Geodesic
Automaticity and Invariant Measures of Linear Cellular Automata
Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1691-1726

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DOI

We show that spacetime diagrams of linear cellular automata $\unicode[STIX]{x1D6F7}:\,\mathbb{F}_{p}^{\mathbb{Z}}\rightarrow \mathbb{F}_{p}^{\mathbb{Z}}$with $(-p)$-automatic initial conditions are automatic. This extends existing results on initial conditions that are eventually constant. Each automatic spacetime diagram defines a $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant subset of $\mathbb{F}_{p}^{\mathbb{Z}}$, where $\unicode[STIX]{x1D70E}$ 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 $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measures on $\mathbb{F}_{3}^{\mathbb{Z}}$. Finally, given a linear cellular automaton $\unicode[STIX]{x1D6F7}$, we construct a nontrivial $(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D6F7})$-invariant measure on $\mathbb{F}_{p}^{\mathbb{Z}}$ for all but finitely many $p$.
DOI : 10.4153/S0008414X19000488
Mots-clés : linear cellular automata, invariant measure, automatic sequence, Christol’s theorem 
Rowland, Eric; Yassawi, Reem. Automaticity and Invariant Measures of Linear Cellular Automata. Canadian journal of mathematics, Tome 72 (2020) no. 6, pp. 1691-1726. doi: 10.4153/S0008414X19000488
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     title = {Automaticity and {Invariant} {Measures} of {Linear} {Cellular} {Automata}},
     journal = {Canadian journal of mathematics},
     pages = {1691--1726},
     year = {2020},
     volume = {72},
     number = {6},
     doi = {10.4153/S0008414X19000488},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008414X19000488/}
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