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Conservation laws in cellular automata

Marcus Pivato

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Recommended by L Bunimovich

If Bbb X is a discrete Abelian group and Script A a finite set, then a cellular automaton (CA) is a continuous map fraktur F:Script ABbb XScript ABbb X that commutes with all Bbb X-shifts. If phi:Script ABbb R, then, for any ainScript ABbb X, we define Σphi(a) = ∑xinBbb Xphi(ax) (if finite); phi is conserved by fraktur F if Σphi is constant under the action of fraktur F.

We characterize such conservation laws in several ways, deriving both theoretical consequences and practical tests, and provide a method for constructing all one-dimensional CA exhibiting a given conservation law.


PACS

05.45.-a Nonlinear dynamics and nonlinear dynamical systems

02.20.-a Group theory

MSC

37B15 Cellular automata

20K45 Topological methods (See also 22A05, 22B05)

37E05 Maps of the interval (piecewise continuous, continuous, smooth)

Subjects

Mathematical physics

Statistical physics and nonlinear systems

Dates

Issue 6 (November 2002)

Received 5 November 2001, in final form 17 June 2002

Published 16 September 2002



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