Franks' Lemma for C^2-Man\'e Perturbations of Riemannian Metrics and Applications to Persistence

Given a compact Riemannian manifold, we prove a uniform Franks' lemma at second order for geodesic flows and apply the result in persistence theory.


INTRODUCTION
One of the most important tools of C 1 generic and stability theories of dynamical systems is the celebrated Franks' Lemma [15]: In a few words, the lemma asserts that given a collection S of m points p i in the manifold M , any isomorphism from Π to Π can be the collection of the differentials of a diffeomorphism g , C 1 close to f , at each point of S provided that the isomorphism is sufficiently close to the direct sum of the maps D p i f , i = 1, . . . , m. The sequence of points is particularly interesting for applications in dynamics when the collection S is a subset of a periodic orbit. The idea of the proof of the lemma is quite elementary: we conjugate the isomorphisms L i by the exponential map of M in suitably small neighborhoods of the points p i 's and then glue (smoothly) the diffeomorphism f outside the union of such neighborhoods with this collection of conjugate-to-linear maps. So the proof strongly resembles an elementary calculus exercise: given a C 1 function h : R −→ R, x 0 ∈ R, and a number c close to h (x 0 ), we can glue h outside a small neighborhood U of x 0 with the linear function in U σ(x) = h(x 0 ) + c(x − x 0 ); so σ(x 0 ) = h(x 0 ) and σ (x 0 ) = c; to get a new function that is C 1 close to h.
Franks' Lemma admits a natural extension to flows, and its important applications in the study of stable dynamics gave rise to versions for more specific families of systems, like symplectic diffeomorphisms and Hamiltonian flows [35,41]. It is clear that for specific families of systems the proof of the lemma should be more difficult than just gluing conjugates of linear maps by the exponential map since this surgery procedure in general does not preserve specific properties of systems, like preserving symplectic forms in the case of symplectic maps. Franks' Lemma was extensively used by R. Mañé in his proof of the C 1 structural stability conjecture [23], and we could claim with no doubts that it is one of the pillars of the proof together with C. Pugh's C 1 closing lemma [30,31] (see Newhouse [27] for the proof of the C 1 structural stability conjecture for symplectic diffeomorphisms).
A particularly challenging problem is to obtain a version of Franks' Lemma for geodesic flows. First of all, a typical perturbation of the geodesic flow of a Riemannian metric in the family of smooth flows is not the geodesic flow of another Riemannian metric. To ensure that perturbations of a geodesic flow are geodesic flows as well the most natural way to proceed is to perturb the Riemannian metric in the manifold itself. But then, since a local perturbation of a Riemannian metric changes all geodesics through a neighborhood, the geodesic flow of the perturbed metric changes in tubular neighborhoods of vertical fibers in the unit tangent bundle. Since local perturbations of the metric are not quite local for the geodesic flow, the usual strategy applied in generic dynamics of perturbing a flow in a flowbox without changing the dynamics outside the box does not work. This poses many interesting, technical problems in the theory of local perturbations of dynamical systems of geometric origin, the famous works of Klingenberg-Takens [18] and Anosov [3] (the bumpy metric theorem) about generic properties of closed geodesics are perhaps the two best known examples. Moreover, geodesics in general have many self-intersections so the effect of a local perturbation of the metric on the global dynamics of perturbed orbits is unpredictable unless we know a priori that the geodesic flow enjoys some sort of stability (negative sectional curvatures, Anosov flows for instance).
The family of metric perturbations which preserves a compact piece of a given geodesic is the most used to study generic theory of periodic geodesics. This family of perturbations is relatively easy to characterize analytically when we restrict ourselves to the category of conformal perturbations or, more generally, to the set of perturbations of Lagrangians by small potentials. Recall that a Riemannian metric h in a manifold M is conformally equivalent to a Riemannian metric g in M if there exists a positive, C ∞ function b : M −→ R such that h x (v, w) = b(x)g x (v, w) for every x ∈ M and v, w ∈ T x M . Given a C ∞ , Tonelli Lagrangian L : T M −→ R defined in a compact manifold M , and a C ∞ function u : M −→ R, the function L u (p, v) = L(p, v) + u(p) gives another Tonelli Lagrangian. The function u is usually called a potential because of the analogy between this kind of Lagrangian and mechanical Lagrangians.
By Maupertuis' principle (see for example [12]), the Lagrangian associated to a metric h in M that is conformally equivalent to g is of the form for some function u. Since the Lagrangian of a metric g is given by the formula . Now, given a compact part γ : [0, T ] −→ M of a geodesic of (M , g ), the collection of potentials u : M → R such that γ[0, T ] is still a geodesic of L(p, v) = L g (p, v) + u(p) contains the functions whose gradients vanish along the subset of T γ(t ) M which are perpendicular to γ (t ) for every t ∈ [0, T ] (see for instance [37,Lemma 2.1]). Lagrangian perturbations of Tonelli Lagrangians of the type L h (p, v) = L g (p, v) + u(p) were used extensively by R. Mañé to study generic properties of Tonelli Lagrangians and applications to Aubry-Mather theory (see for instance [24,25]). Mañé's idea proved to be very fruitful and insightful in Lagrangian generic theory, and opened a new branch of generic theory that is usually called Mañé's genericity. Recently, Rifford-Ruggiero [34] gave a proof of Klingenberg-Takens and Anosov C 1 genericity results for closed geodesics using control theory techniques applied to the class of Mañé type perturbations of Lagrangians. Control theory ideas simplify a great deal the technical problems involved in metric perturbations and at the same time show that Mañé type perturbations attain full Hamiltonian genericity. This result, combined with a previous theorem by Oliveira [28], led to the Kupka-Smale Theorem for geodesic flows in the family of conformal perturbations of metrics.
These promising applications of control theory to the generic theory of geodesic flows motivate us to study Franks' Lemma for conformal perturbations of Riemannian metrics or equivalently, for Mañé type perturbations of Riemannian Lagrangians. Before stating our main theorem, let us recall first some notations and basic results about geodesic flows. The geodesic flow of a Riemannian manifold (M , g ) will be denoted by φ t ; the flow acts on the unit tangent bundle T 1 M ; a point θ ∈ T 1 M has canonical coordinates θ = (p, v), where p ∈ M and v ∈ T p M ; and γ θ denotes the unit speed geodesic with initial conditions γ θ (0) = p, γ θ (0) = v. Let N θ ⊂ T θ T 1 M be the plane of vectors that are perpendicular to the geodesic flow with respect to the Sasaki metric (see for example [38]). The collection of these planes is preserved by the action of the differential of the geodesic flow: D θ φ t (N θ ) = N φ t (θ) for every θ and t ∈ R.
Let us consider a geodesic arc of length T and let Σ 0 and Σ T be local transverse sections for the geodesic flow which are tangent to N θ and N φ T (θ) respectively. Let P g (Σ 0 , Σ T , γ θ ) be a Poincaré map going from Σ 0 to Σ T . In horizontal-vertical coordinates of N θ , the differential is a symplectic endomorphism of R (2n−2) × R (2n−2) . This endomorphism can be expressed in terms of the Jacobi fields of γ θ which are perpendicular to γ θ (t ) for every t : whereJ denotes the covariant derivative along the geodesic. We can identify the set of all symplectic endomorphisms of R 2n−2 × R 2n−2 with the symplectic group where X * denotes the transpose of X and where d g denotes the geodesic distance with respect to g . We note that by compactness of M , there isτ > 0 such that any geodesic of length 2τ has no selfintersection, which makes any cylinder of the form C g γ θ [t 1 , t 2 ] ; ρ a genuine open cylinder provided t 2 − t 1 and ρ are small enough. Our first result is the following.  2.' supp(σ) ⊂ C g γ θ t ,t +τ ; ρ Σ. Theorem 1.1 improves a previous result by Contreras [7,Theorem 7.1] which gives a controllability result at first order under an additional assumption on the curvatures along the initial geodesic. Other proofs of Contreras' Theorem can also be found in [40] and [21]. The Lazrag proof follows already the ideas from geometric control introduced in [34] to study controllability properties at first order. Our new Theorem 1.1 shows that controllability holds at second order without any assumption on curvatures along the geodesic. Its proof amounts to study how small conformal perturbations of the metric g along Γ := γ θ ([0, T ]) affect the differential of P g (Σ 0 , Σ T , γ θ ). This can be seen as a problem of local controllability along a reference trajectory in the symplectic group. As in [34], the idea is to see the Hessian of the conformal factor along the initial geodesic as a control and to obtain Theorem 1.1 as a uniform controllability result at second order for a control system in the symplectic group Sp(n − 1). We apply Franks' Lemma to extend some results concerning the characterization of hyperbolic geodesic flows in terms of the persistence of some C 1 generic properties of the dynamics. These results are based on well known steps towards the proof of the C 1 structural stability conjecture for diffeomorphisms.
Let us first introduce some notations. Given a smooth compact Riemannian manifold (M , g ), we say that a property P of the geodesic flow of (M , g ) is -C k persistent from Mañé's viewpoint if for every C ∞ function f : M −→ R whose C k norm is less than we have that the geodesic flow of the metric (M , (1 + f )g ) has property P as well. By Maupertuis' principle, this is equivalent to the existence of an open C k -ball of radius > 0 of functions q : M −→ R such that for every C ∞ function in this open ball the Euler-Lagrange flow of the Lagrangian L(p, v) = 1 2 g p (v, v) − q(p) in the level of energy equal to 1 has property P . This definition is inspired by the definition of C k−1 persistence for diffeomorphisms: a property P of a diffeomorphism f : M −→ M is called -C k−1 persistent if the property holds for every diffeomorphism in the -C k−1 neighborhood of f . It is clear that if a property P is -C 1 persistent for a geodesic flow then the property P is -C 2 persistent from Mañé's viewpoint for some . An interesting application of Theorem 1.2 is the following extension of Theorem A in [36]: C 1 persistently expansive geodesic flows in the set of Hamiltonian flows of T 1 M are Anosov flows. We recall that a non-singular smooth flow φ t : Q −→ Q acting on a complete Riemannian manifold Q is -expansive if given x ∈ Q we have that for each y ∈ Q such that there exists a continuous surjective function ρ : R −→ R with ρ(0) = 0 satisfying for every t ∈ R then there exists t (y), |t (y)| < , such that φ t (y) (x) = y. A smooth non-singular flow is called expansive if it is expansive for some > 0. Anosov The proof of the above result requires the set of periodic orbits to be dense. Such a result follows from expansiveness on surfaces [36] and from the absence of conjugate points in any dimension. If we drop the assumption of the absence of conjugate points we do not know whether periodic orbits of expansive geodesic flows are dense (and so if the geodesic flow in Theorem 1.3 is Anosov). This is a difficult, challenging problem.
The paper is organized as follows. In the next section, we introduce some preliminaries which describe the relationship between local controllability and some properties of the End-Point mapping and we introduce the notions of local controllability at first and second order. We recall a result of controllability at first order (Proposition 2.1) already used in [34] and state a result (Propositions 2.2) at second order whose proof is given in Section 2.5. In Section 3, we provide the proof of Theorem 1.1, and the proof of Theorems 1.2 and 1.3 are given in Section 4.

PRELIMINARIES IN CONTROL THEORY
Our aim here is to provide sufficient conditions for first and second order local controllability results. These kind of results could be developed for nonlinear control systems on smooth manifolds. For the sake of simplicity, we restrict our attention here to the case of affine control systems on the set of (symplectic) matrices. We refer the interested reader to [1,9,17,20,33] for a further study in control theory.
2.1. The End-Point mapping. Let us a consider a bilinear control system on M 2m (R) (with m, k ≥ 1), of the forṁ where the state X (t ) belongs to M 2m (R), the control u(t ) belongs to R k , t ∈ [0, T ] → A(t ) (with T > 0) is a smooth map valued in M 2m (R), and B 1 , . . . , B k are k matrices in M 2m (R). GivenX ∈ M 2m (R) andū ∈ L 2 [0, T ]; R k , the Cauchy problemẊ possesses a unique solution XX ,ū (·). The End-Point mapping associated withX in time T > 0 is defined as It is a smooth mapping whose differential can be expressed in terms of the linearized control system (see [33]). GivenX ∈ M 2m (R),ū ∈ L 2 [0, T ]; R k , and settingX (·) := XX ,ū (·), the differential of EX atū is given by the linear operator where Y (·) is the unique solution to the linearized Cauchy problem Note that if we denote by S(·) the solution to the Cauchy problem then there holds Let Sp(m) be the symplectic group in M 2m (R) (m ≥ 1), that is, the smooth submanifold of matrices X ∈ M 2m (R) satisfying X * JX = J. Denote by S (2m) the set of symmetric matrices in M 2m (R). The tangent space to Sp(m) at the identity matrix is given by Therefore, if there holds then Sp(m) is invariant with respect to (1), that is, for everyX ∈ Sp(m) and In particular, this means that for everyX ∈ Sp(m), the End-Point mapping EX ,T is valued in Sp(m). Given X ∈ Sp(m) andū ∈ L 2 [0, T ]; R k , we are interested in local controllability properties of (1) aroundū.
Such a property is satisfied as soon as EX ,T is locally open atū. Our aim in the next sections is to give an estimate from above on the size of u L 2 in terms of X − XX ,u (T ) .
In fact, the control system which is relevant in the present paper is not always controllable at first order. For example, it is the case if we consider the control system (see (21)) coming from the canonical flat metric on the torus of dimension ≥ 3. We leave the reader to check that in this case the assumption (8) does not hold which by analyticity implies that controllability of (21) at first order fails (see [9]). So, we need sufficient condition for controllability at second order.
2.3. Second-order controllability results. Using the same notations as above, we say that the control system (1) is controllable at second order aroundū in Controllability at second order is weaker than controllability at first order (compare (6)- (9)). It requires a study of the End-Point mapping at second order.

Recall that given two matrices
The following result is the key point in the proof of Theorem 1.1. It provides sufficient conditions for controllability at second order of (1) aroundū ≡ 0 in a uniform manner. Its proof will be given in Sections 2.5. For the sake of simplicity we restrict here our attention to control systems of the form (1) satisfying (10)- (11). More general results can be found in [20].

PROPOSITION 2.2. Let Θ be a compact set of parameters and T
Define for every θ ∈ Θ the k sequences of smooth mappings B (7) and assume that the properties and are satisfied for every θ ∈ Θ. Assume, moreover, that the sets are compact. Then, for every compact set X ⊂ Sp(m), there are µ, K > 0 such that for every θ ∈ Θ, everyX ∈ X and every X ∈ B X θ (T ), µ ∩ Sp(m) (X θ (T ) denotes the solution at time T of the control system (1) with parameter θ starting fromX everyX ∈ X and every X ∈ B X θ (T ), µ ∩ Sp(m), there are a finite number of controls u 1 , . . . , u S in S such that any u satisfying property (13) has the form Our proof is based on a series of results on openness properties of C 2 mappings near critical points in Banach spaces which was developed by Agrachev and his co-authors [1].

Some sufficient condition for local openness.
Here we are interested in the study of mappings F : U → R N of class C 2 in an open set U in some Banach space X . We call critical point of F any u ∈ U such that D u F : U → R N is not surjective. We call the quantity corank(u) : The following quantitative result whose proof can be found in [1,20,33] provides a sufficient condition at second order for local openness. (We denote by B X (·, ·) the balls in X with respect to the norm · X .) Then there exist¯ , c ∈ (0, 1) such that for every ∈ (0,¯ ) the following property holds: In the above statement, D 2 u F | ker(Dū F ) refers to the quadratic mapping from ker(DūF ) to R N defined by Again, the proof of Theorem 2.3 which follows from previous results by Agrachev-Sachkov [1] and Agrachev-Lee [2] can be found in [20,33].

Proof of Proposition 2.2.
Let us first assume that there are no parameters, that is, that Θ is a singleton. First, we claim that it is sufficient to treat the casē X = I 2m . As a matter of fact, if X u : then for everyX ∈ Sp(m), the trajectory X uX : [0, T ] → M 2m (R) starts atX and satisfies So any trajectory of (1), that is, any control, steering I 2m to some X ∈ Sp(m) gives rise to a trajectory, with the same control, steeringX ∈ Sp(m) to XX ∈ Sp(m).
Since right-translations in Sp(m) are diffeomorphisms, we infer that local controllability at second order aroundū ≡ 0 fromX = I 2m implies controllability at second order aroundū ≡ 0 for anyX ∈ Sp(m). Then the existence of K , µ would follow by compactness of X . So from now on, we assume that X = I 2m (in the sequel we omit the lower index and simply write I ). We recall thatX : stands for a solution of (17) associated with some control u ∈ L 2 [0, T ]; R k . Furthermore, we may also assume that the End-Point mapping E I ,T : is not a submersion at u because it would imply controllability at first order aroundū and so at second order, as desired.
We equip the vector space M 2m (R) with the scalar product defined by Let us fix P ∈ TX (T ) Sp(m) such that P belongs to Im D 0 E I ,T ⊥ {0} with respect to our scalar product (note that Im We infer that We conclude by noticing that We check easily that for every u ∈ L 2 [0, T ]; R k the second derivative of E I ,T at 0 is given by (see [33]) where Then we infer that for every u ∈ L 2 ([0, T ]; R k ), It is useful to work with an approximation of the quadratic form P · D 2 0 E I ,T . For every δ > 0, we see the space L 2 ([0, δ]; R k ) as a subspace of L 2 ([0, T ]; R k ) by the canonical immersion δ] andũ(t ) := 0 otherwise. For the sake of simplicity, we keep the same notation forũ and u. LEMMA 2.5. There is C > 0 such that for every δ ∈ (0, T ), we have Proof of Lemma 2.5. Setting for every i , j = 1, . . . , k, and using (10), we check that for any t , s Then by (18) we infer that for any δ ∈ (0, T ) and any u ∈ L 2 ([0, δ]; R k ), We conclude easily by Cauchy-Schwarz inequality.
Returning to the proof of Proposition 2.2, we now want to show that the assumption (15) of Theorem 2.3 is satisfied. We are in fact going to show that a stronger property holds, namely that the index of the quadratic form in (15) goes to infinity as δ tends to zero. LEMMA 2.6. For every integer N > 0, there are δ > 0 and a space L δ ⊂ L 2 [0, δ]; R k of dimension larger than N such that the restriction of Q δ to L δ satisfies Proof of Lemma 2.6. Using the notation for any pair of continuous functions h 1 , h 2 : [0, δ] → R, we check that for every u ∈ L 2 [0, δ]; R k , The sum of the first two terms in the right-hand side of (19) is given by (we set P := P * S(T )) Using (11) with i =ī and Lemma 2.4 we obtain tr P B 1 i (0), Bī = 0, and consequently In conclusion, the sum of the first two terms in the right-hand side of (19) can be written as By the same arguments as above, the sum of the third and fourth terms in the right-hand side of (19) can be written as We now need the following technical result whose proof is given in Appendix.
Let us now show how to conclude the proof of Lemma 2.6. Recall that P ∈ TX (T ) Sp(m) was fixed in Im D 0 E I ,T ⊥ {0} and that by Lemma 2.4, we know that (taking t = 0) Consequently, using (12) we infer that there areī ,j ∈ {1, . . . , k} withī =j such that belongs to L N δ and by an easy change of variables, Moreover, it satisfies We get the result for δ > 0 small enough.
We can now conclude the proof of Proposition 2.2. First we note that given N ∈ N strictly larger than m(2m + 1), if L ⊂ L 2 [0, T ]; R k is a vector space of dimension N , then the linear operator has a kernel of dimension at least N − m(2m + 1), which means ker D 0 E I ,T ∩ L has dimension at least N −m(2m +1). Then, thanks to Lemma 2.6, for every integer N > 0, there are δ > 0 and a subspace L δ ⊂ L 2 [0, δ]; R k ⊂ L 2 [0, T ]; R k such that the dimension ofL δ := L δ ∩ker D 0 E I ,T is larger than N and the restriction of Q δ toL δ satisfies By Lemma 2.5, we have Then we infer that Note that since E I ,T is valued in Sp(m), which is a submanifold of M 2m (R), assumption (15) . Therefore, by (20), assumption (15) is satisfied. Consequently, thanks to Theorem 2.3 there exist¯ , c ∈ (0, 1) such that for every ∈ (0,¯ ) the following property holds: for every u ∈ U , Z ∈ TX (T ) Sp(m) with there are w 1 , w 2 ∈ L 2 [0, T ]; R k such that u + w 1 + w 2 ∈ U , Apply the above property with u =ū ≡ 0 and X ∈ Sp(m) such that X −X (T ) =: c 2 2 with <¯ .
Set Z := Π(X ), then we have (Π is an orthogonal projection so it is 1-lipschitz) Therefore by the above property, there are w 1 , w 2 ∈ L 2 [0, T ]; R k such thatũ := SinceΠ |B (X (T ),µ)∩Sp(m) is a local diffeomorphism, taking > 0 small enough, we infer that In conclusion, the control system (1) is controllable at second order around u ≡ 0 and we have (13). Assume now that we are given a set S ⊂ L 2 ([0, T ]; R k ) which is dense in  (14)) follows as above. In the case when Θ is a singleton (no parameters), the uniformity of K , µ with respect toX ∈ X follows by compactness of X . In the general case, the result follows by compactness of Θ and of the sets We refer the reader to [20,33] for further details.
and consider local transverse sections Σ 0 ,Σ,Σ, Σ T ⊂ T 1 M respectively tangent to N θ , Nθ, Nθ, N θ T . Then we have Since the sets of endomorphisms of Sp(m) of the form DθP g Σ , Σ T , γ θ and D θ P g Σ 0 ,Σ, γ θ , that is, the differential of Poincaré maps associated with geodesics of lengths T −t −τ andt , are compact and the left and right translations in Sp(m) are diffeomorphisms and since moreover self-intersections are transverse, it is sufficient to prove Theorem 1.1 with T =τ. More exactly, it is sufficient to show that there areδ,K > 0 such that for every δ ∈ (0,δ), the following property holds: let γ = γ θ : The following result follows easily (see also [37, Lemmas 2.1 and 2.2]). It relies on the construction and on well known formulae for the conformal connection and sectional curvatures. The notation ∂ l with l = 0, 1, . . . , n − 1 stands for the partial derivative in coordinates x 0 = t , x 1 , . . . , x n−1 and H σ a,ρ,u denotes the Hessian of σ a,ρ,u with respect to g .
Set m := n − 1 and k := m(m + 1)/2. Let us study the effect of a function u = (u i j ) i ≤ j =1,...,n−1 : [0,τ] → R n(n−1)/2 on the symplectic mapping P h (γ)(τ). By the Jacobi equation, we have where J : [0,τ] → R m is solution to the Jacobi equation where R h (t ) is the m × m symmetric matrix whose coefficients are the R i j (t ).
In other terms, P h (γ)(τ) is equal to the 2m × 2m symplectic matrix X (τ) given by the solution X : [0,τ] → Sp(m) at timeτ of the following Cauchy problem (compare [34, Sect. 3.2] and [21]): where the 2m × 2m matrices A(t ), E (i j ) are defined by where the E (i j ), 1 ≤ i ≤ j ≤ m are the symmetric m × m matrices defined by and E (i j ) k,l = δ i k δ j l + δ i l δ j k ∀i , j = 1, . . . , m.
Since our control system has the form (1), all the results gathered in Section 2 apply. So, Theorem 1.1 will follow from Proposition 2.2. First by compactness of M and regularity of the geodesic flow, the compactness assumptions in Proposition 2.2 are satisfied. It remains to check that assumptions (10), (11), and (12) hold.
First we check immediately that so assumption (10) is satisfied. Since the E (i j ) do not depend on time, we check easily that the matrices B 0 i j , B 1 i j , B 2 i j associated to our system are given by (remember that we use the notation [B, B ] := B B − B B ) , for every t ∈ [0,τ] and any i , j = 1, . . . , m with i ≤ j . An easy computation yields for any i , j = 1, . . . , m with i ≤ j and any t ∈ [0,τ], Then we get for any i , j = 1, . . . , m with i ≤ j , So assumption (11) is satisfied. It remains to show that (12) holds. We first notice that for any i , j , k, l = 1, . . . , m with i ≤ j , k ≤ l , we have where F (pq) is the m × m skew-symmetric matrix defined by It is sufficient to show that the space S ⊂ M 2m (R) given by has dimension p := 2m(2m + 1)/2. First since the set matrices E (i j ) with i , j = 1, . . . , m and i ≤ j forms a basis of the vector space of m ×m symmetric matrices S (m) we check easily by the above formulas that the vector space has dimension 2(m(m +1)/2) = m(m +1). We check easily that the vector spaces are orthogonal to S 1 with respect to the scalar product P · Q = tr(P * Q). So, we need to show that S 2 + S 3 has dimension p − m(m + 1) = m 2 . By the above formulas, we have and in addition S 2 and S 3 are orthogonal. The first space S 2 has the same dimension as S (m), that is, m(m + 1)/2. Moreover, by (22) for every i = j , k = i , and l ∉ {i , j }, we have E (i j ), E (kl ) = F ( j l ). The space spanned by the matrices of the form [0,τ] → R n(n−1)/2 such that any symplectic matrix A close to P g (γ)(τ) has the form X (τ) where X is solution to the Cauchy problem (21) associated with a smooth control of the form S s=1 λ s ω a s u s with |λ| ≤ K |A − P g (γ)(τ)| 1/2 and a 1 , . . . , a S ∈ (0, 1/100). Since the factors σa s , ρ, u s vanish above each intersection of Σ with γ([0,τ]) (by Lemma 3.1.3, we infer easily that σ := S s=1 λ s σ a s ,ρ,u s solves the required properties 1, 2', 3, 4 for ρ > 0 small enough). This concludes the proof of Theorem 1.1.

REMARK 3.2.
By Proposition 2.1, the control system (21) can be shown to be controllable at first order alongū ≡ 0 if (8) holds. This amounts to verifying that some assumption on the curvature along γ is satisfied, see [20,21].

PROOFS OF THEOREMS 1.2 AND 1.3
Let us start with the proof of Theorem 1.2, namely, if the periodic orbits of the geodesic flow of a smooth compact manifold (M , g ) of dimension ≥ 2 are C 2 persistently hyperbolic from Mañé's viewpoint then the closure of the set of periodic orbits is a hyperbolic set. Recall that an invariant set Λ of a smooth flow ψ t : Q −→ Q acting without singularities on a complete manifold Q is called hyperbolic if there exist constants C > 0, λ ∈ (0, 1), and a direct sum decomposition T p Q = E s (p) ⊕ E u (p) ⊕ X (p) for every p ∈ Λ, where X (p) is the subspace tangent to the orbits of ψ t , such that 1. ∥ Dψ t (W ) ∥≤ C λ t ∥ W ∥ for every W ∈ E s (p) and t ≥ 0, 2. ∥ Dψ t (W ) ∥≤ C λ −t ∥ for every W ∈ E u (p) and t ≤ 0.
In particular, when the set Λ is the whole Q the flow is called Anosov. The proof follows the same steps of the proof of Theorem B in [36] where the same conclusion is obtained supposing that the geodesic flow is C 1 persistently expansive in the family of Hamiltonian flows.

Dominated splittings and hyperbolicity.
Let F 2 (M , g ) be the set of Riemannian metrics in M conformal to g endowed with the C 2 topology such that all closed orbits of their geodesic flows are hyperbolic.
The first step of the proof of Theorem 1.2 is closely related with the notion of dominated splitting introduced by Mañé.
The invariant splitting of an Anosov flow is always dominated, but the converse may not be true in general. However, for geodesic flows the following statement holds.

THEOREM 4.2. Any continuous, Lagrangian, invariant, dominated splitting in a compact invariant set for the geodesic flow of a smooth compact Riemannian manifold is a hyperbolic splitting. Therefore, the existence of a continuous Lagrangian invariant dominated splitting in the whole unit tangent bundle is equivalent to the Anosov property in the family of geodesic flows.
This statement is proved in [36] not only for geodesic flows but for symplectic diffeomorphisms. Actually, the statement extends easily to a Hamiltonian flow in a nonsingular energy level (see also Contreras [6]).
The following step of the proof of Theorem 1.2 relies on the connection between persistent hyperbolicity of periodic orbits and the existence of invariant dominated splittings. One of the most remarkable facts about Mañé's work about the stability conjecture (see Proposition II.1 in [22]) is to show that persistent hyperbolicity of families of linear maps is connected to dominated splittings, the proof is pure generic linear algebra (see Lemma II.3 in [22]). Then Mañé observes that Franks' Lemma allows us to reduce the study of persistently hyperbolic families of periodic orbits of diffeomorphisms to persistently hyperbolic families of linear maps. Let us explain briefly Mañé's result and see how its combination with Franks' Lemma for geodesic flows implies Theorem 1.2.
Let GL(n) be the group of linear isomorphisms of R n . Let ψ : Z −→ GL(n) be a sequence of such isomorphisms, we denote by E s j (ψ) the set of vectors v ∈ R n such that sup and by E u j (ψ) the set of vectors v ∈ R n such that Let us say that the sequence ψ is hyperbolic if E s j (ψ) E u j (ψ) = R n for every j ∈ Z. Actually, this definition is equivalent to requiring the above direct sum decomposition for some j . A periodic sequence ψ is characterized by the existence of n 0 > 0 such that ψ j +n 0 = ψ j for every j . It is easy to check that the hyperbolicity of a periodic sequence ψ is equivalent to the classical hyperbolicity of the linear map be a family of periodic sequences of linear maps indexed in a set Λ. Let us define the distance d (ψ, η) between two families of periodic sequences indexed in Λ by We say that the family {ψ α , α ∈ Λ} is hyperbolic if every sequence in the family is hyperbolic. Let us call by periodically equivalent two families ψ α , η α for which given any α, the minimum periods of ψ α and η α coincide. Following Mañé, we say that the family {ψ α , α ∈ Λ} is uniformly hyperbolic if there exists > 0 such that every periodically equivalent family η α such that d (ψ, η) < is also hyperbolic. The main result concerning uniformly hyperbolic families of linear maps is the following symplectic version of Lemma II.3 in [22].
where k is the integer part of n m ; 2. for all α ∈ Λ, j ∈ Z, The proof of Theorem 4.3 is based on Mañé's Lemma II.3 in [22], which is proved for linear isomorphisms without the symplectic assumption. The proof of this lemma is long and involved, and it relies basically on generic properties of linear cocycles. So the proof of Theorem 4.3 consists of following step by step the proof of Lemma II.3 in [22] and checking that it holds for symplectic cocycles. To see a detailed proof we refer to an extended version of this paper in arxiv (Section 4.3) (see also [20]). Now we are ready to combine Franks' Lemma from Mañé's viewpoint and Theorem 4.3 to get a geodesic flow version of Theorem 4.3. Recall that by construction ofτ > 0, every closed geodesic (with respect to g ) has period greater thanτ.
The family ψ g is a collection of periodic sequences, and by Franks' Lemma from Mañé's viewpoint (Theorem 1.1) we have Proof. Letδ,τ > 0 be given by Franks' Lemma (Theorem 1.1 with T =τ). If g is in the interior of F 2 (M , g ) then there exists an open C 2 neighborhood U of g in the set of metrics that are conformally equivalent to g such that every closed orbit of the geodesic flow of (M , h), where h ∈ U , is hyperbolic. In particular, given a periodic point θ ∈ T 1 M for the geodesic flow of (M , g ), the set of metrics h θ ∈ U for which the orbit of θ is still a periodic orbit for the geodesic flow of (M , h θ ) have the property that this orbit is hyperbolic as well for the h θ -geodesic flow. By Theorem 1.1, for any δ ∈ (0,δ), the (K δ)-C 2 open neighborhood of the metric g in the set of its conformally equivalent metrics covers a δ-open neighborhood of symplectic linear transformations of the derivatives of the Poincaré maps between the sections Σ θ s , Σ θ s+1 defined above. Then consider δ > 0 such that the (K δ)-C 2 open neighborhood of the metric g is contained in U , and we get that the family A θ,i ,g is uniformly hyperbolic. Since this holds for every periodic point θ for the geodesic flow of (M , g ) the family ψ g is uniformly hyperbolic.
Therefore, applying Theorem 4.3 to the sequence ψ g we obtain the following.
where E s (τ) ⊕ E u (τ) = N τ is the hyperbolic splitting of the geodesic flow of (M , g ) at a periodic point τ and k = ω/D ; 2. there exists a continuous Lagrangian, invariant, dominated splitting in the closure of the set of periodic orbits of φ t which extends the hyperbolic splitting of periodic orbits: if θ is periodic then G s (θ) = E s (θ), G u (θ) = E u (θ). Theorem 4.5 improves Theorem 2.1 in [36] where the same conclusions are claimed assuming that the geodesic flow of (M , g ) is in the C 1 interior of the set of Hamiltonian flows all of whose periodic orbits are hyperbolic.
Hence, the proof of Theorem 1.2 follows from the combination of Theorems 4.2 and Theorem 4.5.

4.2.
Proof of Theorem 1.3. Let E 2 (M , g ) be the set of Riemannian metrics in M conformally equivalent to g , endowed with the C 2 topology, whose geodesic flows are expansive. The main result of the subsection is an improved version of Proposition 1.1 in [36]. Theorem 1.3 will follow easily from Theorem 4.6 and the results of the previous section. We just give an outline of the proof based on [36]. The argument is by contradiction. Suppose that there exists h in the interior of E 2 (M , g ) whose geodesic flow has a nonhyperbolic periodic point θ. Let Σ be a cross section of the geodesic flow at θ tangent to N θ . The derivative of the Poincaré return map has some eigenvalues in the unit circle. By the results of Rifford-Ruggiero [34] every generic property in the symplectic group is attained by C 2 perturbations by potentials of (M , h) preserving the orbit of θ. This means that there existsh C 2 -close to h and conformally equivalent to it such that the orbit of θ is still a periodic orbit of the geodesic flow of (M ,h) and the derivative of the Poincaré mapP : Σ −→ Σ has generic unit circle eigenvalues. By the Central Manifold Theorem of Hirsch-Pugh-Shub [16] there exists a central invariant submanifold Σ 0 ⊂ Σ such that the return map P 0 of the geodesic flow of (M ,h) is tangent to the invariant subspace associated to the eigenvalues of DP in the unit circle. Moreover, we can suppose by the C k Mañé-generic version of the Klingenberg-Takens Theorem due to Carballo-Gonçalves [5] that the Birkhoff normal form of the Poincaré map at the periodic point θ is generic. So we can apply the Birkhoff-Lewis fixed point Theorem due to Moser [26] to deduce that given δ > 0 there exists infinitely many closed orbits of the geodesic flow of (M ,h) in the δ-tubular neighborhood of the orbit of θ. This clearly contradicts the expansiveness of the geodesic flow of (M ,h) ∈ E 2 (M , g ).
In the case where (M , g ) is a closed surface, we know that the expansiveness of the geodesic flow implies the density of the set of periodic orbits in the unit tangent bundle (see [36] for instance). So if g is in the interior of E 2 (M , g ) the closure of the set of periodic orbits of the geodesic flow of (M , g ) is a hyperbolic set by Theorem 1.2, and since this set is dense its closure is the unit tangent bundle, and therefore the geodesic flow is Anosov. If the dimension of M is arbitrary, then we know that if (M , g ) has no conjugate points the expansiveness of the geodesic flow implies the density of periodic orbits as well, so we can extend the above result for surfaces. α l g l satisfies (23). By bilinearity of the product, this amounts to saying that  For every integer d ≥ 0, let dim(d ) be the dimension of the vector space generated by V (0), . . . ,V (d ). The function d → dim(d ) is nondecreasing and valued in the positive integers. Moreover, it is bounded by 7(d + 1). Thus it is stationnary and in consequence there isd ≥ 0 such that for every q >d , and 2C (d + 2)(p − d − 1)(p + 1) − (d + 3)((2d + 3)p + d + 1)(p + 2) p(p + 1)(p + 2)(d + 2)(d + 3) ap −1 = 0.
Since the sets we are dealing with are algebraic (see [4,10]), we infer that given (A, B,C ) ∈ R 3 , the algebraic set S (A, B,C ) ⊂ R d −N has at most dimension three, which means that S ⊂ R d −N has at most dimension six.
In conclusion, the coefficients (a 0 p ) p∈{0,d } of f 0 have to belong to the intersection of 2 + 5N hyperplanes in R d [X ], and if in addition if there is no vector space of dimension N + 1 in L ∪ {0}, then the (d − P )-tuples (a 0 p ) p∈{P +1,d } must belong to S ⊂ R d −N of dimension ≤ 6. But, for d large enough, the intersection of 2 + 5N hyperplanes in R d [X ] with the complement of an algebraic set of dimension at most 6+P +1 is non-empty. This concludes the proof of Lemma 2.7.