On the vectorial multifractal analysis in a metric space

Multifractal analysis is typically used to describe objects possessing some type of scale invariance. During the last few decades, multifractal analysis has shown results of outstanding significance in theory and applications. In particular, it is widely used to characterize the geometry of the singularity of a measure $\mu$ or to study the time series, which has become an important tool for the study of several natural phenomena. In this paper, we investigate a more general level set studied in multifractal analysis. We use functions defined on balls in a metric space and that are Banach valued which is more general than measures used in the classical multifractal analysis. This is done by investigating Peyrière's multifractal Hausdorff and packing measures to study a relative vectorial multifractal formalism. This leads to results on the simultaneous behavior of possibly many branching random walks or many local Hölder exponents. As an application, we study the relative multifractal binomial measure in symbolic space $\mathrm{\partial }\mathcal{A}$.