Jets of closed orbits of Mañé’s generic Hamiltonian flows

Abstract

We prove a perturbation theorem for the k-jets, k ≥ 2, of the Poincaré map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian H : T * M → R, on a closed manifold M . As a consequence we obtain Mañé generic properties of Hamiltonian and Lagrangian flows.

Waist theorems for Tonelli systems in higher dimensions

“…In such a case, generically, all the periodic orbits (if there are any) must be hyperbolic. This is guaranteed by the following result due to Carballo and Miranda [CM13], which extends a result of Klingenberg and Takens [KT72] for geodesic flows. The notions of hyperbolicity or of twist type for the periodic orbits of a Tonelli Lagrangian are those inherited from the corresponding periodic orbits of the dual Tonelli Hamiltonian.…”

“…This latter property is not generic. However, a theorem of Carballo-Miranda [CM13], which extends a result of Klingenberg-Takens [KT72] for geodesic flows, implies that, by perturbing a Tonelli Lagrangian with a C ∞ generic potential, each periodic orbit in a given energy level is either hyperbolic or of twist type. In the twist case, the celebrated Birkhoff-Lewis Theorem [Kli78, Theorem 3.3.A.1] guarantees that the periodic orbit is an accumulation of periodic orbits on the same energy level.…”

Generic Properties of Geodesic Flows on Analytic Hypersurfaces of Euclidean Space

“…Theorem 2 was originally proved for k-jets of C k+1 -smooth Poincaré maps arising in the classical setting in the C k+1 -topology for any k ∈ N by Klingenberg and Takens [28]. An analogue for the case of Mañé generic Hamiltonians was proved for k = 1 in [38] and for k ≥ 2 in [13].…”

Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence

“…. , B k ∈ M 2m (R) be matrices in M 2m (R) satisfying (5). Define the k sequences of smooth mappings…”

“…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.…”