Reduction of behavior of additive cellular automata on groups

A class of additive cellular automata (ACA) on a finite group is defined by an index-group g and a finite field Fp for a prime modulus p [1]. This paper deals mainly with ACA on infinite commutative groups and direct products of them with some non commutative p-groups. It appears that for all abelian groups, the rules and initial states with finite supports define behaviors which being restricted to some infinite regular series of time moments become significantly simplified. In particular, for free abelian groups with n generators states V [t] of ACA with a ruleR at time moments t = p, k > k0, can be viewed as ||R|| copies of initial state V [0] moving through an n-dimensional Euclidean space. That is the behavior is similar to gliders from J.Conway’s automaton Life. For some other special infinite series of time moments the automata states approximate self-similar structures and the approximation becomes better with time. An infinite class DHC(S, θ) of non-commutative p-groups is described which in particular includes quaternion and dihedral p-groups. It is shown that the simplification of behaviors takes place as well for direct products of non-commutative groups from the class DHC(S, θ) with commutative groups. Finally, an automaton on a non-commutative group is constructed such that its behavior at time moments 2, k ≥ 2, is similar to a glider gun. It is concluded that ACA on non-commutative groups demonstrate more diverse variety of behaviors comparing to ACA on commutative groups. Subj-class: nlin. CG MSC-class: 37B15, 68Q80