Everywhere Zero Pointwise Lyapunov Exponents for Sensitive Cellular Automata
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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.
KeywordsCellular automata Sensitive Lyapunov exponents
The author acknowledges the emmy.network foundation under the aegis of the Fondation de Luxembourg for its financial support.
- 3.Cassaigne, J., Ollinger, N., Torres, R.: A small minimal aperiodic reversible Turing machine. J. Comput. Syst. Sci. 84 (2014). https://doi.org/10.1016/j.jcss.2016.10.004
- 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). https://doi.org/10.1007/978-3-319-58631-1_10CrossRefzbMATHGoogle Scholar
- 8.Jeandel, E.: Computability of the entropy of one-tape Turing machines. Leibniz International Proceedings in Informatics, LIPIcs 25 (2013). https://doi.org/10.4230/LIPIcs.STACS.2014.421
- 11.Lyapunov, A.: General Problem of the Stability of Motion. Control Theory and Applications Series. Taylor & Francis (1992). https://books.google.fi/books?id=4tmAvU3_SCoC