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Multifractal dimensions of product measures

Published online by Cambridge University Press:  24 October 2008

L. Olsen
L. Olsen
Affiliation:
University of St. Andrews, Department of Mathematics, North Haugh, St. Andrews, Fife KY16 9SS, Scotland e-mail: [email protected]
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Abstract

We study the multifractal structure of product measures. for a Borel probability measure μ and q, t Є , let and denote the multifractal Hausdorff measure and the multifractal packing measure introduced in [O11] Let μ be a Borel probability merasure on k and let v be a Borel probability measure on t. Fix q, s, t Є . We prove that there exists a number c > 0 such that for E ⊆k, F ⊆l and Hk+l provided that μ and ν satisfy the so-called Federer condition.

Using these inequalities we give upper and lower bounds for the multifractal spectrum of μ × ν in terms of the multifractal spectra of μ and ν

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 120 , Issue 4 , November 1996 , pp. 709 - 734
Copyright
Copyright © Cambridge Philosophical Society 1996

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References

REFERENCES

[AP] Aebeiter, M. and Patzschke, N.. Random self-similar multifractals, preprint, 1994.Google Scholar
[BM] Besicovitch, A. S. and Moran, P. A. P.. The measure of product and cylinder sets. J. Land. Math. Soc. 20 (1945), 110–120.Google Scholar
[CM] Cawley, R. and Mauldin, R. D.. Multifractal decomposition of Moran fractals. Advances in Mathematics 92 (1992), 196–236.CrossRefGoogle Scholar
[EM] Edgar, G. A. and Mauldin, R. D.. Multifractal decompositions of digraph recursive fractals. Proc. London Math. Soc. 65 (1992), 604–628.CrossRefGoogle Scholar
[EF] Ernst, L. R. and Freilich, G.. A Hausdorff measure inequality. Trans. Amer. Math. Soc. 219 (1976), 361–368.CrossRefGoogle Scholar
[Fa1] Falconer, K. J.. The geometry of fractal sets (Cambridge University Press, 1985).CrossRefGoogle Scholar
[Fa2] Falconer, K. J.. Fractal geometry-mathematical foundations and applications (John Wiley & Sons, 1990).CrossRefGoogle Scholar
[Hal] Haase, H.. Dimension of measures. Acta Universitatis Carolinae – Mathematica et Physica 31 (1990), 29–34.Google Scholar
[Ha2] Haase, H.. On the dimension of product measures. Mathematika 37 (1990), 316–323.CrossRefGoogle Scholar
[HJKPS] Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I. and Shraiman, B. J.. Fractal measures and their singularities: the characterization of strange sets. Phys. Rev. A 33 (1986), 1141–1151.CrossRefGoogle ScholarPubMed
[Ho] Howroyd, J.. On Hausdorff and packing dimension of product spaces. Math. Proc. Camb. Phil. Soc. 119 (1996), 715–727.CrossRefGoogle Scholar
[HT] Hu, X. and Taylor, S. J.. Fractal properties of products and projections of measures in d. Math. Proc. Camb. Phil. Soc. 115 (1994), 527–544.CrossRefGoogle Scholar
[Hu] Hutchinson, J.. Fractals and self-similarity. Indiana Univ. Math. J. 30 (1981), 713–747.CrossRefGoogle Scholar
[Ki] King, J.. The singularity spectrum for general Sierpinski carpets, preprint, 1992.Google Scholar
[KG] King, J. and Geronimo, J. S.. Singularity spectrum for recurrent IFS attractors. Nonlinearity 6 (1992), 337–348.CrossRefGoogle Scholar
[Lo1] Lopes, A. O.. The dimension spectrum of the maximal measure. SIAM jour. Math. Anal. 20 (1989), 1243–1254.CrossRefGoogle Scholar
[Lo2] Lopes, A. O.. Dimension spectra and a mathematical model for phase transition. Adv. Appl. Math. 1 (1990), 475–502.CrossRefGoogle Scholar
[LN] Lau, K.-S. and Ngai, S.-M.. Multifractal measures and a weak separation condition, preprint, 1994.Google Scholar
[Ma] Marstrand, J. M.. The dimension of Cartesian product sets. Proc. Land. Math. Soc. 50 (1954), 198–206.Google Scholar
[MR] Mandelbrot, B. and Riedi, R.. Multifractal formalism for infinite multinomial measures, preprint, 1994.Google Scholar
[Oll] Olsen, L.. A multifractal formalism. Advances in Mathematics, 116 (1995), 82–196.CrossRefGoogle Scholar
[O12] Olsen, L.. Random geometrically graph directed self-similar multifractals. Pitman Research Notes in Mathematics Series, Vol. 307 (Longman, 1994).Google Scholar
[O13] Olsen, L.. Self-affine multifractal Sierpinski sponges in d, preprint, 1994.Google Scholar
[Pe] Peyriére, J.. Multifractal measures. Proceedings of the NATO Advanced Study Institute on Probabilistic and Stochastic Methods in Analysis with Applications, II Ciocco, pp. 175–186, NATO ASI Series, Series C: Mathematical and Physical Sciences, Vol. 372 (Kluwer Academic Press, 1992).Google Scholar
[Ra] Rand, D.. The singularity spectrum f(α) for cookie-cutters. Ergodic Theory and Dynamical Systems 9 (1989), 527–541.CrossRefGoogle Scholar
[RT] Raymond, X. S. and Tricot, C.. Packing regularity of sets in n-space. Math. Proc. Camb. Phil. Soc. 103 (1988), 133–145.CrossRefGoogle Scholar
[Roc] Rockafellar, R. T.. Convex analysis (Princeton University Press, 1970).CrossRefGoogle Scholar
[Rog] Rogers, C. A.. Hausdorff measures (Cambridge University Press, 1970).Google Scholar
[SS] Schmeling, J. and Siegmund-Schultze, R.. The singularity spectrum of self-affine fractals with a Bernoulli measure, preprint, 1992.Google Scholar
[TT] Taylor, S. J. and Tricot, C.. Packing measure, and its evaluation for a Brownian path. Trans. Amer. Math. Soc. 288 (1985), 679–699.CrossRefGoogle Scholar
[Tr] Tricot, C.. Two definitions of fractional dimension. Math. Proc. Camb. Phil. Soc. 91 (1982), 57–74.CrossRefGoogle Scholar