**1. Introduction**

In the last two decades, multifractal properties have been the subject of intense research in very different areas of science [1–13]. The fashion for searching for new areas of multifractality is still ongoing. The shape, location, and spread of the spectrum of dimensions (singularities)—the leading multifractality indicator—provide invaluable information about the layout. We use the formalism [14] that describes not only systems in the state of statistical equilibrium but also stationary states. Furthermore, we indicate that formalism can easily be extended to transient states.

Our approach is complementary to the commonly used multifractal detrended fluctuation analysis (MF-DFA) [1,2]. More precisely, in the presence of state intervention, our concept of using (normalized) market shares for multifractal analysis of the market of competing firms is new. It starts with a partition function expressed directly by shares. Thanks to this, it bypasses the onerous preparation of traditional MF-DFA, based on a fluctuation function built with the help of time series.

We demonstrate how our method works with the example of a competing company market model published previously [15]. In this model, we assume that companies can merge, create spin-offs, and go bankrupt in the presence of state intervention. This tendency for firms to disappear from the market can counterbalance the tendency to design firms, leading to critical phenomena. We examined these phenomena in our previous work [15]. In this work, we explore a different aspect of the market model of competing companies, namely, multifractality.

Moreover, we show that the actual market of *S*&*P* 500 companies is multifractal. Finally, we indicate that this market can be (roughly) described by the multifractal formalism, in which companies are divided into four groups differing significantly in market shares.

The paper consists of two parts. The first part consists of Section 1 (Introduction) together with Section 2 (Theory), which on the example of our critical company market model [15] presents the multifractal approach. The second part presents this multifractal

**Citation:** Chorowski, M.; Kutner, R. Multifractal Company Market: An Application to the Stock Market Indices. *Entropy* **2022**, *24*, 130. https://doi.org/10.3390/e24010130

Academic Editor: Stanisław Drozd˙ z˙

Received: 30 December 2021 Accepted: 12 January 2022 Published: 16 January 2022

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approach to the real market of the *S*&*P* 500 index. Moreover, this part compares the obtained results for the actual market with the four-group market model.
