Multifractal theory within quantum calculus

Abstract

Within framework of the quantum calculus, we represent the partition function and the mass exponent of a multifractal, as well as the average of random variables distributed over self-similar set, on the basis of the deformed expansion in powers of the difference $q-1$. 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 known relation $\tau_q=D_q(q-1)$. We find the physical average related to the escort probability in terms of the deformed expansion as well. It is demonstrated the mass exponent can acquire a singularity that relates to a phase transition of the multifractal set in the course of its deformation.