On Finsler manifolds of negative flag curvature

Yong Fang and Patrick Foulon

D´epartement de Math´ematiques,Universit´ede Cergy-Pontoise, 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France (e-mail : [email protected]) CIRM, Luminy, case 916, 13288 Marseille Cedex 9, France (e-mail: [email protected])

Abstract One of the key differences between Finsler metrics and Rieman- nian metrics is the non-reversibility, i.e. given two points p and q, the Finsler distance d(p, q) is not necessarily equal to d(q, p). In this pa- per, we build the main tools to investigate the non-reversibility in the context of large-scale geometry of uniform Finsler Cartan-Hadamard manifolds. In the second part of this paper, we use the large-scale geometry to prove the following dynamical theorem: Let ϕ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then its cohomological pressure is equal to its Liouville metric entropy. This result generalizes a previous Riemannian result of U. Hamenst¨adt.

1 Introduction

Let M be a C∞ manifold and TM be its tangent bundle. A C∞-Finsler + metric on M is a function F : TM → R satisfying: (a) F (tu) = tF (u) for any u ∈ TM and t ≥ 0; (b) F is strictly positive and C∞ over TM − {0}; (c) In standard local coordinates (xi, yi) on TM, the matrix of partial deriva- 2 2 tives ∂ F is positive definite. ∂yi∂yj A Finsler metric F is said to be Riemannian if there exists a C∞- √ Riemannian metric g such that F = g. We say that F is reversible if

1 1 INTRODUCTION 2 for any u ∈ TM, F (−u) = F (u). If it is not the case, F is said to be non-reversible. Given a Finsler manifold (M,F ) and p, q ∈ M, by taking the infimum of the F -length of piecewise C1 curves joining p to q, we obtain the Finsler distance d(p, q). Since F is generally non-reversible, d(p, q) is not necessarily equal to d(q, p). In Finsler geometry, the so-called flag curvature generalizes the Rieman- nian sectional curvature (see Section 2). We say that (M,F ) is a Finsler Cartan-Hadamard manifold if the following conditions are satisfied: (a) M is connected and simply connected; (b) There existsa ¯ > 0 such that the flag curvatures of (M,F ) are less than or equal to −a¯2; (c) The Finsler distance function d is forward complete, which means that forward Cauchy sequences converge (see Section 2). For any s ∈ M , v ∈ TxM − {0} and u ∈ TxM we define in standard 2 2 local coordinates g (u, u) = 1 P ∂ F (v)uiuj, which is a positive definite v 2 ∂yi∂yj p quadratic form. The associated norm is denoted by k · kv= gv(·, ·). Ac- cording to D. Egloff (see [3]), the uniformity constant of (M,F ) is defined as k u kv1 C = supx∈M {supv1,v2,u∈TxM−{0} }. k u kv2

If C < +∞ then the Finsler Cartan-Hadamard manifold (M,F ) is said to be uniform. Principal examples of uniform Finsler Cartan-Hadamard man- ifolds are universal coverings of closed negatively curved Finsler spaces. We should mention that there exist quite a few interesting non-uniform Finsler Cartan-hadamard manifolds (see [1]).

Non-reversible Finsler metrics are quite different from Riemmanian or re- versible Finsler metrics. For example, let γ be an element in the fundamental group π1(M). The lengths of the closed geodesics representing respectively γ and γ−1 are not necessarily equal. Therefore the marked length spectrum is quite different from the Riemannian or reversible Finsler cases. Similarly, the notions such as cross ratio and geodesic currents must be reformulated due to the non-reversibility. It is important for the whole theory of Finlser manifolds that we do not symmetrize or reduce the general Finsler metric to the special Riemannian or reversible Finsler case. The non-reversibility is essential, which should be preserved.

Here is the organization of the first part of this paper on the large-scale 1 INTRODUCTION 3 geometry of uniform Finsler Cartan-Hadamard manifolds: Given a uniform Finsler Cartan-Hadamard manifold (M,F ), we construct two boundaries at infinity: the stable boundary ∂sM and the unstable boundary ∂uM. We prove a Finlser Morse lemma which is used to construct a geometric u s flip homeomorphism σus : ∂ M → ∂ M. A direct consequence is that the u s geodesic space of (M,F ) is naturally parametrized by ∂ M ×∂ M −Gr(σus), where Gr(σus) denotes the graph of σus. By using our geometric flip, we define the so-called unstable-stable angle. Then we use it to formulate and prove the uniformly visible property in our context. Finally we prove the visibility axiom. In the second part of this paper, we study the geodesic flows of closed neg- atively curved Finsler manifolds. In this part, we denote by (M,F ) a closed Finsler manifold of negative flag curvature. Its universal covering (M,f Fe) is a uniform Finsler Cartan-Hadamard manifold. Let ϕ be the geodesic flow of (M,F ) over the spherebundle SM = {u ∈ TM : F (u) = 1}. Let X be the generating vector field of ϕ. It is well-known that ϕ is Anosov, i.e. the tangent bundle T (SM) admits a ϕ-invariant continuous splitting into subbundles ss su T (SM) = RX ⊕ E ⊕ E , ss su such that vectors of E ( resp. E ) are contracted exponentially by Dϕt in the positive (resp. negative) sense.

In general, the subbundles Ess and Esu are only H¨oldercontinuous and rarely more regular. For any k ≥ 1, we say that the Anosov splitting of ϕ is Ck if Ess and Esu are both Ck subbundles. Recall that the topological entropy htop(ϕ), the cohomological pressure P (ϕ) and the Liouville metric entropy hL(ϕ) are three important dynamical invariants of ϕ. In general we have htop(ϕ) ≥ P (ϕ) ≥ hL(ϕ). The celebrated entropy conjecture asserts the following:

Entropy conjecture. Let ψ be the geodesic flow of a closed negatively curved Riemannian manifold. If htop(ψ) = hL(ψ) then the Riemannian metric is locally symmetric of rank one.

For general Finsler geodesic flows, according to [12] and [6], the coincidence between htop(ϕ) and hL(ϕ) is not the appropriate hypothesis in Entropy conjecture (see Subsection 4.4), which should be replaced by the coincidence between P (ϕ) and hL(ϕ). 2 GEOMETRIC PRELIMINARIES 4

With the help of large-scale geometry, we prove the following result relat- ing the regularity of Anosov splitting with the coincidence between P (ϕ) and hL(ϕ), which generalizes a previous Riemannian result of U. Hamenst¨adt:

Theorem. Let ϕ be the geodesic flow of a closed negatively curved Finsler manifold. If its Anosov splitting is C2, then the cohomological pressure of ϕ is equal to its Liouville metric entropy.

2 Geometric preliminaries

2.1 Non-reversible distance Let (M,F ) be a Finsler manifold and c :[a; b] → M be a piecewise C1 curve. R b 0 The length of c is defined as l(c) = a F (c (t))dt. For p, q ∈ M the Finsler distance between p and q is defined as d(p, q) = infcl(c) where the infimum is taken over all piecewise C1 curves joining p to q. The reversed curve of c is defined asc ¯ :[−b; −a] → M such thatc ¯(t) = c(−t). Since F is generally non-reversible, the length l(¯c) is not necessarily equal to l(c), which implies that the Finsler distance d(p, q) is not necessarily equal to d(q, p). The uniformity constant C of (M,F ) is defined as in Section 1. It is clear that closed Finsler manifolds are uniform. Therefore their universal coverings with lifted Finsler metrics are also uniform. We define the conjugate Finsler metric of F as follows:

F ∗(u) = F (−u), ∀ u ∈ TM.

The non-reversible distance of F ∗ is denoted by d∗. We have the following proposition: Proposition 2.1. Let (M,F ) be a uniform Finsler manifold with uniformity constant C. For any v ∈ TM and any p, q ∈ M, 1 1 F (v) ≤ F ∗(v) ≤ CF (v); d(p, q) ≤ d∗(p, q) = d(q, p) ≤ Cd(p, q). C C

Proof. Let v ∈ TxM − {0}. Since the Finsler metric F is positively 2 homogeneous of degree one, then gv(v, v) = (F (v)) , i.e. k v kv= F (v). Similarly we have

2 ∗ 2 g−v(v, v) = g−v(−v, −v) = (F (−v)) = (F (v)) ,

∗ i.e. k v k−v= F (v). Thus by the definition of the uniformity constant, we 1 F ∗(v) 1 ∗ obtain C ≤ F (v) ≤ C, which implies that C d(p, q) ≤ d (p, q) ≤ Cd(p, q). 2 GEOMETRIC PRELIMINARIES 5

Let c be a piecewise C1 curve in M andc ¯ its reversed curve. By the definition of F ∗ it is clear that l∗(¯c) = l(c). Therefore d∗(p, q) = d(q, p). The proof is complete. 

Let c(t) be a C∞ curve in M. We say that c is a geodesic if it has con- stant speed and verifies the following:

For ∀ t1 < t2 close enough, l(c |[t1,t2]) = d(c(t1), c(t2)). The Finsler manifold (M,F ) is said to be forward complete if any geodesic defined over [a, b[ can be extended to a geodesic over [a, +∞[. Similarly (M,F ) is said to be backward complete if any geodesic defined over ]a, b] can be extended to a geodesic over ] − ∞, b]. We say that (M,F ) is complete if it is both forward complete and backward complete. Corollary 2.2. If a uniform Finsler manifold (M,F ) is forward complete, it is also backward complete, thus complete. Proof. By Proposition 1, any forward Cauchy sequence of d∗ is also forward Cauchy with respect to d. Therefore by the theorem of Hopf-Rinow (see [1]), (M,F ∗) is also forward complete. Now let c be a geodesic of F defined over ]a, b]. By Proposition 2.1, its reversed curvec ¯ is a geodesic of F ∗. We deduce thatc ¯ can be extended to a F ∗-geodesic over [−b, +∞[. Therefore the reversed curve ofc ¯ is a F -geodesic over ] − ∞, b] which extends c. Thus (M,F ) is backward complete. 

2.2 Finsler differential geometry There are several ways to do differential geometry over Finsler manifolds, according to the chosen connection. Well-known Finsler connections are Berwald connection, Cartan connection, Chern connection and dynamical derivative. The reader can choose anyone that he (or she) prefers. Anyway, different choices of connections give rise to the same Riemman curvature tensor R (i.e. the Jacobi endomorphism in [7]), which can be viewed as a family of linear endomorphisms indexed by vectors:

R = {Rv : TxM → TxM : x ∈ M, v ∈ TxM − {0}}

(see [1]). Given a tangent plane P containing v, the flag curvature of P in the direction v is defined as

gv(Rv(u), u) K(P, v) = 2 , gv(v, v)gv(u, u) − (gv(v, u)) 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 6 where the plane P is generated by v and u. Flag curvature is the Finsler generalization of Riemannian sectional curvature. As in Riemannian geome- try, the sign of flag curvature controlls the behaviour of geodesics via Jacobi fields, which are defined as follows: Let c(t) be a geodesic of (M,F ) and J(t) a C∞ vector field along c. Let 0 J (t) = Dc0(t)J(t) be the derivative of J(t) given by the chosen connection. We say that J(t) is a Jacobi field if

00 J (t) + Rc0(t)(J(t)) = 0.

The variation of k J(t) kc0(t) is controlled by the sign of flag curvatures (see [1]).

3 Large-scale geometry in the uniform case

Throughout this section, let (M,F ) be a uniform Finsler Cartan-Hadamard manifold. For any p, q ∈ M, by Corollary 2.2 and the Cartan-Hadamard theorem, there exists a unique unit speed geodesic joining p to q, which is denoted by [p, q]. Let C be the uniformity constant of (M,F ) and leta ¯ > 0 such that the flag curvatures of (M,F ) are less than or equal to −a¯2. Let F ∗ be the conjugate Finsler metric of F , which is also a uniform Finsler Cartan- Hadamard manifold with the same uniformity constant and the same upper curvature bound.

3.1 Finsler Morse lemma The Morse lemma is a large-scale consequence of Jacobi field estimates. Since we do not symmetrize our Finsler metric, our strategy to prove Morse lemma is then to use simultanously the Finlser metric F and its conjugate metric F ∗. Let us begin with several definitions: Definition 3.1. The forward open ball of center x and of radius r is defined as BF (x, r) = {y ∈ M : d(x, y) < r}.

The backward open ball is defined as BF ∗ (x, r). Let K be an open subset of M and K¯ be its closure. We say that K is strictly convex if for any p, q ∈ K¯ , the open geodesic segment ]p, q[ is contained in K. For general Finsler manifolds, J.H.C. Whitehead proved in [14] that small forward (or backward) open balls are strictly convex. In the special case 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 7 of uniform Finsler Cartan-Hadamard manifolds, all forward (or backward) open balls are in fact strictly convex (see [4]). Therefore for any p ∈ M and any geodsic segment c in M, there exists a unique point q ∈ c such that

d(p, q) = inf{d(p, c(t))}.

This point q is said to be the F -projection of p over c and denoted by P rc(p). Since the reversed curvec ¯ is a F ∗-geodesic, there exists a unique point q0 ∈ c¯ such that d∗(p, q0) = inf{d∗(p, c¯(s))}. Therefore by Proposition 1 we have

d(q0, p) = d∗(p, q0) = inf{d∗(p, c¯(s))} = inf{d(c(t), p)}.

This point q0 is said to be the F ∗-projection of p over c and denoted by ∗ P rc (p). Definition 3.2. Let λ, k > 0. A piecewise C1 curve γ : I → M is called a (λ, k)-quasi-geodesic if for any [t0, t1] ⊆ I,

l(γ |[t0,t1]) ≤ λ · d(γ(t0), γ(t1)) + k. For A ⊆ M and p ∈ M the distance from p to A is defined as

d(p, A) = inf{d(p, q): q ∈ A}.

Similarly the distance from A to p is defined as

d(A, p) = inf{d(q, p): q ∈ A}

For A, B ⊆ M the Hausdorff distance from A to B is defined as

dH (A, B) = max{supp∈A{d(p, B)} ; supq∈B{d(A, q}}.

The Finsler Morse lemma can be formulated as follows: Theorem 3.3. (Morse lemma) Let (M,F ) be a uniform Finsler Cartan- Hadamard manifold and γ :[a, b] → M be a (λ, k)-quasi-geodesic. Then there exists a constant D = D(λ, k, C, a¯) such that

dH (γ([a, b]), [γ(a), γ(b)]) ≤ D(λ, k, C, a¯).

The proof will be done via several lemmas by adapting the Riemannian (or reversible Finsler) arguments to our setting (see [11]): 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 8

Lemma 3.4. Let γ :[a, b] → M be a piecewise C1 curve and c :[a0, b0] → M ∗ ∗ be a geodesic segment. Let p = P rc (γ(a)) and q = P rc (γ(b)). If for any s ∈ [a, b], d(γ(s), c) ≥ ρ > 0, then

cosh( aρ¯ ) l(γ) ≥ C d(p, q). C2 Proof. Recall firstly that C is the uniformity constant of (M,F ). With- ∗ out loss of generality we suppose that for any s ∈ [a, b], P rc (γ(s)) ∈ ]c(a0), c(b0)[. Let β(s) be the curve of F ∗-projections of γ(s) over c. For any s ∈ [a, b], let v(s) be the tangent vector at β(s) of the F ∗-geodesic from γ(s) to β(s). By the first variation formula of arc length we have

∗ 0 gv(s)(v(s), β (s)) = 0.

Let X(s) be the vector field along c such that expβ(s)(X(s)) = γ(s). Since the reversed curve of the F ∗-geodesic from γ(s) to β(s) is the F -geodesic from β(s) to γ(s), then X(s) and −v(s) are positively colinear. Moreover, ∗ since g−v(s) = gv(s) then we deduce that

0 gX(s)(X(s), β (s)) = 0, i.e. X(s) is perpendicular to c with respect to gX(s). Take some s0 ∈ [a, b] and let l0 = d(β(s0), γ(s0)). We consider the F -geodesic variation over [a, b] × [0, l0]: t cs(t) = expβ(s)( X(s)). l0 0 Its variation vector field at s0 is a Jacobi field J with J(0) = β (s0) and 0 J(l0) = γ (s0). Now with respect to gc0 we decompose J(t) such that s0

J(t) = AtT + J ⊥,

0 ⊥ 0 where A ∈ , T = c and gc0 (T,J ) = 0. Since gX(s)(X(s), β (s)) = 0, R s0 s0 ⊥ 0 then J (0) = β (s0). Now by Jacobi field estimates we have for any t ∈ [0, l0],

⊥ ⊥ k J (t) kc0 (t)≥ cosh(¯at) k J (0) kc0 (0) . s0 s0 In particular

0 ⊥ 0 k γ (s0) kc0 (l )≥k J (l0) kc0 (l )≥ cosh(¯al0) k β (s0) kc0 (0) . s0 0 s0 0 s0 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 9

Therefore

0 1 0 cosh(¯al0) 0 cosh(¯al0) 0 F (γ (s0)) ≥ k γ (s0) kc0 (l )≥ k β (s0) kc0 (0)≥ F (β (s0)). C s0 0 C s0 C2 Since d(γ(s), c) ≥ ρ for any s, then 1 ρ l = d(β(s ), γ(s )) ≥ d(γ(s ), β(s )) ≥ . 0 0 0 C 0 0 C

aρ¯ 0 cosh( C ) 0 We deduce that F (γ (s0)) ≥ C2 F (β (s0)). So by integrating we get aρ¯ cosh( C ) l(γ) ≥ C2 d(p, q). The proof is complete.  Lemma 3.5. Under the assumptions of Lemma 3.4, if γ is a (λ, k)-quasi- geodesic then there exists ρ0 = ρ0(λ, k, C, a¯) verifying the following: If d(γ(a), c) = d(γ(b), c) = ρ0 and d(γ(t), c) ≥ ρ0 for any t ∈ [a, b], then d(p, q) ≤ 1 Proof. By Lemma 3.4 we have cosh( aρ¯ ) C d(p, q) ≤ l(γ) ≤ λd(γ(a), γ(b)) + k C2 ≤ λ(d(γ(a), p) + d(p, q) + d(q, γ(b))) + k Moreover by Proposition 2.1 we have

∗ ∗ 2 2 d(γ(a), p) ≤ Cd (γ(a), p) = Cd (γ(a), c) ≤ C d(γ(a), c) = C ρ0.

∗ ∗ Similarly d(q, γ(b)) = d (γ(b), q) = d (γ(b), c) ≤ Cd(γ(b), c) = Cρ0. There- fore cosh( aρ¯ ) C d(p, q) ≤ λ(C2ρ + d(p, q) + Cρ ) + k. C2 0 0 Thus it is clear that for ρ0 large enough we have d(p, q) ≤ 1. The proof is complete. 

Proof of Morse lemma. (a) Let c = [γ(a), γ(b)] and suppose that there exists t ∈ [a, b] such that d(γ(t), c) > ρ0. There exists [t0, t1] ⊆ [a, b] such that t ∈ [t0, t1] and

d(γ(t0), c) = d(γ(t1), c) = ρ0; d(γ(s), c) ≥ ρ0, ∀ s ∈ [t0, t1].

∗ Let p and q be the F -projections of γ(t0) and γ(t1) over c. By Lemma 3.5 we have

2 d(γ(t), c) ≤ l(γ |[t0,t1]) + d(γ(t1), q) ≤ λd(γ(t0), γ(t1)) + k + C d(γ(t1), c) 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 10

2 ≤ λ(d(γ(t0), p) + d(p, q) + d(q, γ(t1))) + k + C ρ0 2 2 ≤ λ(C ρ0 + 1 + Cρ0) + k + C ρ0 = D1(λ, k, C, a¯).

(b) Conversely, let a = t0 ··· < tn = b such that d(γ(ti), γ(ti+1)) ≤ 1. Let pi ∗ the F -projection of γ(ti) over c. We have

∗ ∗ d(pi, γ(ti)) = d (γ(ti), pi) = d (γ(ti), c) ≤ Cd(γ(ti), c) ≤ CD1(λ, k, C, a¯).

Let q ∈ c between pi and pi+1 for some i. We have

2 2 d(γ, q) ≤ Cd(q, γ) ≤ C(d(q, pi)+d(pi, γ(ti))) ≤ C d(pi, q)+C D1(λ, k, C, a¯)

2 2 ≤ C (d(pi, γ(ti)) + d(γ(ti), γ(ti+1)) + d(γ(ti+1), pi+1)) + C D1(λ, k, C, a¯)

= D2(λ, k, C, a¯).

The proof is complete. 

3.2 Forward uniform visibility Let SM = {u ∈ TM : F (u) = 1} be the unit bundle. For any p ∈ M, let SpM be the sphere fiber over p. The family of inner products {gv} induces a ∞ C Riemannian metric over SpM, denoted by gp. Let dp the distance over SpM determined by gp. Let x, y, p ∈ M, the forward angle between x and y with respect to p is defined as 0 0 ]p(x, y) = dp(c1(0), c2(0)), where c1 (resp. c2) is the unit speed geodesic from p to x (resp. y). Definition 3.6. We say that (M,F ) is forward uniformly visible if for any  > 0 there exists R = R(, C, a¯) such that for any x, y, p ∈ M,

d(p, [x, y]) ≥ R(, C, a¯) =⇒ ]p(x, y) ≤ . Theorem 3.7. Let (M,F ) be a uniform Finsler Cartan-Hadamard mani- fold. Then (M,F ) is forward uniform visible

The proof of this theorem will be done via several lemmas:

Lemma 3.8. Let p ∈ M and c, γ be two unit speed positive geodesic rays issuing from p. For any α ∈ [0, 1] and a1, a2 > 0 we have

2 d(c(αa1), γ(αa2)) ≤ C · α · d(c(a1), γ(a2)) 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 11

Proof. Let u(s) ∈ TpM such that expp(u(s)) is the unit speed geodesic segment from c(a1) to γ(a2). We consider the geodesic variation cs(t) = expp(tu(s)). Let J(t) be the Jacobi field at s0, which admits the decompo- sition J(t) = AtT + J ⊥(t), 0 ⊥ where A is a constant, T = cs and J (0) = 0. Therefore for any t and α ∈ [0, 1], by Jacobi field estimates, t k J ⊥(t) k ≥ k J ⊥(αt) k . T αt T We deduce that 1 k J(t) k ≥ k J(αt) k . T α T 2 Thus F (J(αt)) ≤ C k J(αt) kT ≤ C ·α k J(t) kT ≤ C ·α·F (J(t)). The proof is complete by integration.  Lemma 3.9. Let p ∈ M and c, γ be two unit speed positive geodesic rays issuing from p. Suppose that c0(0) = v and γ0(0) = u. For any  > 0, there exists a positive constant P (, C, a¯) such that if dp(v, u) >  then for any s, t ≥ 0, d(c(s), γ(t)) ≥ P (, C, a¯)(s + t). Proof. Let us define firstly a C∞ Riemannian metric over M: For any p ∈ M, let gp be the Riemannian metric over SpM defined above. Let νp be its volume form and V ol(SpM) its Riemannian volume. For any u, w ∈ TpM we define 1 Z gi(u, w) = gv(u, w)dνp, V ol(Sp(M)) v∈SpM which is a C∞ Riemannian metric over M satisfying for any u ∈ TM, 1 F (u) ≤ pg (u, u) ≤ CF (u). C i V For any p ∈ M we define a plat Riemannian metric gi over TpM by extending the scalar product (gi)p via translations. Since for any p ∈ M, Dpexp = Id, then there exists an open subset U of TM verifying the fol- lowing conditions: (a) For any p ∈ M, U ∩ TpM is an open neighborhood of the zero vector in TpM; (b) For any p ∈ M, any q ∈ expp(U ∩ TpM) and any u ∈ TqM, 1 q F (v) ≤ gV (exp−1(u), exp−1(u)) ≤ 2C · F (u). 2C i p p 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 12

Now let N be a positive integer large enough such that the geodesic segment s t R [c( N ), γ( N )] is contained in expp(U ∩ T pM). Let dp be the distance over V TpM induced by the flat Riemannian metric gi . Then by Condition (b), we have s t s t 2Cd(c( ), γ( )) ≥ dR(exp−1(c( )), exp−1(γ( ))) N N p p N p N s t 1 = dR( v, u) = dR(sv, tu). p N N N p N s t By Lemma 3.8 we have d(c(s), γ(t)) ≥ C2 d(c( N ), γ( N )). Therefore 1 d(c(s), γ(t)) ≥ dR(sv, tu). 2C3 p

V R Let k · kp be the norm on TpM induced by gi and let ] be the Euclidean V √ √ angle measure defined by gi . Since for any v ∈ TpM − {0}, gv and gi are C-bi-Lipschitz equivalent then fo any v1, v2 ∈ SpM,

1 R v1 v2 dp(v1, v2) ≤ ] ( , ) ≤ Cdp(v1, v2). C k v1 kp k v2 kp

So if d (v, u) > , we have R( v , u ) >  . Now by applying the follow- p ] kvkp kukp C ing elementary lemma: 2 Let ABC be a triangle in R . For any π ≥  > 0 there exists a positive constant ∆() such that if ]B ≥  then AC ≥ ∆() · (AB + BC), we obtain 1 ∆(  ) ∆(  ) d(c(s), γ(t)) ≥ dR(sv, tu) ≥ C (s k v k +t k u k ) ≥ C (s + t). 2C3 p 2C3 p p 2C4 The proof is complete.  Lemma 3.10. Let γ be a geodesic, then its reversed curve γ¯ is a (C2, 0)- quasi-geodesic. Proof. By Proposition 2.1 we have

F (¯γ0(−t)) ≤ CF ∗(¯γ0(−t)) = CF (γ0(t)).

Therefore for any [a, b] contained in the domain of γ, we have

2 l(¯γ |[−b,−a]) ≤ Cd(γ(a), γ(b)) ≤ C d(¯γ(−b), γ¯(−a)).

The proof is complete.  3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 13

Proof of Theorem 3.7. Let  > 0. Let c and γ be two unit speed geodesics from p to x and y. Let c0(0) = v and γ0(0) = u. We suppose that dp(v, u) > . By Lemmas 3.9 and 3.10, there exists a positive constant λ = λ(, C, a¯) such that the curveγ ¯ ∗ c is a (λ(, C, a¯), 0)-quasi-geodesic. Therefore by Morse lemma there exists a positive constant D(, C, a¯) such that dH (¯γ ∗ c, [x, y]) ≤ D(, C, a¯). In paticular, d(p, [x, y]) ≤ D(, C, a¯). We deduce that (M,F ) is forward uniformly visible. 

3.3 Unstable boundary and stable boundary Definition 3.11. Let (M,F ) be a uniform Finsler Cartan-Hadamard man- ifold. Let c1 : [0 + ∞[→ M and c2 : [0, +∞[→ M be two unit speed positive geodesic rays. We say that c1 and c2 are asymptotic if d(c1(t), c2(t)) is bounded over [0, +∞[, which is an equivalence relation by Proposition 1. The stable boundary of (M,F ) is defined as the space of the equivalence classes, denoted by ∂sM. Similarly two unit speed negative geodesic rays γ1 :] − ∞, 0] → M and γ2 :] − ∞, 0] → M are said to be asymptotic if d(γ1(t), γ2(t)) is bounded over ] − ∞, 0]. The unstable boundary denoted by ∂uM is defined similarly.

s For any p ∈ M, let ψp : SpM → ∂ M be the application defined as 0 ψp(u) = [γ(t)], where γ(t) is the positive geodesic ray such that γ (0) = u and [γ] denotes its equivalence class.

s Proposition 3.12. For any p ∈ M, ψp : SpM → ∂ M is bijective.

Proof. Take u, v ∈ SpM such that u 6= v. Let c and γ be the positive geodesic rays issuing respectively from u and v. By Lemma 3.8 we have for any t > 1, t d(c(t), γ(t)) ≥ d(c(1), γ(1)). C2 Since c(1) 6= γ(1), then the positive geodesic rays c and γ are not asymptotic. Thus ψp is injective. Now let γ be an arbitrary unit speed positive geodesic ray issuing from q. We can construct as follows another positive geodesic ray issuing from p: Let un be the tangent vector at p of the unit speed geodesic segment [p, γ(n)]. Without loss of generality we suppose that limn→∞ un = u. Let c(t) be the positive geodesic ray issuing from u. 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 14

Let us verify that c and γ are asymptotic: For any n ∈ N, let γn = [p, γ(n)]. For any t > 0, if n is large enough then d(c(t), γn(t)) < 1. Let ln the F -length of [p, γ(n)]. We have

ln ≤ d(p, q) + l(γ |[0,n]) ≤ d(p, q) + n.

Similarly n ≤ ln + C · d(p, q). Therefore l lim n = 1. n→∞ n ∗ Now we consider the reversed curvesγ ¯n andγ ¯, which are both F - geodesics. By Proposition 2.1 and Lemma 3.8 we have

∗ ∗ ln − t n − t d(γn(t), γ(t)) ≤ C ·d (¯γn(ln −t), γ¯(n−t)) = C ·d (γn( ·ln), γ( ·n)) ln n

∗ ln − t ln − t ∗ ln − t n − t ≤ C[d (¯γn( · ln), γ¯( · n)) + d (¯γ( · n), γ¯( · n))] ln ln ln n l ≤ C3d∗(p, q) + C · t | n − 1 |≤ C3d∗(p, q) + 1, n for n large enough. W deduce that c and γ are asymptotic. The proof is complete.  −1 Proposition 3.13. Let p, q ∈ M, the map ψq ◦ ψp : SpM → SqM is continuous.

Proof. The topology of SpM (resp. SqM) is defined via the distance dp (resp. dq) (see Subsection 3.2). Let n be a large integer and consider the large sphere S(q, n) = {x ∈ M : d(q, x) = n}.

We define a map fn : SqM → S(q, n) as follows: For any u ∈ SqM, let γ be the positive geodesic ray such that γ0(0) = u. We define

fn(u) = γ ∩ S(q, n). ∞ It is clear that fn is a homeomorphism (even C ). By the construction in Proposition 3.12 it is clear that the sequence of −1 −1 maps {fn ◦ψp} converges pointwisely to ψq ◦ψp. Moreover by the forward uniform visibility of (M,F ), the convergence is in fact uniform. Therefore −1 ψq ◦ ψp is also continuous.  By Propositions 3.12 and 3.13, there exists a well-defined topology over s ∂ M such that ψp is a homeomorphism. Since the reversed negative geodesic rays of F are positive geodesc rays of F ∗, then the unstable boundary ∂uM is canonically identified with the stable boundary of F ∗. Therefore by applying Propositions 3.12 and 3.13 to F ∗, ∂uM is also naturally a topological sphere. 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 15

3.4 Geometric flip map Proposition 3.14. Let c :] − ∞, 0] → M be a negative geodesic ray. Then there exists a positive geodesic ray γ such that dH (c, γ) is bounded by a constant depending only on C and a¯.

Proof. By Lemma 3.10, the reversed curvec ¯ : [0, +∞[→ M is a (C2, 0)- quasi-geodesic. Then by Morse lemma there exists a constant D = D(C, a¯) such that for any n ∈ N,

dH (¯c |[0,n], [p, c¯(n)]) ≤ D.

0 Let γn = [p, c¯(n)] and un = γn(0). Without loss of generality we suppose that limn→∞ un = u. Let γ be the unit speed positive geodesic ray issuing from u. We have

dH (c, γ) = dH (¯c, γ) ≤ D = D(C, a¯).

The proof is complete. . If two negative geodesic rays c1 an c2 are asymptotic, then by similar arguments as in Proposition 3.12 their corresponding positive geodesic rays are also asymptotic. Theorefore the following map is well-defined:

u s σus : ∂ M → ∂ M, σsu([c]) = [γ], where γ is the positive geodesic ray constructed in Proposition 3.14. By ∗ s using the conjugate metric F we construct similarly a map σsu : ∂ M → ∂uM such that

σus ◦ σsu = Id∂uM ; σsu ◦ σus = Id∂sM .

Therefore σus and σsu are both bijective. By the construction of σus and the forward uniform visibility, similar arguments as in Proposition 3 prove the following:

Proposition 3.15. Under the notations above, the maps σus and σsu are both continuous.

Therefore the stable and unstable boundaries are natually homeomor- phic. For any p ∈ M and any u ∈ SpM, there exists a unique unit speed negative geodesic ray c :] − ∞, 0] → M such that c0(0) = u. By Propositions 2 and 4 there exists a unique unit speed positive geodesic ray γ issuing from 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 16

0 p such that [γ] = σus([c]). Let v = γ (0). The geometric flip map is defined as follows: σ : SM → SM, σ(u) = v.

Similar construction by using σsu gives rise to a map sending v to u. There- fore σ is homeomorphic.

Recall that the classical flip map I : TM → TM is defined as I(u) = −u. With respect to a Finsler metric F , the classical flip should be modified as follows: The algebraic flip map associated to F , σa : SM → SM is defined as σ (u) = −u . a F (−u) √ In the case that F is Randers, i.e. F = g − θ where g is Riemannian ∞ metric and θ is a C 1-form, we proved in [5] that σ = σa if and only if the 1-form θ is closed. So generally speaking, our geometric flip map is different from the algebraic flip map.

3.5 Unstable-stable angle Let p ∈ M, x ∈ M ∪ ∂uM and y ∈ M ∪ ∂sM. Let v (resp. u) be the tangent vector at p of the unit speed geodesic from x to p (resp. from p to y). The unstable-stable angle of (x, y) with respect to p is defined as

u,s ]p (x, y) = dp(σ(v), u), where σ is the geometric flip. u s u,s For α ∈ ∂ M and β ∈ ∂ M, it is clear that ]p (α, β) ≥ 0 and u,s ]p (α, β) = 0 if and only if σus(α) = β. Definition 3.16. We say that (M,F ) is unstable-stable uniformly visible if for any  > 0 there exists a positive constant R = R(, C, a¯) such that for any x, y, p ∈ M,

u,s d(p, [x, y]) ≥ R =⇒ ]p (x, y) ≤ . Theorem 3.17. Let (M,F ) be a uniform Finsler Cartan-Hadamard mani- fold. Then (M,F ) is unstable-stable uniformly visible. Let us begin the proof with the following lemma:

Lemma 3.18. Under the notations above, let γ1 (resp. γ2) be the unit speed geodesic from x to p (resp. from p to y). Then there exist two positive u,s constants λ = λ(, C, a¯) and k = k(, C, a¯) such that if ]p (x, y) >  then the composition γ1 ∗ γ2 is a (λ, k)-quasi-geodesic. 3 LARGE-SCALE GEOMETRY IN THE UNIFORM CASE 17

Proof. Without loss of generality we suppose that γ1 (resp. γ2) is the unit speed negative (resp. positive) geodesic ray. By Proposition 3.14 there exists a positive geodesic ray c issuing from p such that

dH (γ1, c) ≤ D(C, a¯).

0 0 0 Let v = γ1(0), u = γ2(0) and w = c (0). Then we have w = σ(v) and therefore u,s ]p (x, y) = dp(w, u) > . 0 0 Let s < 0 and t > 0. Let s ≥ 0 such that c(s ) is the F -projection of γ1(s) on the positive geodesic ray c. We have

0 0 0 d(γ1(s), γ2(t)) ≥ d(c(s ), γ2(t))−d(c(s ), γ1(s)) ≥ d(c(s ), γ2(t))−C ·D(C, a¯).

By Lemm 3.9 we have

0 0 d(c(s ), γ2(t)) ≥ P (, C, a¯)(s + t).

By the triangle inequality we get 1 s0 ≥ d(p, γ (s)) − d(c(s0), γ (s)) ≤ | s | −C · D(C, a¯). 1 1 C We deduce that P (, C, a¯) (γ (s), γ (t)) ≥ P (, C, a¯)(s0 +t)−C ·D ≥ (| s | +t)−C ·D(P +1), 1 2 C which means that γ1 ∗ γ2 is a quasi-geodesic. The proof is complete. 

Lemma 3.14 combined with Morse lemma proves Theorem 3.13.

3.6 Visibility axiom Definition 3.19. We say that (M,F ) verifies the visibility axiom if for any u s α ∈ ∂ M and any β ∈ ∂ M such that β 6= σus(α), there exists a unit speed geodesic line c joining α to β, i.e. [c |]−∞,0]] = α and [c |[0,+∞[] = β. Theorem 3.20. Let (M,F ) be a uniform Finsler Cartan-Hadamard mani- fold. u (a) For any α ∈ ∂ M there exists no geodesic line joining α to σus(α); u s (b) For any α ∈ ∂ M and any β ∈ ∂ M such that β 6= σus(α), there exists a unit speed geodesic line c joining α to β, i.e. (M,F ) verifies the visibility axiom. 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 18

Proof. (a) Suppose on the contrary that there exists a geodesic line c joinging α to σus(α). Then we have

dH (c(] − ∞, 0], c([0, +∞[))) ≤ D(C, a¯).

0 0 For any t < 0, let t > 0 such that c(t ) is the F -projection of c(t) on c |[0,+∞[. Thus we get d(c(t), c(t0)) ≤ D(C, a¯). However d(c(t), c(t0)) =| t | +t0 ≥| t |, which gives a contradiction for t large enough.

(b) Let p ∈ M. Let c1 (resp. c2) be the unit speed negative (resp. positive) geodesic ray from α to p (resp. from p to β). Let {pn} ({qn}) be a sequence along c1 (resp. c2) coverging to α (resp. β). Since β 6= σus(α), then for any n, u,s u,s ]p (pn, qn) = ]p (α, β) =  > 0. Therefore by unstable-stable uniform visibility, there exists a positive con- stant R = R(, C, a¯) such that for any n,

d(p, [pn, qn]) ≤ R.

We deduce that there exists mn ∈ [pn, qn] such that d(p, mn) ≤ R. Without loss of generality we suppose that limn→∞ mn = m. Let γ1 (resp. γ2) be the unit speed negative (resp. positive) geodesic ray from α to m (resp. from m to β). For any x ∈ γ1 and y ∈ γ2, since [pn, qn] converges to the curve γ1 ∗ γ2 and mn → m, then we get

d(x, y) = d(x, m) + d(m, y).

Therefore γ1 ∗ γ2 minimizes the distance, which implies that γ1 ∗ γ2 is a geodesic line from α to β. 

4 Finsler geodesic flows in negative curvature

Throughout this section, let (M,F ) be a closed Finsler manifold of strictly negative flag curvature. Let (M,f Fe) be its universal covering, which is a uniform Finsler Cartan-Hadamard manifold. 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 19

4.1 Finsler geodesic flows The geodesic flow ϕ : SM → SM is defined as follows: For any u ∈ SM, there exists a unique unit speed geodesic γ such that 0 γ (0) = u. We define for any t ∈ R,

0 ϕt(u) = γ (t).

The most important feature of ϕ is its uniform hyperbolicity. For the sake of completeness, let us recall the definition of Anosov flows:

Definition 4.1. Let N be a closed C∞ manifold equipped with a Rieman- nian metric and ψ : N → N be a C∞ flow tangent to the vector field X. We say that ψ is Anosov if there exist a, b > 0 and a Dψ-invariant splitting of the tangent bundle TM = X ⊕ Ess ⊕ Esu, such that for any t > 0, u ∈ Ess, v ∈ Esu,

−bt −bt k Dψt(u) k≤ a · e k u k ; k Dψ−t(v) k≤ a · e k v k .

ss su ss su The vector bundles E , E , E ⊕RX and E ⊕RX are called respec- tively strong stable, strong unstable, stable, unstable bundle of ψ. They are all integrable to H¨oldercontinuous foliations with C∞ leaves denoted respectively by W ss, W su, W s and W u. In general, the bundles Ess and Esu are just H¨oldercontinuous and rarely more regular (see [10]).

The following result was proved in [8]:

Theorem 4.2. Let (M,F ) be a closed Finsler manifold of strictly negative flag curvature. Then its geodesic flow ϕ is an Aosov flow. Moreover, the stable and unstable bundles of ϕ are both tranverse to the fibers of the sphere bundle SM → M.

The canonical 1-form of ϕ, denoted by λ, is defined as

λ(X) = 1 and λ(Ess) = λ(Esu) = 0.

It is well-known that the Finsler geodesic flow ϕ is a contact flow with respect to λ, i.e. λ(X) = 1 and dλ(X, ·) = 0.

Let Wfs be the lifted foliation of W s in the covering space SMf → SM. 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 20

Proposition 4.3. Under the notations above, each leaf of Wfs intersects exactly once any fiber of SMf → Mf. s s Proof. Let v ∈ SMf and Wfv be the leaf of Wf containing v. By Theorem 4.2, the foliation Wfs is tranverse to the fibers of the bundle π : SMf → Mf. s Since the fibers are compact, then by [2] the map π | s : Wfv → Mf is a Wfv ∞ covering map. Moreover, since Mf is simply connected, then π | s is a C Wfv diffeomorphism. 

Let Qes (resp. Qeu) be the leaf space of Wfs (resp. Wfu). By Proposition 6 and Subsection 3.3, the stable boundary ∂sM (resp. the unstable bound- ary ∂uM) is canonically identified to Qes (resp. Qeu) as follows: Let v ∈ SMf and π : SMf → Mf be the canonical projection. The leaf s u Wfv (resp. Wfv ) is identified to the equivalence class of the positive (resp. negative) geodesic ray {π ◦ ϕt(v)}t≥0 (resp. {π ◦ ϕt(v)}t≤0).

4.2 Space of lifted geodesics

Proposition 4.4. Under the notations above, for any α ∈ ∂uMf and β ∈ s ∂ Mf such that β 6= σus(α), there exists, up to a time translation, a unique unit speed geodesic line joining α to β Proof. By Theorem 3.16 (b) we need just verify the uniqueness. Let 0 c1 and c2 be two unit speed geodesic lines from α to β. Thus c1(t) and 0 c2(t) are two orbits of the lifted flow ϕe. Since c1 and c2 are both positively asymptotic to β, then thier corresponding orbits are contained in the same leaf of Wfs. 0 0 ss We suppose that c1(t) = ϕet(u), c2(t) = ϕet(v) and u, v ∈ Wfw . We equip SMf with the lifted metric of a C∞ Riemannian metric over SM, which s induces a Riemannian distance dr on the leaf Wfw. s ∞ By Proposition 4.3, the projection π : Wfw → Mf is a C diffeomorphism. Then by the compactness of SM, there exists a positive constant A such s that for any η ∈ T (Wfw), 1 Fe(Dπ(η)) ≤k η k≤ A · Fe(Dπ(η)). A

s Therefore (Wfw, dr) and (M,f de) are bi-Lipschitz equivalent. However by the Anosov property, for any t > 0, 1 d (ϕ (v), ϕ (u)) ≥ · ebtd (v, u), r e−t e−t a r 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 21 which implies that limt→+∞ dr(ϕe−t(v), ϕe−t(u)) = +∞. We deduce that over Mf limt→+∞d(c1(−t), c2(−t)) = +∞,

Thus c1 and c2 are not negatively asymptotic, which is a contradiction. 

Let GMf be the space of lifted unit speed geodesics of (M,f Fe). By Theo- rem 3.16 and Proposition 4.4, GMf can be naturally parametrized by

u s ∂ Mf × ∂ Mf − Gr(σus), where Gr(σus) is the graph of σus. The geometric involution of GMf is as follows: I : GMf → GM,If (α, β) = (σsu(β), σus(α)).

Since GMf is the quotient of SMf under the action of ϕe, then the differential of the lifted canonical 1-form dλe projects to a C∞ symplectic structure over GMf. It seems promising to study the symplectic geometry of (GM,f dλe).

4.3 Fundamental group action on boundaries Let (M,F ) be a closed Finsler manifold of strictly negative flag curvature. Let π1(M) be its fundamental group, which acts naturally on Mf and pre- s u serves the lifted foliations Wf and Wf . Therefore, π1(M) acts naturally on the stable and unstable boundaries of (M,f Fe). Definition 4.5. Let S be a topological space and Φ : S → S be a homeo- morphism. We say that Φ has a north-south dynamic if Φ fixes exactly two points {a, b} ⊆ S and for any x ∈ X − {a, b},

lim Φn(x) = a and lim Φ−n(x) = b. n→+∞ n→+∞

Proposition 4.6. Let γ ∈ π1(M). If γ is not the identity element, then the γ-action over ∂sMf (resp. ∂uMf) has a north-south dynamic. Proof. Let c be the unique unit speed geodesic in the homotopy class of u s γ. Let ec be the lift of c in Mf. There exist α ∈ ∂ Mf and β ∈ ∂ Mf such that lim c(t) = α and lim c(t) = β. t→+∞ e t→−∞ e

0 0 By Theorem 3.16 we have β 6= σus(α). Let α = σsu(β) and β = σus(α). Thus α 6= α0 and β 6= β0. 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 22

Let ϕet(v) be the ϕe-orbit corresponding to ec, which is preserved by the γ- action. Therefore, there exists T 6= 0 such that

γ(v) = ϕeT (v).

s s −1 We deduce that γ(Wfv ) = Wfv and γ(β) = β. By replacing γ by γ if s necessary, we suppose that T > 0. For any w ∈ Wfv − {v},

n s s s γ (Wf ) = Wf n = Wf n . w γ (w) ϕe−nT ◦γ (w)

n n s Since ϕe−nT ◦ γ (v) = v then ϕe−nT ◦ γ (w) ∈ Wfv − {v}. Since γ acts isometricly on SMf with respect to the lifted Riemannian metric, then the Anosov property implies that

n lim ϕ−nT ◦ γ (w) = v. n→+∞ e

n s s s 0 Thus limn→+∞ γ (Wfw) = Wfv . We deduce that for any δ ∈ ∂ Mf − {β, β },

lim γn(δ) = β. n→+∞

0 0 Now let c1 be the unit speed geodesic line from α to β . Similar argumets as above prove that γ(β0) = β0 and that for any δ ∈ ∂sMf − {β, β0},

lim γ−n(δ) = β0. n→+∞

The proof is complete. 

4.4 Hamenst¨adt theorem and its Finsler generalization Let φ : N → N be a topologically transitive C∞ Anosov flow generated by X. For any ϕ-invariant probability measure µ we denote by hµ(φ) the metric entropy of φ with respect to µ. The topological entropy of φ, htop(φ) is given by

htop(φ) = sup{hµ(φ): µ is a φ−invariant probability}.

If φ is volume-preserving, we denote by ν its unique invariant Lebesgue probability measure. We call hν(φ) the Liouville metric entropy of φ, which will also be denoted by hL(φ). 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 23

1 Let H (N, R) be the first de Rham cohomology group of N. Let µ be a φ-invariant probability measure. The µ-winding cycle of Schwartzman is a 1 map Φµ :H (N, R) → R defined as Z Φµ(α) = α(X)dµ, N where α is a closed C∞ 1-form over N. Since µ is a φ-invariant, it is 1 clear that Φµ is a well-defined map. We define Λ : H (N, R) → R by Λ(α) = P(φ, α(X)), i.e. the topological pressure of φ with respect to the function α(X) (see [10]). Immediately from the definition we obtain the relationship

Λ(α) = sup{hµ(ϕ) + Φµ(α): µ is ϕ−invariant}

If df is an exact form then Λ(α) = Λ(α + df). Therefore Λ is well-defined. Following [13], the cohomological pressure of ϕ, P(φ) is defined as

1 P(φ) = inf{Λ(α):[α] ∈ H (N, R)}. It was proved in [6] that for any contact Anosov flow φ : N → N,

htop(φ) ≥ P(φ) ≥ hL(φ).

Proposition 4.7. Let ϕ be the geodesic flow of a closed negatively curved reversible Finsler manifold (M,F ). Then we have htop(ϕ) = P (ϕ).

Proof. There exists a unique ϕ-invariant probability measure µBM such that hµBM (ϕ) = htop(ϕ), which is said to be the Bowen-Margulis measure (see [10]). In [13] it was proved that

P(ϕ) = sup{hµ(ϕ): µ is ϕ − invariant with Φµ ≡ 0}

Therefore the proof will be complete once we prove that the winding cycle

ΦµBM is zero. Let π : SM → M be the natural projection. Since (M,F ) is negatively ∗ 1 1 curved then the induced map π : H (M, R) → H (SM, R) is a group isomorphism. Let θ be a closed C∞ 1-form over M and X be the tangent vector field of ϕ. We have

π∗θ(X) = θ(Dπ(X)) = θ, where θ is viewd as a function over SM. Let σ : SM → SM be the map defined as σ(u) = −u. Since (M,F ) is reversible then σ∗X = −X, which 4 FINSLER GEODESIC FLOWS IN NEGATIVE CURVATURE 24

implies that for any t ∈ R, ϕ ◦ σ = σ ◦ ϕ−t. We deduce that σ∗µBM = µBM . Therefore Z Z Z Z ∗ π θ(X)dµBM = θdµBM = σ∗θd(σ∗µBM ) = (−θ)dµBM , SM SM SM SM

R ∗ which implies that SM π θ(X)dµBM = 0. The proof is complete. 

Therefore the cohomological pressure P (φ) is particularly interesting for non-reversible Finsler metrics, and the non-coincidence between htop(φ) and P (φ) is a manifestation of non-reversibility. The following Riemannian result was proved in [9]:

Theorem 4.8. Let ϕ be the geodesic flow of a closed negatively curved 2 Riemannian manifold. If its Anosov splitting is C , then htop(ϕ) = hL(ϕ). Later, G. Paternain observed in [12] that Theorem 4.4 can not be gener- alized directly to the Finsler case: Let (M, g) be a closed locally symmetric Riemannian space of rank one and θ be a C∞ closed 1-form over M. If θ is √ small enough, the Randers metric F = g − θ is negatively curved. Let ϕ be the geodesic flow of F , which satisfies the following conditions: ∞ Its Anosov splitting is C , but htop(ϕ) > hL(ϕ).

By using Proposition 4.6, we have the following Finsler generalization of Theorem 4.4:

Theorem 4.9. Let ϕ be the geodesic flow of a closed negatively curved 2 Finsler manifold. If its Anosov splitting is C , then P (ϕ) = hL(ϕ). Proof. We mention that a local version of this theorem was proved by the first author in [6], whose proof is valid for any negatively curved Finsler manifold with the help of Proposition 4.6, i.e. any non-trivial element of π(M) has a north-south dynamic over the unstable boundary ∂uMf (see Proposition 9 and Theorem 8.2 in [6]). The main ideas of the proof are as follows: By using the north-south dynamic we can prove that for any C1 1-form α such that dα is ϕ-invariant, there exists a constant a ∈ R such that

dα = a · dλ, where λ denotes the canonical 1-form of ϕ (see Subsection 4.1). Then com- bined with a time change idea we prove the coincidence between P (ϕ) and hL(ϕ).  REFERENCES 25

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