Core Concepts
There is a correspondence between certain multiband linear cellular automata and endomorphisms of algebraic vector groups over finite fields, which allows for the deduction of new results concerning the temporal dynamics of such automata.
Abstract
The paper proposes a correspondence between multiband linear cellular automata, which are models of computation used to describe physical phenomena, and endomorphisms of certain algebraic unipotent groups over finite fields. This correspondence is based on the construction of a universal element that specializes to a normal generator for any finite field.
The authors use this correspondence to derive new results about the temporal dynamics of such automata, leveraging their prior study of the endomorphism ring of vector groups. These results include:
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, where the function is either a rational function or cannot be analytically continued over any point of the circle of convergence.
An asymptotic formula for the number of periodic orbits.
Since multiband linear cellular automata can simulate higher-order linear automata, the results apply to that class as well. The key innovation is the link between points over the algebraic closure of the finite field with Galois action and periodic sequences with entries in the finite field with the action of the shift map.
Stats
logp #Fix(g^n) = na - tn p^vp(n)
ζg(z) = 1 / (1 - p^a z) or cannot be analytically continued over any point of the circle of convergence |z| = 1/p^a
Pℓ ~ p^ℓa - t_ℓ p^vp(ℓ) / ℓ + O(√p^ℓa) as ℓ → +∞