An example of entropy non-expansive KAM-nondegenerate nearly integrable system

In this paper we present 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.