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Automaticity and Invariant Measures of Linear Cellular Automata
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 72 / Issue 6 / December 2020
- Published online by Cambridge University Press:
- 05 September 2019, pp. 1691-1726
- Print publication:
- December 2020
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- Article
- Export citation
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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 
$p$.
