  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.
