March 2011 Multifractal spectra for random self-similar measures via branching processes
J. D. Biggins, B. M. Hambly, O. D. Jones
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Adv. in Appl. Probab. 43(1): 1-39 (March 2011). DOI: 10.1239/aap/1300198510


Start with a compact set KRd. 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).


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J. D. Biggins. B. M. Hambly. O. D. Jones. "Multifractal spectra for random self-similar measures via branching processes." Adv. in Appl. Probab. 43 (1) 1 - 39, March 2011.


Published: March 2011
First available in Project Euclid: 15 March 2011

zbMATH: 1223.28010
MathSciNet: MR2761142
Digital Object Identifier: 10.1239/aap/1300198510

Primary: 28A80 , 60G18 , 60J80

Keywords: general branching process , Hausdorff dimension , local dimension , multifractal spectrum , Self-similar measure

Rights: Copyright © 2011 Applied Probability Trust


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Vol.43 • No. 1 • March 2011
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