Directional Dynamics along Arbitrary Curves in Cellular Automata

This paper studies directional dynamics in cellular automata, a formalism previously introduced by the third author. The central idea is to study the dynamical behaviour of a cellular automaton through the conjoint action of its global rule (temporal action) and the shift map (spacial action): qualitative behaviours 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. Key words: cellular automata, topological dynamics, directional dynamics

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Martin Delacourt, Victor Poupet, Mathieu Sablik, Guillaume Theyssier
• Keywords: Cellular Automata, Cellular Automaton, CORR 2010, Education, Topological Dynamics