The branching measure, Hausdorff and packing measures on the Galton-Watson tree

Abstract

We present some recent results concerning the branching measure, the exact Husdorff measure and the exact packing measure, defined on the boundary of the Caalton-Watson tree. The results show that in good cases, these three measures coincide each other up to a constant, that the branching measure is homogeneous (it has the same local dimension at each point) if and only if a certain simple condition is satisfied, and that it is singular with respect to the equally splitting measure. Similar results on marked trees are also presented, and are applied to the study of flows in networks and to the search of exact gauges for statistically self-similar fractals in ℝn.