Positive metric entropy arises in some nondegenerate nearly integrable systems

TL;DR

This paper demonstrates that certain small perturbations of integrable systems can produce positive metric entropy, indicating chaotic behavior outside invariant tori, which challenges traditional expectations from KAM theory.

Contribution

It constructs a specific smooth perturbation of a geodesic flow on a flat 3-torus that yields positive metric entropy outside KAM tori, revealing new chaotic dynamics.

Findings

Existence of positive metric entropy outside KAM tori in perturbed systems

Construction of a smooth perturbation leading to positive Lyapunov exponents

Chaotic behavior persists despite small perturbations in 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.