# Geometric determination of unstable periodic orbit actions

**Speaker:** Jizhou Li, Washington State University, Pullman (WA), USA

**Title:** Geometric determination of unstable periodic orbit actions

**Abstract: **Semiclassical sum rules, such as the Gutzwiller trace formula, depend on the properties of closed (periodic or homoclinic/heteroclinic) orbits. The interferences between such orbit sums are governed by classical action functions and Maslov indices. We investigate the periodic orbits inside the convergence zone of the normal form transformation, which arise from intersections of Moser invariant curves, topologically forced by homoclinic intersections between the stable and unstable manifolds. We show that a rotary-N periodic orbit can be treated in two ways: the rst is to identify it as self-intersections of a single Moser curve, forced by a rotary-N (therefore non-primary) homoclinic orbit segment; the second is to identify it as mutual-intersections between N Moser curves, forced by N rotary-1 (primary) homoclinic orbit segments. In both cases, the periodic orbit actions are determined by the corresponding homoclinic actions and certain phase space areas bounded the stable and unstable manifolds. Application to determine the small action

di erences between orbits in cycle expansions are developed as an example of the use and power of these relations.

This work is in collaboration with Steven Tomsovic (WSU).