Calculating growth rates and moments for additive cellular automata

Abstract

Suppose that the cellular automaton F is the graph of an additive rule modulo p. It is shown how to compute the growth rates of the numbers of each kind of finite block of symbols, if p is a power of a prime and the initial configuration is finite. All accessible blocks are then proved to have the same growth rate for a fixed F. If the modulus is not a power of a prime, different blocks may have different growth rates. These growth rates are identical with the Hausdorff dimensions of related fractals. The methods generalize further to yield explicit computations of the centers of mass and higher moments for the fractals.