The Annals of Probability

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

Toshiro Watanabe

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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 ϕ according to whether ϕ-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.

Article information

Source
Ann. Probab., Volume 35, Number 3 (2007), 1007-1038.

Dates
First available in Project Euclid: 10 May 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1178804321

Digital Object Identifier
doi:10.1214/009117906000000629

Mathematical Reviews number (MathSciNet)
MR2319714

Zentralblatt MATH identifier
1127.60083

Subjects
Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A78: Hausdorff and packing measures
Secondary: 60G18: Self-similar processes 28A80: Fractals [See also 37Fxx]

Keywords
Galton–Watson tree exact Hausdorff measure shift self-similar additive random sequence boundary branching measure dominated variation b-decomposable distribution

Citation

Watanabe, Toshiro. Exact Hausdorff measure on the boundary of a Galton–Watson tree. Ann. Probab. 35 (2007), no. 3, 1007--1038. doi:10.1214/009117906000000629. https://projecteuclid.org/euclid.aop/1178804321


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