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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. [Copyright &y& Elsevier]