Coarse-graining invariant orbits of one-dimensional Z(p)-linear cellular automata

This paper studies the properties of state-time evolution patterns of one-dimensional linear cellular automata over the field Z(p) (p prime) which are invariant under certain coarse-graining operations. A procedure is developed for finding all solutions to this invariance problem. The resulting patterns display a complexity which may range from periodic over self-similar, quasi-periodic, quasi-randomlike towards randomlike. Conditions for the existence of periodic solutions are derived.