Iterations of linear maps over finite fields

We study the dynamics of the evolution of Ducci sequences and the Martin–Odlyzko–Wolfram cellular automaton by iterating their respective linear maps on . After a review of an algebraic characterization of cycle lengths, we deduce the relationship between the maximal cycle lengths of these two maps from a simple connection between them. For n odd, we establish a conjugacy relationship that provides a more direct identification of their dynamics. We give an alternate, geometric proof of the maximal cycle length relationship, based on this conjugacy and a symmetry property. We show that the cyclic dynamics of both maps in dimension 2n can be deduced from their periodic behavior in dimension n. This link is generalized to a larger class of maps. With restrictions shared by both maps, we obtain a formula for the number of vectors in dimension 2n belonging to a cycle of length q that expresses this number in terms of the analogous values in dimension n.