Around the stability of KAM-tori

Abstract

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 ${ø}_0\in {\mathbb{R}}^d$, $C^{\infty }$ (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.

Cited by in corpus (7)

• On the divergence of Birkhoff Normal Forms
• Double exponential stability of quasi-periodic motion in Hamiltonian systems
• General KAM theorems and their applications to invariant tori with prescribed frequencies
• Normal form à la Moser for diffeomorphisms and generalization of Rüssmann's translated curve theorem to higher dimension
• Towards continuity: Universal frequency-preserving KAM persistence and remaining regularity
• Quasi-analytic properties of the KAM curve
• Positive measure of effective quasi-periodic motion near a Diophantine torus