An extension of Hawkes' theorem on the Hausdorff dimension of a Galton-Watson tree
The genealogical tree of a supercritical multi-type Galton-Watson branching process (simple Galton-Watson process assigned labels from a finite set \(\mathcal L\)) is considered. There is defined the limit set \(\Lambda\) of the simple Galton-Watson process as the set of all infinite descent lines, i.e., the set of infinite sequences \(\xi =\xi_1 \xi_2 \ldots\) of individuals such that each \(\xi_{n+1}\) is an offspring of \(\xi_n\), and such that \(\xi_1\) is a member of the first generation. The limit set \(\Lambda\) is endowed with the metric \(d(\zeta,\xi)=2^{-n}\) where \(n=n(\zeta,\xi)\) is the index of the first generation where \(\zeta\) and \(\xi\) differ. For each \(\xi\in\Lambda\), its pedigree \(\Phi (\xi)= l_1l_2\ldots \in\Omega=\mathcal L^{\mathcal N}\) (labels assigned to the individuals \(\xi_1, \xi_2, \ldots\)) is defined. A subset \(B\) of \(\Omega\) is called non-polar if there is a positive probability such that \(B\cap\Phi(\Lambda)\neq\emptyset\). Let \(\mu\) be the ergodic, shift-invariant probability measure on \(\Omega\), let \(\Omega_\mu\) be the set of all \(\mu\)-generic sequences, let \(q(i,j)\) be the mean number of type-\(j\) offsprings produced by a type-\(i\) individual, and let \(Q\) be the matrix of means. Define \(h(\mu)\) to be the entropy of the measure \(\mu\), and \(c(\mu)=h(\mu)+\int_\Omega \log q (\omega_0,\omega_1)d\mu (\omega)\). The following result is proved: Let all the \(q(i,j)\) be finite and let the matrix \(Q\) be irreducible. If \(c(\mu)<0\), then the set \(\mathcal A(\mu)=\Lambda\cap\Phi^{-1}(\Omega_\mu)\) is almost surely empty. If \(c(\mu)\geq 0\), then the Hausdorff dimension of \(\mathcal A(\mu)\) in the metric \(d\) is \(c(\mu)/\log 2\). This extends a theorem of \textit{J. Hawkes} [J. Lond. Math. Soc., II. Ser. 24, 373-384 (1981; Zbl 0468.60081)], which shows that for one-type Galton-Watson process with Malthusian parameter \(\alpha >1\) the Hausdorff dimension is \(\log_2\alpha\).
supercritical multi-type Galton-Watson branching processes
scientific article; zbMATH DE number 1406473