ON A CLASS OF FRACTAL MATRICES II: DISTRIBUTION AND SCALE-DEPENDENT FRACTAL DIMENSIONS OF THE INTEGER SETS IN EXCESS-MATRICES AND THEIR DISGUISES

Excess-matrices were defined in a previously published Part I as a special class of integer matrices with multifractal features. The present Part II considers compactifications of an excess-matrix E, i.e., matrices obtained by replacing submatrices of specific sizes in E by a single number according to some prescribed rule. It will be shown how these compactifications can be obtained by a direct mapping from E itself, through the derivation of a set of formulas for the associated mapping sequences (as defined in Part I). Then these formulas are used for characterizing the distribution of certain integer sets over these matrices. The evolution of this distribution over all scales of observation is described by two different fractal-dimension functions, which are scale-dependent extensions of classical practical fractal-dimension measures. Expressions for these functions are derived, and their properties discussed. Exploring the limits of these functions which relate them to the classical small-scale fractal dimension reveals some difficulty in the interpretation of the latter when applied to excess-matrices.