Attractors of linear cellular automata

In this paper we study the asymptotic behavior of D-dimensional linear cellular automata over the ring Z_m (D ≥ 1, m ≥ 2). In the first part of the paper we consider nonsubjective cellular automata (CA). We prove that, after a transient phase of length at most [log₂m], the evolution of a linear nonsurjective cellular automata F takes place completely within a subspace Y_F. This result suggests that we can get valuable information on the long term behavior of F by studying its properties when restricted to Y_F. We prove that such study is possible by showing that the system (Y_F, F) is topologically conjugated to a linear cellular automata F^* defined over a different ring Z_m. In the second part of the paper, we study the attractor sets of linear cellular automata. Recently, Kurka has shown that CA can be partitioned into five disjoint classes according to the structure of their attractors. We present a procedure for deciding the membership in Kurka's classes for any linear cellular automata. Our procedure requires only gcd computations involving the coefficients of the local rule associated to the cellular automata.