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.
Highlights
Already in the early 1950’s the study of nearly integrable Hamiltonian systems has drawn the attention of many outstanding mathematicians such as Arnol’d, Kolmogorov and Moser
For any integrable Hamiltonian system the whole phase space is foliated by invariant Lagrangian submanifolds that are diffeomorphic to tori, generally called KAM tori, and on which the dynamics is conjugated to a rigid rotation
In 1954 Kolmogorov [10]—and later Arnol’d [1] and Moser [11] in different contexts—proved that, for small perturbations of an integrable system, it is still possible to find a big measure set of KAM tori
Summary
Already in the early 1950’s the study of nearly integrable Hamiltonian systems has drawn the attention of many outstanding mathematicians such as Arnol’d, Kolmogorov and Moser. In 1954 Kolmogorov [10]—and later Arnol’d [1] and Moser [11] in different contexts—proved that, for small perturbations of an integrable system, it is still possible to find a big measure set of KAM tori This result, commonly referred to as KAM theorem, contributed to raise new interesting questions, for instance about the destiny of the stable motions that are destroyed by effect of the perturbation (in other words, about the dynamics outside KAM tori). The question we address in the present paper is, the following: “How much random can the motion outside KAM tori be?” It is well-known that, C 2-generically, the Hamiltonian flow has positive topological entropy (cf [13], see [6] for an analogous statement for Riemannian geodesic flows). Notice that, by upper semicontinuity (see [14]), the metric entropy we get is microscopic
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