Jets of closed orbits of Mañé's generic Hamiltonian flows
In [Int. Math. Res. Not. 2012, No. 22, 5246--5265 (2012; Zbl 1277.37040)], \textit{L. Rifford} and \textit{R. O. Ruggiero} proved a Kupka-Smale theorem for \(C^2\) Tonelli Hamiltonians \(H:T^{*}M \rightarrow \mathbb{R}\), with \(M\) a closed manifold: For each \(c \in \mathbb{R}\), there is a residual set \(KS(c) \subset C^{\infty}(M)\) such that for every \(u \in KS(c)\), the Hamiltonian \(H_u=H+u\) satisfies: (1) \(H_u^{-1}(c)\) is a regular energy level. (2) all closed orbits in \(H_u^{-1}(c)\) are non-degenerate (hyperbolic or elliptic). (3) all heteroclinic intersections between invariant manifolds associated to the hyperbolic orbits in \(H_u^{-1}(c)\) are transverse. This theorem generalizes a result by \textit{E. R. Oliveira} [Discrete Contin. Dyn. Syst. 21, No. 2, 551--569 (2008; Zbl 1147.53310)] for the two-dimensional case. We recall that a Tonelli Hamiltonian is a function in the cotangent bundle such that its restriction to each fiber is convex and superlinear. The special type of perturbation used the present paper, by adding a smooth potential, was introduced by R. Mañé and this explains the expression ``Mañé's generic'' in the title. It is well known, by a theorem of \textit{E. Zehnder} [Commun. Pure Appl. Math. 26, 131--182 (1973; Zbl 0261.58002)], that with additional hypothesis on higher-order derivatives of the Poincaré map associated to an elliptic orbit, one obtains non trivial hyperbolic sets (and hence positive topological entropy) due to the accumulation of transverse homoclinic intersections of invariant manifolds. The main consequence of the principal theorem proven in the paper under review is the genericity of these conditions, thus completing the above result of Rifford and Ruggiero. The point is to produce perturbations on the \(k\)-jet at \(0\) of the Poincaré map via perturbations of the Hamiltonian function. Following the notation of the authors, let \(J_s^{k}(n)\) be the Lie group of symplectic \(k\)-jets, or the equivalence class of jets at \(0\) of symplectic diffeomorphisms \(f:({\mathbb{R}}^{2n},0) \rightarrow ({\mathbb{R}}^{2n},0)\). Let \(Q \subset J_s^{k}(n)\) be an open subset invariant under conjugacy, \(\sigma \cdot Q \cdot {\sigma}^{-1} =Q\). Suppose that \(\theta(t)\) is a non-trivial closed orbit for the Hamiltonian flow of \(H\) and that the \(k\)-jet of the Poincaré map at \(0\) is in \(\bar{Q}\). Then there exists \(u \in C^{\infty}(M)\), \(C^{r}\) close to \(0\), \(r \geq k+1\), such that (a) \(\theta(t)\) is a closed orbit of the Hamiltonian flow of \(H_u\) and (b) the \(k\)-jet of the Poincaré map of \(\theta\) is in \(Q\).
scientific article; zbMATH DE number 6201539
scientific article