Positive metric entropy arises in some nondegenerate nearly integrable systems

Abstract

The celebrated KAM Theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a huge 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 paper we present a Lagrangian perturbation of the geodesic flow on a flat 3-torus. The perturbation is $C^{\infty }$ small but the flow has a positive measure of trajectories with positive Lyapunov exponent, namely, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori.

15 pages, 4 figures

Cited by in corpus (2)

• Flexibility of sections of nearly integrable Hamiltonian systems
• Key ideas behind perturbing any completely integrable Hamiltonian system obtaining volume entropy non-expansiveness