Jets of closed orbits of Ma\~n\'e generic Hamiltonian flows

TL;DR

This paper establishes a perturbation theorem for the jets of Poincaré maps of closed Hamiltonian orbits, leading to generic properties of Hamiltonian and Lagrangian flows on closed manifolds.

Contribution

It introduces a new perturbation theorem for higher-order jets of Poincaré maps, advancing the understanding of generic properties in Hamiltonian dynamics.

Findings

Proves a perturbation theorem for $k$-jets of Poincaré maps

Derives Mañé generic properties for Hamiltonian flows

Enhances the understanding of stability and genericity in Hamiltonian systems

Abstract

We prove a perturbation theorem for the $k$-jets, $k\geq 2$, of the Poincar\'e map of a closed orbit of the Hamiltonian flow of a Tonelli Hamiltonian $H: T^*M\to \R$, on a closed manifold $M$. As a consequence we obtain Ma\~n\'e generic properties of Hamiltonian and Lagrangian flows.