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Homoclinic and unstable periodic orbits in chaotic systems play central rolesin various semiclassical sum rules. The interferences between terms aregoverned by the action functions and Maslov indices. In this article, weidentify geometric relations between homoclinic and unstable periodic orbits,and derive exact formulae expressing the periodic orbit classical actions interms of corresponding homoclinic orbit actions plus certain phase space areas.The exact relations provide a basis for approximations of the periodic orbitactions as action differences between homoclinic orbits with well-estimatederrors. This make possible the explicit study of relations between periodicorbits, which results in an analytic expression for the action differencesbetween long periodic orbits and their shadowing decomposed orbits in the cycleexpansion.

Exact relations between homoclinic and periodic orbit actions in chaotic systems