsian representations and is preserved under continuous deformation. We abuse terminology by referring to these positive eigenvalues as the eigenvalues of ρ(X).
Let λ 1,j < · · · < λ n,j be the eigenvalues for ρ(A j ), µ 1,j < · · · < µ n,j be the eigenvalues for ρ(B j ) and ν 1,j < · · · < ν n,j be the eigenvalues for ρ(C j ). Then for any k = 1, . . . , n − 1, Bonahon-Dreyer established the following equalities (see Now, pick any curve η in P and let P 1 , P 2 be the two pairs of pants on either side of η. Assume without loss of generality that A 1 and A 2 are the two elements in π 1 (P 1 ) and π 1 (P 2 ) that correspond to η. It is clear that the orientations on η corresponding to A 1 and A 2 are opposite, so log( λ k+1,1 λ k,1 ) = log( λ n−k+1,2 λ n−k,2 ) for all k = 1, . . . , n − 1. This implies the equality Doing this for each curve in P, we have (n − 1)(3g − 3) linear equations involving the shear and triangle invariants. These equations are known as the closed leaf equalities.
Finally, for each η in P, Bonahon-Dreyer also defined (n − 1) different numbers specifying how to "glue" the structures on two pairs of pants together along η if the boundary invariants for these two pairs of pants corresponding to η are compatible. This gives us another (3g − 3)(n − 1) gluing parameters. (In [1], these are known as the shear invariants along closed leaves.) However, we will not describe these parameters here as they will not matter for our purposes.

3.6.
A reparameterization of the Hitchin component. In this section, we give a linear reparameterization of the shear-triangle parameterization. This reparameterization will have exactly (2g−2)(n 2 −1) parameters (instead of (g−1)(2n 2 +3n−5) parameters), and will be more explicitly analogous to the Fenchel-Nielsen coordinates for Hit 2 (S), or the Goldman parameters for Hit 3 (S).
To specify this parameterization, we again choose a pants decomposition P and the same ideal triangulation T for S as described in Section 3.2. On top of that, we choose an orientation on each curve in P. Henceforth, P will be an oriented pants decomposition. Label the oriented curves in P by {η 1 , . . . , η 3g−3 }.
Notation 3.11. Denote the set of group elements in Γ corresponding to oriented closed curves in P by Γ P .
We will have three different kinds of parameters. The first kind is the eigenvalue information of the holonomy about each of the oriented simple closed curves in P. More specifically, for any η u in P, choose any X in Γ that corresponds to η u . Then for any ρ in Hit n (S), let α 1,u < · · · < α n,u be the eigenvalues of ρ(X) and define for any v = 1, . . . , n − 1. These quantities are called the boundary invariants, and there are (3g − 3)(n − 1) of them. Since we can do this for every ρ in Hit n (S), we can view these boundary invariants as functions β v,u : Hit n (S) → R + . The second kind of parameters are what we will call the internal parameters, which are functions associated to each pair of pants P j . In fact, these are a specially chosen subset of the shear and triangle invariants used in the shear-triangle parameterization. For all j = 1, . . . , 2g − 2, these parameters are (1) τ (x,y,z),j for all positive integers x, y, z such that x + y + z = n, (2) τ (x,y,z),j for all positive integers x, y, z such that x + y + z = n and x > 1,  The third and final kind of parameters are the gluing parameters used in the shear-triangle parameterization. As mentioned before, there are (3g − 3)(n − 1) of them.
Together, the boundary invariants, internal parameters and gluing parameters give us (2g − 2)(n 2 − 1) functions on Hit n (S). We claim that these quantities in fact give us a parameterization of Hit n (S).
internal parameters and (3g − 3)(n − 1) gluing parameters described above define a real analytic parameterization Proof. Pick any j = 1, . . . , 2g − 2 and assume without loss of generality that η 1 , η 2 , η 3 are the three boundary components of P j , oriented so that P j is on the left of each η u , u = 1, 2, 3. Let ρ be a representation in Hit n (S). By Theorem 3.10, it is sufficient to show that given the collection of real numbers {τ (x,y,z),j (ρ) : x + y + z = n} ∪ {τ (x,y,z),j (ρ) : x + y + z = n, x > 1} ∪ {σ (x,y,0),j (ρ) : x + y = n, x > 1}, we can use the closed leaf equalities to recover the remaining triangle parameters for T j evaluated at ρ and the remaining shear parameters for P j evaluated at ρ.
The reparameterization of Hit n (S) given in the previous proposition will be called the modified shear-triangle parameterization.

Main results
The main goal of this paper is a generalization of the results in [26] to Hit n (S) for any n ≥ 3. In [26], the author used the Goldman parameterization of Hit 3 (S) to study a large family of sequences called the Goldman sequences. Using the modified shear-triangle parameterization of Hit n (S) in place of the Goldman parameterization, we will now define a family of sequences in Hit n (S) that are a generalization of the Goldman sequences. These are called the internal sequences.
The map π j sends each Hitchin representation to its internal parameters for the jth pair of pants.
One should think of the internal sequences as sequences where we hold the boundary invariants "essentially fixed" and deform the internal parameters "as much as possible". In this definition, we do not impose any condition on the gluing parameters because we do not require such a condition for Theorem 4.5. We are interested in how geometric properties of Hitchin representations degenerate along these internal sequences. Of particular interest are two geometric quantities, Θ and h top , which we will now define. where ω 1 < · · · < ω n are the eigenvalues of ρ(X). Then let Θ, h top : Hit n (S) → R ≥0 be the functions given by Θ(ρ) := min{l ρ (X) : X = A k for any A ∈ Γ P and any k ∈ Z}, where [Γ] is the set of conjugacy classes of Γ. Recall that Γ P is the subset of Γ corresponding to the closed curves in P.
Observe that l ρ : Γ → R is a non-negative function that is invariant under conjugation in P SL(n, R). Hence, Θ and h top depend only on the conjugacy class of ρ. Moreover, the existence of a minimum for the set {l ρ (X) : X = A k for any A ∈ Γ P and any k ∈ Z} is a consequence of the fact that we can find a lower bound for l ρ (X) which depends only on the combinatorics of how a closed curve in S corresponding to X interacts with T (see Lemma 6.4).
Although it is not apparent from its definition, h top (ρ) is the topological entropy of a flow one can associate to any Hitchin representation. In fact, Sambarino proved the following theorem (see Section 7 and Theorem 3.2 of [23]).
It is well known that the geodesic flow of a hyperbolic surface is differentiable and Anosov, so the work of Bowen [2] implies that the flows constructed by Sambarino for representations in the Fuchsian locus admit a Markov coding. Also, it follows by the construction of the Markov coding given by Pollicot (Section 2 of [21]) that any continuous reparameterization of an Anosov flow on a compact manifold also admits a Markov coding. In particular, the flow that Sambarino associates to each Hitchin representation admits a Markov coding, so using arguments given by Parry-Pollicot (Theorem 8 of [21] or the proof of Theorem 2 of [20]), we have that Also, Pollicot-Sharp proved (Theorem 1.3 of [22]) that h top is a real analytic function on Hit n (S).
In the case when n = 2, a representation ρ in Hit 2 (S) is equivalent to a marked hyperbolic surface N homeomorphic to S, and the flow constructed by Sambarino is Livšic equivalent to the geodesic flow of the hyperbolic metric on N . When n = 3, we know by Choi-Goldman [6] that a representation ρ in Hit 3 (S) is equivalent to a marked convex real projective surface N homeomorphic to S. The flow in this case is Livšic equivalent to the geodesic flow of the Hilbert metric on N .
With these definitions, we can now state the main theorem of this paper.
be an internal sequence. Then the following hold: Before we proceed to the proof of Theorem 4.5, we will highlight some corollaries. Let Proof. Choose a Cartan decomposition of the lie algebra sl(n, R) = k + p, where k is a compact Lie algebra. Then let a be the Cartan subalgebra of sl(n, R) in p and a + a choice of positive Weyl chamber in a. For example, we can take k to be the Lie algebra of SO(n) ⊂ SL(n, R), p to be the Lie algebra of the group of upper triangular matrices in SL(n, R), a to be the set of traceless n × n diagonal matrices and a + to be the subset of a where the diagonal entries are strictly decreasing down the diagonal. Let a : P SL(n, R) → a + be the Cartan projection, i.e. the map satisfying the property that for all g in P SL(n, R), g = k · exp(a(g)) · l, where k, l are elements in the maximal compact subgroup K of P SL(n, R) with Lie algebra k. One can then verify that if o is the point in M whose stabilizer is K, then for any g in P SL(n, R), where λ 1 (a(g)) ≤ λ 2 (a(g)) ≤ · · · ≤ λ n (a(g)) are the eigenvalues of a(g) and c n is a constant depending only on n. Thus, for any ρ in Hit n (S), we have log |{X ∈ Γ : ||a(ρ(X))|| < T }|.
Using the fact that n i=1 λ i (a(g))) = 0 for all g in P SL(n, R), we can also deduce ||a(g)|| ≥ c n n λ n (a(g)) − λ 1 (a(g)) , which implies that for any ρ in Hit n (S), The corollary follows immediately from Theorem 4.5.
Using Corollary 4.6, we can also deduce some properties of the minimal immersions that come from Hitchin representations. The following result is a special case of a theorem of Corlette [5] generalizing the work of Donaldson [7]. Since Hitchin representations are discrete and faithful by Labourie (Theorem 1.5 of [16]), this result is also a consequence of a theorem by Eells-Sampson [9].  is an immersion. Also, for any ρ in Hit n (S), there exist conformal structures Σ on S for which f is conformal, i.e. f is a minimal immersion. In this case, f satisfies the following inequality: where Sec is the sectional curvature in ρ(Γ)\M , m is the Riemannian metric on ρ(Γ)\M , dV is the volume measure of f * m and B f is the second fundamental form of f .
By the Gauss equation, one sees (see Section 6.1 of [25]) that if f is a minimal immersion, then for all points p in Σ, This, together with Corollary 4.6, allows us to conclude the following.
where B fi is the second fundamental form of f i , m i is the Riemannian metric on ρ i (Γ)\M , Sec i is the sectional curvature in ρ i (Γ)\M , and the integral is taken using the volume measure of f * i m i . More collaries to Theorem 4.5 in the case when n = 3 are discussed in Section 3.1 of [26]. The rest of the paper is devoted to its proof.

Lower bound for lengths of closed curves
In this section, we will fix ρ in Hit n (S), a non-identity element X in Γ and let ω 1 < · · · < ω n be the eigenvalues of ρ(X). The goal is to first give a combinatorial description of ρ(X), and then use this combinatorial description to obtain the lower bound for l ρ (X) in Theorem 5.19.
We will denote the Frenet curve corresponding to ρ by ξ, and the attracting and repelling fixed points of X by x + and x − respectively. Since we have an orientation on S, we also have an orientation (drawn as clockwise in all figures) on ∂ ∞ Γ. Let s 0 and s 1 be the two closed subsegments of ∂ ∞ Γ with endpoints x − and x + , oriented from x − to x + and such that the orientation on s 0 agrees with the orientation on ∂ ∞ Γ. 5.1. Finite combinatorial description of closed curves. Now, we give a complete description of ρ(X) by finitely many pieces of combinatorial data. This combinatorial data is similar in spirit to the combinatorial data used to describe closed curves in Sections 3.2 and 3.3 of [26].
As mentioned in the introduction, if we choose a hyperbolic metric on S, the combinatorial description of ρ(X) needs to capture how the directed geodesic in S associated to X "winds around" collar neighborhoods of the curves in P and how it "crosses between" these collar neighborhoods. The ideal triangulation is good enough to describe the "crossing", but not the "winding". Hence, to describe the "winding", we need to define some additional structure on ∂ ∞ Γ using ρ.
Pick any oriented closed curve in P, and choose any A in Γ P corresponding to this closed curve. Let a − and a + be the repelling and attracting fixed points of A in ∂ ∞ Γ respectively, and let r 0 and r 1 be the two closed subsegments of ∂ ∞ Γ with endpoints a − and a + , oriented from a − to a + , and so that the orientation on r 0 agrees with the orientation on ∂ ∞ Γ.
By reversing the orientation on ∂ ∞ Γ if necessary, we can assume that there is some x in r 0 \ {a + } such that {x, a − } lies in T . Choose any such x, and choose a normalization so that for all k = 1, . . . , n, we have ξ(a − ) (k) ∩ ξ(a + ) (n−k+1) = [e k ] and ξ(x) (1) = [e 1 + · · · + e n ]. Here, {e 1 , . . . , e n } is the standard basis of R n . This implies that ρ(A) is a diagonal matrix, and that if we denote the eigenvalue of ρ(A) corresponding to e i by λ i , then 0 < λ 1 < · · · < λ n . Since is topologically a circle, there are two closed subsegments of H with endpoints ξ(a − ) (n−1) and ξ(a + ) (n−1) . Pick any z in the interior of r 1 and let t 1 be the unique subsegment of H with endpoints ξ(a − ) (n−1) and ξ(a + ) (n−1) that contains (ξ(a − ) (n−1) ∩ ξ(a + ) (n−1) ) + ξ(z) (1) . Since ξ is Frenet, it is clear that for all z in r 1 , if and only if z = a ± respectively. Thus, the map is a continuous surjection. Also, for any pair of distinct points z, z in ∂ ∞ Γ so that a + , z, z , a − lie in r 1 in that order, one can compute Thus, by Proposition 2.10, f is also injective, and hence is a homeomorphism. Using our normalization, we can write for all z in r 1 . Since f is a homeomorphism, the map is also a homeomorphism. Also, g(A · z) = λn λ1 g(z) for all z in r 1 . Hence, if we let y in r 1 \ {a − } be the point so that g(y) is minimized subject to the conditions that {y, a + } lies in T and g(y) ≥ 1, then we can conclude that (5.1) 1 ≤ g(y) < λ n λ 1 .
Using x and y as described, we can define some additional structure on ∂ ∞ Γ.
Definition 5.1. Let A be an element in Γ P , and choose any x in r 0 so that {x, a − } is an edge in T . Let y be the point in r 1 described as above. A mesh (see Figure  4) of A is the set of pairs {{A k · x, A k · y} : k ∈ Z}.
One can check that if we use any x in A · x in place of x and perform this construction, then the mesh we obtain will be the same. This implies that there are only two possible meshes of A, because the set is the union of two A -orbits. Once and for all, choose one of these meshes, denoted There is a natural ordering on E A induced by the action of A.
Next, we will use the pair (x − , x + ) and the subsegments s 0 , s 1 to define several subsets of ∂ ∞ Γ and ∂ ∞ Γ [2] that we use to give the combinatorial description of X. Some of these are summarized in Notation 5.2 below.
• Let I X = I be the set of edges in T that intersect {x − , x + } and let I X = I be the subset of I that are not closed leaves. Observe Since the accumulation points of I are exactly the closed leaves, we can define a bijective successor map suc : I → I. Moreover, the ordering induces a cyclic order on I, and the successor map suc : I → I descends to a successor map suc : I → I.
From the way T was defined, it is easy to see that every closed leaf in I is binodal. Also, any node is the vertex of exactly two distinct binodal edges, and at most one of these binodal edges is a closed leaf. Thus, for any vertex a of any closed leaf {a, b 0 } in I , there is a unique binodal edge {a, b 1 } that is not a closed leaf and has a as a vertex. Let {a, b 2 } be the unique edge in I that is adjacent to {a, b 1 } and also has a as a vertex. Define N a := {b ∈ ∂ ∞ Γ : {a, b} ∈ I , b = b i for i = 0, 1, 2}, and let W be the set of vertices for the closed leaves in I (see Figure 5). Then define The main payoff we gain from considering V i instead of V i is that V i is discrete, which allows us to define bijective successor functions suc : V i → V i for both i = 0, 1. (See Figure 6.) Let Z be the edges in B that are Z-type and S be the edges in B that are S-type. Since Z and S are X -invariant, we can define Z := Z/ X and S := S/ X .
Note that Z ∪ S = B, and the cyclic order on I induces cyclic orders on Z, S and B. Let e and e be consecutive edges in B with e preceding e , and observe the following (see Figure 6): (1) If e and e are not of the same type, then in B, there are representatives e, e of e, e respectively so that e ≺ e and e, e share a common vertex. (2) If e and e are of the same type, then in B, there are representatives e, e of e, e respectively so that e ≺ e and there is exactly one closed leaf between them (in I ). If e and e are not of the same type, choose a pair e, e as described in (1) and let A( e, e ) be the element in Γ P that has the common vertex of e and e as a fixed point. If e, e are of the same type, choose a pair e, e as described in (2) and let A( e, e ) be the element in Γ P whose attracting and repelling fixed points are the initial and terminal vertices of the oriented closed leaf between e and e . In either case, consider E A( e, e ) .
. This is the combinatorial data we associate to each X in Γ. If we choose a hyperbolic metric on S and let γ be the closed geodesic in S associated to ρ(X), then the cyclic sequence {(suc −1 (e i ), e i , suc(e i ), T i )} m i=1 tells us how γ "crosses between" the collar neighborhoods of curves in P and the cyclic sequence {(t(e i , e i+1 ))} m i=1 tells us how γ "winds around" these collar neighborhoods.
Proof. It is clear that if X 0 and X 1 are conjugate, then ψ(X 0 ) = ψ(X 1 ). We will now show the converse. Choose a hyperbolic metric on S. Then the ideal triangulation T can be viewed as a Γ-invariant ideal triangulation of the Poincaré disc D, so T is an ideal triangulation of the hyperbolic surface S. Also, the union of meshes can be viewed as a Γ-invariant collection of geodesics in D, so the quotient is a collection of 3g − 3 geodesics in the hyperbolic surfaces S. Observe that γ in E has a lift to E that lies in E A if and only if γ intersects the closed geodesic corresponding to A. Moreover, γ intersects P only at this closed geodesic. Let γ X0 , γ X1 be the oriented closed geodesics in S that correspond to X 0 , X 1 respectively. It is sufficient to show that if ψ(X 0 ) = ψ(X 1 ) = {(suc −1 (e i ), e i , suc(e i ), T i , t(e i , e i+1 ))} m i=1 , then γ X0 and γ X1 are homotopic as oriented curves. We will do this by constructing polygons in S along the paths of γ X0 and γ X1 , and show that we can homotope the subsegments of γ X0 and γ X1 that lie in these polygons relative to the edges of the polygons. Pick any A in Γ P and let a − , a + be the repelling and attracting fixed points of A respectively. Also, let r 0 and r 1 be two oriented subsegments of ∂ ∞ Γ = ∂D with endpoints a − and a + , oriented from a − to a + , and such that the orientation on r 0 agrees with the clockwise orientation on ∂D. Let l := {b 0 , b 1 } and l := {b 0 , b 1 } be two consecutive geodesics in E A , with b 0 and b 0 in r 0 and l preceding l . In r 0 , there is a unique c 0 strictly between b 0 and b 0 so that {c 0 , a − } lies in T . Similarly, in r 1 , there is a unique c 1 strictly between b 1 and b 1 so that {c 1 , a + } lies in T . Let H l,l be the closed convex hexagon in D with vertices b 0 , c 0 , b 0 , b 1 , c 1 , b 1 , and observe that π restricted to the interior of H l,l is also injective. (See Figure 8.) For j = 0, 1, let γ Xj be the axis of X j and let e i be a lift of e i that intersects γ Xj . Then define the points p i,−,Xj := suc −1 ( e i ) ∩ γ Xj , p i,+,Xj := suc( e i ) ∩ γ Xj , and let p i,±,Xj = π( p i,±,Xj ). Let α i,Xj be the oriented closed subsegment of γ Xj containing π( e i ∩ γ Xj ) and with endpoints p i,−,Xj , p i,+,Xj , oriented from p i,−,Xj to p i,+,Xj . Also, let β i,Xj be the closed subsegment of γ Xj containing π( e i ∩ γ Xj ) and with endpoints p i,−,Xj , p i+1,+,Xj , oriented from p i,−,Xj to p i+1,+,Xj . Observe that γ Xj can be written as the cyclic concatenation where · is concatenation and the inverse is reversing the parameterization. Since ψ(X 0 ) = ψ(X 1 ), we know that the initial and terminal endpoints of α i,X0 lie on the same edges of T as those of α i,X1 respectively. For the same reasons, the initial and terminal endpoints of β i,X0 lie on the same edges of T as those of β i,X1 . It is thus sufficient to show that for all i = 1, . . . , m, (1) α i,X0 is homotopic to α i,X1 and (2) β i,X0 is homotopic to β i,X1 as oriented curves relative to the edges in T containing their endpoints.
First, we will show that (1) holds. Observe that α 0 := α i,X0 and α 1 := α i,X1 lie in π(Q ei ) for some lift e i of e i . Also, for each vertex of e i , the two edges of Q ei adjacent to this vertex are mapped via π to the same edge in T . (See Figure 7.) Since we know e i is the same type (Z or S) for both X 0 and X 1 , the lifts α 0 , α 1 of α 0 , α 1 respectively that lie in Q ei have their initial endpoints in a common edge of Q ei and their terminal endpoints in a common edge of Q ei . It is thus clear that (1) holds.
We now need to show that for k = 0, . . . , |t(e i , e i+1 )|, the segments β k,0 and β k,1 are homotopic relative to the edges in E A(ei,ei+1) and T that contain their endpoints.
Observe that β k,0 and β k,1 lie in π(H l,l ) for any consecutive pair l, l in E A(ei,ei+1) , with l preceding l . Consider the lift β 0,j of β 0,j that lies in H l,l . Since the initial endpoint of β 0,j is p i,−,Xj , the triple (suc −1 (e i ), e i , suc(e i )) determines the edge of H l,l that p i,−,Xj lies in. Observe then that • if t(e i , e i+1 ) < 0, the terminal endpoint q 1,j of β 0,j lies in l, • if t(e i , e i+1 ) > 0, the terminal endpoint q 1,j of β 0,j lies in l , • if t(e i , e i+1 ) = 0, the triple (suc −1 (e i+1 ), e i+1 , suc(e i+1 )) determines the edge of H l,l containing p i,−,Xj , which is the terminal endpoint of β 0,j .
In any case, this proves that β 0,0 and β 0,1 have initial endpoints on the same edge of H l,l and terminal endpoints on the same edge in H l,l . Similar arguments show the same for β k,0 and β k,1 for k = 1, . . . , |t(e i , e i+1 )|, so (2) holds.

5.2.
Crossing and winding (p)-subsegments of X. For the rest of this section, we will use the combinatorial description ψ(X) of ρ(X) to obtain a lower bound for l ρ (X). Let H be the plane ξ(x − ) (1) + ξ(x + ) (1) in R n . The next two definitions describe two kinds of subsegments of P(H) that we will use to obtain our lower bound. These are the crossing (p)-subsegments and the winding (p)-subsegments The crossing (p)-subsegments are generalizations of the crossing segments used in [26] while the winding (p)-subsegments are analogous to the winding and pantschanging segments used in [26].
If we apply (2) of Lemma 2.5 to this setting for a fixed p = 0, . . . , n − 1, we see that there is some subsegment γ of P(H) with endpoints ξ(x − ) (1) , ξ(x + ) (1) that contains L p,+ ( e), L p,− ( e) for all e in B. In particular, c p ( e) and w p ( e) are welldefined and lie in γ for all e in B. In fact, we can say more. If we orient γ from ξ(x − ) (1) to ξ(x + ) (1) , then we also have orientations induced on c p ( e) and w p ( e). Part (2) of Lemma 2.5 then tells us that the orientation on c p ( e) is from L p,− ( e) to L p,+ ( e) and the induced orientation on w p ( e, e ) is from L p,− ( e) to L p,+ ( e ).
Next, we define a notion of length for subsegments of γ. This gives us a notion of length for each crossing (p)-subsegment and winding (p)-subsegment, which we will use to obtain a lower bound for l ρ (X).
We will now obtain a lower bound for the length of X in terms of the lengths of the crossing (p)-subsegments and winding (p)-subsegments of H. Choose an edge z in Z and let z = X · z. Observe that the set of elements in Z between z and z is finite, so we can enumerate them according to the ordering on Z. In other words, the set of elements between z and z can be written as Lemma 5.9. Fix any p = 0, . . . , n − 1. Then Proof. By Proposition 2.9, we know log Moreover, since suc( z i ) suc −1 ( z i+1 ), we can use (2) of Lemma 2.5 to see that for i = 1, . . . , |Z|. Appying (8) of Proposition 2.7, we then have Taking the logarithm gives us the lemma.
Similarly, if we choose an edge s in S and let s = X · s, then we can label the edges in S between s and s by The same proof as in Lemma 5.9 will give the following lemma.
We now want a similar lower bound for l ρ (X) in terms of the lengths of the winding (p)-subsegments. As before, let b be any edge in B, let b = X · b and label the edges in B between b and b by Also, define that are of the same type}, that are not of the same type}.
Lemma 5.11. Fix p = 0, . . . , n − 1. Then Note that either s i,j = s i ,j or s i,j ∩ s i ,j is at most a single point. Also, it is clear that Observe also that any point in the subsegment between L p,− ( b 1 ) and L p,+ ( b |B|+1 ) is contained in the interior of at most three different winding (p)-segments. Hence, Now, we can give a lower bound for l ρ (X) in terms of the lengths of the crossing (p)-subsegments and the winding (p)-subsegments.
These imply that where the final inequality is a consequence of Proposition 2.12. Proof of (2). Similarly, let s 1 , s 1 be the closed subinterval of s 1 with endpoints b and x + , x − and suc −1 (b) respectively. For p = 0, . . . , n − 2 and any r = 1, . . . , n − p − 1, there exists b in s 1 and b in s 1 (see Figure 9) so that Figure 9.
Hence, we have where the last equality follows from (6) and (7) of Proposition 2.7.
A proof similar to the one for Lemma 5.14 gives the following lemma. Recall that Q = T \ P as defined in Section 3.2. For the remainder of this section, let e and e be consecutive elements in B with e preceding e , and let e, e be the equivalence classes in B that contain e and e . Also, let a − and a + be the repelling and attracting fixed points for A = A( e, e ) respectively. Lemma 5.17. If e and e are not of the same type, then l(w 1 ( e, e )) + l(w n−2 ( e, e )) ≥ max{0, |t(e, e )| − 1} · L.
Proof. Let r 0 , r 1 be the oriented subsegments of ∂ ∞ Γ with endpoints a − , a + , oriented from a − to a + , so that the orientation on r 0 agrees with the clockwise orientation on ∂ ∞ Γ. Since e and e are not of the same type, either both of x − , x + lie in r 0 or both of x − , x + lie in r 1 . If the former holds, we will show that l(w 1 ( e, e )) ≥ max{0, |t(e, e )| − 1} · L and if the latter holds, we will show that l(w n−2 ( e, e )) ≥ max{0, |t(e, e )| − 1} · L. Figure 10. e, e are not of the same type.
Also, if we let 0 < λ 1 < · · · < λ n be the eigenvalues for ρ(A), then for any p = 0, . . . , n − 2, one can compute that where the inequality is a consequence of Proposition 2.12, and the last equality is a computation similar to that done in Proposition 2.9. By taking the product of these inequalities over p = 0, . . . , n − 2, and then taking logarithm, we obtain (n − 1) · l(w 1 ( e, e )) ≥ (t(e, e ) − 1) · log λ n λ 1 .
From here, the proof will proceed in two cases, depending on the sign of t(e, e ).
x + A t(e,e ) · c 1 a + Figure 11. e, e are of the same type and t(e, e ) > 0.
By (2) of Lemma 2.5, we see that α p , α p , β p , β p lie on P(H) in that order. Hence, we can apply Proposition 2.12 to see that Observe that for any integer k, (1) of Proposition 2.12 says that This then implies that δn−p+1 δn−p < 0. Thus, we have By doing this for all p = 1, . . . , n − 1 and taking the product, we get that l(w p ( e, e )) ≥ log δ n δ 1 + (t(e, e ) − 1) · log λ n λ 1 . A t(e,e ) · c 1 Figure 12. e, e are of the same type and t(e, e ) < 0.
As before, (2) of Lemma 2.5, implies that α p , α p , β p , β p lie on P(H) in that order, thus allowing us to use Proposition 2.10 and Proposition 2.12 to compute .
As before, .
The inequality in the lemma follows immediately from this.
As a corollary of the estimates in Proposition 5.12, Proposition 5.16, Lemma 5.17 and Lemma 5.18, we obtain the following theorem.

Degeneration along internal sequences
In this section, we will use the analysis in Section 5 to prove Theorem 4.5.
6.1. Proof of (1) of main theorem. Observe that X is an element of Γ such that X = A k for any A in Γ P and any integer k if and only if r(ψ(X)) = 0. Theorem 5.19 then implies that for any ρ in Hit n (S), K(ρ) is a lower bound for Θ(ρ). Thus, to prove (1) of Theorem 4.5, it is sufficient to prove the following theorem.
We start by using the closed leaf equalities (3.2), (3.3) and (3.4) to prove the following lemma. (2) There is some number z 0 ∈ {1, . . . , n − 2} so that there are triangle invariants τ 1 , τ 2 in {τ (x,y,z),j : z = z 0 } ∪ {τ (x,y,z),j : z = z 0 } satisfying and the triangle invariants in {τ (x,y,z),j : z < z 0 } ∪ {τ (x,y,z),j : z < z 0 } are bounded above and below along {ρ i } ∞ i=1 . Proof. First, we prove the claim that if (1) does not hold, then there must be some triangle invariant τ 2 in A j ∪ A j so that Suppose that every triangle invariant in A j ∪A j is bounded below by a real number when evaluated along Then, by Equation (3.4) and the assumption that every triangle invariant for P j is bounded below by a real number when evaluated along {ρ i } ∞ i=1 , we can deduce that there is some shear invariant σ 2,[cj ,aj ] associated to the edge [c j , a j ] so that We have thus proven that under the hypothesis that every triangle invariant in A j ∪A j is bounded below by a real number when evaluated along {ρ i } ∞ i=1 , the three edges [a j , b j ], [b j , c j ] and [a j , c j ] each have a pair shear invariants associated to them with the property that one of them converges to −∞ while the other converges to ∞. In particular, (1) holds. The contrapositive of this is the claim.
Next, we prove that if (1) does not hold, then (2) must hold. Suppose (1) does not hold. By Equation (3.5), we see that every shear invariant for [a j , b j ] is bounded below along {ρ i } ∞ i=1 if and only if every shear invariant for [a j , b j ] is bounded above along {ρ i } ∞ i=1 . Hence, the shear invariants for the edge [a j , b j ] are bounded both above and below.
Also, we can assume without loss of generality that τ 2 in the above claim lies in {τ (x,y,z),j : z = z 0 } ∪ {τ (x,y,z),j : z = z 0 } for some z 0 so that the triangle invariants in Finally, suppose for contradiction that there is some z 0 < z 0 with the property that there is some triangle invariant τ 1 in {τ (x,y,z),j : z = z 0 } ∪ {τ (x,y,z),j : z = z 0 } such that lim i→∞ τ 1 (ρ i ) = ∞.
We can assume that z 0 is the minimal such number. A similar proof as the one given above then implies that there is some τ 2 in {τ (x,y,z),j : z = z 0 } ∪ {τ (x,y,z),j : z = z 0 } with lim i→∞ τ 2 (ρ i ) = −∞.
However, this contradicts the definition of z 0 .
Armed with Lemma 6.2, we are now ready to prove Theorem 6.1.
Proof of Theorem 6.1. For any subsequence of {ρ i } ∞ i=1 , choose a further subsequence, denoted {ρ i k } ∞ k=1 , so that the shear and triangle invariants converge (possibly to ∞ or −∞) along {ρ i } ∞ i=1 . It is sufficient to show that lim k→∞ K(ρ i k ) = ∞.
For the rest of the proof, we will simplify notation by relabeling the sequence . Let {a, b} be any edge in T that is not a closed leaf, and let c, d be the unique pair of points in ∂ ∞ Γ such that {a, c}, {b, c}, {a, d}, {b, d} are all edges in T . It is sufficient to show that both of the following hold: (a) There is a sequence {M i } ∞ i=1 of (n − 2)-dimensional subspaces in R n so that M i is an element of By Lemma 3.6, we can assume without loss of generality that the edge a = a j , b = b j , c = c j and d = A j · c j for some pair of pants P j . In this setting, either (1) or (2) of Lemma 6.2 must hold.
Next, suppose that (2) holds. Let {e 1 , . . . , e n } be the standard basis for R n and choose normalizations so that ξ i (a j ) (k) ∩ ξ i (b j ) (n−k+1) = [e k ] for k = 1, . . . , n and ξ i (c j ) (1) = [e 1 + e 2 + · · · + e n ]. By Theorem 3.10 and our analysis of the shear and triangle invariants in Section 3.3 and 3.4, we know that there is some representation ρ in Hit n (S) with corresponding Frenet curve ξ such that under a suitably chosen normalization, the following hold.
The above inequality together with inequality (6.3) imply the lemma. Proof. By the definition of an internal sequences, we know that the sequence {L(ρ i )} ∞ i=1 is bounded away from 0 and ∞. This means in particular that is a positive number. Also, by Theorem 6.1, we know that lim i→∞ K(ρ i ) = ∞. This proposition thus follows if we can prove the following statement: For any pair of sequences of positive numbers {T j } ∞ j=1 and {K i } ∞ i=1 such that lim i→∞ T j = ∞ and lim j→∞ K i = ∞, we have that lim i→∞ lim sup j→∞ 1 T j log Tj −Qi,j ·Ki L0 where Q i,j ∈ {0, 1, . . . ,

Tj
Ki } is the integer so that Tj −Qi,j ·Ki L0 This is exactly Proposition 3.21 of [26].