Two types of KAM-nondegenerate nearly integrable systems with positive metric entropy

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, there is still an overwhelming measure of invariant tori with quasi-periodic dynamics in the perturbed system. These invariant tori are known as KAM tori. What happens outside KAM tori draws a lot of attention. In this talk I will present two types of $C^\infty$ small Lagrangian perturbations of the geodesic flow on a flat torus. In both cases the resulting flows have positive metric entropy. This way we get positive metric entropy outside KAM tori. What is special about the second type of perturbations is that positive metric entropy is generated in arbitrarily small tubular neighborhoods of single trajectorys. This is a joint work with D.Burago and S.Ivanov.