GALTON-WATSON TREES WITH THE SAME MEAN HAVE THE SAME POLAR SETS

Evans defined a notion of what it means for a set B to be polar for a process indexed by a tree. The main result herein is that a tree picke d from a Galton-Watson measure whose offspring distribution has mean m and finite variance will almost surely have precisely the same polar sets as a deterministic tree of the same growth rate. This implies tha t deterministic and nondeterministic trees behave identically in a var iety of probability models. Mapping subsets of Euclidean space to tree s and polar sets to capacity criteria, it follows that certain random Canter sets are capacity-equivalent to each other and to deterministic Canter sets. An extension to branching processes in varying environme nt is also obtained.