Key ideas behind perturbing any completely integrable Hamiltonian system obtaining volume entropy non-expansiveness

TL;DR

This paper outlines a method to perturb any completely integrable Hamiltonian system slightly, creating positive metric entropy near a single trajectory while maintaining KAM conditions, highlighting a new way to generate chaos.

Contribution

It introduces a novel perturbation technique that produces positive entropy in integrable systems without losing KAM non-degeneracy.

Findings

Positive metric entropy can be generated in arbitrarily small neighborhoods.

Perturbations can be $C^{ abla}$ small while satisfying KAM conditions.

The approach applies to any completely integrable Hamiltonian system with at least 4 degrees of freedom.

Abstract

This paper is an announcement of a result followed with explanations of some ideas behind. The proofs will appear elsewhere. Our goal is to construct a Hamiltonian perturbation of any completely integrable Hamiltonian system with $2n$ degrees of freedom ($n\geq 2$). The perturbation is $C^{\infty}$ small but the resulting flow has positive metric entropy and it satisfies KAM non-degeneracy conditions. The key point is that positive entropy can be generated in an arbitrarily small tubular neighborhood of one trajectory.