Analytic invariants for the $1:-1$ resonance

Gaivão, José Pedro

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1367-1426.

Etant donnés des champs de vecteurs Hamiltoniens analytiques dans ${ℂ}^4$ ayant un point d’équilibre satisfaisant une résonance $1:-1$ non semisimple, nous construisons deux constantes qui sont invariantes relativement aux changements de coordonnées symplectiques analytiques. Ces invariants sont égaux à zéro lorsque l’Hamiltonien est intégrable. Nous montrons également que ces invariants sont différents de zéro dans un ensemble ouvert et dense.

Associated to analytic Hamiltonian vector fields in ${ℂ}^4$ having an equilibrium point satisfying a non semisimple $1:-1$ resonance, we construct two constants that are invariant with respect to local analytic symplectic changes of coordinates. These invariants vanish when the Hamiltonian is integrable. We also prove that one of these invariants does not vanish on an open and dense set.

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