Solution of some conjectures about topological properties of linear cellular automata

  • Authors:

  • Gianpiero Cattaneo;Alberto Dennunzio;Luciano Margara

  • Affiliations:

  • Dipartimento di Informatica Sistemistica e Comunicazione, Università degli studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy;Dipartimento di Informatica Sistemistica e Comunicazione, Università degli studi di Milano-Bicocca, via Bicocca degli Arcimboldi 8, 20126 Milano, Italy;Dipartimento di Scienze dell'Informazione, Università di Bologna, via Mura Anteo Zamboni 7, Bologna, Italy

  • Venue:
  • Theoretical Computer Science - Special issue: Theoretical aspects of cellular automata
  • Year:
  • 2004

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Abstract

We study two dynamical properties of linear D-dimensional cellular automata over Zm namely, denseness of periodic points and topological mixing. For what concerns denseness of periodic points, we complete the work initiated in (Theoret. Comput. Sci. 174 (1997) 157, Theoret. Comput. Sci. 233 (1-2) (2000) 147, 14th Annual Symp. on Theoretical Aspects of Computer Science (STACS '97), LNCS n. 1200, Springer, Berlin, 1997, pp. 427-438) by proving that a linear cellular automata has dense periodic points over the entire space of configurations if and only if it is surjective (as conjectured in (Cattaneo et al., 2000)). For non-surjective linear CA we give a complete characterization of the subspace where periodic points are dense. For what concerns topological mixing, we prove that this property is equivalent to transitivity and then easily checkable. Finally, we classify linear cellular automata according to the definition of chaos given by Devaney in (An Introduction to Chaotic Dynamical Systems, 2nd ed., Addison-Wesley, Reading, MA, USA, 1989).