Limit Measures for Affine Cellular Automata, II

Abstract

If M is a monoid (e.g. the lattice Z^D), and A is an abelian group, then A^M is a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F:A^M --> A^M that commutes with all shift maps. If F is diffusive, and mu is a harmonically mixing (HM) probability measure on A^M, then the sequence {F^N mu} (N=1,2,3,...) weak*-converges to the Haar measure on A^M, in density. Fully supported Markov measures on A^Z are HM, and nontrivial LCA on A^{Z^D} are diffusive when A=Z/p is a prime cyclic group. In the present work, we provide sufficient conditions for diffusion of LCA on A^{Z^D} when A=Z/n is any cyclic group or when A=[Z/(p^r)]^J (p prime). We show that any fully supported Markov random field on A^{Z^D} is HM (where A is any abelian group).