Thin and thick points for branching measure on a Galton-Watson tree

Suppose that [mu] is the branching measure on the boundary of a supercritical Galton-Watson tree with offspring distribution N such that E[N log+ N]<[infinity]. We determine the dimension spectrum of thin points under the condition P{N[less-than-or-equals, slant]1}=0 and the dimension spectrum of thick points under the condition that either 0<sup{t: E[exp(tN)]<[infinity]}<[infinity] or ess sup N<[infinity]. Our main tool for obtaining lower bounds is to study percolation on the Galton-Watson tree.