Ergodicity, transitivity, and regularity for linear cellular automata over Zm

AbstractWe study the dynamical behavior of D-dimensional linear cellular automata over Zm. We provide an easy-to-check necessary and sufficient condition for a D-dimensional linear cellular automata over Zm to be ergodic and topologically transitive. As a byproduct, we get that for linear cellular automata ergodicity is equivalent to topological transitivity. Finally, we prove that for 1-dimensional linear cellular automata over Zm, regularity (denseness of periodic orbits) is equivalent to surjectivity