Pseudofractals and box counting algorithm

Abstract

We show that for sets with the Hausdorff--Besicovitch dimension equal zero the box counting algorithm commonly used to calculate Renyi exponents ($d_q$) can exhibit perfect scaling suggesting non zero $d_q$'s. Properties of these pathological sets ({\it pseudofractals}) are investigated. Numerical, as well as analytical estimates for $d_q$'s are obtained. A simple indicator is given to distinguish pseudofractals and fractals in practical applications of the box counting method. Histograms made of pseudofractal sets are shown to have Pareto tails.

10 pages, 3 figures, RevTeX

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