Computing fractal dimensions for additive cellular automata

Abstract

Efficient methods are given for finding the statistics of symbol blocks occuring in the graphs of additive cellular automata after large numbers of iterations. In particular, the methods yield fractal dimensions, entropies, and a recurrence relation for the number of nonzero symbols. The methods are based on the construction of a nonnegative integer matrix A closely related to the automation. The largest eigenvalue of A tells the fractal dimension, while the corresponding eigenvector gives the relative frequencies of blocks.