Attractors in Restricted Cellular Automata

The goal of this note is to extend previous results about the dynamics of cellular automata to "restricted cellular automata." Roughly speaking, a cellular automaton is a rule that updates a configuration of "states" that are arranged along the integer lattice in $\mathbb {R}$. In applications one often thinks of one of these states as "blank" or "quiescent," while the other "active" states evolve against a quiescent background. Often the physically relevant configurations are those with only a finite number of active states. If ${X_0}$ is the set of all such states, and if a cellular automaton maps ${X_0}$ to ${X_0}$, then its restriction to ${X_0}$ is a restricted cellular automaton. The main results show that there are rather strong constraints on the collection of attractors for any restricted cellular automaton. These constraints parallel those described in [H1] for the unrestricted case.