Topological Dynamics of 2D Cellular Automata

Mathieu Sablik (1) , Guillaume Theyssier (2)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées
2 LAMA - Laboratoire de Mathématiques

Topological dynamics of cellular automata (CA), inherited from classical dynamical systems theory, has been essentially studied in dimension 1. This paper focus 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.

Mots clés: cellular automata; tilings; decidability; dynamical systems; sensitivity to initial conditions

Domaines: Mathématique discrète [cs.DM]

Mathieu Sablik, Guillaume Theyssier. Topological Dynamics of 2D Cellular Automata. 2007. ⟨hal-00175396v1⟩