Flexibility of sections of nearly integrable Hamiltonian systems

Given any symplectomorphism on D2n(n ≥ 1) which is C∞ close to the identity, and any completely integrable Hamiltonian system φHt in the proper dimension, we construct a C∞ 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). We use some results in Berger-Turaev [On Herman's positive entropy conjecture, Adv. Math. 349 (2019) 1234 - 1288], though in higher dimensions we could simply apply a construction from [D. Burago and S. Ivanov, Boundary distance, lens maps and entropy of geodesic ows of Finsler metrics, Geom. & Topol. 20 (2016) 469-490].