  We study the dynamics of (synchronous) one-dimensional cellular automata with
cyclical boundary conditions that evolve according to the majority rule with
radius $ r $. We introduce a notion that we term cell stability with which we
express the structure of the possible configurations that could emerge in this
setting. Our main finding is that apart from the configurations of the form $
(0^{r+1}0^* + 1^{r+1}1^*)^* $, which are always fixed-points, the other
configurations that the automata could possibly converge to, which are known to
be either fixed-points or 2-cycles, have a particular spatially periodic
structure. Namely, each of these configurations is of the form $ s^* $ where $
s $ consists of $ O(r^2) $ consecutive sequences of cells with the same state,
each such sequence is of length at most $ r $, and the total length of $ s $ is
$ O(r^2) $ as well. We show that an analogous result also holds for the
minority rule.
