Exploring multifractal moment measures and scaling functions in Moran structures
Abstract
The $L^q$-spectrum of a Borel measure is a fundamental concept in multifractal analysis. It is widely recognized that the $L^q$-spectrum associated with a fractal measure provides significant insights into its underlying dynamics and geometry. Consequently, the study of the $L^q$-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 $L^q$-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.
