Note on the Multifractal Formalism of Covering Number on the Galton-Watson Tree

Abstract

We consider, for t in the boundary of Galton-Watson tree (∂T), the covering number Nn(t) by cylinder of generation n. For a suitable set I and a sequence (sn,γ), we establish almost surely, and uniformly on γ, the Hausdorff and packing dimensions of the set {t ∈ ∂T : Nn(t) − nb ∼ sn,γ} for b ∈ I.