An extension of Hawkes' theorem on the Hausdorff dimension of a Galton-Watson tree

Let J be the genealogical tree of a supercritical multitype Galton-Watson p rocess, and let Lambda be the limit set of J, i.e., the set of all infinite self-avoiding paths (called ends) through J that begin at a vertex of the first generation. The limit set A is endowed with the metric d (zeta, xi) = 2(-n) where n = n (zeta, xi) is the index of the first generation where ze ta and xi differ. To each end zeta is associated the infinite sequence Phi (zeta) of types of the vertices of zeta. Let Omega be the space of all such sequences. For any ergodic, shift-invariant probability measure mu on Omeg a, define Omega(mu) to be the set of all mu-generic sequences, i.e., the se t of all sequences omega epsilon Omega such that each finite sequence v occ urs in omega with limiting frequency mu(Omega(v)), where Omega(v) is the se t of all omega' epsilon Omega that begin with the word v. Then the Hausdorf f dimension of Lambda boolean AND Phi(-1) (Omega(mu)) in the metric d is (h (mu) + integral(Omega) log q (omega(0), omega(1)) d mu (omega)) (+) / lo g 2 , almost surely on the event of nonextinction, where h (mu) is the entropy of the measure mu and q (i, j) is the mean number of type-j offspring of a ty pe-i individual. This extends a theorem of HAWKES [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log(2) alpha whe re alpha is the Malthusian parameter.