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Multifractal spectra for random self-similar measures via branching processes

Part of: Classical measure theory Markov processes Stochastic processes

Published online by Cambridge University Press:  01 July 2016

J. D. Biggins ,
B. M. Hambly  and
O. D. Jones
J. D. Biggins*
Affiliation:
University of Sheffield
B. M. Hambly*
Affiliation:
University of Oxford
O. D. Jones*
Affiliation:
University of Melbourne
*
∗ Postal address: School of Mathematics and Statistics, University of Sheffield, Sheffield S3 7RH, UK. Email address: j.biggins@sheffield.ac.uk
∗∗ Postal address: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, UK.
∗∗∗ Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville, VIC 3010, Australia.
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Abstract

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Start with a compact set K ⊂ Rd. This has a random number of daughter sets, each of which is a (rotated and scaled) copy of K and all of which are inside K. The random mechanism for producing daughter sets is used independently on each of the daughter sets to produce the second generation of sets, and so on, repeatedly. The random fractal set F is the limit, as n goes to ∞, of the union of the nth generation sets. In addition, K has a (suitable, random) mass which is divided randomly between the daughter sets, and this random division of mass is also repeated independently, indefinitely. This division of mass will correspond to a random self-similar measure on F. The multifractal spectrum of this measure is studied here. Our main contributions are dealing with the geometry of realisations in Rd and drawing systematically on known results for general branching processes. In this way we generalise considerably the results of Arbeiter and Patzschke (1996) and Patzschke (1997).

Keywords

Self-similar measureHausdorff dimensiongeneral branching processmultifractal spectrumlocal dimension

MSC classification

Primary: 28A80: Fractals 60J80: Branching processes (Galton-Watson, birth-and-death, etc.) 60G18: Self-similar processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 43 , Issue 1 , March 2011 , pp. 1 - 39
Copyright
Copyright © Applied Probability Trust 2011 

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