Asymptotic randomization of sofic shifts by linear cellular automata

Pivato, Marcus and Yassawi, Reem (2006). Asymptotic randomization of sofic shifts by linear cellular automata. Ergodic Theory and Dynamical Systems, 26(4) pp. 1177–1201.



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 : AMAM is called a linear cellular automaton if there is a finite subset FM and non-zero coefficients φf ∈ Z so that, for any aAM, 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* − limJj,→∞ Φjµ = η, where η is the Haar measure on AM, and JN 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.

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