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By Tullio Ceccherini-Silberstein

Cellular automata have been brought within the first 1/2 the final century by way of John von Neumann who used them as theoretical types for self-reproducing machines. The authors current a self-contained exposition of the speculation of mobile automata on teams and discover its deep connections with fresh advancements in geometric team idea, symbolic dynamics, and different branches of arithmetic and theoretical laptop technology. the subjects handled comprise particularly the backyard of Eden theorem for amenable teams, and the Gromov-Weiss surjunctivity theorem in addition to the answer of the Kaplansky conjecture at the reliable finiteness of staff earrings for sofic teams. the amount is completely self-contained, with 10 appendices and greater than three hundred workouts, and appeals to a wide viewers together with experts in addition to beginners within the box. It presents a finished account of modern development within the concept of mobile automata according to the interaction among amenability, geometric and combinatorial workforce conception, symbolic dynamics and the algebraic idea of staff earrings that are taken care of the following for the 1st time in publication form.

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Let y ∈ AG/N , g ∈ G, and g = ρ(g). We have τ (y)(g) = τ (y ◦ ρ)(g) = μ((g −1 (y ◦ ρ))|S ) = μ((g −1 y)|S ). Thus τ is a cellular automaton with memory set S and local defining map μ : AS → A. 12). 2. The map Φ : CA(G; A) → CA(G/N ; A) is a monoid epimorphism. Proof. Let σ : AG/N → AG/N be a cellular automaton over G/N with memory set T ⊂ G/N and local defining map ν : AT → A. Let S ⊂ G be a finite set such that ρ induces a bijection φ : S → T . Consider the map μ : AS → A defined by μ(y) = ν(y ◦ φ−1 ) for all y ∈ AS .

A) Let τ1 , τ2 : AG → AG be two cellular automata. Show that the set {x ∈ AG : τ1 (x) = τ2 (x)} ⊂ AG is a subshift of finite type. (b) Deduce from (a) that if τ : AG → AG is a cellular automaton then the set Fix(τ ) = {x ∈ AG : τ (x) = x} ⊂ AG is a subshift of finite type. (c) Conversely, show that if X ⊂ AG is a subshift of finite type then there exists a cellular automaton τ : AG → AG such that X = Fix(τ ). 27. Suppose that a group Γ acts continuously on a topological space Z. One says that the action of Γ on Z is topologically transitive if for any pair of nonempty open subsets U and V of Z there exists an element γ ∈ Γ such that U ∩ γV = ∅.

Recall that any word w ∈ A∗ can be uniquely Exercises 35 written in the form w = a1 a2 · · · an , where n ≥ 0 and ai ∈ A for 1 ≤ i ≤ n. The integer n is called the length of the word w and it is denoted by (w). In the sequel, we shall identify the word w = a1 a2 · · · an with the pattern p : {1, 2, . . , n} → A defined by p(i) = ai for 1 ≤ i ≤ n. Given a subshift X ⊂ AZ and an integer n ≥ 0, we denote by Ln (X) ⊂ A∗ the set consisting of all words w ∈ A∗ for which there exists an element x ∈ X such that w = x(1)x(2) · · · x(n).

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