The origin of an increasing or decreasing multifractal spectrum

The multifractal dimensionality Dq as a function of q expresses the distribution of measure over space. When all the moments scale with resolution in exactly the same way, we have a flat spectrum, and a single monofractal dimensionality. We argue that for multifractal spectra the scaling of the moments of the measure distribution should be considered, rather than the usual definition, which considers only the limit at high resolution. A difference in scaling of the zeroth- and second-order moment must lead to zero variance at some resolution. At that point the scaling must stop, as the variance cannot become negative. For discrete systems that happens at the particle level, with the scaling region above and D2>D0, or, in general, with an increasing spectrum. The variance increases with increasing box size. If, on the other hand, the variance of the distribution decreases with increasing box size the spectrum Dq is the standard non-increasing function of q. Then scaling must stop at the level where homogeneity is reached. We present a string of bits, either unity or zero, which has uniform scaling properties over many levels, with the number of unity bits in a substring taken as the measure. For substrings of size up to 2n the inhomogeneity of the measure increases monotonously, while for larger substrings it decreases again. Thus, we have a multifractal set, which at small sizes has an increasing spectrum, while for large size the scaling is identical to that of a bimodal Cantor set, with a decreasing spectrum.