Local dimensions of the branching measure on a Galton–Watson tree

Abstract

Let μ= μ ω be the branching measure on the boundary ∂ T of a supercritical Galton–Watson tree T = T (ω) . Denote by d(μ,u) and d(μ,u) the lower and upper local dimensions of μ at u∈∂ T . It is well known that almost surely, d(μ,u)= d(μ,u)= logm for μ-almost all u∈∂ T , where m is the expected value of the offspring distribution. Here we find exactly when the result holds for all u∈∂ T , and obtain some limit theorems about the uniform local dimensions of μ. We also find the exact local dimension of μ at u∈∂ T for μ-almost all u.