Some remarks on the optimality of the Bruno-Rüssmann condition

Abed Bounemoura (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

We prove that the Bruno-Rüssmann condition is optimal for the analytic preservation of a quasi-periodic invariant curve for an analytic twist map. The proof is based on Yoccoz's corresponding result for analytic circle diffeomorphisms and the uniqueness of invariant curves with a given irrational rotation number. We also prove a similar result for analytic Tonelli Hamiltonian flow with n = 2 degrees of freedom; for n ≥ 3 we only obtain a weaker result which recovers and slightly improves a theorem of Bessi.

Domaines: Systèmes dynamiques [math.DS]

Abed Bounemoura. Some remarks on the optimality of the Bruno-Rüssmann condition. Bulletin de la société mathématique de France, 2019, 147 (2), pp.341-353. ⟨hal-01679497⟩