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Automaticity and invariant measures of linear cellular automata (1811.01256v3)

Published 3 Nov 2018 in cs.FL, cs.DM, and math.DS

Abstract: We show that spacetime diagrams of linear cellular automata $\Phi : {\mathbb F}_p{\mathbb Z} \to {\mathbb F}_p{\mathbb Z}$ with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions which are eventually constant. Each automatic spacetime diagram defines a $(\sigma, \Phi)$-invariant subset of ${\mathbb F}_p{\mathbb Z}$, where $\sigma$ is the left shift map, and if the initial condition is not eventually periodic then this invariant set is nontrivial. For the Ledrappier cellular automaton we construct a family of nontrivial $(\sigma, \Phi)$-invariant measures on ${\mathbb F}_3{\mathbb Z}$. Finally, given a linear cellular automaton $\Phi$, we construct a nontrivial $(\sigma, \Phi)$-invariant measure on ${\mathbb F}_p{\mathbb Z}$ for all but finitely many $p$.

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