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Multifractal spectra of branching measure on a Galton-Watson tree

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  14 July 2016

Narn-Rueih Shieh  and
S. James Taylor
Narn-Rueih Shieh*
Affiliation:
National Taiwan University
S. James Taylor*
Affiliation:
University of Sussex
*
∗ Postal address: Department of Mathematics, National Taiwan University, Taipei, Taiwan. Email address: [email protected]
∗∗ Postal address: School of Mathematics, University of Sussex, Falmer, Brighton, UK.
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Abstract

If Z is the branching mechanism for a supercritical Galton-Watson tree with a single progenitor and E[ZlogZ] < ∞, then there is a branching measure μ defined on ∂Γ, the set of all paths ξ which have a unique node ξ|n at each generation n. We use the natural metric ρ(ξ,η) = e−n, where n = max{k : ξ|k = η|k}, and observe that the local dimension index is d(μ,ξ) = limn→∞ log(μB(ξ|n))/(-n) = α = logm, for μ-almost every ξ. Our objective is to consider the exceptional points where the above display may fail. There is a nontrivial ‘thin’ spectrum for ̄d(μ,ξ) when p1 = P{Z = 1} > 0 and Z has finite moments of all positive orders. Because ̱d(μ,ξ) = a for all ξ, we obtain a ‘thick’ spectrum by introducing the ‘right’ power of a logarithm. In both cases, we find the Hausdorff dimension of the exceptional sets.

Keywords

Branching measureGalton-Watson treeHausdorff dimensionpacking dimensionmultifractal spectrum

MSC classification

Primary: 60G17: Sample path properties 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Papers
Information
Journal of Applied Probability , Volume 39 , Issue 1 , March 2002 , pp. 100 - 111
Copyright
Copyright © Applied Probability Trust 2002 

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