For some automata it can be seen that states repeat themselves; indeed this behaviour is a very prominent feature of Wolfram's Class II automata. Formally, a automaton with transition rule is idempotent if
Even though the algebra is messy, it can be verified that for such a transition rule, a cell remains unchanged from the second generation onward.
It is a temptation to state the idempotency condition in the form ; in a sense it is true that iterated acts like ; but has three arguments and only one value, and so it is better to give the precise definition shown above.
It is too strong a requirement to insist that always, because only the identity rule fulfils this requirement. There are also many evident variations on the theme -- evolution could stagnate after the third generation rather than the second, for example. Likewise, the states could shift sideways rather than stagnating:
for example. Other variations would have the states undergoing permutations leaving the entire configuration stagnant only in those generations in which the full cycle of the permutation had run its course.
The evolution of idempotent automata, or those equivalent by one of the variations, is not particularly interesting from a dramatic point of view. Nevertheless, they form a class susceptible to a closed analytical treatment, and they are of rather common occurrence when all the possible types of rules are taken into account. There is an interesting borderline class of automata whose rules are not idempotent, but monotonic. For example, one of the automata has the rule (& is boolean and)
which is the logical and of the three binary neighbors. The rule is Wolfram's # 128, and has the property that zeroes persist. Unless one began with an infinite or cyclic chain of ones, all ones must eventually disappear, and the rule looks idempotent to the zeroes left behind. It is convenient to call these rules asymptotically idempotent and classify them together with idempotent rules.