Local dimensions of the branching measure on a Galton-Watson tree
Let \(\mu =\mu _{\omega}\) be the branching measure on the boundary \(\partial {\mathbf T}\) of a supercritical Galton-Watson tree \({\mathbf T}={\mathbf T}(\omega)\). Denote by \(\overline{d}(\mu, u)\) and \(\underline{d}(\mu, u)\) the upper and lower local dimensions of \(\mu \) at \(u\in \partial {\mathbf T}\), respectively. It is known that \(\overline{d}(\mu, u) = \underline{d}(\mu, u)=\log m\) for \(\mu\)-almost all \(u\in \partial {\mathbf T}\), where \(m=E{\mathbf N}\) is the expexted value of the offspring distribution \({\mathbf N}\). Assuming that \(E{\mathbf N}^{1+\delta}<\infty\) for some \(\delta >0\) the author finds necessary and sufficient conditions when the mentioned result holds for all \(u\in \partial {\mathbf T}\) and proves limit theorems concerning the local dimension of \(\mu\). He also finds the exact local dimensions of \(\mu\) at \(u\in \partial {\mathbf T}\) for \(\mu\)-almost all \(u\).
scientific article; zbMATH DE number 1619322