Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation

TL;DR

This paper proves that for certain analytic Hamiltonian systems near an invariant torus, the convergence of the Birkhoff normal form guarantees the existence of an actual convergent normalizing transformation.

Contribution

It establishes that convergence of the Birkhoff normal form under specific conditions implies the convergence of the normalizing transformation itself.

Findings

Convergent Birkhoff normal form leads to an actual convergent normalizing transformation.

The result applies to Hamiltonian systems near invariant tori with zero frequency.

The normal form's specific analytic structure is crucial for the proof.

Abstract

Consider an analytic Hamiltonian system near its analytic invariant torus $\mathcal T_0$ carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at $\mathcal T_0$ is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation -- not just a formal power series -- bringing the Hamiltonian into its Birkhoff normal form.