Directional dynamics for cellular automata: A sensitivity to initial condition approach

A cellular automaton is a continuous function F defined on a full-shift AZ which commutes with the shift σ. Often, to study the dynamics of F one only considers implicitly σ. However, it is possible to emphasize the spatio-temporal structure produced by considering the dynamics of the Z×N-action induced by (σ,F).In this purpose we study the notion of directional dynamics. In particular, we are interested in directions of equicontinuity and expansivity, which generalize the concepts introduced by Gilman [Robert H. Gilman, Classes of linear automata, Ergodic Theory Dynam. Systems 7 (1) (1987) 105–118] and P. Kůrka [Petr Kůrka, Languages, equicontinuity and attractors in cellular automata, Ergodic Theory Dynam. Systems 17 (2) (1997) 417–433]. We study the sets of directions which exhibit this special kind of dynamics showing that they induce a discrete geometry in space-time diagrams.