Journal of Modern Dynamics (JMD)

Positive metric entropy arises in some nondegenerate nearly integrable systems

Pages: 43 - 56, Volume 11, 2017      doi:10.3934/jmd.2017003

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Dong Chen - Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, United States (email)

Abstract: The celebrated KAM theory says that if one makes a small perturbation of a non-degenerate completely integrable system, we still see a large 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. The measure of this set is of course extremely small. Still, the flow has positive metric entropy. From this result we get positive metric entropy outside some KAM tori.

Keywords:  KAM theory, Finsler metric, dual lens map, Hamiltonian flow, perturbation, metric entropy.
Mathematics Subject Classification:  Primary: 37A35, 37J40; Secondary: 53C60.

Received: May 2016;      Revised: October 2016;      Available Online: December 2016.