Advertisement

A Class of Multinomial Multifractal Measures with Negative (Latent) Values for the “Dimension” f(α)

  • Benoit B. Mandelbrot
Chapter
Part of the Ettore Majorana International Science Series book series (EMISS)

Abstract

As is well known, fractals are sets of points that possess is the property of being invariant by dilation. When a fractal set is exactly self-similar, or is self-similar in a statistical sense, a central role is played by a positive quantity called fractal dimension, which need not be an integer, and which generalizes the “ordinary” dimension.

In this paper, a further generalization of dimension is introduced and motivated. When its value is positive, it effectively falls back on known definitions of fractal dimension. But its motivating virtue is that its value can be negative, in which case it quantifies and measures usefully the loose idea of “degree of emptiness” of an empty set.

Self-similar multifractals are also geometric objects invariant by dilation, but they are not sets. They are measures, a notion well illustrated by the concrete distributions of probability, of mass or of turbulent dissipation. They are described by a function f(α).

This paper shows that negative dimensions are needed to investigate the statistical properties of certain random self-similar multifractals, namely those for which f(α) < 0 for some α’s, called latent (= present, but hidden). The positive f(α)’s define a “typical” distribution for the measure, while the negative f(α)’s rule the variability between different samples from the same ensemble or population. Moments whose order lies beyond certain thresholds q* and q*min are called latent. Their sample values are extremely sample dependent. Negative dimensions are best investigated using “supersamples.”

In addition to negative f’s, the multifractals investigated here involve a critical exponent q bottom < 0, such that population moments of exponent q less than q bottom are infinite. The corresponding sample moments are extremely “ill-behaved.”

The “ordinary” multifractals, whose simplest example is the binomial, are called “manifest” by the author. Compared to them, the multifractals studied in this paper exhibit two “anomalies” that play a central role in the approach to multifractals the author has been developing since 1968. Their study requires nothing but elementary manipulations, making them the simplest examples of the anomalies in question.

Currently, the most active applications of negative dimensions are to the distribution of the dissipation of turbulence and to the distribution of the hitting probability along the boundary of a DLA cluster, which is an example of harmonic measure.

Keywords

Fractal Dimension Harmonic Measure Sample Variability Negative Dimension Cascade Stage 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Blumenfeld, R. and Aharony, A. 1989. Breakdown of multifractal behavior in diffusion limited aggregates. Phys. Rev. Lett. 62, 2977–2980.PubMedCrossRefGoogle Scholar
  2. Bramson, M. D. 1978. Maximal displacement of branching Brownian motions. Communication of Pure and Applied Mathematics 31, 531–581.CrossRefGoogle Scholar
  3. Bramson, M. D. 1983. Convergence of solutions of the Kolmogorov equation to travelling waves. Memoirs of the American Mathematical Society 44, No. 285.Google Scholar
  4. Gates M.E. and Deutsch, J.M. 1987. Spatial correlations in multifractals. Phys. Rev. A 35, 4907.CrossRefGoogle Scholar
  5. Chhabra, A., and Jensen, R.V. 1989. Direct determination of the f(α) singularity spectrum. Phys. Rev. Lett. 62, 1327.PubMedCrossRefGoogle Scholar
  6. Deutschel, J.D. and Stroock D.W. 1989. Large Deviations. New York: Academic Press.Google Scholar
  7. Fourcade, B., Breton, P. and Tremblay, A.-M.S. 1987. Multifractals and critical phenomena in percolating networks: Fixed point, gap scaling and universality. Phys. Rev. B 36, 8925.CrossRefGoogle Scholar
  8. Fourcade, B. and Tremblay, A.-M.S. 1987. Anomalies in the multifractal analysis of self-similar resistor networks. Phys. Rev. A 36, 2352.PubMedCrossRefGoogle Scholar
  9. Frisch, U. and Parisi, G. 1985. Fully developed turbulence and intermittency in Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, International School of Physics “Enrico Fermi,” Course 88, edited by M. Ghil. North-Holland, Amsterdam, p 84.Google Scholar
  10. Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I. and Shraiman, B.I. 1986. Fractal measure and their singularities: The characterization of strange sets. Phys. Rev. A 33, 1141.PubMedCrossRefGoogle Scholar
  11. Hentschel, H.G.E. and Procaccia, I. 1983. The infinite number of generalized dimensions of fractals and strang attractors. Physica (Utrecht) 8D, 435.Google Scholar
  12. Mandelbrot, B.B. 1974. Intermittent turbulence in self-similar cascades; divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331; also Comptes Rendus 278A, 289, 355.CrossRefGoogle Scholar
  13. Mandelbrot, B.B. 1982. The Fractal Geometry of Nature. New York: W.H. Freeman.Google Scholar
  14. Mandelbrot, B.B. 1984. Fractals in physics: squig clusters, diffusions, fractal measures and the unicity of fractal dimension. J. Stat. Phys. 34, 895.CrossRefGoogle Scholar
  15. Mandelbrot, B.B. 1988 An introduction to multifractal distribution functions, in Fluctuations and Pattern Formation (Cargèse, 1988). H.E. Stanley and N. Ostrowsky, Dordrecht-Boston: Kluwer, 1988, 345–360. NOTE: This paper is entirely superseded by Mandelbrot 1989b. In particular, the treatment of Section 3.5 is impossibly hasty and contains an error; most of its material is presented better and more accurately in Mandelbrot 1989b and the present text.Google Scholar
  16. Mandelbrot, B.B. 1989a. Multifractal measures, especially for the geophysicist. Pure and Applied Geophysics 131, nos. 1/2. NOTE: The treatment of αH in Section 8 of this paper is in error, and Figure 5 is incomplete. Updated presentations are given in Mandelbrot 1989b, which the reader is advised to study before Section 8 of Mandelbrot 1989a.Google Scholar
  17. Mandelbrot, B.B. 1989b. Limit lognormal multifractal measures, in: Frontiers of Physics: Landau Memorial conference (Tel Aviv, 1988). Edited by Errol Gotsman, New York: Pergamon.Google Scholar
  18. Mandelbrot, B.B. 1989c. Negative fractal dimensions and multifractals, in: Proceedings of Stat Phys 17. Edited by Constantino Tsallis. Physica A.Google Scholar
  19. Mandelbrot, B.B. (forthcoming). Fractals and Multifractals: Noise, Turbulence and Galaxies (Selecta, Vol. 1). New York: Springer.Google Scholar
  20. McKean, H. P. 1975. Application of Brownian motion to the equation of Kolmogorov-Petrovskii-Piskunov. Communication of Pure and Applied Mathematics, 28, 323Google Scholar
  21. Meneveau, C. and Sreenivasan, K.R. 1987. Simple multifractal cascade model for fully developed turbulence. Phys. Rev. Lett. 59, 1424.PubMedCrossRefGoogle Scholar
  22. Meneveau, C. and Sreenivasan, K.R. 1989. Measurement of f(α) from scaling of histograms, and applications to dynamic systems and fully developed turbulence. Physics Letters A 137, 103–112.CrossRefGoogle Scholar
  23. Prasad R.R., Meneveau, C. and Sreenivasan, K.R. 1988. Multifractal nature of the dissipation field of passive scalars in fully turbulent flaws. Phys. Rev. Lett. 61, 74.PubMedCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1989

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
    • 2
  1. 1.Physics DepartmentIBM T.J. Watson Research CenterYorktown HeightsUSA
  2. 2.Mathematics DepartmentYale UniversityNew HavenUSA

Personalised recommendations