Key ideas behind perturbing any completely integrable Hamiltonian system obtaining volume entropy non-expansiveness

This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with $2n$ degrees of freedom ($n\geq 2$). The perturbation is $C^{\infty}$ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.