Exploring Multifractal Moment Measures and Scaling Functions in Moran Structures

The Lq-spectrum of a Borel measure is a fundamental concept in multifractal analysis. It is widely recognized that the Lq-spectrum associated with a fractal measure provides significant insights into its underlying dynamics and geometry. Consequently, the study of the Lq-spectrum is crucial for understanding dynamical systems and fractal measures. Our objective in this paper is to determine the exact rate of convergence of the Lq-spectra for Moran measures satisfying the Set Strong Separation Condition. As an application, we demonstrate that the empirical multifractal moment measures converge weakly to the normalized multifractal measures. Finally, we reexamine the analysis using tube formulas, and we try to show that the multifractal and fractal dimensions of the overlaps in a Moran set satisfying the Strong Open Set Condition are strictly smaller than the dimension of the set itself.