Limit measures for affine cellular automata II

Abstract

If $\mathbb{M}$ is a monoid and $\mathcal{A}$ is an Abelian group, then $\mathcal{A}^{\mathbb{M}}$ is a compact Abelian group; a linear cellular automaton (LCA) is a continuous endomorphism $\mathfrak{F}:\mathcal{A}^{\mathbb{M}}\longrightarrow \mathcal{A}^{\mathbb{M}}$ that commutes with all shift maps. If $\mathfrak{F}$ is diffusive, and $\mu$ is a harmonically mixing (HM) probability measure on $\mathcal{A}^{\mathbb{M}}$, then the sequence $\{\mathfrak{F}^N \mu\}_{N=1}^{\infty}$ weak* converges to the Haar measure on $\mathcal{A}^{\mathbb{M}}$ in density. Fully supported Markov measures on $\mathcal{A}^{\mathbb{Z}}$ are HM and non-trivial LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ are diffusive when $\mathcal{A}=\mathbb{Z}_{/p}$ is a prime cyclic group.

In the present work, we provide sufficient conditions for diffusion of LCA on $\mathcal{A}^{(\mathbb{Z}^D)}$ when $\mathcal{A}=\mathbb{Z}_{/n}$ is any cyclic group or when $\mathcal{A}=(\mathbb{Z}_{/p^r})^J$ (p prime). We also show that any fully supported Markov random field on $\mathcal{A}^{(\mathbb{Z}^D)}$ is HM (where $\mathcal{A}$ is any Abelian group).