Everywhere Zero Pointwise Lyapunov Exponents for Sensitive Cellular Automata

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12286)


Lyapunov exponents are an important concept in differentiable dynamical systems and they measure stability or sensitivity in the system. Their analogues for cellular automata were proposed by Shereshevsky and since then they have been further developed and studied. In this paper we focus on a conjecture claiming that there does not exist such a sensitive cellular automaton, that would have both the right and the left pointwise Lyapunov exponents taking the value zero, for each configuration. In this paper we prove this conjecture false by constructing such a cellular automaton, using aperiodic, complete Turing machines as a building block.


Cellular automata Sensitive Lyapunov exponents 



The author acknowledges the foundation under the aegis of the Fondation de Luxembourg for its financial support.


  1. 1.
    Blondel, V.D., Cassaigne, J., Nichitiu, C.M.: On the presence of periodic configurations in Turing machines and in counter machines. Theor. Comput. Sci. 289, 573–590 (2002)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bressaud, X., Tisseur, P.: On a zero speed sensitive cellular automaton. Nonlinearity 20(1), 1–19 (2006). Scholar
  3. 3.
    Cassaigne, J., Ollinger, N., Torres, R.: A small minimal aperiodic reversible Turing machine. J. Comput. Syst. Sci. 84 (2014).
  4. 4.
    D’amico, M., Manzini, G., Margara, L.: On computing the entropy of cellular automata. In: Larsen, K.G., Skyum, S., Winskel, G. (eds.) ICALP 1998. LNCS, vol. 1443, pp. 470–481. Springer, Heidelberg (1998). Scholar
  5. 5.
    Finelli, M., Manzini, G., Margara, L.: Lyapunov exponents vs expansivity and sensitivity in cellular automata. In: Bandini, S., Mauri, G. (eds.) ACRI 1996, pp. 57–71. Springer, London (1997). Scholar
  6. 6.
    Greiner, W.: Lyapunov exponents and chaos. In: Greiner, W. (ed.) Classical Mechanics, pp. 503–516. Springer, Heidelberg (2010). Scholar
  7. 7.
    Guillon, P., Salo, V.: Distortion in one-head machines and cellular automata. In: Dennunzio, A., Formenti, E., Manzoni, L., Porreca, A.E. (eds.) Cellular Automata and Discrete Complex Systems, pp. 120–138. Springer International Publishing, Cham (2017). Scholar
  8. 8.
    Jeandel, E.: Computability of the entropy of one-tape Turing machines. Leibniz International Proceedings in Informatics, LIPIcs 25 (2013).
  9. 9.
    Kůrka, P.: Topological dynamics of cellular automata. In: Meyers, R.A. (ed.) Encyclopedia of Complexity and Systems Science, pp. 9246–9268. Springer, New York (2009). Scholar
  10. 10.
    Kůrka, P.: On topological dynamics of Turing machines. Theor. Comput. Sci. 174(1), 203–216 (1997). Scholar
  11. 11.
    Lyapunov, A.: General Problem of the Stability of Motion. Control Theory and Applications Series. Taylor & Francis (1992).
  12. 12.
    Shereshevsky, M.A.: Lyapunov exponents for one-dimensional cellular automata. J. Nonlinear Sci. 2(1), 1–8 (1992). Scholar
  13. 13.
    Tisseur, P.: Cellular automata and Lyapunov exponents. Nonlinearity 13(5), 1547–1560 (2000). Scholar
  14. 14.
    Wolfram, S.: Universality and complexity in cellular automata. Phys. D Nonlinear Phenom. 10(1), 1–35 (1984). Scholar
  15. 15.
    Wolfram, S.: Twenty problems in the theory of cellular automata. Phys. Scr. T9, 170–183 (1985). Scholar

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Authors and Affiliations

  1. 1.University of TurkuTurkuFinland

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