Multifractal spectra for random self-similar measures via branching processes

• Published in: Advances in applied probability 2011-03, Vol.43 (1), p.1-39
• Main Authors: Biggins, J. D., Hambly, B. M., Jones, O. D.
• Format: Article
• Language: English
• Subjects: 28A80; 60G18; 60J80; Average linear density; Convexity; Dealing; Division; Fractal analysis; Fractals; general branching process; Geometry; Hausdorff dimension; Hausdorff dimensions; local dimension; Martingales; Mathematical analysis; multifractal spectrum; Null set; Power laws; Probability; Random numbers; Random variables; Random walk; Reproduction; Self-similar measure; Self-similarity; Spectra; Stochastic Geometry and Statistical Applications; Studies; Unions
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1239/aap/1300198510

Start with a compact set K ⊂ R d . 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 R d 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).