Topological Dynamics of Cellular Automata: Dimension Matters

Abstract : 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 Pi_2 in dimension 1, but becomes Sigma_3-hard for dimension 3.
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Theory of Computing Systems, Springer Verlag, 2009
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Soumis le : jeudi 30 avril 2009 - 18:05:43
Dernière modification le : jeudi 18 janvier 2018 - 02:00:46
Document(s) archivé(s) le : mercredi 22 septembre 2010 - 12:57:09

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  • HAL Id : hal-00339354, version 2
  • ARXIV : 0811.2731

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Mathieu Sablik, Guillaume Theyssier. Topological Dynamics of Cellular Automata: Dimension Matters. Theory of Computing Systems, Springer Verlag, 2009. 〈hal-00339354v2〉

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