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Simultaneous Multifractal Analysis of the Branching and Visibility Measure on a Galton-Watson Tree

Part of: Markov processes Classical measure theory

Published online by Cambridge University Press:  01 July 2016

Adam L. Kinnison  and
Peter Mörters
Adam L. Kinnison*
Affiliation:
University of Bath
Peter Mörters*
Affiliation:
University of Bath
*
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
∗ Postal address: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.
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Abstract

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On the boundary of a Galton-Watson tree we can define the visibility measure by splitting mass equally between the children of each vertex, and the branching measure by splitting unit mass equally between all vertices in the nth generation and then letting n go to infinity. The multifractal structure of each of these measures is well studied. In this paper we address the question of a joint multifractal spectrum, i.e. we ask for the Hausdorff dimension of the boundary points which simultaneously have an unusual local dimension for both these measures. The resulting two-parameter spectrum exhibits a number of surprising new features, among them the emergence of a swallowtail-shaped spectrum for the visibility measure in the presence of a nontrivial condition on the branching measure.

Keywords

Multifractal spectrumtwo-parameter spectrummixed spectrumtwo-dimensional multifractal analysisrandom treeself-similar fractalbranching processpercolation

MSC classification

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 28A80: Fractals
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 42 , Issue 1 , March 2010 , pp. 226 - 245
Copyright
Copyright © Applied Probability Trust 2010 

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