Multiband linear cellular automata and endomorphisms of algebraic vector groups

TL;DR

This paper establishes a novel link between multiband linear cellular automata and algebraic endomorphisms of vector groups, enabling new insights into their dynamics and fixed point counts through algebraic methods.

Contribution

It introduces a universal construction connecting cellular automata with algebraic group endomorphisms, leading to new dynamical formulas and properties for these automata.

Findings

Formula for the number of fixed points based on p-adic valuation

Dichotomy in the Artin-Mazur dynamical zeta function

Asymptotic count of periodic orbits

Abstract

We propose a correspondence between certain multiband linear cellular automata - models of computation widely used in the description of physical phenomena - and endomorphisms of certain algebraic unipotent groups over finite fields. The correspondence is based on the construction of a universal element specialising to a normal generator for any finite field. We use this correspondence to deduce new results concerning the temporal dynamics of such automata, using our prior, purely algebraic, study of the endomorphism ring of vector groups. These produce 'for free' a formula for the number of fixed points of the $n$-iterate in terms of the $p$-adic valuation of $n$, a dichotomy for the Artin-Mazur dynamical zeta function, and an asymptotic formula for the number of periodic orbits. Since multiband linear cellular automata simulate higher order linear automata (in which states depend on finitely many prior temporal states, not just the direct predecessor), the results apply equally well to that class.