  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.
