Positive metric entropy arises in some nondegenerate nearly integrable systems

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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 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.

Original languageEnglish (US)
Pages (from-to)43-56
Number of pages14
JournalJournal of Modern Dynamics
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • Analysis
  • Algebra and Number Theory
  • Applied Mathematics


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