Around the stability of KAM tori

Abstract

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 T 0 with Diophantine frequency ω 0 is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T 0 satisfies a Rüssmann transversality condition, the torus T 0 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 ω 0 . For frequency vectors ω 0 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 T 0 , then T 0 is accumulated by KAM tori of positive total measure. In four degrees of freedom or more, we construct for any ω 0 ∈ R d , C ∞ (Gevrey) Hamiltonians H with a smooth invariant torus T 0 with frequency ω 0 that is not accumulated by a positive measure of invariant tori.