Directional Dynamics along Arbitrary Curves in Cellular Automata

Details Directional Dynamics along Arbitrary Curves in Cellular Automata Directional Dynamics along Arbitrary Curves in Cellular Automata

2010-01-29

arxiv.org/abs/1001.5470v2

Computer Science

Discrete Mathematics

Scientific paper

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.

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