Around the stability of KAM tori

Abstract

Abstract. We study the accumulation of an invariant quasi-periodic torus of a hamiltonian flow by other quasi-periodic invari-ant 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 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 non degeneracy condition at T0, then T0 is accumulated by KAM tori of positive total measure.