Theory of Computing Systems

, Volume 48, Issue 3, pp 693–714 | Cite as

Topological Dynamics of Cellular Automata: Dimension Matters

  • Mathieu Sablik
  • Guillaume Theyssier


Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focuses on higher dimensional CA and aims at showing that the situation is different and more complex starting from dimension 2. The main results are the existence of non sensitive CA without equicontinuous points, the non-recursivity of sensitivity constants, the existence of CA having only non-recursive equicontinuous points and the existence of CA having only countably many equicontinuous points. They all show a difference between dimension 1 and higher dimensions. Thanks to these new constructions, we also extend undecidability results concerning topological classification previously obtained in the 1D case. Finally, we show that the set of sensitive CA is only \(\varPi _{2}^{0}\) in dimension 1, but becomes \(\varSigma _{3}^{0}\)-hard for dimension 3.


Multidimensional cellular automata Topological dynamics Complexity of decision problem 


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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.LATP, UMR 6632—CNRS, Université de Provence, CMI, Université de ProvenceTechnopôle Château-GombertMarseille Cedex 13France
  2. 2.LAMA, UMR 5127—CNRSUniversité de SavoieLe Bourget-du-lac CedexFrance

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