  We consider backward filtrations generated by processes coming from
deterministic and probabilistic cellular automata. We prove that these
filtrations are standard in the classical sense of Vershik's theory, but we
also study them from another point of view that takes into account the
measurepreserving action of the shift map, for which each sigma-algebra in the
filtrations is invariant. This initiates what we call the dynamical
classification of factor filtrations, and the examples we study show that this
classification leads to different results.
