Topological Dynamics of 2D Cellular Automata

  • Mathieu Sablik
  • Guillaume Theyssier
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5028)


Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on 2D CA and aims at showing that the situation is different and more complex. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants and the existence of CA having only non-recursive equicontinuous points. They all show a difference between the 1D and the 2D case. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case.


Cellular Automaton Turing Machine Cellular Automaton Recursive Function Topological Dynamics 
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  1. 1.
    Kůrka, P.: Languages, equicontinuity and attractors in cellular automata. Ergodic Theory and Dynamical Systems 17, 417–433 (1997)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Blanchard, F., Maass, A.: Dynamical properties of expansive one-sided cellular automata. Israel J. Math. 99 (1997)Google Scholar
  3. 3.
    Blanchard, F., Tisseur, P.: Some properties of cellular automata with equicontinuity points. Ann. Inst. Henri Poincaré, Probabilités et statistiques 36, 569–582 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fagnani, F., Margara, L.: Expansivity, permutivity, and chaos for cellular automata. Theory of Computing Systems 31(6), 663–677 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Durand, B., Formenti, E., Varouchas, G.: On undecidability of equicontinuity classification for cellular automata. In: Morvan, M., Rémila, É. (eds.) DMCS 2003. Volume AB of DMTCS Proceedings, pp. 117–128 (2003)Google Scholar
  6. 6.
    Kari, J.: Reversibility and Surjectivity Problems of Cellular Automata. Journal of Computer and System Sciences 48(1), 149–182 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Bernardi, V., Durand, B., Formenti, E., Kari, J.: A new dimension sensitive property for cellular automata. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 416–426. Springer, Heidelberg (2004)Google Scholar
  8. 8.
    Berger, R.: The undecidability of the domino problem. Mem. Amer. Math Soc. 66 (1966)Google Scholar
  9. 9.
    Hedlund, G.A.: Endomorphisms and Automorphisms of the Shift Dynamical Systems. Mathematical Systems Theory 3(4), 320–375 (1969)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Wang, H.: Proving theorems by pattern recognition ii. Bell System Tech. Journal 40(2) (1961)Google Scholar
  11. 11.
    Myers, D.: Nonrecursive tilings of the plane. ii. The Journal of Symbolic Logic 39(2), 286–294 (1966)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Mathieu Sablik
    • 1
  • Guillaume Theyssier
    • 2
  1. 1.UMPA, (UMR 5669 — CNRS, ENS Lyon), 46, allée d’Italie 69364 Lyon cedex 07, France and LATP, (UMR 6632 — CNRS, Université de Provence), CMI, Université de Provence, Technopôle Château-Gombert, 39, rue F. Joliot Curie, 13453 Marseille Cedex 13France
  2. 2.LAMA(UMR 5127 — CNRS, Université de Savoie)Le Bourget-du-lac cedexFrance

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