On the Dynamical Behavior of Cellular Automata on Finite Groups

During the last few decades, significant efforts have been devoted to analyze the dynamical behavior of cellular automata (CAs) on cyclic groups, their Cartesian power (referred to as linear cellular automata), and on general abelian groups (referred to as additive cellular automata). Many fundamental properties describing the dynamical behavior of a system such as injectivity, surjectivity, sentitivity to the initial conditions, topological transitivity, ergodicity, positive expansivity, denseness of periodic orbits, and chaos have been fully characterized for these classes of cellular automata, i.e., the relation between the cellular automaton (CA) local rule and the CA global behavior was made explicit, being this task a challenging and important problem in CA general theory. A natural step forward leads to investigate the dynamical behavior of group cellular automata, i.e., cellular automata defined on (not necessarily abelian) finite groups. Despite the work recently carried out by some authors, none of the previously mentioned properties has yet been fully characterized in the case of general finite groups. In this paper, we study the dynamical behavior of cellular automata on a number of classes of finite groups such as simple, symmetric, alternating, dihedral, quaternion and decomposable groups and we provide exact characterizations for some of the above mentioned properties. To do this, in each of those classes, we focus our attention to the non-abelian scenarios. Some results are quite surprising because they show that the non-abelianness of the group imposes strong limitations on defining the local rule of the cellular automaton, making the class of group cellular automata very constrained. Finally, we also introduce a graph allowing one to build and study the local rules of any group cellular automaton.