Multi-multifractal decomposition of digraph recursive fractals

In many situations, both deterministic and probabilistic, one is interested in measure theory in local behaviours, for example in local dimensions, local entropies or local Lyapunov exponents. It has been relevant to study dynamical systems, since the study of multifractal can be further developed for a large class of measures invariant under some map, particularly when there exist strange attractors or repelers (hyperbolic case). Multifractal refers to a notion of size, which emphasizes the local variations of the weight of a measure, of the entropy or the Lyapunov exponents. All these notions are explicited in the case of digraph recursive fractal studied by Edgar & Mauldin where some questions are given. We study the extremal measures and introduce the notion of multi-multifractality which may be useful in problems of rigidity.