Spectral Domain Boundaries in Cellular Automata

• Authors: Marcus Pivato
• Venue: Fundamenta Informaticae - Special issue on DLT'04
• Year: 2007

Let $A^{Z^D}$ be the Cantor space of $Z^D$-indexed configurations in a finite alphabet A, and let σ be the $Z^D$-action of shifts on $A^{Z^D}$. A cellular automaton is a continuous, σ-commuting self-map Φ of $A^{Z^D}$, and a Φ-invariant subshift is a closed, (Φ, σ)-invariant subset u ⊂ $A^{Z^D}$. Suppose a ∈ $A^{Z^D}$ is u-admissible everywhere except for some small region we call a defect. It has been empirically observed that such defects persist under iteration of Φ, and often propagate like 'particles' which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under Φ, and partly explain the outcomes of their collisions.