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# Semi-classical Integrability,Hyperbolic Flows and the Birkhoff Normal Form

We prove that a Hamiltonian $p\in C^\infty(T^*{\bf R}^n)$ is locally integrable near a non-degenerate critical point $\rho_0$ of the energy, provided that the fundamental matrix at $\rho_0$ has rationally independent eigenvalues, none purely imaginary. 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 complex case, showing that when $p$ is holomorphic near $\rho_0\in T^*{\bf C}^n$, then $\re p$ becomes integrable in the complex domain for real times, while the Birkhoff series and the Birkhoff transforms may not converge, {\em i.e.,} $p$ may not be integrable. These normal forms also hold in the semi-classical frame.