## 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)

### MSC:

 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