Convergence or generic divergence of the Birkhoff normal form (Q1425318)

The author proves that the Birkhoff normal form of a Hamiltonian flow at a nonresonant point with given quadratic part is convergent or generically divergent. This result clarifies some confusion in the literature on the distinction between the problem of convergence of the Birkhoff normal form and Birkhoff transformation. This confusion motivated the author to give a brief but interesting historical overview at the beginning of the paper.