A family of sand automata

TL;DR

This paper investigates the dynamical behaviors of a family of two-dimensional sand automata derived from one-dimensional rules, identifying properties like equicontinuity points and non-surjectivity, and generalizing these results.

Contribution

It introduces a specific algebraic sand automaton and classifies its dynamical properties, extending the analysis to a broader class of sand automata.

Findings

Identified a simple algebraic sand automaton G with specific dynamical properties.

Proved G has equicontinuity points but is not fully equicontinuous.

Showed G is not surjective and generalized these results to other sand automata.

Abstract

We study some dynamical properties of a family of two-dimensional cellular automata: those that arise from an underlying one dimensional sand automaton whose local rule is obtained using a latin square. We identify a simple sand automaton G whose local rule is algebraic, and classify this automaton as having equicontinuity points, but not being equicontinuous. We also show it is not surjective. We generalise some of these results to a wider class of sand automata.