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Hausdorff dimension of a Galton–Watson tree



Abstract. Let T be the genealogical tree of a supercritical multitype Galton–Watson process, and let � be the limit set of T, i.e., the set of all infinite self-avoiding paths (called ends) through T that begin at a vertex of the first generation. The limit set � 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 sequence �(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 � ∩ �−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 log 2 α, where α is the Malthusian parameter. 1

Year: 1998
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