On the order of lazy cellular automata

We study the most elementary family of cellular automata defined over an arbitrary group universe $G$ and an alphabet $A$: the lazy cellular automata, which act as the identity on configurations in $A^G$, except when they read a unique active transition $p \in A^S$, in which case they write a fixed symbol $a \in A$. As expected, the dynamical behavior of lazy cellular automata is relatively simple, yet subtle questions arise since they completely depend on the choice of $p$ and $a$. In this paper, we investigate the order of a lazy cellular automaton $\tau : A^G \to A^G$, defined as the cardinality of the set $\{ \tau^k : k \in \mathbb{N} \}$. In particular, we establish a general upper bound for the order of $\tau$ in terms of $p$ and $a$, and we prove that this bound is attained when $p$ is a quasi-constant pattern.