  Dynamical transitions, such as a change from bound to unbound motion, often
occur as post-adiabatic crossings of a time-dependent separatrix. Whether or
not any given orbit will include such a crossing transition typically depends
sensitively on initial conditions, but a simple estimate for the fraction of
orbits which will cross the separatrix, based on Liouville's theorem, has
appeared several times in the literature. Post-adiabatic dynamical transitions
have more recently been reconsidered as a control problem rather than an
initial value problem: what forms of time-dependent Hamiltonian can most
efficiently induce desired transitions, or prevent unwanted ones? We therefore
apply the Liouvillian estimate for the transition fraction to show how
engineering separatrix volumes in phase space can be a control technique for
dynamical transitions.
