Introduction.- Vectorial multifractal measures and dimensions.- A relative vectorial multifractal formalism of the level sets X (α).- A relative vectorial multifractal formalism of the level sets X (α, β).- Relative multifractal formalism of branching random walk with random metric.- Appendix.

This book introduces a new multifractal vectorial formalism based on Hewitt-Stromberg measures, with particular emphasis on its application to branching random walks on the Galton-Watson tree. This formalism relies on the use of vector-valued functions defined on balls in a metric space and taking values in a Banach space, thus offering a generalization of classical multifractal analysis. These measures lie between Hausdorff and packing measures and then, the authors’ study is specially imported especially when the classical multifractal formalism does not hold. The authors investigate the fractal dimension of the sets of infinite branches of the boundary of a super-critical Galton-Watson tree (endowed with a random metric) along which the averages of a valued branching random walk, have a given set of limit points.

Furthermore, the authors examine additional general sets of levels in multifractal analysis, leading to the development of a relative multifractal vectorial formalism. They explore this relative formalism within the framework of the branching random walk.