  In this paper we identify the geometric structures that restrict transport
and mixing in perturbations of integrable volume-preserving systems with
nonzero net flux. Unlike KAM tori, these objects cannot be continued to the
tori present in the integrable system but are generated by resonance and have a
contractible direction. We introduce a remarkably simple algorithm to analyze
the behavior of these maps and obtain quantitative properties of the tori. In
particular, we present assertions regarding the distribution of the escape
times of the unbounded orbits, the abundance of tori, and the size of the
resonant regions.
