Spectral properties of reversible one-dimensional cellular automata
Mora, J. C. , Vergara, S. V. , Martinez, G. J. and McIntosh, H. V. (2003) Spectral properties of reversible one-dimensional cellular automata. International Journal of Modern Physics C, 14 (3). pp. 379-395. ISSN 0129-1831
Publisher's URL: http://dx.doi.org/10.1142/S0129183103004541
Reversible cellular automata are invertible dynamical systems characterized by discreteness, determinism and local interaction. This article studies the local behavior of reversible one-dimensional cellular automata by means of the spectral properties of their connectivity matrices. We use the transformation from every one-dimensional cellular automaton to another of neighborhood size 2 to generalize the results exposed in this paper. In particular we prove that the connectivity matrices have a single positive eigenvalue equal to 1; based on this result we also prove the main result of this paper: the idempotent behavior of these matrices. This property is an important feature for detecting which one-dimensional cellular automata are reversible. Hence we present a procedure using the eigenvectors of these matrices to find the inverse rule for a given reversible one-dimensional cellular automaton. Finally illustrative examples are provided.
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