One-dimensional cellular automata with a unique active transition
Alonso Castillo-Ramirez, Maria G. Magaña-Chavez, Luguis de los Santos Baños
TL;DR
This work classifies one-dimensional cellular automata on the full shift $A^\mathbb{Z}$ that have a unique active transition, defined by a fixed pattern $p$ on an interval neighborhood $S$ with $0\in S$. It establishes a dichotomy: such CA are either idempotent or strictly almost equicontinuous, with the type determined by a translational symmetry present in a subpattern of $p$, and it provides a fast test via a subpattern on $S+S$ to decide idempotence. The results connect dynamical behavior to algebraic structure in the CA monoid and clarify when $p$ can persist under iteration. The paper also clarifies the necessity of $S$ being an interval for the main theorem and suggests directions for extending the theory to non-interval neighborhoods and higher dimensions, with implications for modeling discrete complex systems.
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.