Directional Dynamics along Arbitrary Curves in Cellular Automata

Martin Delacourt (1, 2) , Victor Poupet (2) , Mathieu Sablik (3) , Guillaume Theyssier (4)

1 LIF - Laboratoire d'informatique Fondamentale de Marseille
2 ESCAPE - Systèmes complexes, automates et pavages
3 LATP - Laboratoire d'Analyse, Topologie, Probabilités
4 LAMA - Laboratoire de Mathématiques

This paper studies directional dynamics on one-dimensional cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behavior of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviors inherited from topological dynamics (equicontinuity, sensitivity, expansivity) are thus considered along arbitrary curves in space-time. The main contributions of the paper concern equicontinuous dynamics which can be connected to the notion of consequences of a word. We show that there is a cellular automaton with an equicontinuous dynamics along a parabola, but which is sensitive along any linear direction. We also show that real numbers that occur as the slope of a limit linear direction with equicontinuous dynamics in some cellular automaton are exactly the computably enumerable numbers.

Mots clés: Automates cellulaires; dynamique directionnelle

Domaines: Systèmes dynamiques [math.DS]

Martin Delacourt, Victor Poupet, Mathieu Sablik, Guillaume Theyssier. Directional Dynamics along Arbitrary Curves in Cellular Automata. Theoretical Computer Science, 2011, 412, pp.3800-3821. ⟨10.1016/j.tcs.2011.02.019⟩. ⟨hal-00451729v3⟩