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 T-0 with Diophantine frequency omega(0) is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at T-0 satisfies a Rassmann 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 omega(0). For frequency vectors coo 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 omega(0) is an element of R-d, C-infinity (Gevrey). Hamiltonians H with a smooth invariant torus T-0 with frequency omega(0) that is not accumulated by a positive measure of invariant tori.