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 → +∞

Quotes

None.

Key Insights Distilled From

by Jakub Byszew... at **arxiv.org** 04-22-2024

Deeper Inquiries

The correspondence between multiband linear cellular automata and endomorphisms of unipotent algebraic groups can be generalized to more general algebraic groups by considering a broader class of algebraic structures. Instead of restricting the study to vector groups or unipotent groups, one can explore the dynamics of endomorphisms on a wider range of algebraic groups. This extension would involve analyzing the properties and behaviors of endomorphisms on diverse algebraic structures beyond just unipotent groups. By investigating the interactions between cellular automata and endomorphisms in these more general algebraic settings, one can uncover new insights into the relationship between computational models and algebraic structures.

The results obtained in the context of finite fields F_p can potentially be extended to cases where the finite field is replaced by an arbitrary field K of characteristic p > 0. This extension would involve adapting the concepts and techniques used in the study of multiband linear cellular automata and endomorphisms to the setting of arbitrary fields. By considering the dynamics of endomorphisms on fields other than finite fields, one can explore how the behaviors and patterns observed in cellular automata models translate to more general algebraic structures. This extension would broaden the applicability of the results to a wider range of mathematical contexts beyond finite fields.

Studying the dynamics of endomorphisms of general unipotent algebraic groups can provide valuable insights into the behavior of algebraic structures under repeated transformations. By analyzing how endomorphisms act on unipotent groups, researchers can gain a deeper understanding of the intrinsic properties and symmetries of these algebraic structures. This analysis can also shed light on the stability, periodicity, and complexity of the dynamics induced by endomorphisms on unipotent groups. Furthermore, exploring the relationship between the dynamics of endomorphisms and the structures of unipotent groups can lead to new discoveries in the field of algebraic geometry and computational mathematics. This study may also pave the way for generalizing the concepts of cellular automata to more diverse algebraic settings, providing a broader framework for analyzing computational models in algebraic contexts.

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