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Convergence or generic divergence of the Birkhoff normal form. (English) Zbl 1038.37048

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.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
70H05 Hamilton’s equations
37-03 History of dynamical systems and ergodic theory
01A72 Schools of mathematics
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