×

Random self-similar multifractals. (English) Zbl 0873.28003

This extensive paper starts with technical preparations concerning random self-similar measures \(\Phi\). The setup of the multifractal formalism follows an outline of Falconer. A strong open set condition for the random setting is derived. Under this strong open set condition the authors give a complete description of the behavior of the multifractal spectrum of \(\Phi\) generalizing results of Cawley, Mauldin, Falconer, and Olsen. The authors are also concerned with generalized dimension and the tangential distribution of \(\Phi\).

MSC:

28A80 Fractals
60D05 Geometric probability and stochastic geometry
60G57 Random measures
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arbeiter, Random Recursive Construction of Self-Similar Fractal Measures. The Noncompact Case, Probab. Th. Rel. Fields 88 pp 497– (1990) · Zbl 0723.60040
[2] Bandt , C. 1992
[3] Cawley, Multifractal Decomposition of Moran Fractals, Adv. in Math. 92 pp 196– (1992) · Zbl 0763.58018
[4] Falconer, Cambridge Tracts in Mathematics 85 (1985)
[5] Falconer, Fractal Geometry - Mathematical Foundations and Applications (1990)
[6] Falconer, The Multifractal Spectrum of Statistically Self-Similar Measures, J. Theor. Prob. 7 pp 681– (1994) · Zbl 0805.60034
[7] Geronimo, An Exact Formula for the Measure Dimension Associated with a Class of Picewise Linear Maps, Constr. Approx. 5 pp 89– (1989)
[8] Graf , S. 1993
[9] Hentschel, The Infinite Number of Generalised Dimensions of Fractals and Strange Attractors, Physica 8D pp 435– (1985)
[10] Lau, Mean Quadratic Variations and Fourier Asymptotics of Self Similar Measures, Mh. Math. 115 pp 99– (1993) · Zbl 0778.28005
[11] Olsen, A Multifractal Formalism, Adv. in Math. 116 pp 82– (1995)
[12] Olsen, Random Geometrically Graph Directed Self-Similar Multifractals, Pitman Research Notes in Mathematics Series 307 (1994) · Zbl 0801.28002
[13] Patzschke, Self-Similar Random Measures are Locally Scale Invariant Probab., Th. Rel. Fields 97 pp 559– (1993) · Zbl 0794.60045
[14] Patzschke, Self-Similar Random Measures IV. - The Recursive Construction Model of Falconer, Graf and Mauldin and Williams Math. Nachr. 149 pp 285– (1990) · Zbl 0719.60001 · doi:10.1002/mana.19901490122
[15] Pesin, On Rigorous Mathematical Definitions of Correlation Dimension and Generalized Spectrum for Dimensions., J. Statist. Phys. 71 pp 529– (1993) · Zbl 0916.28006
[16] Riedi , R. 1993
[17] Riedi, An Improved Multifractal Formalism and Self-Similar Measures, J. Math. Anal. Appl. 189 pp 462– (1995) · Zbl 0819.28008
[18] Schief, Separation Properties of Self-Similar Sets, Proc. Amer. Math. Soc. 122 pp 111– (1994) · Zbl 0807.28005 · doi:10.1090/S0002-9939-1994-1191872-1
[19] Strichartz, Self-Similar Measures and Their Fourier Transforms III, Indiana Univ. Math. J. 42 pp 367– (1993) · Zbl 0790.28003
[20] Zähle, Self-Similar Random Measures I. Notion, Carrying Hausdorff Dimension and Hyperbolic Distribution, Probab. Th. Rel. Fields 80 pp 79– (1988) · Zbl 0638.60064
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.