Maximal temporal period of a periodic solution generated by a one-dimensional cellular automaton

Abstract

We study one-dimensional cellular automata evolutions with both temporal and spatial periodicity. The main objective is to investigate the longest temporal periods among all two-neighbor rules, with a fixed spatial period $\sigma$ and number of states $n$. When $\sigma =2,3,4$, or $6$, and we restrict the rules to be additive, the longest period can be expressed as the exponent of the multiplicative group of an appropriate ring. We also construct non-additive rules with temporal period on the same order as the trivial upper bound $n^{\sigma }$. Experimental results, open problems, and possible extensions of our results are also discussed.