Drift by coupling to an anti-integrable limit

A symplectic twist map near an anti-integrable limit has an invariant set that is conjugate to a full shift on a set of symbols. We couple such a system to another twist map in such a way that the resulting system is symplectic. At the anti-integrable limit we construct a set of nonzero measure of orbits of the second map that drifts arbitrarily far, even when the coupling is arbitrarily small. Moreover, these drifting orbits persist near the anti-integrable limit.