AUTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA

Eric Rowland (1) , Reem Yassawi (2)

1 Hofstra University [Hempstead]
2 CTN - Combinatoire, théorie des nombres

We show that spacetime diagrams of linear cellular automata with (-p)-automatic initial conditions are automatic. This extends existing results on initial conditions which are eventually constant. Each automatic spacetime diagram defines a jointly invariant subset of $\F_p^\Z$, and if the initial condition is not eventually periodic then this invariant set is nontrivial. We construct, for the Ledrappier cellular automaton, a family of nontrivial jointly-invariant measures on the space of configurations with entries from the finite field with 3 elements.. Finally, given a linear cellular automaton, we construct a nontrivial jointly-invariant measure on the space of configurations with entries from the finite field with p for all but finitely many p.

Domaines: Mathématiques [math]; Systèmes dynamiques [math.DS]

Eric Rowland, Reem Yassawi. AUTOMATICITY AND INVARIANT MEASURES OF LINEAR CELLULAR AUTOMATA. Canadian Journal of Mathematics, 2019, 72 (6), pp.1691 - 1726. ⟨10.4153/S0008414X19000488⟩. ⟨hal-02086887⟩