Generic properties of Lagrangians on surfaces: The Kupka-Smale theorem

Abstract

We consider generic properties of Lagrangians. Our main result is the Theorem of Kupka-Smale, in the Lagrangian setting, claiming that, for a convex and superlinear Lagrangian defined in a compact surface, for each k ∈ R, generically, in Mañé's sense, the energy level, k, is regular and all periodic orbits, in this level, are nondegenerate at all orders, that is, the linearized Poincaré map, restricted to this energy level, does not have roots of the unity as eigenvalues. Moreover, all heteroclinic intersections in this level are transversal. All the results that we present here are true in dimension n ≥ 2, except one (Theorem 18), whose proof we are able to obtain just for dimension 2. * Supported by CAPES, scholarship.Lagrangian submanifolds in order to get the transversality of heteroclinic intersections.The Kupka-Smale Theorem , in this formulation, resembles the Bumpy Metrics Theorem, for geodesic flows, formulated by R. Abraham in 1968, and proved by D. V. Anosov in 1983 (Anosov, [4]).The work of W. Klingenberg and F. Takens [12] in the Bumpy Metrics Theorem proof was corrected by Anosov [4] using an induction method similar to the one used by M. Peixoto [16] in the proof of the classical Kupka-Smale Theorem.In this work, we will employ the same techniques used by Anosov [4], adapted to perturbations by potentials. In order to apply to the case of perturbation by potentials, it is necessary to introduce a modification of the standard argument in the Control Theory for differential equations, initially used by Klingenberg [11], for geodesic flow perturbation setting, by J. A. Miranda [14] for magnetic flows on surfaces and by Contreras [7] for the proof of Franks' Lemma for geodesic flows. In this case we do not have especial coordinates, like Fermi coordinates, as in [11], [14] and [7], thus we introduce a new method without the use of tubular neighborhoods, that solves this trouble. In the beginning of the proof we use an argument similar to the one used by Robinson [17], Lemma 19, Pg. 592, but the proof is quite different.Observe that the generic properties in Mañé's sense cannot be obtained from the pioneering work of Robinson in the general Hamiltonian setting (see [17] and [18]) because the set of all Hamiltonians is bigger than the set of all potentials in M .The transversality is the easy part. Here we follow the approach of Contreras & Paternain [9], Lemma 2.6 or J.A. Miranda [14], Lemma 3.9. The problem in this case is to construct an explicit potential that represents the perturbation.The main obstacle in the proof of the Kupka-Smale Theorem in dimension n > 2 is the nature of the perturbation constructed. As in Contreras [7], Lemma 7.3 and 7.4, we need to solve a matrix equation in the Lie algebra of the symplectic group S p (n) = {Symplectic matrices 2n × 2n}. The solubility of this equation is strongly related with the existence of repeated eigenvalues of the matrix H pp in local coordinates. The problem is that we can not change this characteristic by adding a potential. Moreover, the equati...

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

“…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.…”

Non-degeneracy of extremal points in multi-dimensional space

“…A special case has been proved recently if the configuration manifold is a surface (see [3]). Various non-degeneracy of periodic orbits in systems with two degrees of freedom is also obtained in [6].…”

Generic Properties of Closed Orbits of Hamiltonian Flows from Mañé's Viewpoint

“…The simplicity of the idea of perturbing just by adding small potentials, and the deep preliminary results obtained by Mañé about generic properties of Aubry-Mather sets (invariant objects of Lagrangian systems with action minimizing properties) had a strong appeal to researchers in classical mechanics and dynamical systems. Since Mañé's initial paper [15], a great deal of work has been devoted to understand generic properties of systems from this point of view (see Massart [16], Contreras [6], Contreras and Iturriaga [7], Contreras and Paternain [8], Oliveira [19], Ruggiero [23], etc..), specially to solve his famous conjecture about the generic uniqueness of the Aubry-Mather set in a specified homology class (see Bernard and Contreras [5] or Figalli and Rifford [11,12]). However, the apparent simplicity of this sort of perturbations is in contrast with the highly technical difficulties arising from the fact that this family of perturbations is a very restricted one.…”

“…The problem has been already considered by Oliveira [19] who shows the Kupka-Smale theorem for energy levels of Hamiltonians in compact surfaces from the point of view of Mañé, (assuming that genericity for potentials means density). Actually, in [19] the Mañé C ∞ genericity of the transversal intersections of stable and unstable submanifolds of closed orbits is proved for n-dimensional compact manifolds So item (2) holds in fact for any dimension.…”

“…The proof of item (1) in [19] has two steps as in the usual proof of Kupka-Smale theorem for diffeomorphisms and flows. First of all, it is proved a transversality result, which holds in any dimension, called in [19] the reduction lemma: given an energy level, and given T > 0, the set of Hamiltonians whose Hamiltonian flows in the level have a finite number of closed orbits of period at most T is Mañé C ∞ generic. The reduction lemma implies that item (1) follows from the proof of the genericity of the required spectral properties in each single orbit.…”