Algebraic invariants for crystallographic defects in cellular automata

Abstract

Let L:= Z^D be the D-dimensional lattice and let A^L be the Cantor space of L-indexed configurations in some finite alphabet A, with the natural L-action by shifts. A `cellular automaton' is a continuous, shift-commuting self-map F of A^L, and an `F-invariant subshift' is a closed, F-invariant and shift-invariant subset X of A^L. Suppose x is a configuration in A^L that is X-admissible everywhere except for some small region we call a `defect'. It has been empirically observed that such defects persist under iteration of F, and often propagate like `particles' which coalesce or annihilate on contact. We construct algebraic invariants for these defects, which explain their persistence under F, and partly explain the outcomes of their collisions. Some invariants are based on the cocycles of multidimensional subshifts; others arise from the higher-dimensional (co)homology/homotopy groups for subshifts, obtained by generalizing the Conway-Lagarias tiling groups and the Geller-Propp fundamental group.