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Conservation Laws and Invariant Measures in Surjective Cellular Automata

Abstract : We discuss a close link between two seemingly different topics studied in the cellular automata literature: additive conservation laws and invariant probability measures. We provide an elementary proof of a simple correspondence between invariant full-support Bernoulli measures and interaction-free conserved quantities in the case of one-dimensional surjective cellular automata. We also discuss a generalization of this fact to Markov measures and higher-range conservation laws in arbitrary dimension. As a corollary, we show that the uniform Bernoulli measure is the only shift-invariant, full-support Markov measure that is invariant under a strongly transitive cellular automaton.
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Jarkko Kari, Siamak Taati. Conservation Laws and Invariant Measures in Surjective Cellular Automata. 17th International Workshop on Celular Automata and Discrete Complex Systems, 2011, Santiago, Chile. pp.113-122, ⟨10.46298/dmtcs.2968⟩. ⟨hal-01196135⟩

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