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 ${\mathcal{T}}_0$ with Diophantine frequency ${\omega }_0$ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ${\mathcal{T}}_0$ satisfies a Rüssmann transversality condition, the torus ${\mathcal{T}}_0$ 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 ${\omega }_0$ 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 ${\mathcal{T}}_0$, then ${\mathcal{T}}_0$ is accumulated by KAM tori of positive total measure.

In four degrees of freedom or more, we construct for any ${\omega }_0\in {\mathbb{R}}^d$, $C^{\infty }$ (Gevrey) Hamiltonians $H$ with a smooth invariant torus ${\mathcal{T}}_0$ with frequency ${\omega }_0$ that is not accumulated by a positive measure of invariant tori.