Realization of tangent perturbations in discrete and continuous time conservative systems

We prove that any perturbation of the symplectic part of the derivative of a Poisson diffeomorphism can be realized as the derivative of a $C^1$-close Poisson diffeomorphism. We also show that a similar property holds for the Poincar\'e map of a Hamiltonian on a Poisson manifold. These results are the conservative counterparts of the Franks lemma, a perturbation tool used in several contexts most notably in the theory of smooth dynamical systems.


Introduction
The so-called Franks lemma [11,Lemma 1.1] states that any perturbation of the derivative of a diffeomorphism at a finite set can be realized as the derivative of a nearby diffeomorphism in the C 1 topology. This perturbative result has been crucially used to produce dynamical results out of related properties for linear systems. Many different dynamical behaviors can then be proved to hold in dense or even residual sets of diffeomorphisms. In the case of flows similar perturbation techniques are contained in [7,18].
Based on the usefulness of the Franks lemma, it is natural to ask if it still holds by restricting its focus to certain subgroups of diffeomorphisms. In the volumepreserving context Bonatti, Díaz and Pujals [6,Proposition 7.4] proved a version of the Franks lemma for diffeomorphisms and Bessa and Rocha [4,Lemma 3.2] for flows. In the symplectic case some authors have stated and used it (see e.g. [2, Lemma 12] and [13,Lemma 5.1]), but up to our knowledge no proof is available in the literature.
In this paper we present a complete proof of the symplectic perturbative result version, as a particular case of a slightly more general setting concerning Poisson diffeomorphisms. More specifically, we show that a perturbation of the symplectic part of the derivative of a Poisson diffeomorphism at a point p can be realized as the derivative of a nearby Poisson diffeomorphism which differs from the original map only at a small neighborhood of p.
We also show the similar result for general Hamiltonians in Poisson manifolds as a simple application of the ideas used for symplectomorphisms (this simplifies considerably the methods described in the manuscript [24]). In fact, we show that a linear perturbation of the derivative of the Poincaré map is realizable as the derivative of the Poincaré map of a nearby Hamiltonian. When considering geodesic flows see [9] for surfaces and [8] for a higher dimensional manifold (see also [23]).
The fact that every Poisson diffeomorphism has to preserve the symplectic foliation of a Poisson manifold is an obstruction to state a general Franks lemma for Poisson maps. However, the type of perturbation we study here includes time one maps of Hamiltonian flows on Poisson manifolds. The Hamiltonian dynamical systems on Poisson manifolds, sometimes referred as generalized Hamiltonian systems, arise naturally in problems of celestial mechanics, mean field theory, ecology populations, among many others (cf. e.g. [5,17,22] and references therein).
1.1. Organization of the paper. In section 2 we provide basic definitions on Poisson manifolds and state our main results, Theorems 2.1 and 2.2, on Poisson diffeomorphisms and the Poincaré map of Hamiltonians, respectively. Section 3 contains the key technical lemma regarding the Hamiltonian function that we will use to achieve perturbations for the special case of rotations. The linear symplectic geometry techniques to reduce a general case to rotations will be provided in the last part of section 3. We will prove Theorem 2.1 in section 4. At the beginning of section 5 we will show Theorem 2.2, first in a simpler case and later, through a Poisson flowbox theorem (Theorem 5.4), for the general setting.

Statement of results
A Poisson manifold is a pair (M, π) where M is a manifold and π a Poisson structure on M. Recall that a Poisson structure is a bivector field π ∈ X 2 (M) with the property that [π, π] = 0, where [·, ·] is the Schouten bracket (cf. e.g. [14,21]). The bivector field π provides a vector bundle map ♯ π : T * M → T M. The image of this map is an integrable singular distribution which integrates to a symplectic foliation, i.e. a foliation whose leaves have a symplectic structure induced by the Poisson structure. The rank of Poisson structure at p ∈ M is half of the dimension of the symplectic leaf passing through p.
A regular Poisson manifold is a Poisson manifold with constant rank. We now restrict our attention to regular Poisson manifolds (M, π) with rank d and dimension 2d + n. By the splitting theorem, the Poisson version of the Darboux theorem (see e.g. [14]), there is an atlas is the canonical Poisson structure. Here, (x 1 , .., x d , y 1 , ..., y d , z 1 , .., z n ) stands for the coordinates of R 2d × R n . We will always use local coordinates given by the splitting theorem so that the derivative of f ∈ Pois 1 (M) at p ∈ M, where A ∈ Sp(2d, R) is a symplectic matrix, i.e. A ∈ GL(2d, R) such that A T JA = J, a is any 2d × n real matrix, b ∈ GL(n, R) and Theorem 2.1. Let ε > 0, f ∈ Pois 1 (M) and p ∈ M. Then, there is δ > 0 such that for every neighborhood V of p, We now focus on the Hamiltonian flow case. Consider (M, π) to be a regular Poisson manifold with rank d + 1 and dimension 2(d + 1) + n, and H ∈ C 2 (M) a Hamiltonian function. The map ♯ π : T * M → T M associates a Hamiltonian H : M → R to a Hamiltonian vector field by X H = ♯ π (dH), which generates the Hamiltonian flow ϕ t H in M. Let E be the energy surface passing through a regular point p ∈ M, i.e. the connected component of H −1 ({H(p)}) containing p. Take S to be the symplectic leaf passing through p. Around p the set E ∩ S is a 2d + 1 dimensional submanifold of M. A transversal Σ to the flow at p in E ∩ S is a 2d-dimensional smooth submanifold verifying Note that Σ is a symplectic submanifold of S. Now, take p ′ = ϕ T H (p) for some T > 0, a transversal Σ ′ to the flow at p ′ and some neighborhood U ⊂ M of p. The Poincaré map of H at p is defined to be the Notice that U is assumed to be sufficiently small such that, by the implicit function theorem, τ is C 1 and τ (U) is bounded. The linear Poincaré map of H at p is the derivative of the Poincaré map at p, which can be seen as an element of Sp(2d, R) using local coordinates.
For H ∈ C 2 (M), ε > 0 and D ⊂ M, consider the set Since we want to realize perturbations by Hamiltonians inside B ε (H, Γ) for an orbit segment Γ of ϕ t H , we fix the transversals Σ and Σ ′ taken at p and p ′ both in Γ.
Then, there is δ > 0 such that for every tubular neighborhood W of Γ, and α i,j = 0 for the remaining cases. Notice that π k (I) = I. We can easily check that as the set of the symplectic matrices that rotate only in two conjugate directions.
Recall the form of the one-parameter group of rotations SO(2, Consider also the embedding is given by and α i,j = 0 for the remaining cases. We can easily check that it is also a homomor- 0 ≤ ℓ(r) ≤ 1, ℓ = 1 and ℓ C 2 bounded by a universal constant (we fix this value in the following).

Hamiltonian perturbation.
For r > 0 define B r ⊂ R 2d+n to be the Euclidean open ball centered at the origin with radius r. We write · 2 for the Euclidean norm.
We will also be using the following notations: It is easy to check that d dt ρ = d dt ρ i = 0. This in turn means that ϑ i and ϑ j are constants of motion as well. So, the Hamiltonian flow is as stated in the first claim since it is made of two-dimensional rotations between the symplectic conjugated coordinates. Now, it is simple to check that The second order derivatives are the following: where w stands for x i or y i , u and v replace x j , y j or z k with j = i and u = v. So, for some constant c > 0, Proof. Without loss of generality, set i = 1.
The implicit function theorem now proves the lemma.
Consider the canonical symplectic form respectively. The eigenvalues can be made distinct as long as A is sufficiently close to L. Thus, A is diagonalizable by a matrix S. It remains to show that S is symplectic.
Since v i and w i are close to e i and e j , respectively, the scalar ω 0 (v i , w i ) is close to one. Dividing the eigenvectors v i by ω 0 (v i , w i ) gives us a symplectic basis of eigenvectors close to the canonical one. This matrix forms the columns of S −1 which is therefore close to the identity.
Lemma 3.5. There exist ǫ, c > 0 such that any A ∈ Sp(2d, R) with A − I < ǫ can be written as Proof. Our goal is first to write the matrix A as a product of diagonal and diagonalizable matrices. Then we will show that diagonal matrices can be written as the product of symplectic rotations in Rot 2 (2d, R) up to symplectic linear conjugacy.
Any invertible diagonal matrix L = diag(η 1 , . . . , η d , η −1 1 , . . . , η −1 d ) ∈ Sp(2d, R) can be written as the product of d diagonal symplectic matrices each being essentially two-dimensional: This means that we can reduce our setting to two-dimensions to deal with such decompositions for L 1 and L 2 as given above (corresponding to a total of 2d diagonal matrices).
That is, P i is close to ( 1 0 1 1 ) so P ± i is bounded from above by some constant c. The conjugating matrices P m are of the form P i P where P is the close to identify symplectic matrix given by Lemma (3.4), so P m ∈ Sp(2d, R) and P ± m ≤ c. Notice that (η i + η −1 i )/2 ≥ 1 for every η i > 0, so 1 − cos ξ < 1 − cos θ < |η i − 1|. Finally, there is constant c ′ such that for any k = 1, . . . , d,

Proof of Theorem 2.1
We want to construct a perturbation g of f around p ∈ M which realizes a matrix A π D p f close to D p f . We choose where (U, ϕ) is a local chart (from the splitting theorem) at f (p) with ϕ(f (p)) = 0, and h is a Poisson diffeomorphism of R 2d+n that fixes the origin. Therefore,

By Lemma 3.5 we write
k . We want to use Lemma 3.1 for each k by constructing K k as in (3.3) for α k corresponding to a rotation in the coordinates i(k) and d + i(k). This guarantees the existence of Poisson map Notice that, by the integral formula ϕ t We obtain the following estimates from Lemmas 3.1 and 3.5, as long as δ is small enough. Notice that |α k | ≤ c 4 R α k − I for some constant c 4 > 0 whenever |α k | is close to zero. Finally, This is less than ε as long as δ is small enough.
where ℓ is the bump function defined at (3.2). We write the function K i as K i in (3.3) rotating the coordinates i and d + i.
Finally, we need to estimate the C 2 -norm of the perturbation. It is simple to check that Writing ρ = (x,ŷ, z) 2 /2 and ρ i = (x 2 i + y 2 i )/2, the second order derivatives are the following: where w stands for x i or y i , u and v replace x j , y j or z k with j = i and u = v. So, for some constant c > 0, in which together with (5.2) yields part (3) of the lemma.
Consider a finite set of matrices A k = P k R k P −1 k ∈ Sp(2d, R), k = 1, . . . , N, with P k ∈ Sp(2d, R), R k = π i(k) (R α k ), α k ∈ R and 1 ≤ i(k) ≤ d. We write H i(k) for the Hamiltonian in (5.1) for a function K i(k) . Lemma 5.3. There is 0 < r < 1 such that where T k (x, y) = (Nx 1 − k + 1, x 2 , . . . , x d , y 1 , . . . , y d ) and Φ is defined at (3.1), verifies Proof. For each k = 1, . . . , N + 1 let So, using Lemma 5.2 and taking r sufficiently small, the Poincaré map of H between the transversals to Γ 0 given by P H : Recall that for any f ∈ C 2 (R 2d+n ) we have that ϕ t f •P = P −1 • ϕ t f • P where P is a linear map. The Poincaré map for the transversals at the edges of Γ 0 , P H : Σ 0 → Σ N +1 = Σ ′ 0 , is the composition of the above maps. That is, Finally, the norm can be estimated also by using Lemma 5.2, for some constant c > 0. In particular, if M = R 2d+n with coordinates (y 1 , ..., y 2d+n ) and the standard Poisson structure
The following result is the version of a straightening theorem in the Poisson context, (cf. [1,3] for the symplectic case).
Theorem 5.4 (Poisson flowbox coordinates). Let (M 2d+n , π) be a C s -Poisson manifold, a Hamiltonian H ∈ C s (M, R), s ≥ 2 or s = ∞, and x ∈ M such that the rank of π is constant in a neighborhood of x. If X H (x) = 0, there exist a neighborhood U ⊂ M of x and a local C s−1 -Poisson diffeomorphism g : (U, π) → (R 2d+n , π 0 ) such that H = H 0 • g on U.
Proof. Fix e = H(x). Since X H (x) = 0 one can find a coordinate patch (U, (q 1 , ..., q 2d+n )) centered at x, such that X H = ∂ ∂q 1 . In the neighborhood U we have: {H, q 1 } = π(dH, dq 1 ) = X H (q 1 ) = ∂q 1 ∂q 1 = 1 We will denote q 1 by G and the neighborhood U will be allowed to remain as small as needed. For small enough U one can define the transversal Σ at point x by which is a C s regular connected submanifold of dimension 2d + n − 1. Equivalently, g * X H = H H 0 . Furthermore, notice that the map F → X F from C s (M) to the set of C s−1 vector fields X s−1 (M) is a Poisson map, i.e.