Around the stability of KAM-tori

الأصل · EN

We show that an analytic invariant torus ₀ with Diophantine frequency ø₀ is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at ₀ satisfies a Rüssmann transversality condition, the torus 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 ø₀. For frequency vectors ø₀ 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 ₀, then ₀ is accumulated by KAM tori of positive total measure. In 4 degrees of freedom or more, we construct for any ø₀ ∈ ᵈ, C∞ (Gevrey) Hamiltonians H with a smooth invariant torus ₀ with frequency ø₀ that is not accumulated by a positive measure of invariant tori.