Linear Dynamical Systems over Finite Rings

We present the theory of linear systems over various kinds of finite commutative rings. Since these systems are finite, the trajectories have to run into repeating cycles eventually. This periodic behavior is the main interest of this topic. Using an approach similar to Fitting's lemma, a bijective-nilpotent decomposition of the system can be achieved, which in some cases even gives a decomposition of the system matrix. In particular, this allows us, to apply results about invertible system matrices, where all trajectories are purely periodic, to the more general setting. Finally, the algorithmic potential of the theory is discussed.