  To characterize transport in a deterministic dynamical system is to compute
exit time distributions from regions or transition time distributions between
regions in phase space. This paper surveys the considerable progress on this
problem over the past thirty years. Primary measures of transport for
volume-preserving maps include the exiting and incoming fluxes to a region. For
area-preserving maps, transport is impeded by curves formed from invariant
manifolds that form partial barriers, e.g., stable and unstable manifolds
bounding a resonance zone or cantori, the remnants of destroyed invariant tori.
When the map is exact volume preserving, a Lagrangian differential form can be
used to reduce the computation of fluxes to finding a difference between the
action of certain key orbits, such as homoclinic orbits to a saddle or to a
cantorus. Given a partition of phase space into regions bounded by partial
barriers, a Markov tree model of transport explains key observations, such as
the algebraic decay of exit and recurrence distributions.
