Positive metric entropy arises in some nondegenerate nearly integrable systems

2016/04/26 by Dong Chen, Chen, Dong · 1 citation
Mathematics · Physics and Astronomy · #37A35 #37J40 #53C60 #Chaos control and synchronization #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.1604.07483

openalex publication_date 2016/04/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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^∞ 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.

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