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. * 283 Fractals 1999.07:283-299. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/12/15. For personal use only. 2. FRACTAL DIMENSIONS, PROBABILITY AND ENTROPY: OVERVIEW This section provides an overview of the box-count method for computation of fractal dimensions for substitution systems having a fixed and finite alphabet. This includes MSS and SSS in which all sums are taken mod(p) for some finite p. Consideration of more complicated situations which can arise is deferred to Sec. 3. Fractals 1999.07:283-299. Downloaded from www.worldscientific.com by YALE UNIVERSITY on 07/12/15. For personal use only.