Spectral domain boundaries in cellular automata

Abstract

Let L:=Z^D be a D-dimensional lattice. Let A^L be the Cantor space of L-indexed configurations in a finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F:A^L-->A^L. An `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is an element of A^L that is X-admissible everywhere except for some small region of L which we call a `defect'. Such defects are analogous to `domain boundaries' in a crystalline solid. It has been empirically observed that these defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We use spectral theory to explain the persistence of some defects under F, and partly explain the outcomes of their collisions.