Figure 15: Mean field probability distributions for
the first five generations of evolution of Rule 110. The uniform 50%
distribution tends toward superstability.
As Figure 15 shows, the distribution of cell states for Rule 110 tends quickly toward 50%, even though the a priori estimate favors 1's, and any strict alternation of 0's and 1's is ruled out. Note that the curves shown in the diagram are rigorously calculated mean field probabilities for successive composites, or generations of evolution, but that they tend to behave as iterates would.
Although they don't adhere to the first generation fixed point, they tend toward a superstable fixed point at 50%, while the fixed point at zero becomes ever more unstable.
Figure 16: Two-block probabilities, showing a linear
(rather than quadratic) relation valid at low densities, slope 1/2, holding
over an extensive range.
In the contour map of pair probabilities, shown in Figure 16, single 1's, whose frequency forms the horizontal coordinate, tend to occur with double the frequency of pairs of 1's, whose frequency defines the vertical coordinate. That would indicate that the 1's are more likely to form blocks than to be isolated, a predilection compatible with the scarcity of sequences in which 0's and 1's alternate.