Multifractal spectra of branching measure on a Galton-Watson tree

Abstract

IfZis the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|nat each generationn. We use the natural metric ρ(ξ,η) = e−n, wheren= max{k: ξ|k= η|k}, and observe that the local dimension index isd(μ,ξ) = limn→∞log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) whenp1= P{Z= 1} > 0 andZhas finite moments of all positive orders. Because ̱d(μ,ξ) =afor all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.