Multiband linear cellular automata and endomorphisms of algebraic vector groups
Jakub Byszewski, Gunther Cornelissen
TL;DR
This work builds a bridge between finite-field cellular automata and endomorphisms of unipotent algebraic groups in positive characteristic by constructing a trace-based map $\iota$ that identifies fixed-point data with shift-periodic sequences. It proves a bijection $\sigma\mapsto g_\sigma$ between endomorphisms of $\mathbf{G}_a^r$ over $\mathbf{F}_p$ and one-sided multiband linear cellular automata, with $\mathrm{Fix}(\sigma^{\circ n})$ corresponding to $\mathrm{Fix}(g_\sigma^{\circ n})\cap\mathcal{P}$, enabling a transfer of algebraic-dynamics results to temporal dynamics. For confined systems, it derives a precise growth law $\log_p \#\mathrm{Fix}(g^{\circ n}) = n a - t_n p^{v_p(n)}$, a dynamical zeta-function dichotomy, and asymptotics for the number of periodic orbits $P_\ell \sim p^{\ell a - t_\ell p^{v_p(\ell)}}/\ell$, with invariants computable from eigenvalues of $m_{\tilde{\sigma}}\phi^M$. The results generalize to multiband and higher-order linear automata, providing a powerful framework to translate purely algebraic dynamics of additive groups into the temporal dynamics of cellular automata in characteristic $p$.
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.