Encyclopedia of Complexity and Systems Science

2009 Edition
| Editors: Robert A. Meyers (Editor-in-Chief)

Dynamics of Cellular Automata in Non-compact Spaces

  • Enrico Formenti
  • Petr Kůrka
Reference work entry
DOI: https://doi.org/10.1007/978-0-387-30440-3_138

Definition of the Subject

In topological dynamics, the assumption of compactness is usually adopted as it has farreaching consequences. Each compact dynamical system has an almost periodic point, containsa minimal subsystem, and each trajectory has a limit point. Nevertheless, there areimportant examples of non‐compact dynamical systems like linear systems on\( { \mathbb{R}^n }\)

This is a preview of subscription content, log in to check access.

Notes

Acknowledgments

We thank Marcus Pivato and Francois Blanchard for careful reading of the paper and many valuable suggestions. The research waspartially supported by the Research Program Project “Sycomore” (ANR-05-BLAN-0374).

Bibliography

Primary Literature

  1. 1.
    Besicovitch AS (1954) Almost periodic functions. Dover, New YorkGoogle Scholar
  2. 2.
    Blanchard F, Formenti E, Kůrka P (1999) Cellular automata in the Cantor, Besicovitch and Weyl spaces. Complex Syst 11(2):107–123Google Scholar
  3. 3.
    Blanchard F, Cervelle J, Formenti E (2005) Some results about the chaotic behaviour of cellular automata. Theor Comput Sci 349(3):318–336MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cattaneo G, Formenti E, Margara L, Mazoyer J (1997) A shift‐invariant metric on \( { S^{\mathbb{z}} } \) inducing a nontrivial topology. Lecture Notes in Computer Science, vol 1295. Springer, BerlinGoogle Scholar
  5. 5.
    Formenti E, Kůrka P (2007) Subshift attractors of cellular automata. Nonlinearity 20:105–117Google Scholar
  6. 6.
    Hedlund GA (1969) Endomorphisms and automorphisms of the shift dynamical system. Math Syst Theory 3:320–375MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hurd LP (1990) Recursive cellular automata invariant sets. Complex Syst 4:119–129MathSciNetzbMATHGoogle Scholar
  8. 8.
    Iwanik A (1988) Weyl almost periodic points in topological dynamics. Colloquium Mathematicum 56:107–119MathSciNetzbMATHGoogle Scholar
  9. 9.
    Kamae J (1973) Subsequences of normal sequences. Isr J Math 16(2):121–149MathSciNetGoogle Scholar
  10. 10.
    Knudsen C (1994) Chaos without nonperiodicity. Am Math Mon 101:563–565MathSciNetzbMATHGoogle Scholar
  11. 11.
    Kůrka P (1997) Languages, equicontinuity and attractors in cellular automata. Ergod Theory Dyn Syst 17:417–433Google Scholar
  12. 12.
    Kůrka P (2003) Cellular automata with vanishing particles. Fundamenta Informaticae 58:1–19Google Scholar
  13. 13.
    Kůrka P (2005) On the measure attractor of a cellular automaton. Discret Continuous Dyn Syst 2005(suppl):524–535Google Scholar
  14. 14.
    Marcinkiewicz J (1939) Une remarque sur les espaces de a.s. Besicovitch. C R Acad Sc Paris 208:157–159Google Scholar
  15. 15.
    Sablik M (2006) étude de l'action conjointe d'un automate cellulaire et du décalage: une approche topologique et ergodique. Ph D thesis, Université de la MediterranéeGoogle Scholar

Books and Reviews

  1. 16.
    Besicovitch AS (1954) Almost periodic functions. Dover, New YorkGoogle Scholar
  2. 17.
    Kitchens BP (1998) Symbolic dynamics. Springer, BerlinzbMATHGoogle Scholar
  3. 18.
    Kůrka P (2003) Topological and symbolic dynamics. Cours spécialisés, vol 11 . Société Mathématique de France, ParisGoogle Scholar
  4. 19.
    Lind D, Marcus B (1995) An introduction to symbolic dynamics and coding. Cambridge University Press, CambridgezbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Enrico Formenti
    • 1
  • Petr Kůrka
    • 2
  1. 1.Laboratoire I3S – UNSA/CNRS UMR 6070Université de Nice Sophia AntipolisSophia AntipolisFrance
  2. 2.Center for Theoretical StudyAcademy of Sciencesand Charles UniversityPragueCzech Republic