Pre-Expansivity in Cellular Automata

G Gajardo (1) , G Nesme (2) , Guillaume Theyssier (3)

1 Departamento de Ingeniería Matemática
2 LIG - Laboratoire d'Informatique de Grenoble
3 I2M - Institut de Mathématiques de Marseille

We introduce the property of pre-expansivity for cellular automata (CA): it is the property of being expansive on asymptotic pairs of configurations (i.e. configurations that differ in only finitely many positions). Pre-expansivity therefore lies between expansivity and pre-injectivity, two important notions of CA theory. We show that there exist one-dimensional positively pre-expansive CAs which are not positively expansive and they can be chosen reversible (while positive expansivity is impossible for reversible CAs). We show however that no bi-dimensional CA which is linear over an Abelian group can be pre-expansive. We also consider the finer notion of k-expansivity (expansivity over pairs of configurations with exactly k differences) and show examples of linear CA in dimension 2 and on the free group that are k-expansive depending on the value of k, whereas no (positively) expansive CA exists in this setting.

Mots clés: cellular automata; linear cellular automata; 2-dimensional cellular; automata; expansivity; chaos; directional dynamics

Domaines: Mathématique discrète [cs.DM]; Systèmes dynamiques [math.DS]

G Gajardo, G Nesme, Guillaume Theyssier. Pre-Expansivity in Cellular Automata. 2017. ⟨hal-01286018v3⟩