Non-degenerate Liouville tori are KAM stable

Abed Bounemoura (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

In this short note, we prove that a quasi-periodic torus, with a non-resonant frequency (that can be Diophantine or Liouville) and which is invariant by a sufficiently regular Hamiltonian flow, is KAM stable provided it is Kolmogorov non-degenerate. When the Hamiltonian is smooth (respectively Gevrey-smooth, respectively real-analytic), the in-variant tori are smooth (respectively Gevrey-smooth, respectively real-analytic). This answers a question raised in a recent work by Eliasson, Fayad and Krikorian ([EFK]). We also take the opportunity to ask other questions concerning the stability of non-resonant invariant quasi-periodic tori in (analytic or smooth) Hamiltonian systems.

Domaines: Systèmes dynamiques [math.DS]

Abed Bounemoura. Non-degenerate Liouville tori are KAM stable. Advances in Mathematics, 2016, 292, pp.42-51. ⟨hal-01089410⟩