Exact Hausdorff measure on the boundary of a Galton--Watson tree

TL;DR

This paper establishes conditions for the existence of an exact Hausdorff measure on the boundary of a Galton--Watson tree and classifies gauge functions, solving Hawkes's conjecture and related problems.

Contribution

It provides a necessary and sufficient condition for the existence of an absolutely continuous exact Hausdorff measure and classifies gauge functions, addressing longstanding conjectures.

Findings

Established a criterion for the existence of an absolutely continuous exact Hausdorff measure.

Classified gauge functions based on their Hausdorff measure of the boundary.

Solved Hawkes's conjecture and determined the local dimension of the branching measure.

Abstract

A necessary and sufficient condition for the almost sure existence of an absolutely continuous (with respect to the branching measure) exact Hausdorff measure on the boundary of a Galton--Watson tree is obtained. In the case where the absolutely continuous exact Hausdorff measure does not exist almost surely, a criterion which classifies gauge functions $\phi$ according to whether $\phi$-Hausdorff measure of the boundary minus a certain exceptional set is zero or infinity is given. Important examples are discussed in four additional theorems. In particular, Hawkes's conjecture in 1981 is solved. Problems of determining the exact local dimension of the branching measure at a typical point of the boundary are also solved.