additive cellular automata and volume growth加法细胞自动机和体积增长.pdfVIP

Entropy 2000, 2, 142-167 entropy ISSN 1099-4300 © 2000 by MDPI /entropy/ Additive Cellular Automata and Volume Growth Thomas B. Ward School of Mathematics, University of East Anglia, Norwich, UK E-mail: t.ward@uea.ac.uk, URL: http://www.mth.uea.ac.uk/people/tw.html Received: 23 June 2000 / Accepted: 29 July 2000 / Published: 24 August 2000 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 alpha- bet, including simple formulæ for the topological entropy and the number of periodic con- figurations. For these systems the periodic points are uniformly distributed along some sub- sequence 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 dynami- cal 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 auto- mata directly in terms of volume growth in th