Jets of closed orbits of Mañé's generic Hamiltonian flows
type: Item
instance of: scholarly article
title: Jets of closed orbits of Mañé's generic Hamiltonian flows
zbMATH Open document ID: 1279.37041
author: C. M. Carballo
author: José A. Miranda
published in: Bulletin of the Brazilian Mathematical Society. New Series
publication date: 2013-08-26T00:00:00Z
full work available at URL: https://arxiv.org/abs/1106.3500
review text: 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 o…
reviewed by: Mário Jorge Dias Carneiro
Mathematics Subject Classification ID: 37J50
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 37C27
Mathematics Subject Classification ID: 37C20
Mathematics Subject Classification ID: 37B40
zbMATH DE Number: 6201539
zbMATH Keywords: Tonelli Hamiltonians
zbMATH Keywords: Poincaré map
zbMATH Keywords: Generic Dynamics
MaRDI profile type: Publication
OpenAlex ID: W2038325112
arXiv ID: 1106.3500
cites work: Q3657094
cites work: A generic property of families of Lagrangian systems
cites work: Convex Hamiltonians without conjugate points
cites work: Geodesic flows with positive topological entropy, twist maps and hyperbolicity
cites work: Connecting orbits between static classes for generic Lagrangian systems
cites work: Q4181002
cites work: Q4693714
cites work: Generic properties of geodesic flows
cites work: Generic properties and problems of minimizing measures of Lagrangian systems
cites work: On Aubry sets and Mather's action functional
cites work: Action minimizing invariant measures for positive definite Lagrangian systems
cites work: Generic properties for magnetic flows on surfaces
cites work: Generic properties of Lagrangians on surfaces: the Kupka-Smale theorem
cites work: Generic Properties of Closed Orbits of Hamiltonian Flows from Mañé's Viewpoint
DOI: 10.1007/S00574-013-0010-1
description: scientific article; zbMATH DE number 6201539