  Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the
properties of periodic, closed, or homoclinic (heteroclinic) orbits. The
interferences embedded in such orbit sums are governed by classical action
functions and Maslov indices. For chaotic systems, the relative actions of such
orbits can be expressed in terms of phase space areas bounded by segments of
stable and unstable manifolds, and Moser invariant curves. This also generates
direct relations between periodic orbits and homoclinic (heteroclinic) orbit
actions. Simpler, explicit approximate expressions following from the exact
relations are given with error estimates. They arise from asymptotic scaling of
certain bounded phase space areas. The actions of infinite subsets of periodic
orbits are determined by their periods and the locations of the limiting
homoclinic points on which they accumulate.
