# Geometric determination of heteroclinic and unstable periodic orbit classical actions

## TL;DR

This paper presents a geometric method to determine actions of heteroclinic and unstable periodic orbits in chaotic systems, linking phase space areas to classical actions and providing explicit approximate formulas.

## Contribution

It introduces a geometric approach to compute orbit actions using phase space areas, offering explicit formulas and error estimates for chaotic dynamical systems.

## Key findings

- Actions of infinite periodic orbits depend on their periods and homoclinic points.
- Explicit approximate expressions for orbit actions are derived with error bounds.
- Phase space areas bounded by manifolds relate to classical orbit actions.

## Abstract

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.