Controllability of twist maps

We study controllability of a class of discrete-time systems arising in Hamiltonian dynamics. In particular, we consider a two-dimensional integrable twist map with time-dependent (control) perturbation. In contrast to the time independent-perturbation case of the Kolmogorov-Arnold-Moser theorem, no invariant structure survives if the perturbation is made a function of time. Using results from ergodic theory, we show that global controllability is obtained using arbitrary small control input.