Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds
Alexander Cantoral Sergio Romaña

		
	Ruelle's inequality and Pesin's formula]Ruelle's inequality and Pesin's formula for Anosov geodesic flows in non-compact manifolds
	
	Keywords: Anosov geodesic flow, Jacobi field, Lyapunov exponents, Ruelle's inequality, Pesin's formula.
	
	Mathematics Subject Classification (2010): 37D40, 53C20. 
	
	
	Instituto de Matemática, Universidade Federal do Rio de Janeiro, CEP 21941-909, Rio de Janeiro, Brazil
	alexander.vidal@im.ufrj.br
	
	
	Instituto de Matemática, Universidade Federal do Rio de Janeiro, CEP 21941-909, Rio de Janeiro, Brazil
	sergiori@im.ufrj.br
	§ ABSTRACT  In this paper we prove Ruelle's inequality for the geodesic flow in non-compact manifolds with Anosov geodesic flow and some assumptions on the curvature. In the same way, we obtain the Pesin's formula for Anosov geodesic flow in non-compact manifolds with finite volume.
	
	
	[
    Sergio Romaña
    Received 16 July 2024; accepted 04 September 2024
=====================================================



§ INTRODUCTION
Ruelle in <cit.> proved an important result in ergodic theory relating entropy and Lyapunov exponents. More precisely, if f:M→ M is a C^1-diffeomorphism on a compact manifold and μ is an f-invariant probability measure on M, then 

    h_μ(f)≤∫∑_𝒳_i(x)>0𝒳_i(x)·(H_i(x))dμ(x),

where h_μ(f) is the entropy, {𝒳_i(x)} is the set of Lyapunov exponents at x∈ M and (H_i(x)) is the multiplicity of 𝒳_i(x).
In situations involving non-compact manifolds, Ruelle's inequality may be compromised. For example, Riquelme in <cit.> constructed diffeomorphisms defined on non-compact manifolds with an invariant measure with positive entropy and the sum of the positive Lyapunov exponents was equal to zero. However, in recent years, certain findings have been achieved that, in particular situations, offer the possibility of verifying Ruelle's inequality in non-compact contexts. Liao and Qiu in <cit.> showed Ruelle's inequality for general Riemannian manifolds under an integrable condition. Riquelme in <cit.> showed Ruelle's inequality for the geodesic flow in manifolds with pinched negative sectional curvature with some condition about the derivatives of the sectional curvature.

The main goal of this work is to prove Ruelle's inequality for the geodesic flow on the unit tangent bundle of a non-compact manifold with Anosov geodesic flow and some assumptions on the curvature. More precisely,

Let M be a complete Riemannian manifold with Anosov geodesic flow. Assume that the curvature tensor and the derivative of the curvature tensor are both uniformly bounded. Then, for every ϕ^t-invariant probability measure μ on SM, we have

    h_μ(ϕ)≤∫_SM∑_𝒳_i(θ)>0𝒳_i(θ)· (H_i(θ))dμ(θ).


We can see that this result generalizes what Riquelme demonstrated in <cit.> since the Anosov geodesic flows encompass the manifolds with pinched negative curvature.

The question arises as to under what conditions equality can be achieved in (<ref>). For example, when the manifold is compact, the diffeomorphism is C^1+α and the measure is absolutely continuous with respect	to the Lebesgue measure, Pesin showed in <cit.> that (<ref>) is actually an equality, called Pesin's formula. Our second result deals with the equality case of Theorem 1.1. In this case, we suppose that the manifold has finite volume.

Let M be a complete Riemannian manifold with finite volume and Anosov geodesic flow, where the flow is C^1-Hölder. Assume that the curvature tensor and the derivative of the curvature tensor are both uniformly bounded. Then, for every ϕ^t-invariant probability measure μ on SM which is absolutely continuous relative to the Lebesgue measure, we have

    h_μ(ϕ)=  ∫_SM∑_𝒳_i(θ)>0𝒳_i(θ)· (H_i(θ))dμ(θ).




 §.§ Structure of the Paper:
 
In section 2, we introduce the notations and geometric tools that we use in the paper. In section 3, we prove the existence of Oseledec's decomposition for the flow at time t=1. In section 4, we explore certain results that will allow us to deal with the challenge of non-compactness of the manifold. Using the strategies exhibited in <cit.> to prove the Ruelle's inequality for diffeomorphisms in the compact case, we prove Theorem 1.1 in section 5. Finally, in section 6 we prove Theorem 1.2 using techniques applied by Mañe in <cit.>.
	


§ PRELIMINARIES AND NOTATION

Throughout this paper, M=(M,g) will denote a complete Riemannian manifold without boundary of dimension n≥ 2, TM is the tangent bundle, SM its unit tangent bundle and π:TM→ M will denote the canonical projection, that is, π(x,v)=x for (x,v)∈ TM.
	

 §.§ Geodesic flow

	Given θ=(x,v)∈ TM, we denote by γ_θ the unique geodesic with initial conditions γ_θ(0)=x and γ'_θ(0)=v. The geodesic flow is a family of C^∞-diffeomorphisms ϕ^t:TM→ TM, where t∈ℝ, given by
	
    ϕ^t(θ)=(γ_θ(t), γ'_θ(t)).

	Since geodesics travel with constant speed, we have that ϕ^t leaves SM invariant. The geodesic flow generates a vector field G on TM given by
	
    G(θ)=. ddt|_t=0ϕ^t(θ)=. ddt|_t=0( γ_θ(t),γ'_θ(t)) .

	For each θ=(x,v)∈ TM, let V be the vertical subbundle of TM whose fiber at θ is given by V_θ= dπ_θ. Let K: TTM→ TM be the connection map induced by the Riemannian metric (see <cit.>) and denotes by H the horizontal subbundle of TM whose fiber at θ is given by H_θ= K_θ. The maps . d π_θ|_H_θ:H_θ→ T_xM and . K_θ|_V_θ: V_θ→ T_xM are linear isomorphisms. This implies that T_θ TM=H_θ⊕ V_θ and the map j_θ:T_θ TM→ T_xM× T_xM given by

    j_θ(ξ)=(dπ_θ(ξ), K_θ(ξ))

	is a linear isomorphism. Furthermore, we can identify every element ξ∈ T_θ TM with the pair j_θ(ξ). Using the decomposition T_θ TM=H_θ⊕ V_θ, we endow the tangent bundle TM with a special Riemannian metric that makes H_θ and V_θ orthogonal. This metric is called the Sasaki metric and it's given by
	
    ⟨ξ, η⟩_θ=⟨ dπ_θ(ξ), dπ_θ(η)⟩_x + ⟨ K_θ(ξ),K_θ(η)⟩_x .

From now on, we work with the Sasaki metric restricted to the unit tangent bundle SM. To begin with, it is valid to ask if SM is a complete Riemannian manifold with this metric. 
 Let M be a complete Riemannian manifold. Then SM is a complete metric space with the Sasaki metric.


Let θ, ω∈ SM and γ:[0,1]→ SM be a curve joining θ and ω. By the identification (<ref>) we can write

    l(γ)   =∫_0^1 γ'(t)dt 
       =∫_0^1 (  dπ_γ(t)( γ'(t))^2 +  K_γ(t)( γ'(t))^2)^1/2  dt
       ≥∫_0^1   dπ_γ(t)( γ'(t)) dt
       =∫_0^1 ( π∘γ)'(t) dt
       =l(π∘γ).

This implies that

    d(θ,ω)≥ d(π(θ),π(ω))

for any two points θ,ω∈ SM. Let { (p_n,v_n)}_n∈ℕ be a Cauchy sequence in SM. By (<ref>) we have that { p_n}_n∈ℕ is a Cauchy sequence in M. Since M is complete, there is p∈ M such that lim_n→ +∞p_n=p. If we consider the compact set X={ (q,v)∈ SM: d(q,p)≤ 1}, for n≥ n_0 we have that (p_n,v_n)∈ X and therefore the Cauchy sequence converges in SM.

The sectional curvature of SM with the Sasaki metric can be calculated from the curvature tensor and the derivative of the curvature tensor of M as explained in <cit.>: Let Π be a plane in T_(x,v)SM and choose an orthonormal basis { (v_1,w_1), (v_2,w_2)} for Π satisfying v_i^2+ w_i^2=1, for i=1,2, and ⟨ v_1,v_2 ⟩ =⟨ w_1,w_2⟩=0. Then the Sasaki sectional curvature of Π is given by

    K_Sas(Π)=   ⟨ R_x(v_1,v_2)v_1,v_2 ⟩  +3⟨ R_x(v_1,v_2)w_1,w_2 ⟩ 
    	+ w_1^2  w_2^2 
       -34 R_x(v_1,v_2)v^2+14R_x(v,w_2)v_1 ^2+14R_x(v,w_1)v_2 ^2 
       +12⟨ R_x(v,w_1)w_2,R_x(v,w_2)v_1 ⟩ - ⟨ R_x(v,w_1)v_1,R_x(v,w_2)v_2 ⟩
       +⟨ (∇_v_1R)_x(v,w_2)v_2,v_1 ⟩ + ⟨ (∇_v_2R)_x(v,w_1)v_1,v_2 ⟩   .

This equality shows that if the curvature tensor of M and its derivatives are bounded, then the sectional curvature of SM with the Sasaki metric is also bounded. This property is crucial as it allows us to compare volumes between subsets of TSM and subsets of SM using the exponential map of SM (see Lemma 5.3).


The types of geodesic flows discussed in this paper are the Anosov geodesic flows, whose definition follows below.

We say that the geodesic flow ϕ^t:SM→ SM is of Anosov type if T(SM) has a continuous splitting T(SM)=E^s⊕⟨ G⟩⊕ E^u such that
 
    dϕ^t_θ (E^s(u)(θ))    =    E^s(u)(ϕ^t(θ)),
    dϕ^t_θ|_E^s   ≤    C λ^t,
    dϕ^-t_θ|_E^u   ≤    C λ^t,






	for all t≥ 0 with C>0 and λ∈ (0,1), where G is the geodesic vector field. It's known that if the geodesic flow is Anosov, then the subspaces E^s(θ) and E^u(θ) are Lagrangian for every θ∈ SM (see <cit.> for more details).


 §.§ Jacobi fields
 To study the differential of the geodesic flow with geometric arguments, let us recall the definition of a Jacobi field. A vector field J along a geodesic γ of M is a Jacobi field if it satisfies the Jacobi equation

    J”(t)+R(γ'(t),J(t))γ'(t)=0,

where R denotes the curvature tensor of M and "'" denotes the covariant derivative along γ. A Jacobi field is determined by the initial values J(t_0) and J'(t_0), for any given t_0∈ℝ. If we denote by S the orthogonal complement of the subspace spanned by G, for every θ∈ SM, the map ξ→ J_ξ defines an isomorphism between S(θ) and the space of perpendicular Jacobi fields along γ_θ, where J_ξ(0)=dπ_θ(ξ) and J'_ξ(0)=K_θ(ξ).
The differential of the geodesic flow is determined by the behavior of the Jacobi fields and, therefore, by the curvature. More precisely, for θ∈ SM and ξ∈ T_θ SM we have (in the horizontal and vertical coordinates)

    dϕ^t_θ(ξ)=(J_ξ(t), J'_ξ(t)),  t∈ℝ.

In the context of an Anosov geodesic flow, if ξ∈ E^s(θ) (respectively, ξ∈ E^u(θ)), the Jacobi field associated J_ξ(t) is called a stable (respectively, unstable) Jacobi field along γ_θ(t).

The following proposition allows us to uniformly limit the derivative of the exponential map from certain conditions on the curvature of the manifold.

Let N be a complete Riemannian manifold and suppose that the curvature tensor is uniformly bounded. Then there exists t_0>0 such that for all x∈ N and for all v,w∈ T_xN with v = w =1 we have
	
    d(exp_x)_tvw≤52, ∀| t| ≤ t_0.



If w∈⟨ v⟩, then w=v or w=-v. In both cases, by Gauss Lemma (see <cit.>) we have that

    d(exp_x)_tvw^2   =⟨ d(exp_x)_tvw, d(exp_x)_tvw⟩
       =⟨ d(exp_x)_tvv, d(exp_x)_tvv⟩
       =1t^2⟨ d(exp_x)_tvtv, d(exp_x)_tvtv⟩
       =1t^2⟨ tv,tv⟩
       =1.

Now assume that w∈⟨ v⟩^⊥. Consider the Jacobi field

    J(t)=d(exp_x)_tvtw,  t∈ [-1,1]

with initial conditions J(0)=0 and J'(0)=w. By Lemma 8.3 of <cit.> there exists t_0>0, independent of the point x, such that

    d(exp_x)_tvw =J(t) | t| ≤32, ∀ t∈(-t_0,t_0)∖{ 0 }.

As T_xN=⟨ v⟩ + ⟨ v⟩^⊥, the last inequality completes the proof.



 §.§ No conjugate points

Let γ be a geodesic joining p,q∈ M, p≠ q. We say that p,q are conjugate along γ if there exists a non-zero Jacobi field along γ vanishing at p and q. A manifold M has no conjugate points if any pair of points are not conjugate. This is equivalent to the fact that the exponential map is non-singular at every point of M. There are examples of manifolds without conjugate points obtained from the hyperbolic behavior of the geodesic flow. In <cit.>, Klingenberg proved that a compact Riemannian manifold with Anosov geodesic flow has no conjugate points. Years later, Mañé (see <cit.>) generalized this result to the case of manifolds of finite volume. In the case of infinite volume, Melo and Romaña in <cit.> extended the result of Mañé over the assumption of sectional curvature bounded below and above. These results show the relationship that exists between the geometry and dynamic of an Anosov geodesic flow.

	
Let M be a complete Riemannian manifold without conjugate points and sectional curvature bounded below by -c^2, for some c>0. When the geodesic flow ϕ^t:SM→ SM is of Anosov type, Bolton in <cit.> showed that there is a positive constant δ such that, for every θ∈ SM, the angle between E^s(θ) and E^u(θ) is greater than δ. Moreover, Eberlein in <cit.> showed that 

	
  1. K_θ(ξ)≤ c dπ_θ(ξ) for every ξ∈ E^s(θ) or E^u(θ), where K:TTM → TM is the connection map.
	
  2. If ξ∈ E^s(θ) or E^u(θ), then 	J_ξ(t)≠ 0 for every t∈ℝ.



 §.§ Lyapunov exponents

Let (M.g) be a Riemannian manifold and f:M→ M a C^1-diffeomorphism. The point x is said to be (Lyapunov-Perron) regular if there exist numbers {𝒳_i(x) }_i=1^l(x), called Lyapunov exponents, and a decomposition of the tangent space at x into T_xM=⊕_i=1^l(x)H_i such that for every vector v∈ H_i(x)∖{ 0}, we have  

    lim_n→±∞1nlogdf^n_xv =𝒳_i(x)

and

    lim_n→±∞1nlog| ( df^n_x) |=∑_i=1^l(x)𝒳_i(x)· (H_i(x)).

Let Λ be the set of regular points. By Oseledec's Theorem (see <cit.>), if μ is an f-invariant probability measure on M such that log^+  df^± 1 is μ-integrable, then the set Λ has full μ-measure. Moreover, the functions x→𝒳_i(x) and x→(H_i(x)) are μ-measurable and f-invariant. In particular, if μ is ergodic, they are μ-almost everywhere constant.


§ EXISTENCE OF LYAPUNOV EXPONENTS

In this section, we will prove that when the geodesic flow is Anosov and the sectional curvature is bounded below, the norm dϕ^± 1_θ is bounded by a positive constant independent of θ. This boundedness is crucial as it ensures, for a given probability measure, the existence of Lyapunov exponents by Oseledec's Theorem. More precisely,

Let M be a complete Riemannian manifold without conjugate points, sectional curvature bounded below by -c^2, for some c>0, and μ an ϕ^t-invariant probability measure in SM. If the geodesic flow is of Anosov type, then log dϕ^± 1∈ L^1(μ).

Before giving a proof of Theorem 3.1, it is essential to establish the following two lemmas.

Let M be a complete Riemannian manifold without conjugate points, sectional curvature bounded below by -c^2, for some c>0, and geodesic flow of Anosov type. For every θ∈ SM, there exists a constant P>0 such that for every ξ∈ E^s(θ), η∈ E^u(θ) with ξ=η =1, we have

    J_η(1) ≤ P  and J_ξ(-1) ≤ P.

	

Fix θ∈ SM and let η∈ E^u(θ) with η =1. Consider a stable Jacobi field J_s along γ_θ such that J_η(0)=J_s(0) and put ω=(J_s(0),J'_s(0)). By item 1 of section 2.3 we have

    J'_s(0)≤ c J_s(0) =  c J_η(0) ≤ c

and

    ω^2= J_s(0)^2  +J_s'(0) ^2≤ 1+c^2.

Define the Jacobi field J(t)= J_η(t)-J_s(t). We can see that J is a perpendicular Jacobi field along γ_θ satisfying J(0)=0. By Rauch's comparison Theorem (see <cit.>) we have that

    J(1)≤sinh ccJ'(0) .
 	
Since the geodesic flow is Anosov and ω∈ E^s(θ),

    J_s(1)≤dϕ^1_θ (ω) ≤ Cλω≤ Cλ√(1+c^2).

From (<ref>) and (<ref>) we have that

    J_η(1)   ≤ J(1) + J_s(1) 
       ≤sinh cc J'(0) + Cλ√(1+c^2)
       ≤sinh cc( J'_η(0) + J'_s(0) )  +Cλ√(1+c^2)
       ≤( 1+cc) sinh c +Cλ√(1+c^2):=P_1.

Using the same technique for the stable case, there exists P_2>0 such that 

    J_ξ(-1)≤ P_2

for every ξ∈ E^s(θ) with ξ =1. Considering P=max{ P_1,P_2}, the conclusion of the lemma follows.

We know that, with the hypothesis of Theorem 3.1, there exists a constant δ>0 such that the angle between the stable and unstable subspaces is uniformly bounded below by δ. As a direct consequence of this result, we have the following lemma.

Let M be a complete Riemannian manifold without conjugate points, sectional curvature bounded below by -c^2, for some c>0, and geodesic flow of Anosov type. Define the function f:SM→ℝ as

    f(θ)=sup{|⟨ξ,η⟩|: ξ∈ E^s(θ), η∈ E^u(θ), ξ=η =1  } .

Then there exists Q>0 such that

    sup_θ∈ SM f(θ)≤ Q<1.


Proof of Theorem 3.1. Fix θ∈ SM and consider ξ∈ T_θ SM with ξ =1. Since the geodesic flow is of Anosov type, we can write

    ξ=sξ_1+rξ_2+ξ_3,

where ξ_1∈ E^s(θ), ξ_2∈ E^u(θ) and ξ_3∈⟨ G(θ)⟩ with ξ_1  =ξ_2 =1. Then

    1=sξ_1+rξ_2 ^2 + ξ_3^2.
 
This implies that ξ_3≤ 1 and sξ_1+rξ_2≤ 1. We have

    sξ_1+rξ_2^2=s^2+r^2+2sr⟨ξ_1,ξ_2 ⟩≤ 1.

It follows from Lemma 3.3 that the regions 

    E_β={ (s,r):s^2+r^2+2srβ≤ 1}

with -Q≤β≤ Q are bounded ellipses. If we consider L=diam(E_Q)/2+1>0, the ball B centered in 0 and radius L contains these ellipses (see Figure 1). In particular, the parameters s,r are bounded, that is | s|, | r|≤ L. By Lemma 3.2 we have that

    dϕ^1_θ(ξ_2)    =√( J_ξ_2(1)^2 + J'_ξ_2(1) ^2  )
       ≤√(1+c^2) J_ξ_2(1)
       ≤√(1+c^2)P.



Then

    dϕ^1_θ(ξ)    ≤| s|dϕ^1_θ(ξ_1)  + | r|dϕ^1_θ(ξ_2)  + dϕ^1_θ(ξ_3) 
       ≤| s| Cλ + | r|√(1+c^2)P+1 
       ≤ LCλ + L√(1+c^2)P+1

for every ξ∈ T_θ SM with ξ  =1. This implies that dϕ^1_θ is bounded and therefore the function logdϕ^1 is μ-integrable, since the constants L and P are independent of the point θ. Using the second inequality of Lemma 3.2 we obtain that log dϕ^-1 is μ-integrable.□


§ CONSEQUENCES OF A GEODESIC FLOW BEING OF ANOSOV TYPE

In this section, we explore some results, based on the hyperbolicity of a geodesic flow, that will allow us to address the challenge of the non-compactness of the manifold in the proof of Ruelle's inequality. 


From now on, let us assume that M is a complete Riemannian manifold without conjugate points, sectional curvature bounded below by -c^2, for some c>0, and the geodesic flow ϕ^t:SM→ SM is of Anosov type. For every ω∈ T_θ SM we can write

    ω=ω^s+ω^u+ω^c,

where ω^s∈ E^s(θ), ω^u∈ E^u(θ) and ω^c∈⟨ G(θ)⟩.

For m∈ℕ large enough, there is τ_1>1 such that for every θ∈ SM

    dϕ^m_θ≤τ_1 dϕ^m_θ(η)

for some η∈ E^u(θ) with η =1.


Fix θ∈ SM and let ω=ω^s+ω^u+ω^c∈ T_θ SM with ω =1. This implies that ω^s+ω^u ≤ 1 and ω^c ≤ 1. Moreover, we know that ω^s≤ L and ω^u≤ L (see Section 3). Consider m∈ℕ large enough such that Cλ^m<1/2.

Case 1: ω^u=0.

Since the geodesic flow is Anosov we have that

    dϕ^m_θ(ω)    ≤ dϕ^m_θ(ω^s) +dϕ^m_θ(ω^c) 
       ≤  Cλ^m + 1
       <C^-1λ^-m
       ≤ dϕ^m_θ(η)

for every η∈ E^u(θ) with η =1.

Case 2: ω^u≠ 0.

Since the geodesic flow is Anosov we have that

    dϕ^m_θ(ω^s)≤ Cλ^m L<C^-1λ^-mL≤ L  dϕ^m_θ(ω^u)ω^u .

Then

    dϕ^m_θ(ω)   ≤ dϕ^m_θ(ω^s) +  dϕ^m_θ(ω^u)+ dϕ^m_θ(ω^c)
       ≤ L  dϕ^m_θ(ω^u)ω^u  +  L  dϕ^m_θ(ω^u)ω^u +1
       < (2L+1) dϕ^m_θ(ω^u)ω^u .

If we consider τ_1=2L+1, in both cases we have that

    dϕ^m_θ(ω)≤τ_1 . d ϕ^m_θ|_E^u(θ)

for every ω∈ T_θ SM with ω =1. Since the norm is always attained in a finite-dimensional space, we conclude the proof of the lemma.


For m∈ℕ large enough, there is τ_2∈ (0,1) such that for every θ∈ SM

    dϕ^m_θ^*≥τ_2  dϕ^m_θ(ξ)

for some ξ∈ E^s(θ) with ξ =1, where dϕ^m_θ^*=inf_v  =1dϕ^m_θ(v).

 Let ε>0 and consider m∈ℕ large enough such that ε≥ (L+1)Cλ^m and √(1-ε^2) >ε Cλ^m, where L comes from Section 3. Fix θ∈ SM and define the following set

    Γ_θ,ε,m:={ω∈ T_θ SM: ω =1,  ω=ω^s+ω^u+ω^c and dϕ^m_θ(ω^u+ω^c) ≥ε} .

Case 1: ω∈Γ_θ,ε,m with ω^s≠ 0.

Since the geodesic flow is Anosov,

    dϕ^m_θ(ω)    ≥ dϕ^m_θ(ω^u+ω^c)  - dϕ^m_θ(ω^s) 
       ≥ dϕ^m_θ(ω^u+ω^c)  - Cλ^m ω^s.

As ω∈Γ_θ,ε,m we have that

    dϕ^m_θ(ω^u+ω^c) ≥ε≥ (L+1)Cλ^m≥ (ω^s +1)Cλ^m.

Then from (<ref>) and (<ref>)

    dϕ^m_θ(ω) ≥ dϕ^m_θ(ω^u+ω^c) -Cλ^m ω^s≥ Cλ^m ≥ dϕ^m_θ(ω^s) ω^s .

Case 2: ω∈Γ_θ,ε,m with ω^s=0.

Since the geodesic flow is Anosov,

    dϕ^m_θ(ω)  =  dϕ^m_θ(ω^u+ω^c) ≥ε>Cλ^m≥dϕ^m_θ(ξ)

for every ξ∈ E^s(θ) with ξ =1.

Case 3: ω∉Γ_θ,ε,m and w=1.

We have that

    ε^2   >dϕ^m_θ(ω^u+ω^c) ^2= dϕ^m_θ(ω^u) ^2+dϕ^m_θ(ω^c) ^2=dϕ^m_θ(ω^u) ^2+ω^c ^2.

Then ω^c <ε and dϕ^m_θ(ω^u) <ε. Since the geodesic flow is Anosov,

    C^-1λ^-mω^u≤ dϕ^m_θ(ω^u)<ε.

This implies that ω^u <ε Cλ^m. On the other hand, as w=1, then

    ω^u + ω^s≥ω^u+ω^s=√(1-ω^c^2)>√(1-ε^2 ) .

Furthermore

    L≥ω^s > √(1-ε^2) - ε Cλ^m>0.

In particular, ω^s≠ 0. Denote by

    E^cu(θ):=E^u(θ)⊕⟨ G(θ)⟩

and define the following linear map

    P_θ: T_θ SM→ E^s(θ)

as the parallel projection onto E^s(θ) along E^cu(θ). Since the angle between the stable and unstable subspaces is uniformly away from 0 for every θ∈ SM, then there is δ≥ 1 such that

    P_θ(ω) ≤δω

for every θ∈ SM and ω∈ T_θ SM (see Theorem 3.1 in <cit.>). Then

    dϕ^m_θ(ω^s) =P_ϕ^m(θ)(dϕ^m_θ(ω)) ≤δdϕ^m_θ(ω) .

By (<ref>), if we choose ε>0 such that ω^s≥ 1/2, we have that

    dϕ^m_θ(ω)≥12δ dϕ^m_θ(ω^s)ω^s.

If we consider τ_2= 1/2δ, in all cases we have that for all for every ω∈ T_θ SM with ω =1 there is ξ∈ E^s(θ), with ξ=1 such that 

    d_θϕ^m(ω)≥τ_2   d_θϕ^m(ξ).

Since the infimum is always attained in a finite-dimensional space, the last inequality concludes the proof of the lemma.














Clearly dϕ^m_θ^* ≤ dϕ^m_θ for every θ∈ SM. From Lemmas 4.1 and 4.2, we can obtain a positive constant, independent of θ, such that the direction of the inequality changes.

For m∈ℕ large enough, there is κ>1, depending on m, such that

    dϕ^m_θ≤κ dϕ^m_θ^*

for every θ∈ SM.

 From Lemma 4.1 we have that

    d ϕ^m_θ≤τ_1  d ϕ^m_θ(η)

for some η∈ E^u(θ) with η=1. Denote by J_η the Jacobi field associated to η. Since the geodesic flow is of Anosov type and dϕ^t_θ(η)=(J_η(t),J'_η(t)), we have from item 1 of Section 2.3 that

    d ϕ^m_θ≤τ_1  d ϕ^m_θ(η)=τ_1 √(J_η(m) ^2 +J'_η(m) ^2  )≤τ_1√(1+c^2) J_η(m) .

In the same way, by Lemma 4.2 we have that

    dϕ^m_θ^*≥τ_2  dϕ^m_θ(ξ)=τ_2√(1+ J'_ξ(m)^2  J_ξ(m) ^2 ) J_ξ(m) ,

for some ξ∈ E^s(θ) with ξ=1, where J_ξ is the Jacobi field associated to ξ. Moreover,

    √(1+c^2)≤√(1+c^2)√(1+ J'_ξ(m)^2  J_ξ(m) ^2 ).

Define the function

    r:   [0,+∞) →ℝ
       t→λ^-t J_ξ(t)λ^t  J_η(t).

This function is well-defined because the stable and unstable Jacobi fields are never zero since the manifold has no conjugate points (see Section 2). We have that

    r'(t)=r(t)(-2logλ + ⟨ J'_ξ(t),J_ξ(t) ⟩⟨ J_ξ(t),J_ξ(t) ⟩ - ⟨ J'_η(t),J_η(t) ⟩⟨ J_η(t),J_η(t) ⟩).

Also

    A(t)=⟨ J'_ξ(t),J_ξ(t) ⟩⟨ J_ξ(t),J_ξ(t) ⟩∈ [-c,c] andB(t)=⟨ J'_η(t),J_η(t) ⟩⟨ J_η(t),J_η(t) ⟩∈ [-c,c].

Since the curvature is bounded below by -c^2, then λ≥ e^-c (see <cit.>). Therefore

    -2logλ-2c≤ -2logλ +A(t) - B(t)≤ -2logλ + 2c.

This implies that 

    -2logλ - 2c≤r'(t)r(t)≤ -2logλ + 2c

and

    r(0)· e^(-2logλ-2c)t≤ r(t)≤ r(0)· e^(-2logλ+2c)t.

Therefore

    r(0)^-1· e^(2logλ - 2c)t≤1r(t)≤ r(0)^-1· e^(2logλ+2c)t.

For t=m we have that

    J_η(m) ≤ r(0)^-1· e^(2logλ+2c)m·λ^-2m J_ξ(m) =r(0)^-1· e^2cm J_ξ(m) .

From (<ref>), (<ref>), (<ref>) and (<ref>)

    dϕ^m_θ   ≤τ_1 √(1+c^2)J_η(m) 
       ≤τ_1√(1+c^2)· r(0)^-1· e^2cm J_ξ(m) 
       ≤τ_1√(1+c^2)√(1+ J'_ξ(m)^2  J_ξ(m) ^2 )· r(0)^-1· e^2cm J_ξ(m) .

From item 1 of Section 2.3 we have that

    1=ξ^2=dπ_θ(ξ) ^2 +  K_θ(ξ)^2≤ (1+c^2)  dπ_θ(ξ) ^2.

Since 1=η^2=dπ_θ(η) ^2 +  K_θ(η)^2, the last inequality implies that

    r(0)^-1=dπ_θ(η) dπ_θ(ξ) ≤1dπ_θ(ξ) ≤√(1+c^2).

Therefore, substituting in (<ref>) and using (<ref>)

    dϕ^m_θ≤κ dϕ^m_θ^*,

where κ=τ_1·τ_2^-1· (1+c^2)· e^2cm>1. 

On the other hand, since the geodesic flow is of Anosov type, we have that the norm dϕ^m_θ is bounded between two positive constants.

For m∈ℕ large enough, there are constants K_1, K_2>0, K_1 depending on m, such that

    K_2< dϕ^m_θ <K_1

for every θ∈ SM.

 Fix θ∈ SM. Since the geodesic flow is of Anosov type, for η∈ E^u(θ) with η =1 we have that

    dϕ^m_θ≥ dϕ^m_θ(η)≥ C^-1λ^-m>C^-1,

then K_2=1/C. On the other hand, from (<ref>) we have that

    dϕ^1_θ   ≤ LCλ + L√(1+c^2)( ( 1+cc) sinh c+Cλ√(1+c^2)) +1
       ≤ LCλ + L√(1+c^2)( 1+cc) sinh c + LCλ(1+c^2)+1
       ≤ 2LCλ + LCλ c^2 +L√(1+c^2)( 1+cc) sinh c + 1:=h(c).

Then, we can consider K_1=h(c)^m.

A direct consequence of Proposition 4.4 is the following result.

Given ε>0, there is β∈ (0,1), depending on m, such that

    β dϕ^m_θ̃<  dϕ^m_θ, ∀θ̃∈ SM: d(θ,θ̃)<ε

for every θ∈ SM.


By Proposition 4.4 we have that

    K_2K_1< dϕ^m_θ dϕ^m_θ̃ <K_1K_2

Considering β=K_2K_1=C^-1h(c)^m the conclusion of the corollary follows. 
 


§ RUELLE'S INEQUALITY

In this section, we will prove Theorem 1.1. For this, we will adapt the idea of the proof of Ruelle's inequality for diffeomorphisms in the compact case exhibited in <cit.>. 


Let M be a complete Riemannian manifold satisfying all the hypotheses of Theorem 1.1 and μ an ϕ^t-invariant probability measure on SM. By simplicity, we consider μ an ergodic ϕ^t-invariant probability measure on SM. In this case, we denote by {𝒳_i} the Lyapunov exponents and { k_i} their respective multiplicities. The proof in the non-ergodic case is a consequence of the ergodic decomposition of such a measure. We can also assume that ϕ=ϕ^1 is an ergodic transformation with respect to μ. If it is not the case, we can choose an ergodic-time τ for μ and prove the theorem for the map ϕ^τ. The proof of the theorem for the map ϕ^τ implies the proof for the map ϕ because the entropy of ϕ^τ and the Lyapunov exponents are τ-multiples of the respective values of ϕ.


Fix ε>0 and m∈ℕ large enough. There exists a compact set K⊂ SM such that μ(K)>1-ε.
Based on the results in Section 4, we present the following theorem, which constitutes a similar version to the inclusion (10.3) described in <cit.>. 
Consider the constants κ>1 and 0<β<1 given by Proposition 4.3 and Corollary 4.5 respectively.

Let M be a complete Riemannian manifold without conjugate points and sectional curvature bounded below by -c^2, for some c>0. If the geodesic flow is of Anosov type, then for every θ∈ K there exists ϱ:=ϱ(K)∈ (0,1) such that

    ϕ^m(exp_θ(B(0,βκ^-1ϱ)))⊆ exp_ϕ^m(θ) (d ϕ^m_θ(B(0,ϱ)).


 We will proceed by contradiction. Suppose that for every n∈ℕ, there are θ_n∈ K and v_n∈ T_θ_nSM with v_n=βκ^-1n such that

    ϕ^m(exp_θ_n(v_n))=exp_ϕ^m(θ_n)(dϕ^m_θ_n(w_n))

where w_n =1n. Since K is compact and w_n→ 0, then dϕ^m_θ_n(w_n) is less than injectivity radius of the
exponential map restricted to the compact set K, for n large enough by Proposition 4.4. Therefore

    dϕ^m_θ_n(w_n)   =d(ϕ^m(θ_n),exp_ϕ^m(θ_n)(dϕ^m_θ_n(w_n)))
       =d(ϕ^m(θ_n),ϕ^m(exp_θ_n(v_n)))
       ≤∫_0^1(ϕ^m∘ c_n)'(t)  dt ,
 
where c_n(t)=exp_θ_n(tv_n). Then

    dϕ^m_θ_n(w_n)   ≤sup_t∈ [0,1]dϕ^m_c_n(t)∫_0^1  c'_n(t)  dt
       =sup_t∈ [0,1]dϕ^m_c_n(t)· v_n .

For n large enough, by Corollary 4.5 we have that

    κβ^-1 dϕ^m_θ_n(w_n) w_n    =  w_n v_n dϕ^m_θ_n(w_n) w_n
       ≤sup_t∈ [0,1]d_c_n(t)ϕ^m 
       <β^-1dϕ^m_θ_n.

Therefore

    κβ^-1dϕ^m_θ_n^*<β^-1dϕ^m_θ_n

which contradicts the Proposition 4.3. 

Now, denote by ϱ_m=βκ^-1ϱ<1, where the constants β, κ and ϱ come from Theorem 5.1.  Using the techniques of separate sets applied in <cit.> we define a finite partition 𝒫=𝒫_K∪{ SM∖ K} of SM in the following way:

	
  . 𝒫_K is a partition of K such that for every X∈𝒫_K, there exist balls B(x,r') and B(x,r) such that the constants satisfy 0<r'<r<2r'≤ϱ_m2 and 
    B(x,r')⊂ X⊂ B(x,r).

	
  . There exists a constant ζ>0 such that the cardinal of 𝒫_K, denoted by | 𝒫_K|, satisfies 
    | 𝒫_K|≤ζ·(ϱ_m)^-(SM).

	
  . h_μ(ϕ^m,𝒫)≥ h_μ(ϕ^m)-ε.

By definition of entropy,

    h_μ(ϕ^m,𝒫)   =lim_k→ +∞ H_μ( . 𝒫|ϕ^m𝒫∨…∨ϕ^km𝒫) 
       ≤ H_μ( . 𝒫| ϕ^m𝒫) 
       ≤∑_D∈ϕ^m𝒫μ(D)·logcard{ X∈𝒫: X∩ D≠∅}.

Denote by φ=sup_θ∈ SMdϕ_θ>1. First, we estimate the number of elements X∈𝒫 that intersect a given element D∈ϕ^m𝒫.

	There exists a constant L_1>0 such that if D∈ϕ^m𝒫 then
	
    card{ X∈𝒫: X∩ D≠∅}≤ L_1·max{φ^m· (SM), (ϱ_m)^-(SM)} .



Consider D∈ϕ^m𝒫, then D=ϕ^m(X') for some X'∈𝒫.

Case I: X'∈𝒫_K.

By the mean value inequality

    diam(D)   =diam(ϕ^m(X'))
       ≤sup_θ∈ SMdϕ_θ^m·diam(X')
       ≤φ^m· 4r',

since X'⊂ B(x,2r'). If X∈𝒫_K satisfies X∩ D≠∅, then X is contained in a 4r'-neighborhood of D, denoted by W. Since φ^m>1 we have that

    diam(W)   ≤φ^m· 4r' + 8r'
       =4r'·( φ^m + 2)
       <12r'·φ^m.
 
Hence

    ∑_{ X∈𝒫_K:X∩ D≠∅}vol(X)≤vol(W)≤ A_1 · (r')^ (SM)·φ^m· (SM),

where A_1>0. Since X∈𝒫_K contains a ball of radius r', the volume of X is bounded below by

    A_2· (r')^ (SM)≤vol(X),

where A_2>0. From (<ref>) and (<ref>) we have that

    card{ X∈𝒫:X∩ D≠∅}   ≤A_1A_2·φ^m· (SM) +1
       ≤(A_1A_2+1) ·φ^m· (SM) .

Case II: X'=SM∖ K.

In this case, we have that 

    card{ X∈𝒫:X∩ D≠∅}   ≤| 𝒫_K| +1
       ≤ (ζ+1)(ϱ_m)^-(SM).

Considering L_1=max{A_1A_2+1, ζ+1} we obtain the desired result.

Now we will get a finer exponential bound for the number of those sets D∈ϕ^m𝒫_K that contain regular points. For this, let Λ_m be the set of regular points θ∈ SM which satisfy the following condition: for k≥ m and ξ∈ T_θ SM

    e^k( 𝒳(θ,ξ)-ε) ξ≤ dϕ^k_θ (ξ) ≤ e^k( 𝒳(θ,ξ)+ε) ξ,

where 𝒳(θ,ξ)=lim_n→±∞1nlogd ϕ^n_θ(ξ). 

If D∈ϕ^m𝒫_K has non-empty intersection with Λ_m, then there is a constant L_2>0 such that

    card{ X∈𝒫: X∩ D≠∅}≤ L_2· e^mε∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i.



 Let X'∈𝒫_K such that ϕ^m(X')=D and suppose that X'∩Λ_m≠∅. Pick a point θ∈ X'∩Λ_m and consider the ball B=B(0,ϱ)⊂ T_θ SM. We claim that 
    X'⊆ exp_θ(B(0,ϱ_m)),

where exp_θ denotes the exponential map defined on the tangent plane T_θ SM. In fact, let z∈ X'. Since SM is complete with the Sasaki metric (see Lemma 2.1) we can choose w∈ T_θ SM such that γ(t)=exp_θ(tw), where γ is a geodesic with γ(0)=θ and γ(1)=exp_θ(w)=z. As diam 𝒫_K < ϱ_m then 
    d(θ,z)=l(γ)< ϱ_m.


Similar to the proof of Proposition 2.2, we obtain that

    ϱ_m> ∫_0^1γ'(s) ds
    
    
    
    	= w .

Then w∈ B(0,ϱ_m) and hence

    z=exp_θ(w)∈ exp_θ(B(0,ϱ_m)).

Since z∈ X' was arbitrary, the claim is proven. Therefore, from Theorem 5.1 we have that

    D=ϕ^m(X')⊆ B_0:=exp_ϕ^m(θ)(B̃_0),

where B̃_0=dϕ^m_θ(B) is an ellipsoid. Since the curvature tensor and the derivative of the curvature tensor of M are both uniformly bounded, we have that the Sasaki sectional curvature of SM is uniformly bounded (see (<ref>)). This implies that the curvature tensor of SM is uniformly bounded. Applying Proposition 2.2 to SM, there exists t_0>0 such that  

    d (exp_ϕ^m(θ))_tv≤52
 
for every | t|≤ t_0 and v∈ T_ϕ^m(θ)SM with v =1. Then, for m large enough, we have that

    diam (D)   ≤ h(c)^m·diam (X')
       ≤ h(c)^m·ϱ_m
       =1C·τ_2τ_1·11+c^2· e^-2cm·ϱ
       <t_02,

where h(c) is the expression that bounds the derivative of  ϕ (see Proposition 4.4). Therefore, we can choose B_0 that satisfies D⊂ B_0 and diam(B_0)<t_0. We know that diam𝒫_K< ϱ_m<ϱ, then if X∈𝒫_K intersects D, it lies in the set

    B_1={Ψ∈ SM: d(Ψ,B_0)<ϱ}.
 
Since X⊂ B(x,r) and 2r<ϱ_m<ϱ, then B(x,ϱ/2)⊂ B_1 and 

    card{ X∈𝒫_K: X∩ D≠∅}≤ b·vol(B_1)·ϱ^-(SM),

for some b>0, where vol(B_1) denotes the volume of B_1 induced by the Sasaki metric. Consider a subset B̃^*_0⊂B̃_0 such that exp_ϕ^m(θ) is a diffeomorphism between B̃^*_0 and B_0. Since

    |  d (exp_ϕ^m(θ))_v| ≤d (exp_ϕ^m(θ))_v ^(SM)

for every v∈B̃^*_0, from (<ref>) we have that

    vol(B_0)≤( 52) ^(SM)·vol(B̃_0).

This implies that the volume of B_1 is bounded, up to a bounded factor, by the product of the lengths of the axes of the ellipsoid B̃_0. Those corresponding to non-positive Lyapunov exponents are at most sub-exponentially large. The remaining ones are of size at most e^m(𝒳_i+ε), up to a bounded factor, for all sufficiently large m. Thus 

    vol(B_1)   ≤ A· e^mε·(diam(B))^ (SM)∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i
       ≤ A· e^mε·(2ϱ)^ (SM)∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i
        =Ã· e^mε·ϱ^ (SM)∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i,

where Ã=A· 2^ (SM), for some A>0. Then substituting in (<ref>) we have that

    card{ X∈𝒫: X∩ D≠∅}   ≤ b·vol(B_1)·ϱ^- (SM)+1
       ≤ b·Ã· e^mε∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i+1
       ≤ (b·Ã+1)· e^mε∏_i:𝒳_i>0e^m(𝒳_i+ε)k_i.

Considering L_2=b·Ã+1 we obtain the desired result.

Proof of Theorem 1.1. We have that μ(SM∖ K)<ε. From (<ref>), Lemmas 5.2 and 5.3 we obtain

    mh_μ(ϕ)-ε   =h_μ(ϕ^m)-ε
       ≤ h_μ(ϕ^m,𝒫) 
       ≤∑_D∈ϕ^m𝒫μ(D)·logcard{ X∈𝒫: X∩ D≠∅}
       ≤∑_D∈ϕ^m𝒫_K, D∩Λ_m=∅μ(D)·logcard{ X∈𝒫: X∩ D≠∅}
        +	∑_D∈ϕ^m𝒫_K, D∩Λ_m≠∅μ(D)·logcard{ X∈𝒫: X∩ D≠∅}
       +μ(ϕ^m(SM∖ K))·logcard{ X∈𝒫: X∩ϕ^m(SM∖ K)≠∅}
       ≤∑_D∈ϕ^m𝒫_K, D∩Λ_m=∅μ(D)( log (L_1) + (SM)·max{ mlog(φ),-log(ϱ_m) }) 
       + ∑_D∈ϕ^m𝒫_K, D∩Λ_m≠∅μ(D)( log(L_2) +mε  +m∑_i:𝒳_i>0(𝒳_i+ε)k_i) 
        + μ(SM∖ K)·( log (L_1) + (SM)·max{ mlog(φ),-log(ϱ_m) })

 
    ≤( log (L_1) + (SM)·max{ mlog(φ) ,-log(ϱ_m) })·μ(SM∖Λ_m) 
        + log(L_2) +mε  +m∑_i:𝒳_i>0(𝒳_i+ε)k_i 
       + ε·( log (L_1) + (SM)·max{ mlog(φ) ,-log(ϱ_m) }).

By Oseledec's Theorem we have that μ(SM∖Λ_m)→ 0 as m→∞. Moreover,

    lim_m→ +∞1mlog(ϱ_m) = -log(h(c))-2c,

where h(c) is the expression that bounds the derivative of ϕ (see Proposition 4.4). Then, dividing by m in (<ref>) and taking m→ +∞ we obtain

    h_μ(ϕ)≤ε + ∑_i:𝒳_i>0(𝒳_i+ε)k_i +ε· (SM)·max{log(φ) ,log(h(c))+2c }.

Letting ε→ 0 we have

    h_μ(ϕ)≤∑_i:𝒳_i>0𝒳_ik_i,

which is the desired upper bound. □



§ PESIN'S FORMULA

In this section, we aim to prove Theorem 1.2. To achieve this goal, we will use the techniques applied by Mañé in <cit.> which don't use the theory of stable manifolds. Adopting this strategy greatly simplifies our proof since we only need to corroborate that all the technical hypotheses used by Mañé continue to be satisfied under the condition of the geodesic flow being Anosov. To simplify notation, we write

    𝒳^+(θ)=∑_𝒳_i(θ)>0𝒳_i(θ)· (H_i(θ)).

We start introducing some notations. Set g:SM→ SM a map and ρ:SM→ (0,1) a function. For θ∈ SM and n≥ 0, define

    S_n(g,ρ,θ)={ω∈ SM: d(g^j(θ),g^j(ω))≤ρ(g^j(θ)), 0≤ j≤ n} .

If μ is a measure on SM and g and ρ are measurable, define

    h_μ(g,ρ,θ)=lim sup_n→∞-1nlogμ(S_n(g,ρ,θ)).

Let E be a normed space and E=E_1⊕ E_2 a splitting.
We say that a subset W⊂ E is a (E_1,E_2)-graph if there exists an open set U⊂ E_2 and a C^1-map ψ:U→ E_1 such that W={ (ψ (x),x): x∈ U}. The number 
    sup{ψ(x)-ψ(y)  x-y: x,y∈ U, x≠ y  }
 is called the dispersion of W.


Let M be a complete Riemannian manifold and μ an ϕ^t-invariant probability measure on SM satisfying the assumptions of Theorem 1.2. Denote by ν the Lebesgue measure on SM. Since the geodesic flow is of Anosov type, consider	
    E^cs(θ)=⟨ G(θ)⟩⊕ E^s(θ)
 for every θ∈ SM. From Theorem 3.1 there is a set Λ⊂ SM such that μ(SM∖Λ)=0 and the Lyapunov exponents of ϕ exist for every θ∈Λ.  Fix any ε>0. By Egorov's and Oseledec's Theorems, there is a compact set K⊂Λ with μ(K)≥ 1-ε such that the splitting T_θ SM=E^cs(θ)⊕ E^u(θ) is continuous when θ varies in K and, for some N>0, there are constants α>β>1 such that, if g=ϕ^N, the inequalities

    d g^n_θ(η)    ≥α^nη
    
     .d g^n_θ|_E^cs(θ)   ≤β^n 
    
     log| ( . d g^n_θ|_E^u(θ)) |    ≥ Nn(𝒳^+(θ)-ε)

hold for all θ∈ K, n≥ 0 and η∈ E^u(θ).


In the same way as in <cit.>, in the remainder of this section, we will treat SM as if it were an Euclidean space. The arguments we use can be formalized without any difficulty by the direct use of local coordinates. Since the geodesic flow is C^1-Hölder, we have the following result proved by Mañe in <cit.>.

For every σ>0 there is ξ>0 such that, if θ∈ K and g^m(θ)∈ K for some m>0, then if a set W⊂ SM is contained in the ball B_ξ^m(θ) and is a (E^cs(θ),E^u(θ))-graph with dispersion ≤σ, then g^m(W) is a (E^cs(g^m(θ)),E^u(g^m(θ)))-graph with dispersion ≤σ.

Fix the constant σ>0 of the statement of Lemma 6.1 small enough such that exists a∈ (0,1), a≤ t_0/2, where t_0 comes from Proposition 2.2 applied to SM, with the following property: if θ∈ K, ω∈ SM and d(θ,ω)<a, then for every subspace E⊂ T_ω SM which is a (E^cs(θ), E^u(θ))-graph with dispersion ≤σ we have

    | log|( . d g_ω|_E) |-log| ( . d g_θ|_E^u(θ) )  |  | ≤ε.

We proved in Theorem 3.1 that the norm of the derivative of ϕ is bounded, then denote 
    P=sup{log|. ( d ϕ_θ|_E)   | : θ∈ SM, E⊂ T_θ SM}.

The following proposition is an adaptation of Mañe's result in <cit.> applied to the case of Anosov geodesic flow for non-compact manifolds. To ensure a comprehensive understanding of our arguments, we chose to include the full proof provided by Mañé. 

For every small ε>0, there exist a function ρ:SM→ (0,1) with logρ∈ L^1(SM,μ), an integer N>0 and a compact set K'⊂ SM with μ(SM∖ K')≤ 2√(ε) such that

    h_ν(ϕ^N,ρ,θ)≥ N( 𝒳^+(θ)-ε-εN-4P√(ε))

for every θ∈ K'.

 For θ∈ K, define L(θ) as the minimum integer ≥ 1 such that g^L(θ)(θ)∈ K. This function is well defined for μ-almost every θ∈ K and it is integrable. Extend L to SM, putting L(θ)=0 when θ∉ K and at points of K that do not return to this set. Define ρ:SM→ (0,1) as

    ρ(θ)=min{ a, ξ^L(θ)},

where a∈ (0,t_0/2) comes from property (<ref>) and ξ>0 comes from Lemma 6.1. Since L is integrable then clearly logρ is also integrable. 
On the other hand, by Birkhoff's ergodic theorem, the function

    Ψ(θ)=lim_n→ +∞1ncard{ 0≤ j< n: g^j(θ)∈Λ∖ K}

is defined for μ-almost every θ∈Λ. Then

    ε≥μ(Λ∖ K)   =∫_ΛΨ dμ
       ≥∫_{θ∈Λ: Ψ(θ)>√(ε)}Ψ dμ
       >√(ε)·μ( {θ∈Λ: Ψ(θ)>√(ε)}).

Therefore,

    μ( {θ∈Λ: Ψ(θ)≤√(ε)})≥ 1-√(ε).

By Egorov's Theorem, there exists a compact set K'⊂ K with μ(K')≥ 1-2√(ε) and N_0>0 such that, if n≥ N_0,

    card{ 0≤ j< n: g^j(θ)∈Λ∖ K}≤ 2n√(ε)

for all θ∈ K'. Since the subspaces E^cs(θ) and E^u(θ) are not necessary orthogonal, there exists B>0 such that

    ν(S_n(g,ρ,θ))≤ B∫_E^cs(θ)ν(( ω+E^u(θ)) ∩ S_n(g,ρ,θ) ) dν(ω)

for every θ∈ K' and n≥ 0, where ν also denotes the Lebesgue measure in the subspaces E^cs(θ) and ω+E^u(θ). For ω∈ E^cs(θ), denote by

    Ω_n(ω)=(ω+E^u(θ)) ∩ S_n(g,ρ,θ).

Take D>0 such that D> vol(W) for every (E^cs(θ),E^u(θ))-graph W with dispersion ≤σ contained in B_ρ(θ)(θ), where θ∈ K' and ρ is the function defined in (<ref>). This constant exists because the domain of the graphs is contained in a ball of radius <1 and the derivatives of the functions defining the graphs are uniformly bounded in norm by σ. 

If g^n(θ)∈ K' and ω∈ E^cs(θ), from Lemma 5 of <cit.> we have that g^n(Ω_n(ω)) is a (E^cs(g^n(θ)),E^u(g^n(θ)))-graph with dispersion ≤σ and

    D>vol(g^n(Ω_n(ω)))=∫_Ω_n(ω)| . dg^n_z|_T_z Ω_n(ω)| dν(z).

Fix any θ∈ K' and let S_n={ 0≤ j<n: g^j(θ)∈ K'}. If n≥ N_0, it follows from (<ref>), (<ref>) and (<ref>) that for ω∈ E^cs(θ) we have

    log| ( . dg^n_z|_T_zΩ_n(ω)) |   =∑_j=0^n-1log| ( . dg_g^j(z)|_T_g^j(z)g^j(Ω_n(ω))) | 
       ≥∑_j∈ S_nlog| ( . dg_g^j(z)|_T_g^j(z)g^j(Ω_n(ω))) |-NP(n-cardS_n)
       ≥∑_j∈ S_nlog| ( . dg_g^j(θ)|_E^u(g^j(θ))) |-ε n-NP(n-cardS_n)
       ≥∑_j=0^n-1log| ( . dg_g^j(θ)|_E^u(g^j(θ))) |-ε n-2NP(n-cardS_n)
       =log| (. dg^n_θ|_E^u(θ))|  -ε n-2NP(n-cardS_n)
       ≥ nN(𝒳^+(θ)-ε)-ε n-2NP(n-cardS_n)
       ≥  nN(𝒳^+(θ)-ε)-ε n-4NPn√(ε).

From (<ref>) we obtain that

    D>ν(Ω_n(ω))·exp( nN(𝒳^+(θ)-ε)-ε n-4NPn√(ε))

for every θ∈ K' and ω∈ E^cs(θ). It follows from (<ref>) that

    ν(S_n(g,ρ,θ))≤ B· D·exp( -nN(𝒳^+(θ)-ε)+ε n+4NPn√(ε)) .

Therefore, for every θ∈ K',

    h_ν(g,ρ,θ)=lim sup_n→∞-1nlogν(S_n(g,ρ,θ))≥ N( 𝒳^+(θ)-ε-εN-4P√(ε)).

This completes the proof of the proposition.


We will show that the function ρ of Proposition 6.2 allows us to find a lower bound for the entropy of ϕ^N. To prove this, Mañé constructed a partition of the manifold with certain properties using strongly the compactness condition (see Lemma 2 of <cit.>). Since the manifold SM is not necessarily compact in our case, we will use another technique to construct a partition that satisfies the same properties. Consider the constant a∈ (0,1), a<t_0/2, used in property (<ref>).


Let M be a complete Riemannian manifold and suppose that the curvature tensor and the derivative of the curvature tensor are both uniformly bounded. For every θ∈ SM we have that

    diam exp_θ U≤52·diam  U,

where U⊂ B(0,a)⊂ T_θ SM. 


Fix θ∈ SM and consider U⊂ B(0,a)⊂ T_θ SM. We need to prove that 

    d(exp_θ u, exp_θ v)≤52 u-v

for every u,v∈ U. Consider the segment q(t)=tu + (1-t)v and the curve γ(t)=exp_θ q(t) that joins exp_θ u with exp_θ v. Then

    l(γ)   =∫_0^1γ'(t)
       =∫_0^1 d(exp_θ)_q(t) (u-v) dt.

For each t∈ [0,1], there are w(t)∈ T_θ SM with w(t)=1 and s(t)∈ℝ with |s(t)|≤ t_0 such that

    q(t)=s(t)w(t).

Since a≤ t_0/2, from Proposition 2.2 we have that

    d(exp_θ)_q(t) (u-v)   = d(exp_θ)_s(t)w(t) (u-v)
       ≤52u-v.

Therefore in (<ref>)

    d(exp_θ u, exp_θ v)≤ l(γ)≤52u-v

completing the proof. 

Consider the function ρ:SM→ (0,1) defined in (<ref>).

There exists a countable partition 𝒫 of SM with finite entropy such that, if 𝒫(θ) denotes the atom of 𝒫 containing θ, then
	
    diam 𝒫(θ)≤ρ(θ)

	for μ-almost every θ∈ SM.

 
For each n≥ 0, define

    U_n={θ∈ SM: e^-(n+1)<ρ(θ)≤ e^-n} .

Since logρ∈ L^1(SM,μ), we have that

    ∑_n=0^∞ n μ(U_n)≤ -∑_n=0^∞  ∫_U_nlogρ(θ)dμ(θ)= -∫_SMlogρ(θ)dμ(θ)<∞.

Then, by Lemma 1 of <cit.> we obtain

    ∑_n=0^∞μ(U_n)logμ(U_n)<∞.

For θ∈ SM∖ K' we have that ρ(θ)=a. Then there exists n_0≥ 0 such that 
    e^-(n_0+1)<a≤ e^-n_0
 and U_n∩ (SM∖ K')=∅ for every n≠ n_0. This implies that U_n⊂ K' for every n≠ n_0. Define

    U_n_0^*=U_n_0∩ K'.
 
Since K' is compact, there exist A>0 and r_0>0 such that for all 0<r≤ r_0, there exists a partition 𝒬_r of K' whose atoms have diameter less than or equal to r and such that the number of atoms in 𝒬_r, denoted by | 𝒬_r|, satisfies

    | 𝒬_r|≤ A( 1r)^ (SM).
 
Define 𝒬 as the partition of K' given by

    
  . Sets X∩ U_n, for n≥ 0, n≠ n_0, where X∈𝒬_r_n and r_n=e^-(n+1) such that μ(X∩ U_n)>0.
    
  . Sets X∩ U_n_0^*, where X∈𝒬_r_n_0 and r_n_0=e^-(n_0+1) such that μ(X∩ U_n_0^*)>0.

On the other hand, consider 0<ε'<a/10 such that, we can choose a measurable set (like a “ring" covering SM∖ K') 

    V_1⊆{θ∈ SM∖ K': d(θ,K')≤ε'}:=E_1

that satisfies 
    μ(V_1)≤√(ε).

Define K'_1=K'∪ V_1 and choose a measurable set (like a “ring" covering SM∖ K'_1)

    V_2⊆{θ∈ SM∖ K'_1: d(θ,K'_1)≤ε'}:=E_2

that satisfies 
    μ(V_2)≤√(ε)2.
 
Proceeding inductively, we define bounded measurable sets 

    V_n⊆{θ∈ SM∖ K'_n-1: d(θ,K'_n-1)≤ε'}:=E_n,

where K'_n-1=K'∪ V_1…∪ V_n-1, with measure 
    μ(V_n)≤√(ε)2^n-1.

Since

    ∑_n=1^∞ nμ(V_n)≤∑_n=1^∞n2^n-1·√(ε)<∞,

by Lemma 1 of <cit.> we have that

    ∑_n=1^∞μ(V_n)logμ(V_n)<∞.

Let k be the number of balls of radius a/10 which cover E_1 and denote by B(θ_1,a/10),…,B(θ_k,a/10) this covering. We claim that

    E_2⊆⋃_i=1^k B(θ_i,a/5).

In fact, suppose that exists θ∈ E_2 such that d(θ,θ_i)≥ a/5, for every i=1,…,k. By construction, there is ω∈ E_1 such that 

    d(θ,ω)≤ε'<a10.

Since we cover E_1 by balls, ω∈ B(θ_i_0,a/10) for some i_0∈{ 1,…,k}. Therefore,

    d(θ,ω)   ≥ d(θ,θ_i_0)-d(θ_i_0,ω)
       > a5 -a10
       =a10,

which is a contradiction with (<ref>). This proves the claim. Since SM is complete (see Lemma 2.1), for each i∈{1,…,k}, there is an open ball B^i(0,a/5)⊂ T_θ_iSM such that 
    exp_θ_i(B^i(0,a/5))=B(θ_i,a/5).
 By <cit.> there exists N_1:=N_1(a)>0, which depends on the dimension of SM and a, such that the minimal number of balls of radius a/10 which can cover B^i(0,a/5) is bounded by N_1. Suppose that

    B^i_1,…, B^i_N_1

are balls of radius a/10 that cover B^i(0,a/5). From Lemma 6.3, if we project these balls to the manifold SM by the exponential map we have that exp_θ_iB^i_j are sets of diameter

    diam  exp_θ_i B^i_j≤52diam B^i_j=52·2a10=a2.

Then we can cover E_2 by k N_1 sets of diameter ≤ a/2. Since every set of diameter ≤ a/2 is contained in a ball of radius a/2, we can cover E_2 by kN_1 balls B(ω_1,a/2),…, B(ω_kN_1,a/2). Analogously, since ε'<a/10 we have that

    E_3⊂⋃_i=1^kN_1 B(ω_i,6a/10).

For each i∈{1,…,kN_1}, there is an open ball B^i(0,6a/10)⊂ T_ω_iSM such that 
    exp_ω_i(B^i(0,6a/10))=B(ω_i,6a/10).
 By <cit.> there exists N_2:=N_2(a)>0, which depends on the dimension of SM and a, such that the minimal number of balls of radius a/10 which can cover B^i(0,6a/10) is bounded by N_2 and repeating the previous process we have that we can cover E_3 by kN_1N_2 balls of radius a/2. Continuing inductively, we obtain that E_n can be covered by kN_1N_2^n-2 balls of radius a/2. Therefore, for every n≥ 1, define a partition 𝒫̂_n of V_n whose atoms have diameter ≤ a and the number of atoms satisfies

    | 𝒫̂_1 |	≤ k, |𝒫̂_n |≤ kN_1N_2^n-2, ∀ n≥ 2.

Finally, define the partition of SM as 
    𝒫=𝒬∪⋃_n≥ 1𝒫̂_n.

Recalling the well-known inequality

    -∑_i=1^m x_ilog x_i≤(∑_i=1^m x_i)( log m -log∑_i=1^m x_i)

which holds for any set of real numbers 0<x_i≤ 1, i=1,…,m. We claim that H(𝒫)<+∞. In fact, from (<ref>) and (<ref>) we obtain that

    H(𝒫)   =∑_n≥ 0, n≠ n_0( -∑_P∈𝒬,P⊂ U_nμ(P)logμ(P))+( -∑_P∈𝒬,P⊂ U_n_0^*μ(P)logμ(P))
       + ∑_n≥ 1( -∑_P∈𝒫̂_nμ(P)logμ(P))
       ≤∑_n≥ 0, n≠ n_0μ(U_n)[ log| 𝒬_r_n| - logμ(U_n) ] +μ(U_n_0^*)[ log | 𝒬_r_n_0 | - logμ(U_n_0^*) ]
       +∑_n≥ 1,μ(V_n)[ log | 𝒫̂_n | - logμ(V_n) ]



    ≤∑_n≥ 0, n≠ n_0μ(U_n)[log A +   (SM)(n+1) - logμ(U_n)]  
        +μ(U_n_0^*)[log A+   (SM)(n_0+1)- logμ(U_n_0^*)]+μ(V_1)[log k -logμ(V_1)]  
       + ∑_n≥ 2μ(V_n)[log k + log N_1 + (n-2)log N_2- logμ(V_n)]
       <∞.

Moreover, if θ∈ U_n, for n≥ 0, n≠ n_0, then 𝒫(θ) is contained in an atom of 𝒬_r_n and

    diam 𝒫(θ)≤ r_n= e^-(n+1)<ρ(θ).

 If θ∈ U_n_0^*, then 𝒫(θ) is contained in an atom of 𝒬_r_n_0 and

    diam 𝒫(θ)≤ r_n_0= e^-(n_0+1)<ρ(θ).

In another case, if θ∈ V_n, for n≥ 1, then 𝒫(θ) is contained in an atom of 𝒫̂_n and diam𝒫(θ)≤ a=ρ(θ). 

Given that M has finite volume, it follows that SM also has finite volume. Lemma 6.4, together with the Radon-Nikodym Theorem and Shannon-McMillan-Breiman Theorem, allow us to obtain the following result proved in <cit.>.
  If μ≪ν, where ν denotes the Lebesgue measure on SM, then 

    h_μ(ϕ^N)≥∫_SM h_ν(ϕ^N,ρ,θ)dμ(θ).
  

Proof of Theorem 1.2. We just need to prove that

    h_μ(ϕ) ≥∫_SM𝒳^+(θ)dμ(θ).

Consider Υ=sup_θ∈ SM{dϕ^1_θ, dϕ^-1_θ}. Then

    ∫_SM∖K'𝒳^+(θ)dμ(θ)   ≤μ(SM∖ K')· (SM)·logΥ
       ≤ 2√(ε)· (SM)·logΥ.

From Propositions 6.2 and 6.5 we have that

    h_μ(ϕ^N)   ≥∫_SMh_ν(ϕ^N,ρ,θ)dμ(θ)
       ≥∫_K' h_ν(ϕ^N,ρ,θ)dμ(θ)
       ≥ N∫_K'𝒳^+(θ)dμ(θ)-Nε-ε-4NP√(ε)
       ≥ N∫_SM𝒳^+(θ)dμ(θ)-2√(ε)N· (SM)·logΥ-Nε-ε-4NP√(ε).

Hence,

    h_μ(ϕ)≥∫_SM𝒳^+(θ)dμ(θ)-2√(ε)· (SM)·logΥ- ε-εN-4P√(ε).

Letting ε→ 0 we obtain the desired lower bound.□


 §.§ Acknowledgments
 Alexander Cantoral thanks FAPERJ for partially supporting the research (Grant  E-26/202.303/2022). Sergio Romaña thanks “Bolsa Jovem Cientista do Nosso Estado No. E-26/201.432/2022",  NNSFC 12071202, and NNSFC 12161141002 from China. The second author thanks the Department of Mathematics of the SUSTech- China for its hospitality during the execution of this work.
	
	
	
	abbrv
	
	
	
	
	
	
	
	
	 
	
