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.

DOI: https://doi.org/10.1017/S0143385706000228

Abstract

Abstract. Let M = Z^D be a D-dimensional lattice, and let (A, +) be an abelian group. A^M is then a compact abelian group under componentwise addition. A continuous function Φ : A^M → A^M 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 ∈ A^M, Φ(a) = Σ_f∈Fφf · σ^f(a). Suppose that µ is a probability measure on A^M 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* − lim_J∋j,→∞ Φ^jµ = η, where η is the Haar measure on A^M, 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.

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About

• Item ORO ID
• 67104
• Item Type
• Journal Item
• ISSN
• 0143-3857
• Project Funding Details

• Funded Project Name	Project ID	Funding Body
  Not Set	Not Set	NSERC Canada

• Academic Unit or School
• Faculty of Science, Technology, Engineering and Mathematics (STEM) > Mathematics and Statistics
  Faculty of Science, Technology, Engineering and Mathematics (STEM)
• Copyright Holders
• © 2006 Cambridge University Press
• Depositing User
• Reem Yassawi