Integrability, Hyperbolic Flows and the Birkhoff Normal Form

We prove that a Hamiltonian p ∈ C(T R) is locally integrable near a nondegenerate critical point ρ0 of the energy, provided that the fundamental matrix at ρ0 has no purely imaginary eigenvalues. This is done by using Birkhoff normal forms, which turn out to be convergent in the C sense. We also give versions of the Lewis-Sternberg normal form near a hyperbolic fixed point of a canonical transformation, using a recent result of A.Banyaga, R.de la Llave and C.Wayne. Then we investigate the complex case, showing that when p is holomorphic near ρ0 ∈ T C, then Re p becomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge, i.e. p may not be integrable.