FRACTAL DIMENSION, PROBABILITY AND ENTROPY IN MATRIX AND SUMMATION SUBSTITUTION SYSTEMS

Abstract

The box-count method is used to obtain fractal dimensions, limiting probabilities and relative entropies for fractal and fractal-like objects generated by matrix and summation substitution systems. Results are based on the fact that a transition matrix formalism can be developed for these systems, yielding box counts at each iteration. The transition matrix is non-negative, and many results on non-negative matrices can be applied. The formalism developed is, up to a multiplicative constant, equivalent to a Markov process.