Asymptotic randomization of sofic shifts by linear cellular automata

Let ${\mathbb{M}}={\mathbb{Z}}^D$ be a $D$ -dimensional lattice, and let $({\mathcal{A}},+)$ be an abelian group. ${\mathcal{A}}^{\mathbb{M}}$ is then a compact abelian group under componentwise addition. A continuous function $\Phi:{\mathcal{A}}^{\mathbb{M}}\longrightarrow{\mathcal{A}}^{\mathbb{M}}$ is called a linear cellular automaton if there is a finite subset ${\mathbb{F}}\subset{\mathbb{M}}$ and non-zero coefficients $\varphi_{\textsf{f}}\in{\mathcal{Z}}$ so that, for any ${\bf{a}}\in{\mathcal{A}}^{\mathbb{M}}\, \Phi({\bf{a}}) = \sum_{{\textsf{f}}\in{\mathbb{F}}}\varphi_{\textsf{f}}\cdot\sigma^{{\textsf{f}}}({\bf{a}})$ . Suppose that $\mu$ is a probability measure on ${\mathcal{A}}^{\mathbb{M}}$ whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on $\Phi$ and $\mu$ ) under which $\Phi$ asymptotically randomizes $\mu$ , meaning that $\mathrm{wk}^*-\lim_{{\mathbb{J}}\ni j\rightarrow\infty} \Phi^j\mu = \eta$ , where $\eta$ is the Haar measure on ${\mathcal{A}}^{\mathbb{M}}$ , and ${\mathbb{J}}\subset{\mathbb{N}}$ has Cesàro density one. In the case when $\Phi=1+\sigma$ and ${\mathcal{A}}=({{\mathbb{Z}}_{/p}})^s$ ( $p$ prime), we provide a condition on $\mu$ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by $1+\sigma$ .

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