Some results about the chaotic behavior of cellular automata

  • Authors:

  • F. Blanchard;J. Cervelle;E. Formenti

  • Affiliations:

  • Institut de Mathématique de Luminy, CNRS, Campus de Luminy, Marseille Cedex, France;Institut Gaspard Monge, Université de Marne-la-Vallée, France;Université de Nice-Sophia Antipolis, Laroratoire, Sophia Antipolis, France

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2005

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Abstract

We study the behavior of cellular automata (CA for short) in the Cantor, Besicovitch and Weyl topologies. We solve an open problem about the existence of transitive CA in the Besicovitch topology. The proof of this result has some interest of its own since it is obtained by using Kolmogorov complexity. To our knowledge it is the first result about discrete dynamical systems obtained using Kolmogorov complexity. We also prove that in the Besicovitch topology every CA has either a unique periodic point (thus a fixed point) or an uncountable set of periodic points. This result underlines the fact that CA have a great degree of stability; it may be considered a further step towards the understanding of CA periodic behavior.Moreover, we prove that in the Besicovitch topology there is a special set of configurations, the set of Toeplitz configurations, that plays a role similar to that of spatially periodic configurations in the Cantor topology, that is, it is dense and has a central role in the study of surjectivity and injectivity. Finally, it is shown that the set of spatially quasi-periodic configurations is not dense in the Weyl topology.