One-dimensional cellular automata with a unique active transition

Abstract

A one-dimensional cellular automaton $τ: A^\mathbb{Z} \to A^\mathbb{Z}$ is a transformation of the full shift defined via a finite neighborhood $S \subset \mathbb{Z}$ and a local function $μ: A^S \to A$. We study the family of cellular automata whose finite neighborhood $S$ is an interval containing $0$, and there exists a pattern $p \in A^S$ satisfying that $μ(z) = z(0)$ if and only if $z \neq p$; this means that these cellular automata have a unique \emph{active transition}. Despite its simplicity, this family presents interesting and subtle problems, as the behavior of the cellular automaton completely depends on the structure of $p$. We show that every cellular automaton $τ$ with a unique active transition $p \in A^S$ is either idempotent or strictly almost equicontinuous, and we completely characterize each one of these situations in terms of $p$. In essence, the idempotence of $τ$ depends on the existence of a certain subpattern of $p$ with a translational symmetry.