Exemples de flots hamiltoniens dont aucune perturbation en topologie \(C^{\infty{}}\) n’a d’orbites périodiques sur un ouvert de surfaces d’énergies. (Examples of Hamiltonian flows such that no \(C^{\infty{}}\) perturbation has a periodic orbit on an open set of energy surfaces). (French. Abridged English version) Zbl 0759.58016

This note constructs counterexamples to the Closing Lemma for Hamiltonian vector fields on the torus of dimension \(2n+2\) in the \(C^{k_ 0+1}\) topology for \(k_ 0>2n+1\). On the torus, fix a constant symplectic form involving a certain diophantine condition. Then a \(C^ \infty\) function \(H_ 0\) on the torus and a neighborhood of \(H_ 0\) are given, with the following properties: For every \(H\) in the neighborhood of \(H_ 0\), each \(c\in[-1/2,1/2]\) is a regular value whose level hypersurface consists of tori of dimension \(2n+1\) such that the Hamiltonian vector fields given by the restriction of \(H\) to these tori are differentiably conjugate to a constant vector field without any closed orbits. The orders of differentiability that are involved depend on the diophantine condition.
Reviewer: D.Erle (Dortmund)


37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
37G99 Local and nonlocal bifurcation theory for dynamical systems
37C10 Dynamics induced by flows and semiflows


Zbl 0759.58018