2023/09/29 by Luca Biasco, Biasco, Luca, Luigi Chierchia +1 · 1 citation
Mathematics · Physics and Astronomy · #37J05 #37J35 #37J40 #70H05 #70H08 #70H15 #Dynamical Systems (math.DS) #FOS: Mathematics #Geometry and complex manifolds #Nuclear physics research studies #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.2309.17041
openalex publication_date 2023/09/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The question of the total measure of invariant tori in analytic, nearly--integrable Hamiltonian systems is considered. In 1985, Arnol'd, Kozlov and Neishtadt, in the Encyclopaedia of Mathematical Sciences \citeAKN1, and in subsequent editions, conjectured that in n=2 degrees of freedom the measure of the non torus set of general analytic nearly--integrable systems away from critical points is exponentially small with the size \e of the perturbation, and that for n≥ 3 the measure is, in general, of order \e (rather than √\e as predicted by classical KAM Theory). In the case of generic natural Hamiltonian systems, we prove lower bounds on the measure of primary and secondary invariant tori, which are in agreement, up to a logarithmic correction, with the above conjectures. The proof is based on a new \sl singular KAM theory, particularly designed to study analytic properties in neighborhoods of the secular separatrices generated by the perturbation at simple resonances.