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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  • HAL Id : hal-00451729, version 3
  • ARXIV : 1001.5470

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

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