Additive cellular automata and volume growth
type: Item
instance of: scholarly article
title: Additive cellular automata and volume growth
zbMATH Open document ID: 0999.68130
published in: Entropy
publication date: 2002-09-10T00:00:00Z
full work available at URL: list00.htm
review text: Summary: A class of dynamical systems associated to rings of S-integers in rational function fields is described. General results about these systems give a rather complete description of the well-known dynamics in one-dimensional additive cellular automata with prime alphabet, including simple formulæ for the topological entropy and the number of periodic configurations. For these systems the periodic points are uniformly distributed along some subsequence with respect to the maximal measure, and in particular are dense. Periodic points may be constructed arbitrarily close to a given configuration, and rationality of the dynamical zeta function is characterized. Throughout the emphasis is to place this particular family of cellular automata into the wider context of S-integer dynamical systems, and to show how the arithmetic of rational function fields determines their behaviour. Using a covering space the dynamics of additive cellular automata are related to a form of hyperbolicity in completions of rational function fields. This expresses the topological entropy of the automata directly in terms of volume growth in the covering space.
Mathematics Subject Classification ID: 68Q80
zbMATH DE Number: 1796656
zbMATH Keywords: cellular automata
zbMATH Keywords: entropy
zbMATH Keywords: rational function field
zbMATH Keywords: adele ring
zbMATH Keywords: hyperbolic dynamics
author: Thomas B. Ward
MaRDI profile type: Publication
OpenAlex ID: W2112878337
DOI: 10.3390/E2030142
description: scientific article; zbMATH DE number 1796656