  Using the Attouch-Th\'era duality, we study the cycles, gap vectors and fixed
point sets of compositions of proximal mappings. Sufficient conditions are
given for the existence of cycles and gap vectors. A primal-dual framework
provides an exact relationship between the cycles and gap vectors. We also
introduce the generalized cycle and gap vectors to tackle the case when the
classical ones do not exist. Examples are given to illustrate our results.
