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Journal Articles Theoretical Computer Science Year : 2011

Directional Dynamics along Arbitrary Curves in Cellular Automata

Abstract

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.
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Dates and versions

hal-00451729 , version 1 (29-01-2010)
hal-00451729 , version 2 (19-08-2010)
hal-00451729 , version 3 (28-06-2013)

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Cite

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⟩
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