Relative multifractal box-dimensions

Abstract

Given two probability measures $\mu$ and $\nu$ on $\mathbb{R}^n$, $E \subset \mathbb{R}^n$ and $q\in \mathbb{R}$, we define in two different ways the lower (resp. the upper ) relative multifractal $q$-box dimension $\underline{C}_{\mu,\nu}^q(E)$ and $\underline{L}_{\mu,\nu}^q(E) $ (resp. $\overline{C}_{\mu,\nu}^q(E)$ and $\overline{L}_{\mu,\nu}^q(E)$ ). Some relationship between these quantities are given, especially when all this quantities are equal to $\Delta_{\mu,\nu}^q(E)$ : the relative multifractal pre-packing
dimension. We also, calculate the relative multifractal spectrum and prove the validity of multifractal formalism. As an application, we study the behavior of projections of measures obeying to the relative multifractal formalism