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 with Diophantine frequency is never isolated due to the following alternative. If the Birkhoff normal form of the Hamiltonian at satisfies a Rüssmann transversality condition, the torus 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 that is foliated by analytic invariant tori with frequency .
For frequency vectors having a finite uniform Diophantine exponent (this includes a residual set of Liouville vectors), we show that if the Hamiltonian satisfies a Kolmogorov nondegeneracy condition at , then is accumulated by KAM tori of positive total measure.
In four degrees of freedom or more, we construct for any , (Gevrey) Hamiltonians with a smooth invariant torus with frequency that is not accumulated by a positive measure of invariant tori.
"Around the stability of KAM tori." Duke Math. J. 164 (9) 1733 - 1775, 15 June 2015. https://doi.org/10.1215/00127094-3120060