Parallel dynamical systems over directed dependency graphs

In this work, we analyze the orbit structure of parallel discrete dynamical systems, over directed dependency graphs, with Boolean functions as global functions. In this sense, for the cases corresponding to the simplest Boolean functions AND and OR, it is proved that only fixed or eventually fixed points appear, as occurs over undirected dependency graphs. However, for general Boolean functions, it is shown that any period can appear, so breaking the pattern found for the undirected case where only (eventually) fixed points or 2-periodic orbits can exist.