Flexibility of sections of nearly integrable Hamiltonian systems

Given any symplectomorphism on $D^{2n} (n\geq 1)$ which is $C^{\infty}$ close to the identity, and any completely integrable Hamiltonian system $Φ^t_H$ 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).

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