Cellular Automata with an Infinite Number of Subshift Attractors

The concept of an attractor is essential for understanding the dynamics of cellular automata. An attractor is defined as a limit set (intersection of forward images) of a nonempty clopen (closed and open) invariant set. Classification of cellular automata based on the system of their attractors has been considered in Hurley [1] or Kůrka [2]. The basic distinction is between cellular automata which possess disjoint attractors and those with attractors that all have nonempty intersections. Another distinction investigated in Formenti and Kůrka [3] is between attractors which are subshifts and those which are not. For example, each clopen set is an attractor for the identity map, since it is invariant and its limit set is itself. Attractors of this kind, however, do not correspond very well with the intuitive idea of attraction. Much more interesting are attractors which are subshifts. It turns out that the limit set of a clopen invariant set is a subshift if and only if the set is spreading, that is, if it propagates both to the left and to the right. This is much closer to the spirit of attraction and gives more relevant information on the dynamics of the cellular automaton in question. The number of attractors may be either finite or countable. Cellular automata with an infinite number of subshift attractors have much higher complexity because they must perform actions at a distance. Their spreading sets may have the same structure but they must have arbitrarily large sizes. Some of these cellular automata perform a kind of self-reproduction when a seed pattern (of arbitrary size) spreads all over the cellular space. We construct two basic examples of such behavior. In Example 8 there is a stationary particle which is never created and sometimes it is destroyed. The nth spreading set consists of patterns (words) in which