Integrability, hyperbolic flows and the Birkhoff normal form

Abstract

We prove that a Hamiltonian $p\in C^{\infty }(T^{\ast }{\mathbf{R}}^n)$ is locally integrable near a non-degenerate critical point ${\rho }_0$ of the energy, provided that the fundamental matrix at ${\rho }_0$ has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the $C^{\infty }$ sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation. Then we investigate the almost holomorphic case.

34 pages