Around the stability of KAM tori

We study the accumulation of an invariant quasi-periodic torus of a Hamiltonian flow by other quasi-periodic invariant tori. We show that an analytic invariant torus T₀ with Diophantine frequency ω₀ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T₀ satisfies a Rüssmann transversality condition, the torus T₀ is accumulated by Kolmogorov-Arnold-Moser (KAM) tori of positive total measure. If the Birkhoff normal form is degenerate, there exists a subvariety of dimension at least d + 1 that is foliated by analytic invariant tori with frequency ω₀. For frequency vectors ω₀ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian H satisfies a Kolmogorov nondegeneracy condition at T₀, then T₀ is accumulated by KAM tori of positive total measure. In four degrees of freedom or more, we construct for any ω₀ ∈ R^d , C^∞ (Gevrey) Hamiltonians H with a smooth invariant torus T₀ with frequency ω₀ that is not accumulated by a positive measure of invariant tori.