Around the stability of KAM-tori

We show that an analytic invariant torus $\cT_0$ with Diophantine frequency $ø_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at $\cT_0$ satisfies a Rüssmann transversality condition, the torus $\cT_0$ 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 $ø_0$. For frequency vectors $ø_0$ having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian $H$ satisfies a Kolmogorov non degeneracy condition at $\cT_0$, then $\cT_0$ is accumulated by KAM tori of positive total measure. In $4$ degrees of freedom or more, we construct for any $ø_0 \in \R^d$, $C^\infty$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus $\cT_0$ with frequency $ø_0$ that is not accumulated by a positive measure of invariant tori.

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