The Classical Kam Theory at the Dawn of the Twenty-first Century

We survey several recent achievements in KAM theory. The achievements chosen pertain to Hamiltonian systems only and are closely connected with the content of Kolmogorov’s original theorem of 1954. They include weak non-degeneracy conditions, Gevrey smoothness of families of perturbed invariant tori, “exponential condensation” of perturbed tori, destruction mechanisms of resonant unperturbed tori, excitation of the elliptic normal modes of the unperturbed tori, and “atropic” invariant tori (i. e., tori that are neither isotropic nor coisotropic). The exposition is informal and nontechnical, and, as a rule, the methods of proofs are not discussed. 2000 Math. Subj. Class. Primary 37J40; Secondary 26E10, 58A10, 70H08.