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 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.
Cite
@article{arxiv.1603.07215,
title = {Pre-Expansivity in Cellular Automata},
author = {A. Gajardo and V. Nesme and Guillaume Theyssier},
journal= {arXiv preprint arXiv:1603.07215},
year = {2019}
}