Enumerating periodic orbits in sequential dynamical systems over graphs

It is well known that periodic orbits with any period can appear in sequential dynamical systems over undirected graphs with a Boolean maxterm or minterm function as global evolution operator. Indeed, fixed points cannot coexist with periodic orbits of greater periods, while periodic orbits with different periods greater than 1 can coexist. Additionally, a fixed point theorem about the uniqueness of a fixed point is known. In this paper, we provide an m-periodic orbit theorem (for m>1) and we give an upper bound for the number of fixed points and periodic orbits of period greater than 1, so completing the study of the periodic structure of such systems. We also demonstrate that these bounds are the best possible ones by providing examples where they are attained.