An Extension of Hawkes' Theorem on the Hausdorff Dimension of a Galton-watson Tree

Let T be the genealogical tree of a supercritical multitype Galton–Watson process, and let 3 be the limit set of T, i.e., the set of all infinite self-avoiding paths (called ends) throughT that begin at a vertex of the first generation. The limit set3 is endowed with the metric d(ζ, ξ) = 2−n where n = n(ζ, ξ) is the index of the first generation where ζ and ξ differ. To each end ζ is associated the infinite sequence8(ζ) of types of the vertices of ζ . Let  be the space of all such sequences. For any ergodic, shift-invariant probability measure μ on , define μ to be the set of all μ-generic sequences, i.e., the set of all sequences ω ∈  such that each finite sequence v occurs in ω with limiting frequency μ((v)), where (v) is the set of all ω′ ∈  that begin with the word v. Then the Hausdorff dimension of 3 ∩8−1(μ) in the metric d is (h(μ)+ ∫  log q(ω0, ω1)dμ(ω))+/ log 2 , almost surely on the event of nonextinction, where h(μ) is the entropy of the measure μ and q(i, j) is the mean number of type-j offspring of a type-i individual. This extends a theorem of Hawkes [5], which shows that the Hausdorff dimension of the entire boundary at infinity is log2 α, where α is the Malthusian parameter.