Integrable symplectic maps and their Birkhoff normal forms (Q1360371)

Let \(f\) be a germ of a real analytic symplectic diffeomorphism near the fixed point \(0\). The author proves that, when \(0\) is non-resonant or simply resonant, \(f\) can be put in Birkhoff normal form through an analytic Birkhoff transformation if and only if \(f\) is integrable in the Liouville sense, that is iff \(f\) possesses \(n\) analytic integrals \(G_1(z),\ldots,G_n(z)\) (germs of analytic functions in a neighborhood of \(0\in\mathbb{C}^{2n}),\) that are functionally independent and for which \(\{G_i,G_j\}\equiv 0\) for any \(i,j,\) where \(\{\cdot,\cdot\}\) denotes the Poisson bracket.