Levi-flat invariant sets of holomorphic symplectic mappings (Q5928024)

Let \(\phi\) be a symplectic biholomorphic mapping defined near the origin of \(\mathbb C^{2n}\) with fixed point \(\phi(0) = 0\). G. D. Birkhoff showed that, under a non-resonance condition, such a mapping could be converted to a certain normal form (called Birkhoff normal form) by formal symplectic transformations based on a certain formal power series. Siegel showed that, in the generic case, the formal series involved diverges. Another way to say this is that the Birkhoff normal form is not realizable by convergent symplectic transformations. However, by works of \textit{J. Vey} [Am. J. Math. 100, 591-614 (1978; Zbl 0384.58012)] and \textit{H. Ito} [Comment. Math. Helv. 64, No. 3, 412-461 (1989; Zbl 0686.58021)], it is known that, if a Hamiltonian system is analytic and integrable (i.e. has enough first integrals), then the power series represents an analytic function, thus giving convergence of the normal form. In the works of Vey and Ito, certain foliations of \(\mathbb C^{2n}\) by codimension \(n\) invariant sets arise. In the present paper, the author shows that the existence of a single (real analytic) invariant set (of certain types) which is Levi-flat implies that \(\phi\) can be transformed into Birkhoff normal form by convergent symplectic transformations. The basic idea is that the assumption of Levi-flatness allows the techniques of the author's paper with \textit{D. Burns} [Am. J. Math. 121 No. 1, 23-53 (1999; Zbl 0931.32009)] to be used to show that enough first integrals exist. The details are far more complicated, however.