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Entropy 2000, 2(3), 142-167; doi:10.3390/e2030142
Article

Additive Cellular Automata and Volume Growth

Received: 23 June 2000 / Accepted: 29 July 2000 / Published: 24 August 2000
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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 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.
Keywords: cellular automata; entropy; rational function field; Adele ring; hyperbolic dynamics cellular automata; entropy; rational function field; Adele ring; hyperbolic dynamics
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Ward, T.B. Additive Cellular Automata and Volume Growth. Entropy 2000, 2, 142-167.

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