*3.3. Conclusions*

It is worth realizing how distributions induce common multifractal structures. Therefore, it is not so much about searching for such structures, but about the possibility of comparing them with each other, i.e., answering the question of which structures are more multifractal and which are less. For this, they must first be classified according to their symmetry and degeneration. The larger the logarithm of these steps, the higher these elevations are.

The degree of asymmetry in the multifractal structure is determined by the *γ* asymmetry coefficient. If *γ* = 1, we have a symmetric multifractal structure. If *γ* > 1, we have left asymmetry, while for *γ* < 1, we have right asymmetry.

The degree of degeneration of the marginal shares determines the elevation of the edges of the spectral dimensions: the left one depends on the degree of degeneration of the maximum share, and the right one depends on the degree of degeneration of the minimum share.

In this way, we have divided multifractal structures into nine groups, where both asymmetries and degenerations match themselves like the symmetry of the left and right hands (see Figure 16 for illustration, there, for example, the first plot in the first column and the last plot in the third column). Only within each group can we introduce a measure that allows us to organize the multifractal structure. The above classification is possible due to the fact that asymmetry and degeneration are independent of each other.

**Figure 16.** Schematic classification of spectrum of dimensions due to asymmetry *γ* and degeneration (*<sup>M</sup>*, *<sup>L</sup>*).

Suppose two multifractal structures have the same span of the spectrum of dimensions and location. One is more multifractal than the other if its degeneration levels are less than the corresponding other.

Another special case is when both multifractal structures' degeneracy levels are equal, while the structures differ in span. Then the more multifractal structure is for, the larger span structure plus *f* 1.

We introduce a precise definition of the linear multifractal capacity, M, utilizing a definition based on Figure 15 and Equation (24),

$$
\mathcal{M} = \Delta \mathfrak{x} + f^1 + M^{-1} + L^{-1}.\tag{29}
$$

Notably, there is no differentiation of multifractality due to location *α*0. The proposed phenomenological measure of multifractal capacity, M, is a partial in the sense that it does not take into account the entire fine structure of the spectrum of dimension *f* .

In conclusion, in this paper, we examine the multifractality/multiscaling coming from shares and not from correlations. In this sense, this work is complementary to our previous one [15]. As a reference case, we have discussed the instructive example of the four-group company market. We have shown that (within the zero-order approximation) each market can be reduced to a four-group company market, which should facilitate market analysis.

Finally, we can say that this is the first time such a multifractal analysis of the market of competing companies has been performed.

Notably, we can apply the approach to any series of shares, e.g., shares of turnover volumes on the stock exchange and shares of companies' quotations on the stock exchange. In short, the approach can be applied to any normalized series of positively defined elements. Moreover, our approach makes it possible to examine the evolution of multifractality of company market especially in the vicinity of crash regions. That is why it is so important to study in the near future the relationship between multifractality and criticality suggested by Figure 7.

**Author Contributions:** Conceptualization, R.K.; data curation, M.C.; formal analysis, M.C.; methodology, R.K.; resources, M.C.; software, M.C.; supervision, R.K.; validation, M.C.; visualization, M.C.; writing—original draft, R.K.; writing—review & editing, R.K. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Data available in a publicly accessible repository. The data presented in this study are openly available in "*S*&*P* 500 Companies by Weight" at https://www.slickcharts. com/sp500.

**Conflicts of Interest:** The authors declare no conflict of interest.
