On the Kolmogorov-like generalization of Tsallis entropy, correlation entropies and multifractal analysis

The generalization à la Kolmogorov of Tsallis entropy, introduced by the authors in a previous work [J. Math. Phys. 37, 4480 (1996)], is revisited. Invariance properties are pointed out under weaker conditions than before. This result leads us to wonder if Tsallis entropy at the Kolmogorov abstraction level brings new information with respect to the generalization that Kolmogorov did of Shannon entropy. The negative answer motivates us to look for other generalizations of Tsallis entropy in order to avoid the lack of new information. Correlation entropies seem to be good candidates for this purpose. The relationship of this kind of entropy with the multifractal analysis is studied with the help of the thermodynamic formalism. We also outline its usefulness to generalize properties of Tsallis entropy.