Franks Lemma for C 2 Mane Perturbations of Riemannian Metrics and Applications to Persistence

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1 Franks Lemma for C Mane Perturbations of Riemannian Metrics and Applications to Persistence A Lazrag, Ludovic Rifford, Rafael Ruggiero To cite this version: A Lazrag, Ludovic Rifford, Rafael Ruggiero. Franks Lemma for C Mane Perturbations of Riemannian Metrics and Applications to Persistence. Journal of modern dynamics, American Institute of Mathematical Sciences, 6,, pp <.3934/jmd >. <hal-786> HAL Id: hal Submitted on 3 Jan 5 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 FRANKS LEMMA FOR C -MAÑÉ PERTURBATIONS OF RIEMANNIAN METRICS AND APPLICATIONS TO PERSISTENCE A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Abstract. 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 generic and stability theories of dynamical systems is the celebrated Franks Lemma [5: Let M be a smooth i.e. of class C compact manifold of dimension n and let f : M M be a C diffeomorphism. Consider a finite set of points S = {p, p,.., p m }, let Π = m i= T p i M, Π = m i= T fp i M. Then there exist ɛ > such that for every < ɛ ɛ there exists δ = δɛ > such that the following holds: Let L = L, L,.., L m : Π Π be an isomorphism such that Li D pi f < δ i =,..., m, then there exists a C diffeomorphism g : M M satisfying gp i = fp i for every i =,..., m, D pi g = L i for each i =,..., m, 3 the diffeomorphim g is in the ɛ neighborhood of f in the C topology. 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 close to f, at each point of S provided that the isomorphism is sufficiently close to the direct sum of the maps D pi f, i =,..., 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

3 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO neighborhoods of the points p i s and then glue smoothly the diffeomorphism f outside the union of such neighborhoods with these collection of conjugate-to-linear maps. So the proof strongly resembles an elementary calculus exercise: we can glue a C function h : R R outside a small neighborhood U of a point x with the linear function in U whose graph is the line through x, hx with slope h x and get a new function that is C close to h. The 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, 4. It is clear that for specific families of systems the proof of the lemma should be more difficult that 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. The Frank s Lemma was extensively used by R. Mañé in his proof of the C structural stability conjecture [3, and we could claim with no doubts that it is one of the pillars of the proof together with C. Pugh s C closing lemma [3, 3 see Newhouse [7 for the proof of the C structural stability conjecture for symplectic diffeomorphisms. A particularly challenging problem is to obtain a version of Frank s 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. So 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 [8 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.

4 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 3 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 = bxg x v, w for every x M and v, w T x M. Given a C, Tonelli Lagrangian L : T M T M R defined in a compact manifold M, and a C function u : M R, the function L u p, v = Lp, v + up 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 [, the Lagrangian associated to a metric h in M that is conformally equivalent to g is of the form Lp, v = g pv, v + up for some function u. Since the Lagrangian of a metric g is given by the formula L g p, v = g pv, v, we get L h p, v = L g p, v + up. Now, given a compact part γ : [, T M of a geodesic of M, g, the collection of potentials u : M R such that γ[, T is still a geodesic of Lp, v = L g p, v + up contains the functions whose gradients vanish along the subset of T γt M which are perpendicular to γ t for every t [, T see for instance [37, Lemma.. Lagrangian perturbations of Tonelli Lagrangians of the type L h p, v = L g p, v + up were used extensively by R. Mañé to study generic properties of Tonelli Lagrangians and applications to Aubry-Mather theory see for instance [4, 5. 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 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 [8 led to the Kupka-Smale Theorem for geodesic flows in the family of conformal perturbations of metrics. These promissing applications of control theory to the generic theory of geodesic flows motivate us to study Frank s 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

5 4 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO denoted by φ t, the flow acts on the unit tangent bundle T M, a point θ T M has canonical coordinates θ = p, v where p M, v T p M, and γ θ denotes the unit speed geodesic with initial conditions γ θ = p, γ θ = v. Let N θ T θ T M be the plane of vectors which 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 φtθ for every θ and t R. Let us consider a geodesic arc, of length T γ θ : [, T M, and let Σ and Σ T be local transverse sections for the geodesic flow which are tangent to N θ and N φt θ respectively. Let P g Σ, Σ T, γ be a Poincaré map going from Σ to Σ T. In horizontal-vertical coordinates of N θ, the differential D θ φ T that is the linearized Poincaré map P g γt := D θ P g Σ, Σ T, γ is a symplectic endomorphism of R n R n. This endomorphism can be expressed in terms of the Jacobi fields of γ θ which are perpendicular to γ θ t for every t: P g γt J, J = JT, JT, where J denotes the covariant derivative along the geodesic. We can identify the set of all symplectic endomorphisms of R n R n with the symplectic group { } Spn := X R n n ; X JX = J, where X denotes the transpose of X and J = [ In I n Given a geodesic γ θ : [, T M, an interval [t, t [, T and ρ >, we denote by C g γθ [t, t ; ρ the open geodesic cylinder along γ θ [t, t of radius ρ, that is the open set defined by C g γθ [t, t ; ρ := { p M t t, t with d g p, γθ t < ρ and d g p, γθ [t, t = d g p, γθ t }, where d g denotes the geodesic distance with respect to g. Our main result is the following..

6 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 5 Theorem. Franks Lemma. Let M, g be a smooth compact Riemannian manifold of dimension. For every T > there exist δ T, τ T, K T > such that the following property holds: For every geodesic γ θ : [, T M, there are t [, T τ T and ρ > with C g γ θ [ t, t + τ T ; ρ γ θ [, T = γ θ t, t + τ T, such that for every δ, δ T, for each symplectic map A in the open ball in Spn centered at P g γt of radius δ and for every ρ, ρ, there exists a C metric h in M that is conformal to g, h p v, w = + σpg p v, w, such that: the geodesic γ θ : [, T M is still a geodesic of M, h, Suppσ C g γθ [ t, t + τ T ; ρ, 3 P h γ θ T = A, 4 the C norm of the function σ is less than K T δ. Theorem. improves a previous result by Contreras [7, Theorem 7. 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 [4 and [. The Lazrag proof follows already the ideas from geometric control introduced in [34 to study controllability properties at first order. Our new Theorem. 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[, T affect the differential of P g Σ, Σ T, γ. 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. as a uniform controllability result at second order for a control system of the form Ẋt = AtXt + u i tb i Xt, for a.e. t, i= in the symplectic group Spn. We apply Franks Lemma to extend some results concerning the characterization of hyperbolic geodesic flows in terms of the persistence of some C generic properties of the dynamics. These results are based on well known steps towards the proof of the C structural stability conjecture for diffeomorphisms.

7 6 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO 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 ɛ-c k - persistent from Mañé s viewpoint if for every C function f : M R whose C k norm is less than ɛ we have that the geodesic flow of the metric M, + fg has property P as well. By Maupertuis principle, this is equivalent to the existence of an open C k -ball of radius ɛ > of functions q : M R such that for every C function in this open ball the Euler-Lagrange flow of the Lagrangian Lp, v = g pv, v qp in the level of energy equal to has property P. This definition is inspired by the definition of C k persistence for diffeomorphisms: a property P of a diffeomorphism f : M M is called ɛ-c k persistent if the property holds for every diffeomorphism in the ɛ-c k neighborhood of f. It is clear that if a property P is ɛ-c persistent for a geodesic flow then the property P is ɛ -C persistent from Mañé s viewpoint for some ɛ. Theorem.. Let M, g be a smooth compact Riemannian manifold of dimension such that the periodic orbits of the geodesic flow are C -persistently hyperbolic from Mañé s viewpoint. Then the closure of the set of periodic orbits of the geodesic flow is a hyperbolic set. An interesting application of Theorem. is the following extension of Theorem A in [36: C persistently expansive geodesic flows in the set of Hamiltonian flows of T M are Anosov flows. We recall that a non-singular smooth flow φ t : Q Q acting on a complete Riemannian manifold Q is ɛ-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 = satisfying d φ t x, φ ρt y ɛ t R, for every t R then there exists ty, ty < ɛ such that φ ty x = y. A smooth non-singular flow is called expansive if it is expansive for some ɛ >. Anosov flows are expansive, and it is not difficult to get examples which show that the converse of this statement is not true. Theorem. yields the following. Theorem.3. Let M, g be a smooth compact Riemannian manifold, suppose that either M is a surface or dim M 3 and M, g has no conjugate points. Assume that the geodesic flow is C persistently expansive from Mañé s viewpoint, then the geodesic flow is 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

8 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 7 points we do not know whether periodic orbits of expansive geodesic flows are dense and so if the geodesic flow in Theorem.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. already used in [34 and state results Propositions. and.4 at second order whose long and technical proofs are given in Sections.5 and.6. In Section 3, we provide the proof of Theorem. and the proof of theorems.,.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. This kind of results could be developed for nonlinear control systems on smooth manifolds. For 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 [, 9,, 7, 33 for a further study in control theory... The End-Point mapping. Let us a consider a bilinear control system on M m R with m, k, of the form Ẋt = AtXt + u i tb i Xt, for a.e. t, i= where the state Xt belongs to M m R, the control ut belongs to R k, t [, T At with T > is a smooth map valued in M m R, and B,..., B k are k matrices in M m R. Given X M m R and ū L [, T ; R k, the Cauchy problem Ẋt = AtXt + ū i tb i Xt for a.e. t [, T, X = X, i= possesses a unique solution X X,ū. The End-Point mapping associated with X in time T > is defined as E X,T : L [, T ; R k M m R u X X,u T.

9 8 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO It is a smooth mapping whose differential can be expressed in terms of the linearized control system see [33. Given X M m R, ū L [, T ; R k, and setting X := X X,ū, the differential of E X at ū is given by the linear operator DūE X,T : L [, T ; R k M m R v Y T, where Y is the unique solution to the linearized Cauchy problem { Ẏ t = AtY t + k i= v itb i t Xt for a.e. t [, T, Y =. Note that if we denote by S the solution to the Cauchy problem { Ṡt = AtSt 3 t [, T, S = I m then there holds 4 DūE X,T v = for every v L [, T ; R k. i= T ST v i tst B i Xt dt, Let Spm be the symplectic group in M m R m, that is the smooth submanifold of matrices X M m R satisfying [ X Im JX = J where J :=. I m Denote by Sm the set of symmetric matrices in M m R. The tangent space to Spm at the identity matrix is given by { } T Im Spm = Y M m R JY Sm. Therefore, if there holds 5 JAt, JB,..., JB k Sm t [, T, then Spm is invariant with respect to, that is for every X Spm and ū L [, T ; R k, X X,u t Spm t [, T. In particular, this means that for every X Spm, the End-Point mapping E X,T is valued in Spm. Given X Spm and ū L [, T ; R k, we are interested

10 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 9 in local controllability properties of around ū. The control system is called controllable around ū in Spm in time T if for every final state X Spm close to X X,ū T there is a control u L [, T ; R k which steers X to X, that is such that E X,T u = X. Such a property is satisfied as soon as E X,T is locally open at ū. Our aim in the next sections is to give an estimate from above on the size of u L in terms of X X X,u T... First order controllability results. Given T >, X Spm, a mapping t [, T At M m R, k matrices B,..., B k M m R satisfying 5, and ū L [, T ; R k, we say that the control system is controllable at first order around ū in Spm if the mapping E X,T : L [, T ; R k Spm is a submersion at ū, that is if the linear operator DūE X,T : L [, T ; R k T XT Spm, is surjective with XT := X X,ū T. The following sufficient condition for first order controllability is given in [34, Proposition. see also [,. Proposition.. Let T >, t [, T At a smooth mapping and B,..., B k M m R be matrices in M m R satisfying 5. Define the k sequences of smooth mappings by {B j },..., {B j k } : [, T T I m Spm 6 { B i t := B i B j i t := Ḃj i t + B j tat AtB j t, i i for every t [, T and every i {,..., k}. Assume that there exists some t [, T such that { } 7 Span B j i t i {,..., k}, j N = T Im Spm. Then for every around ū. X Spm, the control system is controllable at first order The control system which is relevant in the present paper is not always controllable at first order. We need sufficient condition for controllability at second order.

11 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO.3. Second-order controllability results. Using the same notations as above, we say that the control system is controllable at second order around ū in Spm if there are µ, K > such that for every X B XT, µ Spm, there is u L [, T ; R k satisfying E X,T u = X and u L K X XT /. Obtaining such a property requires a study of the End-Point mapping at second order. Recall that given two matrices B, B M m R, the bracket [B, B is the matrix of M m R defined as [B, B := BB B B. The following results are the key points in the proof of our main theorem. Their proofs will be given respectively in Sections.5 and.6. Proposition.. Let T >, t [, T At a smooth mapping and B,..., B k M m R be matrices in M m R satisfying 5 such that 8 B i B j = i, j =,..., k. Define the k sequences of smooth mappings {B}, j..., {B j k } : [, T T I m Spm by 6 and assume that the following properties are satisfied with t = : [ B j i t, } 9 B i Span {B r t s r =,.., k, s i =,..., k, j =,, and { } Span B j i t, [Bi t, Bl t i, l =,.., k and j =,, = T Im Spm. Then, for every X Spm, the control system is controllable at second order around ū. Remark.3. For sake of simplicity we restrict here our attention to control systems of the form satisfying 8-9. More general results can be found in [. To prove Theorem., we will need the following parametrized version of Proposition. which will follow from the fact that smooth controls with support in, T are dense in L [, T ; R k and compactness. Proposition.4. Let T >, and for every θ in some set of parameters Θ let t [, T A θ t be a smooth mapping and B θ,..., B θ k M mr be matrices in M m R satisfying 5 with At = A θ such that B θ i B θ j = i, j =,..., k.

12 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS Define for every θ Θ the k sequences of smooth mappings {B θ,j },..., {B θ,j k } : [, T T Im Spm as in 6 and assume that the following properties are satisfied with t = for every θ Θ: [ and 3 B θ,j i { t, Bi θ Span } r t r =,.., k, s B θ,s i =,..., k, j =,, { } Span B θ,j i t, [B θ, i t, B θ, l t i, l =,.., k and j =,, = T Im Spm. Assume moreover, that the sets { } Bi θ i =,..., k, θ Θ M m R and { } t [, T A θ t θ Θ C [, T ; M m R are compact. Then, there are µ, K > such that for every θ Θ, every X Spm and every X B Xθ T, µ Spm X θ T denotes the solution at time T of the control system with parameter θ starting from X, there is u C [, T ; R k with support in [, T satisfying E X,T θ E X,T θ u = X and u C K X XT / denotes the End-Point mapping associated with the control system with parameter θ. Our proof is based on a series of results on openness properties of C mappings near critical points in Banach spaces which was developed by Agrachev and his co-authors, see [..4. Some sufficient condition for local openness. Here we are interested in the study of mappings F : U R N of class C 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 corank of u, the quantity coranku := N dim Im D u F. If Q : U R is a quadratic form, its negative index is defined by { } ind Q := max diml Q L\{} <. The following non-quantitative result whose proof can be found in [,, 33 provides a sufficient condition at second order for local openness.

13 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Theorem.5. Let F : U R N be a mapping of class C on an open set U X and ū U be a critical point of F of corank r. If ind λ DūF r λ Im DūF 4 KerDūF \ {}, then the mapping F is locally open at ū, that is the image of any neighborhood of ū is an neighborhood of F ū. In the above statement, DūF KerDūF refers to the quadratic mapping from KerDūF to R N defined by D ūf v := KerDūF D ūf v, v v KerDūF. The following result is a quantitative version of the previous theorem. We denote by B X, the balls in X with respect to the norm X. Theorem.6. Let F : U R N be a mapping of class C on an open set U X and ū U be a critical point of F of corank r. Assume that 4 holds. Then there exist ɛ, c, such that for every ɛ, ɛ the following property holds: For every u U, z R N with 5 u ū X < ɛ, z F u < c ɛ, there are w, w X such that u + w + w U, 6 and 7 w Ker D u F, z = F u + w + w, w X < ɛ, w X < ɛ. Again, the proof of Theorem.6 which follows from previous results by Agrachev- Sachkov [ and Agrachev-Lee [ can be found in [, 33. A parametric version of Theorem.6 that will be useful in the proof of Proposition.4 is provided in [..5. Proof of Proposition.. Without loss of generality, we may assume that X = I m. As a matter of fact, if X u : [, T Spm M m R is solution to the Cauchy problem 8 Ẋ u t = AtX u t + u i tb i X u t for a.e. t [, T, X u = I m, i=

14 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 3 then for every X Spm, the trajectory X u X : [, T Mm R starts at X and satisfies d Xu t dt X = At X u t X + u i tb i Xu t X for a.e. t [, T. i= So any trajectory of, that is any control, steering I m to some X Spm gives rise to a trajectory, with the same control, steering X Spm to X X Spm. Since right-translations in Spm are diffeomorphisms, we infer that local controllability at second order around ū from X = I m implies controllability at second order around ū for any X Spm. So from now we assume that X = I m in the sequel we omit the lower index and simply write I. We recall that X : [, T Spm M m R denotes the solution of 8 associated with u = ū while X u : [, T Spm M m R stands for a solution of 8 associated with some control u L [, T ; R k. Furthermore, we may also assume that the End-Point mapping E I,T : L [, T ; R k Spm is not a submersion at ū because it would imply controllability at first order around ū and so at second order, as desired. We equip the vector space M m R with the scalar product defined by P Q = tr P Q P, Q M m R. Let us fix P T XT Spm such that P belongs to Im D E I,T \ {} with respect to our scalar product note that Im D E I,T \ {} is nonempty since D E I,T : L [, T ; R k T XT Spm is assumed to be not surjective. Lemma.7. For every t [, T, we have tr [P ST St B ji tst = j, i =,..., k. Proof of Lemma.7. Recall remember 4 that for every u L [, T ; R k, T D E I,T u = ST St u i tb i Xt dt, where S denotes the solution of the Cauchy problem 3. Thus if P belongs to ImD E I,T, we have tr [ P ST T St i= u i tb i Xt dt = u L [, T ; R k. i=

15 4 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO This can be written as T [ u i t tr P ST St B i St dt = u L [, T ; R k. i= We infer that [ tr P ST St B i St = i {,..., k}, t [, T. We conclude by noticing that d j dt j St B i St = St B j i tst t [, T. Let u L [, T ; R k be fixed, for every ɛ R small we define δ ɛ : [, T M m R by δ ɛ t := E I,t ɛu t [, T. By regularity of the End-Point mapping see [33, we have formally for every t [, T, δ ɛ t = Xt + δ ɛ t + δ ɛ t + oɛ, where δ ɛ is linear in ɛ and δ ɛ quadratic. Then we have for every t [, T, δ ɛ t = Xt + δɛ t + δɛ t + oɛ t = I + Asδ ɛ s + ɛ u i sb i δ ɛ s ds = Xt + + t t i= Asδ ɛ s + Asδ ɛ s + ɛ u i sb i Xs ds i= ɛ u i sb i δɛ s ds + oɛ. Consequently, the second derivative of E I,T at is given by the solution times at time T of the Cauchy problem { Żt = AtZt + k i= u itb i Y t, Z =, i=

16 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 5 where Y : [, T M m R is solution to the linearized Cauchy problem 3. Therefore we have T DE I,T u = ST St u i tb i ϕt dt, i= where T ϕt := ST St u i tb i Xt dt. i= Then we infer that for every u L [, T ; R k, 9 P DE I,T u = i,j= T t [ u i tu j str P ST St B i StSs B j Ss ds dt. It is useful to work with an approximation of the quadratic form P D E I,T. For every δ >, we see the space L [, δ; R k as a subspace of L [, T ; R k by the canonical immersion u L [, δ; R k ũ L [, T ; R k, with { ut if t [, δ ũt := for a.e. t [, T. otherwise. For sake of simplicity, we keep the same notation for ũ and u. Lemma.8. There is C > such that for every δ, T, we have P DE I,T u Q δ u Cδ 4 u L u L [, δ; R k L [, T ; R k, where Q δ : L [, δ; R k R is defined by δ t Q δ u := u i tu j sp i,j t, s ds dt u L [, δ; R k, i,j= with [ P i,j t, s = tr P ST sb i B j + tb i B j + s B ibj + t B i B j + tsbi Bj,

17 6 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO for any t, s [, T. Proof of Lemma.8. Setting for every i, j =,..., k, B i t := B i + t B i + t B i t [, T and using 8, we check that for any t, s [, T, B i tb j s = P i,j t, s + i,j t, s, with i,j t, s := t s B i B j + ts B i B j + t s 4 B i B j. Moreover, remembering that we have d j St B dt j i St = St B j i tst t [, T, St B i St = B i t + O t 3. Then by 9 we infer that for any δ, T and any u L [, δ; R k, P DE I,T u Q δ u δ t = u i tu j str [P ST B i t + O t 3 B j s + O s 3 = i,j= i,j= δ t P i,j t, s ds dt [ u i tu j str P ST O t 3 B j s + B i to s 3 + O t 3 O s 3 + i,j t, s ds dt.

18 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 7 But for every nonnegative integers p, q with p + q 3, we have δ t u i tu j st p s q ds dt i,j= δ t = u i tt p u j ss q ds dt i= j= δ t u i t t p u j s s q ds dt δ i= i= j= t u i t t p+q i= j= u j s ds dt, which by Cauchy-Schwarz inequality yields δ t u i tu j st p s q ds dt i,j= δ u i t t p+q dt δ t u j s ds dt δ k We conclude easily. kδ p+q t p+q i= δ u i t dt δ t t u i t dt δ t i= δ j= u j s ds dt j= j= δ k δ 3 u L k u L t dt = k δ 4 u L. u j s ds dt Returning to the proof of Proposition., we now want to show that the assumption 4 of Theorems.5-.6 is satisfied. We are indeed going to show that a stronger property holds, namely that the index of the quadratic form in 4 goes to infinity as δ tends to zero.

19 8 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Lemma.9. For every integer N >, there are δ > and a subspace L δ L [, δ; R k of dimension larger than N such that the restriction of Q δ to L δ satisfies Proof of Lemma.9. Using the notation Q δ u C u L δ4 u L δ. h h = h t h s := δ t h th s ds dt, for any pair of continuous functions h, h u L [, δ; R k, : [, δ R, we check that for every Q δu = i,j= [ u i su j tr P ST B i Bj i,j= i,j= i,j= i,j= tu i u j tr [P ST B i B j u i s u [ j tr P ST B i Bj t u i u j tr [P ST B i B j [ tu i su j tr P ST Bi Bj. Fix ī, j {,..., k} with ī j and take v = v,..., v k L [, δ; R k such that v i t = t [, δ, i {,..., k} \ {ī, j}.

20 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 9 The sum of the first two terms in the right-hand side of is given by i,j= { [ } v i sv j tr P ST B i Bj + tv i v j tr [P ST B i B j = v ī sv j tr [P ST B ī B j + v j sv ī tr [P ST B jb ī + v ī sv ī tr [P ST B ī B ī + v j sv j tr [P ST B jb j [ tr tr [P ST B j Bī + tv ī v j P ST B ī B j + tv j v ī + tv ī v ī tr [P ST B ī Bī + tv j v j By integration by parts, we have δ δ v ī sv ī = v ī s ds So v ī sv ī tr [P ST B ī B ī δ = [ tr P ST B j B j sv ī s ds tv ī v ī. + tv ī v ī tr [P ST B ī Bī δ v ī s ds sv ī s ds tr [P ST B ī B ī [ + tv ī v ī tr But according to 9 with i = ī remember that t =, we have [ { } B ī, B ī Span Br s r =,.., k, s, then by Lemma.7 we obtain [ tr P ST [ B ī, Bī =, and consequently, v ī sv ī tr [P ST B ī B ī δ δ = v ī s ds + tv ī v ī tr [P ST B ī Bī. P ST [ B ī, Bī. sv ī s ds tr [P ST B ī B ī.

21 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Similarly, we obtain v j sv j tr[p ST B jb j δ = + [ tv j v j tr δ v js ds sv js ds P ST B j B j tr [P ST B jb j. In conclusion, the sum of the first two terms in the right-hand side of can be written as { [ } v i sv j tr P ST B i Bj + tv i v j tr [P ST B i B j i,j= = v ī sv j tr [P ST B ī B j δ δ + v j sv ī tr [P ST B jb ī tr [P ST B ī B ī + v ī s ds sv ī s ds + [ tv ī v j tr P ST B ī B j δ δ + v js ds sv js ds + tv j v ī tr [P ST B j Bī tr [P ST B jb j. By the same arguments as above, the sum of the third and fourth terms in the right-hand side of can be written as i,j= = + { v i v ī s v [ j tr P ST B i Bj t v i + v j tr [P } ST B i B j s v j s tr [P ST B ī B j v + v j ī δ δ s v + v ī s ds ī s ds tr [P ST B ī B ī [ t tr P ST B v j ī B j + v ī δ + t v ī v j δ s v js v js ds ds tr [P ST B jb ī tr [P ST B j Bī tr [P ST B jb j,

22 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS the fifth and last part of Q δv is given by i,j= { [ } tv i sv j tr P ST Bi Bj = tvī sv j tr[p ST B ī B j + tv ī sv ī tr + tv j sv ī tr [P ST B j Bī [P ST B ī + tv j v j tr [P ST B j. By integration by parts, we have tv ī sv ī = δ sv ī s ds, tv j sv j = δ sv js ds, δ and tv j sv ī = Therefore the last part of Q δv can be written as δ sv ī s ds sv js ds tv ī sv j. i,j= { [ } tv i sv j tr P ST Bi Bj = δ + tvī sv j tr[p ST [ B ī, B j δ sv ī s ds sv js ds tr [P ST B j Bī + δ sv ī s ds tr [P ST B ī + δ sv js ds tr [P ST B j.

23 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO To summarize, we have Q δv = vī sv j tr[p ST B ī B j δ v ī s ds + v j sv ī tr [P ST B jb ī δ + sv ī s ds tr [P ST B ī B ī + [ tv ī v j tr P ST B ī B j + tv j v ī tr [P ST B j Bī δ δ + v js ds sv js ds tr [P ST B jb j s v j s + v ī tr [P ST B ī B j v + v j ī tr [P ST B jb ī δ δ s v + v ī s ds ī s ds tr [P ST B ī B ī t v [ + ī t v j tr P ST B v j ī B j + v ī tr [P ST B j Bī δ δ + v js ds ds tr [P ST B jb j s v js δ δ + sv ī s ds sv js ds tr [P ST B j Bī + sv ī s ds tr [P ST B ī + sv js ds tr [P ST B j + tv ī sv j [ tr P ST [ B ī, B j. We now need the following technical result whose proof is given in Appendix. Lemma.. Denote by L ī, j the set of v = v,..., v k L [, ; R k

24 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 3 such that v i t = t [,, i {,..., k} \ {ī, j}, and v ī s ds = sv ī s ds = v js ds = sv js ds =, v ī sv j = v j sv ī = v ī s v j = v j s v ī =, tv ī sv j >. Then, for every integer N >, there are a vector space L N ī, j Lī, j {} of dimension N and a constant KN > such that tvī sv j KN v L v LN ī, j. Let us now show how to conclude the proof of Lemma.9. Recall that P T XT Spm was fixed such that P belongs to Im D E I,T \ {} and that by Lemma.7, we know that taking t = P ST B j i = j, i,.., k. By t =, we also have { } Span ST B j i, ST [B i, Bs i, s,.., k, j =,, = T XT Spm. Consequently, we infer that there are ī, j {,..., k} with ī j such that tr P ST [ B ī, B j <. Let N > an integer be fixed, L N ī, j Lī, j {} of dimension N and the constant KN > given by Lemma., for every δ, t denote by L N δ the vector space of u L [, δ; R k L [, T ; R k such that there is v L ī, j satisfying ut = vt/δ t [, δ. For every v L ī, j, the control u δ : [, T R k defined by u δ t := vt/δ t [, δ belongs to L N δ and by an easy change of variables, T u δ = u δ t δ dt = u δ t dt = δ vt dt = δ v.

25 4 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Moreover it satisfies Q δ u δ = tv ī sv j δ 4 tr P ST [ B ī, B j. Then we infer that Q δ u δ = tvī sv j u δ L δ 4 We get the result for δ > small enough. tr P ST [ ī B, B j δ v L δkn tr P ST [ B ī, B j. We can now conclude the proof of Proposition.. First we note that given N N strictly larger than mm +, if L L [, T ; R k is a vector space of dimension N, then the linear operator D E I,T L : L T XT Spm M m R has a kernel of dimension at least N mm +, which means that Ker D E I,T L has dimension at least N mm+. Then, thanks to Lemma.9, for every integer N >, there are δ > and a subspace L δ L [, δ; R k L [, T ; R k such that the dimension of L δ := L δ Ker D E I,T is larger than N and the restriction of Q δ to L δ satisfies By Lemma.8, we have Then we infer that Q δ u C u L δ4 u L δ. P D E I,T u Q δ u + Cδ 4 u L u L δ. P D E I,T u Cδ 4 u L < u L δ. Note that since E I,T is valued in Spm which is a submanifold of M m R, assumption 4 is not satisfied and Theorems.5 and.6 do not apply. Let Π : M m R T XT Spm be the orthogonal projection onto T XT Spm. Its restriction to Spm, Π := Π Spm, is a smooth mapping whose differential at XT is equal to the identity of T XT Spm so it is an isomorphism. Thanks to

26 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 5 the Inverse Function Theorem for submanifolds, Π is a local C -diffeomorphism at XT. Hence there exist µ > such that the restriction of Π to B XT, µ Spm Π B XT,µ Spm : B XT, µ Spm Π B XT, µ Spm is a smooth diffeomorphism. The map E I,T is continuous so U := E I,T B XT, µ Spm is an open set of L [, T ; R k containing ū =. Define the function F : U T XT Spm by F := Π E I,T = Π E I,T. The mapping F is C and we have F ū = XT, DūF = DūE I,T and D ūf = Π D ūe I,T. Let us check that F satisfies assumption 4. For every P T XT Spm such that P belongs to Im DūF \ {} and every v L [, T ; R k, we have P DūE I,T v = P Π DūE I,T u + P DūE I,T u Π DūE I,T u. But hence D ūe I,T u Π D ūe I,T u P D ūe I,T u = P D ūf u. T XT Spm, Therefore, by, assumption 4 is satisfied. Consequently, thanks to Theorem.6 there exist ɛ, c, such that for every ɛ, ɛ the following property holds: For every u U, Z T XT Spm with u ū L < ɛ, Z F u < c ɛ, there are w, w L [, T ; R k such that u + w + w U, and w Ker D u F, Z = F u + w + w, w L < ɛ, w L < ɛ. Apply the above property with u = ū and X Spm such that X XT =: cɛ with ɛ < ɛ.

27 6 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO Set Z := ΠX, then we have Π is an orthogonal projection so it is -lipschitz Z F ū = ΠX Π XT X XT cɛ = < cɛ. Therefore by the above property, there are w, w L [, T ; R k such that ũ := ū + w + w U satisfies and Z = F ũ, ũ L w L + w L ɛ + ɛ. Since Π B XT,µ Spm is a local diffeomorphism, taking ɛ > small enough, we infer that X = E I,T ũ and ũ L ɛ = X XT /. c In conclusion, the control system is controllable at second order around ū, which concludes the proof of Proposition...6. Proof of Proposition.4. As in the proof of Proposition., we may assume without loss of generality that X = I m. Recall that for every θ Θ, E I,T θ : L [, T ; R k Spm M m R denotes the End-Point mapping associated with with parameter θ starting at I = I m. Given θ Θ two cases may appear, either E I,T θ is a submersion at ū or is not submersion at ū. Let us denote by Θ Θ the set of parameters θ where E I,T θ is submersion at ū and by Θ its complement in Θ. By continuity of the mapping θ D E I,T θ the set Θ is open in Θ while Θ is compact. For every θ Θ, since E I,T θ is submersion at ū, we have uniform controllability at first order around ū for a set of parameters close to θ. So we need to show that we have controllability at second order around ū for any parameter in some neighborhood of Θ. By the proof of Proposition. see, for every θ Θ, every P in the nonempty set Im D E I,T θ \ {} and every integer N > there exists a finite dimensional subspace L θ,p,n L [, T ; R k with such that D := dim L θ,p,n > N, P D E I,T θ u < u L θ,p,n \ {}

28 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 7 and dim L θ,p,n Ker D E I,T θ N mm +. By bilinearity of u P DE I,T θ u and compactness of the sphere in L θ,p,n, there is C θ,p,n > such that P D E I,T θ u C θ,p,n u L u L θ,p,n. Let u,..., u D L [, T ; R k be a basis of L θ,p,n such that u i L = i =,..., D. Since the set of controls u C [, T, R k with Suppu, T is dense in L [, T, R k, there is a linearly independent family ũ,..., ũ D in C [, T, R k with Suppu, T from now we will denote by C [, T, R k the set of functions in C [, T, R k with support in, T such that P DE I,T θ u C θ,p,n u L u L θ,p,n := Span { ũ i i =,..., D }. Moreover by continuity of the mapping P, θ P DE I,T θ, we may also assume that the above inequality holds for any θ close to θ and P close to P. Let an integer N > be fixed, we check easily that the set A := { θ, P Θ M m R P =, P Im D E I,T θ } is compact. Therefore, by the above discussion there is a finite family {θ a, P a } a=,...,a in A together with a finite family of open neighborhoods {V a } a=,...,a of the pairs θ a, P a a =,..., A in A such that A = A a= and there is a finite family of { L a } a=,...,a of finite dimensional subspaces in C [, T, R k such that P D E I,T θ u < u L a \ {}, for every a {,..., A} and any θ, P in V a. Then set LN := V a A L a C [, T, R k, a=

29 8 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO pick a basis ũ,..., ũ B of LN and define F N : Θ R B Spm by B Fθ N λ := E I,T θ λ b ũ b λ = λ,..., λ B R B, θ Θ. b= By construction, F N is at least C and for every θ Θ and every P Im D Fθ N {}, there is a subspace L N θ,p LN such that dim L N θ,p > N, P D F N θ u < u L N θ,p \ {} and dim L N θ,p Ker D Fθ N N mm +. As in the proof of Proposition., we need to be careful because F N is valued in Spm. Given θ, we denote by Π θ : M m R T X θt Spm the orthogonal projection onto T X θt Spm and observe that the restriction of Π to T Xθ T Spm is an isomorphism for θ W θ an open neighborhood of θ. Then we define G N, θ : Θ R B T X θt Spm by G N, θ θ λ := Π θ F N θ λ λ R B, θ W θ. Taking N large enough, by compactness of Θ, a parametric version of Theorem.6 see [ yields ɛ, c, such that for every ɛ, ɛ and for any θ Θ the following property holds: For every θ W θ, λ R B, Z T Xθ T Spm with λ L < ɛ, Z G N, θ λ < c ɛ, there are β, β R B such that and Note that any β Ker Z = G N, θ θ D λ G N, θ θ θ λ + β + β,, β < ɛ, β < ɛ. B λ b ũ b with λ = λ,..., λ B R B b= is a smooth control whose support is strictly contained in [, T. Then proceeding as in the proof of Proposition. we conclude easily. \

30 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 9 3. Proof of Theorem. We recall that given a geodesic γ θ : [, T M, an interval [t, t [, T and ρ >, C g γθ [t, t ; ρ stands for the open geodesic cylinder along γ θ [t, t of radius ρ, that is the open set defined by C g γθ [t, t ; ρ := { p M t t, t with d g p, γθ t < ρ and d g p, γθ [t, t = d g p, γθ t }. The following holds: Lemma 3.. Let M, g be a compact Riemannian manifold of dimension. Then for every T >, there exists τ T, T such that for every θ T M, there are t [, T τ T and ρ > such that C g γ θ [ t, t + τ T ; ρ γ θ [, T = γ θ t, t + τ T. Proof of Lemma 3.. Let r g > be the injectivity radius of M, g, that is the supremum of r > such that any geodesic arc of length r is minimizing between its end-points. We call self-intersection of the geodesic curve γ θ [, T any p M such that there are t t in [, T such that γ θ t = γ θ t = p. We claim that for every integer k > the number of self-intersection of a non-periodic geodesic of length k r g is bounded by k Nk := i = i= kk. We prove it by induction. Since any geodesic of length r g has no self-intersection, the result holds for k =. Assume that we proved the result for k and prove it for k +. Let γ : [, k + r g M be a unit speed geodesic of length k + r g. The geodesic segment γ[kr g, k + r g has no self-intersection but it could intersect the segment γ[, kr g. If the number of intersection of γ[kr g, k + r g with γ[, kr g is greater or equal than k +, then there are t t [kr g, k + r g, i {,..., k }, and s, s [ir g, i + r g such that γt = γs and γt = γs. Since γ is not periodic, this means that two geodesic arcs of length r g join γt to γt, a contradiction. We infer that the number of self-intersection of γ is bounded by Nk + k, and in turn that it is bounded by Nk +. We deduce that for every

31 3 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO integer k, all the disjoint open intervals I i := i kr g Nk, i + kr g Nk i =,..., Nk can not contain a point of self-intersection of a unit speed geodesic γ : [, kr g M. Hence for every unit speed geodesic γ : [, kr g M there is i {,..., Nk } such that no self-intersection of γ is contained in the closed interval We conclude easily. [ i kr g Nk, i + kr g Nk Let T > be fixed, τ T, T given by Lemma 3. and γ θ : [, T M be a unit speed geodesic of length T. Then there are t [, T τ T and ρ > such that C g γ θ [ t, t + τ T ; ρ γ θ [, T = γ θ t, t + τ T.. Set θ = p, v := γ θ t, γ θ t θ = p, ṽ := γθ t, γ θ t + τ T, θ T = p T, v T := γ θ T, γ θ T, and consider local transverse sections Σ, Σ, Σ, Σ T T M respectively tangent to N θ, N θ, N θ, N θt. Then we have P g γt = D θ P g Σ, Σ T, γ = D θp g Σ, ΣT, γ D θp g Σ, Σ, γ Dθ P g Σ, Σ, γ. Since the sets of symplectic endomorphism of Spn of the form D θp g Σ, ΣT, γ and D θ P g Σ, Σ, γ that is the differential of Poincaré maps associated with geodesics of lengths T t τ T and t are compact and the left and right translations in Spn are diffeomorphisms, it is sufficient to prove Theorem. with τ T = T. More exactly, it is sufficient to show that there are δ T, K T > such that for every δ, δ T and every ρ >, the following property holds: Let γ θ : [, τ T M be a geodesic in M, U be the open ball centered at P g γτ T of radius δ in Spn. Then for each symplectic map A U there exists a C metric h in M that is conformal to g, h p v, w = + σpg p v, w, such that The geodesic γ θ : [, τ T M is still a geodesic of M, h, Suppσ C g γ θ [, τ T ; ρ, 3 P h γ θ τ T = A, 4 the C norm of the function σ is less than K T δ.

32 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 3 Set τ := τ T and let γ : [, τ M a geodesic in M be fixed, we consider a Fermi coordinate system Φt, x, x,.., x n, t, τ, x, x,.., x n δ, δ n along γ[, τ, where t is the arc length of γ, and the coordinate vector fields e t,..., e n t of the system are orthonormal and parallel along γ. Let us consider the family of smooth functions {P ij } i,j=,...,n : R n R defined by and P ij y, y,.., y n := yi y j Q y i j {,..., n } P ii y, y,.., y n := y i Q y i {,..., n }, for every y = y, y,.., y n R n where Q : [, + [, + is a smooth cutoff function satisfying { Qλ = if λ /3 Qλ = if λ /3. Given a radius ρ > with C g γ [, τ ; ρ Φ, τ δ, δ n and a family of smooth function u = u ij i j=,...,n : [, τ R such that Supp u ij, τ i j {,..., n }, we define a family of smooth perturbations { } σ ρ,u ij : M R i j=,...,n with support in Φ, τ δ, δ n by σ ρ,u ij Φ t, x, x,.., x n := ρ x u ij tp ij ρ, x ρ,.., x n, ρ for every p = Φ t, x, x,.., x n Φ, τ δ, δ n and we define σ ρ,u : M R by σ ρ,u := n i,j=,i j σ ρ,u ij. The following result follows by construction, its proof is left to the reader. The notation l with l =,,..., n stands for the partial derivative in coordinates x = t, x,..., x n and Hσ ρ,u denotes the Hessian of σ ρ,u with respect to g. Lemma 3.. The following properties hold: Suppσ ρ,u C g γ [, τ ; ρ, σ ρ,u γt = for every t, τ, 3 l σ ρ,u γt = for every t, τ and l =,,..., n,

33 3 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO 4 Hσ ρ,u i, γt = for every t, τ and i =,..., n, 5 Hσ ρ,u i,j γt = u ij t for every t, τ and i, j =,..., n, 6 σ ρ,u C C u C for some universal constant C >. Set m = n and k := mm + /. Let u = u ij i j=,...,n : [, τ R be a family of smooth functions with support strictly contained in, τ and ρ, ρ be fixed, using the previous notations we set the conformal metric h := + σ ρ,u g. We denote by,,, Γ, H, Rm respectively the scalar product, gradient, Christoffel symbols, Hessian and curvature tensor associated with g. With the usual notational conventions of Riemannian geometry as in [, in components we have { Γ k ij = i g jm + j g im m g ij g mk Hf ij = ij f Γ k ij k f, where g kl stands for the inverse of g kl, and we use Einstein s convention of summation over repeated indices. We shall use a superscript h to denote the same objects when they are associated with the metric h. As usual δ ij = δ ij = δ j i will be if i = j, and otherwise. The Christoffel symbols are modified as follows by a conformal change of metrics: if h = e f g then see for example [9 Γ h k ij = Γ k ij + i fδ k j + j fδ k i m fg ij g mk. Thus, since f = ln + σ ρ,u and its derivatives f, f,..., n f vanish along γ[, τ by Lemma 3. -3, the Christoffel symbols of h and g coincide along γ. Then the family of tangent vectors e t = γt, e t,..., e n t is still a family which is orthonormal and parallel along γ[, τ. Moreover, if h = e f g then the curvature tensor Rm h, Rm respectively of h and g satisfy e f Rm h u, v v, w h = Rmu, v v, w Hfu, w, at any p M where f vanishes and any tangent vectors u, v, w T p M such that u, w v and Hfv, =. By Lemma 3. -5, we infer that along γ[, τ, we have for every i, j {,..., n } and every t [, τ, R h ijt := Rm h γt e i t, γt γt, e j t h γt = Rm γt e i t, γt γt, e j t γt u ijt = R ij t u ij t,

34 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 33 with 3 R ij t := Rm γt e i t, γt γt, e j t γt. By the above discussion, γ is still a geodesic with respect to h and by construction Lemma 3. the support of σ ρ,u is contained in a cylinder of radius ρ, so properties and above are satisfied. it remains to study the effect of σ ρ,u on the symplectic mapping P h γτ. By the Jacobi equation, we have P h γτj, J = Jτ, Jτ, where J : [, τ R m is solution to the Jacobi equation Jt + R h tjt = t [, τ, where R h t is the m m symmetric matrix whose coefficients are given by. In other terms, P h γτ is equal to the m m symplectic matrix Xτ given by the solution X : [, τ Spm at time τ of the following Cauchy problem compare [34, Sect. 3. and [: m 4 Ẋt = AtXt + u ij teijxt t [, τ, X = I m, i j= where the m m matrices At, Eij are defined by Rt is the m m symmetric matrix whose coefficients are given by 3 Im At := t [, τ Rt and Eij :=, Eij where the Eij, i j m are the symmetric m m matrices defined by and Eij k,l = δ ik δ jl + δ il δ jk i, j =,..., m. Since our control system has the form, all the results gathered in Section apply. So, Theorem. will follow from Proposition.4. First by compactness of M and regularity of the geodesic flow, the compactness assumptions in Proposition.4 are satisfied. It remains to check that assumptions, and 3 hold. First we check immediately that EijEkl = i, j, k, l {,..., m} with i j, k l.

35 34 A. LAZRAG, L. RIFFORD, AND R. RUGGIERO So, assumption is satisfied. Since the Eij do not depend on time, we check easily that the matrices B ij, B ij, B ij associated to our system are given by remember that we use the notation [B, B := BB B B B ijt = B ij := Eij B ijt = [Eij, At B ijt = [[Eij, At, At, for every t [, τ and any i, j =,..., m with i j. An easy computation yields for any i, j =,..., m with i j and any t [, τ, Eij Bijt = [Eij, At = Eij and Bijt Eij = [[Eij, At, At =. EijRt RtEij Then we get for any i, j =,..., m with i j, [ { } B ij, B ij = Eij Span B rs r s and [ B Eij { } ij, B ij = Eij Span Brs r s. So assumption is satisfied. It remains to show that 3 holds. We first notice that for any i, j, k, l =,..., m with i j, k l, we have [ B ij, Bkl [ = [Eij, A, [Ekl, A [Eij, Ekl =, [Eij, Ekl with 5 [Eij, Ekl = δ il F jk + δ jk F il + δ ik F jl + δ jl F ik, where F pq is the m m skew-symmetric matrix defined by F pq rs := δ rp δ sq δ rq δ sp. It is sufficient to show that the space S M m R given by S := Span {B ij, B ij, B ij, [B kl, } B rr i, j, k, l, r, r T Im Spm

36 FRANK S LEMMA FOR C -MAÑÉ PERTURBATIONS 35 has dimension p := mm + /. First since the set matrices Eij with i, j =,..., m and i j forms a basis of the vector space of m m symmetric matrices Sm we check easily by the above formulas that the vector space { } { } S := Span Bij, Bij i, j = Span Eij, [[Eij, At, At i, j has dimension mm + / = mm +. We check easily that the vector spaces { } { } S := Span Bij i, j = Span [Eij, A i, j and { [B S 3 := Span ij, Bkl } i, j, k, l = {[ } Span [Eij, A, [Ekl, A i, j, k, l are orthogonal to S with respect to the scalar product P Q = trp Q. So, we need to show that S +S 3 has dimension p mm+ = m. By the above formulas, we have { } Eij S := Span i, j Eij and { } [Eij, Ekl S 3 := Span i, j, k, l, [Eij, Ekl and in addition S and S 3 are orthogonal. The first space S has the same dimension as Sm, that is mm+/. Moreover, by 5 for every i j, k = i, and l / {i, j}, we have [Eij, Ekl = F jl. The space spanned by the matrices of the form F jl, F jl with j < l m has dimension mm /. This shows that S 3 has dimension at least mm / and so S S 3 has dimension m. This concludes the proof of Theorem..