Normalized multifractal spectra within the box-counting method

In the classical box-counting method a partition sum is used for calculation of multifractal spectra. It is shown that differently normalized partition sums yield multifractal spectra that are not equivalent. Therefore, their direct comparison is not possible unless convenient transformations are found. This fact is illustrated with conventional and unconventional multifractal spectra of the dyadic Cantor cluster. So far the crucial influence of normalization procedure on multifractal spectra of the box-counting method has not been recognized. To remove this deficiency a generalized partition sum has been proposed. Common partition sums of the box-counting method are included in the generalized one as special normalization events. The multifractal spectra of the generalized partition sum are classified according to the normalization parameters. Transformation relations enabling mutual transitions and correct comparison among the generalized spectra of different normalization are also presented.