Some rigorous results for the Greenberg-Hastings Model

In this paper, we obtain some rigorous results for a cellular automaton known as the Greenberg-Hastings Model. The state space is {0, 1, 2}^Zd. The dynamics are deterministic and discrete time. A site which is 1 changes to 2, a site which is 2 changes to 0, and a site which is 0 changes to a 1 if one of its 2d neighbors is a 1. In one dimension, we compute the exact asymptotic rate at which the system dies out when started at random and compute the topological entropy. In two or more dimensions we show that starting from a nontrivial product measure, the limit exists as 3m and is Bernoulli shift. Finally, we investigate the behavior of the system on a large finite box.