2024/11/06 by Alonso Castillo-Ramirez, Castillo-Ramirez, Alonso, Maria G. Magaña-Chavez +3 · 1 citation
Computer Science · #Cellular Automata and Applications
paper · pdf · doi:10.48550/arxiv.2411.03601
A one-dimensional cellular automaton τ: A^ℤ → A^ℤ is a transformation of the full shift defined via a finite neighborhood S ⊂ ℤ and a local function μ: AS → A. We study the family of cellular automata whose finite neighborhood S is an interval containing 0, and there exists a pattern p ∈ AS satisfying that μ(z) = z(0) if and only if z ≠ p; this means that these cellular automata have a unique 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 ∈ AS 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.