On Pattern Density and Sliding Block Code Behavior for the Besicovitch and Weyl Pseudo-distances

  • Silvio Capobianco
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5901)


Initially proposed by Formenti et al. for bi-infinite sequences, the Besicovitch and Weyl pseudo-distances express the viewpoint of an observer moving infinitely far from the grid, rather than staying close as in the product topology. We extend their definition to a more general setting, which includes the usual infinite hypercubic grids, and highlight some noteworthy properties. We use them to measure the “frequency” of occurrences of patterns in configurations, and consider the behavior of sliding block codes when configurations at pseudo-distance zero are identified. One of our aims is to get an alternative characterization of surjectivity for sliding block codes.

Mathematics Subject Classification 2000: 37B15, 68Q80.


Pattern pseudo-distance sliding block code 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Blanchard, F., Formenti, E., Kůrka, P.: Cellular Automata in Cantor, Besicovitch, and Weyl Topological Spaces. Complex Systems 11(2), 107–123 (1999)Google Scholar
  2. 2.
    Capobianco, S.: Multidimensional Cellular Automata and Generalization of Fekete’s Lemma. Disc. Math. Theor. Comp. Sci. 10(3), 95–104 (2008)MathSciNetGoogle Scholar
  3. 3.
    Capobianco, S.: On the Induction Operation for Shift Subspaces and Cellular Automata as Presentations of Dynamical Systems. Inform. Comput. 207(11), 1169–1180 (2009)zbMATHCrossRefGoogle Scholar
  4. 4.
    Capobianco, S.: Some Notes on Besicovitch and Weyl Distances over Higher-Dimensional Configurations. In: de Oliveira, P.P.B., Kari, J. (eds.) Proceedings of Automata 2009: 15th International Workshop on Cellular Automata and Discrete Complex Systems, Universidade Presbiteriana Mackenzie, São Paulo, SP, Brazil, Section 2: Short Papers, pp. 300–308 (2009)Google Scholar
  5. 5.
    Capobianco, S.: Surjunctivity for Cellular Automata in Besicovitch Spaces. J. Cell. Autom. 4(2), 89–98 (2009)zbMATHMathSciNetGoogle Scholar
  6. 6.
    de la Harpe, P.: Topics in Geometric Group Theory. The University of Chicago Press (2000)Google Scholar
  7. 7.
    Fiorenzi, F.: Cellular Automata and Strongly Irreducible Shifts of Finite Type. Theor. Comp. Sci. 299, 477–493 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Lind, D., Marcus, B.: An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, Cambridge (1995)zbMATHGoogle Scholar
  9. 9.
    Namioka, I.: Følner’s Condition for Amenable Semi Groups. Math. Scand. 15, 18–28 (1962)MathSciNetGoogle Scholar
  10. 10.
    Toffoli, T., Capobianco, S., Mentrasti, P.: When—and How—Can a Cellular Automaton be Rewritten as a Lattice Gas? Theor. Comp. Sci. 403, 71–88 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Weiss, B.: Sofic Groups and Dynamical Systems. Sankhyā: Indian J. Stat. 62, 350–359 (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Silvio Capobianco
    • 1
  1. 1.Institute of Cybernetics at Tallinn University of TechnologyTallinnEstonia

Personalised recommendations