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Additive cellular automata and volume growth.

Ward, T. (2000) 'Additive cellular automata and volume growth.', Entropy., 2 (3). pp. 142-167.

Abstract

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 formulae 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.

Item Type:Article
Keywords:Cellular automata, Entropy, Rational function field, Adele ring, Hyperbolic dynamics.
Full text:PDF - Published Version (261Kb)
Status:Peer-reviewed
Publisher Web site:http://dx.doi.org/10.3390/e2030142
Record Created:18 Oct 2012 15:35
Last Modified:19 Oct 2012 14:27

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