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arXiv · 1604.07483

Positive metric entropy arises in some nondegenerate nearly integrable systems

Dong Chen
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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.

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Dong Chen. 2016-04-26. Positive metric entropy arises in some nondegenerate nearly integrable systems. https://arxiv.org/abs/1604.07483

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