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A coupled cell system is a network of dynamical systems, or 'cells', coupled together. Such systems can be represented schematically by a directed graph whose nodes correspond to cells and whose edges represent couplings. A symmetry of a coupled cell system is a permutation of the cells that preserves all internal dynamics and all couplings. Symmetry can(More)
Let M be a monoid (e.g. N, Z, or Z D), and A an abelian group. A M is then a compact abelian group; a linear cellular automaton (LCA) is a continuous endomorphism F : A M −→ A M that commutes with all shift maps. Let µ be a (possibly nonstationary) probability measure on A M ; we develop sufficient conditions on µ and F so that the sequence {F N µ} ∞ N=1(More)
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 : A M −→ A M is called a linear cellular automaton if there is a finite subset F ⊂ M and non-zero coefficients ϕ f ∈ Z so that, for any a ∈ A M , (a) = f∈F ϕ f · σ f (a). Suppose that µ is a(More)
A right-sided, nearest neighbour cellular automaton (RNNCA) is a continuous transformation Φ : A Z −→A Z determined by a local rule φ : A {0,1} −→A so that, for any a ∈ A Z and any z ∈ Z, Φ(a) z = φ(a z , a z+1). We say that Φ is bipermutative if, for any choice of a ∈ A, the map A ∋ b → φ(a, b) ∈ A is bijective, and also, for any choice of b ∈ A, the map A(More)
If A is a finite alphabet, U ⊂ Z D , and µ U is a probability measure on A U that " looks like " the marginal projection of a stationary stochastic process on A Z D , then can we " extend " µ U to such a process? Under what conditions can we make this extension ergodic, (quasi)periodic, or (weakly) mixing? After surveying classical work on this problem when(More)
If X is a discrete abelian group and A a finite set, then a cellular automaton (CA) is a continuous map F : A X −→ A X that commutes with all X-shifts. If φ : A −→ R, then, for any a ∈ A X , we define Σφ(a) = x∈X φ(a x) (if finite); φ is conserved by F if Σφ is constant under the action of F. We characterize such conservation laws in several ways, deriving(More)
Let A Z be the Cantor space of bi-infinite sequences in a finite alphabet A, and let σ be the shift map on A Z. A cellular automaton is a continuous, σ-commuting self-map Φ of A Z , and a Φ-invariant subshift is a closed, (Φ, σ)-invariant subset S ⊂ A Z. Suppose a ∈ A Z is S-admissible everywhere except for some small region we call a defect. It has been(More)