Summary
In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. S. Afraimovich and M. A. Shereshevsky (1990) Cellular automata as dynamical systems,Research Reports in Physics. Nonlinear Waves 3 (ed. by A. V. Gaponov-Grekhov, M. I. Rabinovich and J. Engelbrecht), Springer-Verlag, Berlin 296–300.
R. Bowen (1975)Equilibrium States and the Ergodic Theory of Anosov Diffeomorphisms, Lecture Notes in Math.,470, Springer-Verlag, New York.
M. Brin and A. Katok (1983) On local entropy,Lecture Notes in Math.,470, Springer-Verlag, New York, 30–38.
E. M. Coven and M. Paul (1974) Endomorphisms of irreducible shifts of finite type,Math. Syst. Theory,8, 165–177.
M. Creuz (1986) Deterministic Ising dynamics,Ann. Phys.,167, 62–72.
J.-P. Eckmann and D. Ruelle (1985) Ergodic theory of chaos and strange attractors,Rev. Modern Phys.,53, 617–656.
R. H. Gilman (1987) Classes of linear automata,Ergod. Theory and Dyn. Syst.,7, 105–118.
J. M. Greenberg, B. D. Hassard and S. P. Hastings (1978) Pattern formation and periodic structures in systems modelled by reaction-diffusion equation,Bull. Amer. Math. Soc.,84, 1296–1327.
G. A. Hedlund (1969) Endomorphisms and automorphisms of the shift dynamical systems,Math. Syst. Theory.,3, 320–375.
M. Hurley (1990) Attractors in cellular automata,Ergod. Theory and Dyn. Syst.,10, 131–140.
J. P. C. Kingman (1973) Subadditive ergodic theory,Ann. Probab.,1, 883–909.
T. Kitagawa (1974) Cell space approaches in biomathematics,Math. Biosci.,19, 27–71.
W. Li., N. Packard and C. Langton (1990) Transition phenomena in cellular automata rule space,Physica 45D, 77–94.
D. A. Lind (1984) Applications of ergodic theory and sofic systems to cellular automata,Physica 10D, 36–44.
N. Margolus (1984) Physics-like models of computation,Physica 10D, 81–95.
J. von Neumann (1966)Theory of Self-Reproducing Automata (ed. by A. W. Burks), University of Illinois, Urbana.
D. Ruelle (1979) Ergodic theory of differentiable dynamical systems,Publ. Math. IHES,50, 275–306.
M. A. Shereshevsky (1991)Ergodic Properties of Certain Surjective Cellular Automata. University of Warwick, Preprint.
M. Shirvani and T. D. Rogers (1991) On ergodic one-dimensional cellular automata,Commun. Math. Phys.,136, 599–605.
S. Wolfram (1984) Universality and complexity in cellular automata,Physica 10D, 1–35.
S. Wolfram (1985) Twenty problems in the theory of cellular automata,Physica Scripta,9, 1–35.
L.-S. Young (1983) Entropy, Lyapunov exponents and Hausdorff dimension in differentiable dynamics,Trans. Circuits and Syst.,30, 599–607.
Author information
Authors and Affiliations
Additional information
Communicated by Michail Rabinovich
Rights and permissions
About this article
Cite this article
Shereshevsky, M.A. Lyapunov exponents for one-dimensional cellular automata. J Nonlinear Sci 2, 1–8 (1992). https://doi.org/10.1007/BF02429850
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02429850