Generalization of multifractal theory within quantum calculus

TL;DR

This paper extends multifractal theory using quantum calculus, introducing generalized partition functions and mass exponents that incorporate deformations and Tsallis entropies, with applications to various self-similar sets.

Contribution

It provides a novel generalization of multifractal formalism through quantum calculus, connecting deformations with multifractal measures and entropies.

Findings

Generalized partition function determined by binomial-type Tsallis entropies

Mass exponent expansion extends known relations to deformed cases

Applications to Cantor set, currency series, and porous surfaces

Abstract

On the basis of the deformed series in quantum calculus, we generalize the partition function and the mass exponent of a multifractal, as well as the average of a random variable distributed over self-similar set. For the partition function, such expansion is shown to be determined by binomial-type combinations of the Tsallis entropies related to manifold deformations, while the mass exponent expansion generalizes the known relation $\tau_q=D_q(q-1)$. We find equation for set of averages related to ordinary, escort, and generalized probabilities in terms of the deformed expansion as well. Multifractals related to the Cantor binomial set, exchange currency series, and porous surface condensates are considered as examples. Keywords:Multifractal set; Deformation; Power series.