Pre-expansivity in cellular automata

Abstract

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 bidimensional 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.

Characterizing asymptotic randomization in abelian cellular automata

“…The results of [26] can be useful for that line of research. For reference, positive pre-expansivity, introduced in [9], is the property of being positively expansive on diamonds. A reversible CA can be positively pre-expansive (but never positively expansive) like the example F 2 of this paper.…”