One-dimensional cellular automata with a unique active transition

Abstract

A one-dimensional cellular automaton $\tau :A^{\mathbb{Z}}\to A^{\mathbb{Z}}$ is a transformation of the full shift defined via a finite neighborhood $S\subset \mathbb{Z}$ and a local function $\mu :A^S\to A$. 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 $\mu (z)=z(0)$ if and only if $z\ne p$; 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 $\tau$ 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 $\tau$ depends on the existence of a certain subpattern of $p$ with a translational symmetry.

14 pages

References in corpus (1)

• Idempotent cellular automata and their natural order