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Special subsets of orbits in chaotic systems, e.g. periodic orbits,heteroclinic orbits, closed orbits, can be considered as skeletons or scaffoldsupon which the full dynamics of the system is built. In particular, asdemonstrated in previous publications [Phys. Rev. E 95, 062224 (2017), Phys.Rev. E 97, 022216 (2018)], the determination of homoclinic orbits is sufficientfor the exact calculation of classical action functions of unstable periodicorbits, which have potential applications in semiclassical trace formulas. Herethis previous work is generalized to the calculation of classical actionfunctions of arbitrary trajectory segments in multidimensional chaoticHamiltonian systems. The unstable trajectory segments' actions are expandedinto linear combinations of homoclinic orbit actions that shadow them in apiece-wise fashion. The results lend themselves to an approximation withcontrollable exponentially small errors, and which demonstrates anexponentially rapid loss of memory of a segment's classical action to its pastand future. Furthermore, it does not require an actual construction of thetrajectory segment, only its Markov partition sequence. An alternative point ofview is also proposed which partitions the trajectories into short segments oftransient visits to the neighborhoods of successive periodic orbits, givingrise to a periodic orbit expansion scheme which is equivalent to the homoclinicorbit expansion. This clearly demonstrates that homoclinic and periodic orbitsare equally valid skeletal structures for the tessellation of phase-spacedynamics.

Homoclinic orbit expansion of arbitrary trajectories in chaotic systems: classical action function and its memory