Ergodic Theory of Cellular Automata

Glossary

Configuration space and the shift :

Let $\mathbb{M}$ be a finitely generated group or monoid (usually abelian). Typically, $\mathbb{M}=\mathrm{\mathbb{N}}:= \left\{0,1,2,\dots \right\}$ or $\mathbb{M}=\mathrm{\mathbb{Z}}:= \left\{\dots, -1,0,1,2,\dots \right\},$or $\mathbb{M}={\mathrm{\mathbb{N}}}^E$, ℤ^D, or ℤ^D × ℕ^E for some D, E ∈ ℕ. In some applications, $\mathbb{M}$ could be nonabelian (although usually amenable), but to avoid notational complexity we will generally assume $\mathbb{M}$ is abelian and additive, with operation ‘+’.

Let $\mathcal{A}$ be a finite set of symbols (called an alphabet). Let ${\mathcal{A}}^{\mathbb{M}}$ denote the set of all functions $\mathbf{a}:\mathbb{M}\to \mathcal{A}$, which we regard as $\mathbb{M}$-indexed configurations of elements in $\mathcal{A}$. We write such a configuration as $\mathbf{a}={\left[{a}_{\mathrm{m}}\right]}_{\mathrm{m}\in \mathbb{M}}$, where $a_{\mathrm{m}}\in \mathcal{A}$ for all $\mathrm{m}\in \mathbb{M}$, and refer to ${\mathcal{A}}^{\mathbb{M}}$ as configuration space.

Treat $\mathcal{A}$as a discrete topological space;...