Flexibility of sections of nearly integrable Hamiltonian systems

TL;DR

This paper demonstrates how small smooth perturbations of nearly integrable Hamiltonian systems can realize arbitrary symplectomorphisms locally, leading to systems with complex dynamics such as positive entropy and hyperbolic behavior.

Contribution

It constructs specific perturbations of integrable Hamiltonian systems to embed arbitrary local symplectic maps, revealing flexibility in their dynamical structures.

Findings

Existence of perturbations realizing any nearby symplectomorphism

Construction of local Poincaré sections with prescribed dynamics

Presence of hyperbolic behavior and positive entropy in perturbed systems

Abstract

Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close to the identity, and any completely integrable Hamiltonian system $\Phi^t_H$ in the proper dimension, we construct a $C^{\infty}$ perturbation of $H$ such that the resulting Hamiltonian flow contains a "local Poincar\'{e} section" that "realizes" the symplectomorphism. As a (motivating) application, we show that there are arbitrarily small perturbations of any completely integrable Hamiltonian system which are entropy non-expansive (and, in particular, exhibit hyperbolic behavior on a set of positive measure).