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Pivato, Marcus and Yassawi, Reem
(2006).
DOI: https://doi.org/10.1017/S0143385706000228
Abstract
Abstract. Let M = ZD be a D-dimensional lattice, and let (A, +) be an abelian group. AM is then a compact abelian group under componentwise addition. A continuous function Φ : AM → AM is called a linear cellular automaton if there is a finite subset F ⊂ M and non-zero coefficients φf ∈ Z so that, for any a ∈ AM, Φ(a) = Σf∈Fφf · σf(a). Suppose that µ is a probability measure on AM whose support is a subshift of finite type or sofic shift. We provide sufficient conditions (on Φ and µ) under which Φ asymptotically randomizes µ, meaning that wk* − limJ∋j,→∞ Φjµ = η, where η is the Haar measure on AM, and J ⊂ N has Cesàro density one. In the case when Φ = 1 + σ and A = (Z/p)s (p prime), we provide a condition on µ that is both necessary and sufficient. We then use this to construct zero-entropy measures which are randomized by 1 + σ.