Flexibility of sections of nearly integrable Hamiltonian systems

Abstract

Given any symplectomorphism on $D^{2n}(n\ge 1)$ which is $C^{\infty }$ close to the identity, and any completely integrable Hamiltonian system ${\Phi }_H^t$ in the proper dimension, we construct a $C^{\infty }$ perturbation of $H$ such that the resulting Hamiltonian flow contains a "local Poincaré 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).

Minor modification. arXiv admin note: text overlap with arXiv:2009.10143

References in corpus (3)

• Positive metric entropy arises in some nondegenerate nearly integrable systems
• Boundary distance, lens maps and entropy of geodesic flows of Finsler metrics
• The essential coexistence phenomenon in Hamiltonian dynamics