Positive metric entropy in nondegenerate nearly integrable systems

Dong Chen

doi:10.3934/jmd.2017003

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

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Positive metric entropy in nondegenerate nearly integrable systems

Dong Chen

Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA

Received: May 24, 2016

Revised: October 05, 2016

Published: November 2016

The author is supported by Dmitri Burago's department research fund 42844-1001.

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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. 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.

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Mathematics Subject Classification: Primary: 37A35, 37J40; Secondary: 53C60.

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Figure 1. A non-ergodic DBG torus

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Figure 2. Graphs of $u_S$, $u_C$ and $u$

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Figure 3. Graph of $\rho$

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Figure 4. Construction of $\phi_1$

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