Abstract
This text is addressed to both the beginner and the seasoned professional, geology being used as the main but not the sole illustration. The goal is to present an alternative approach to multifractals, extending and streamlining the original approach inMandelbrot (1974). The generalization from fractalsets to multifractalmeasures involves the passage from geometric objects that are characterized primarily by one number, namely a fractal dimension, to geometric objects that are characterized primarily by a function. The best is to choose the function ϱ(α), which is a limit probability distribution that has been plotted suitably, on double logarithmic scales. The quantity α is called Hölder exponent. In terms of the alternative functionf(α) used in the approach of Frisch-Parisi and of Halseyet al., one has ϱ(α)=f(α)−E for measures supported by the Euclidean space of dimensionE. Whenf(α)≥0,f(α) is a fractal dimension. However, one may havef(α)<0, in which case α is called “latent.” One may even have α<0, in which case α is called “virtual.” These anomalies' implications are explored, and experiments are suggested. Of central concern in this paper is the study of low-dimensional cuts through high-dimensional multifractals. This introduces a quantityD q, which is shown forq>1 to be a critical dimension for the cuts. An “enhanced multifractal diagram” is drawn, includingf(α), a function called τ(q) andD q.
Similar content being viewed by others
References
Azencott, R., Guivarc'h, Y., andGundy, R. F.,Ecole de Saint-Flour 1978, Lecture Notes in Mathematics (Saint Flour, 1978) Vol.774 (Springer, New York 1980).
Billingsley, P.,Ergodic Theory and Information (J. Wiley, New York 1967 p. 139.
Book, S. A. (1984),Large deviations and applications, InEncyclopedia of Statistical Sciences 4, 476.
Cates, M. E., andDeutsch, J. M. (1987),Spatial Correlations in Multifractals, Phys. Rev.A 35, 4907.
Chernoff, H. (1952),A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations, Ann. Math. Stat.23, 493.
Chhabra, A., andJensen, R. V. (1989),Direct Determination of the f(α) Singularity Spectrum, Phys. Rev. Lett.62, 1327.
Dacunha-Castelle, D.,Grandes Déviations et Applications Statistiques (Astérisque 68) (Societé Mathématique de France, Paris 1979).
Daniels, H. E. (1954),Saddlepoint Approximations in Statistics, Ann. Math. Stat.25, 631.
Daniels, H. E. (1987),Tail Probability Approximations, International Statistical Review55, 37.
Ellis, R. S. (1984),Large Deviations for a General Class of Random Vectors, The Annals of Probability12, 1.
Feder, J.,Fractals (Plenum, New York 1988).
Fourcade, B., Breton, P., andTremblay, A.-M. S. (1987),Multifractals and Critical Phenomena in Percolating Networks: Fixed Point, Gap Scaling and Universality, Phys. Rev.B36, 8925.
Fourcade, B., andTremblay, A.-M. S. (1987),Anomalies in the Multifractal Analysis of Self-similar Resistor Networks, Phys. Rev.A36, 2352.
Frisch, U., andParisi, G.,Fully develped turbulence and intermittency, InTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics (International School of Physics “Enrico Fermi”, Course 88) (eds. Ghil, M.) (North-Holland, Amsterdam, 1985) p. 84.
Grassberger, P. (1983),Generalized Dimensions of Strange Attractors, Phys. Lett.97A, 227.
Gutzwiller, M. C., andMandelbrot, B. B., (1988),Invariant Multifractal Measures in Chaotic Hamiltonian Systems, and Related Structures, Phys. Rev. Lett.60, 673.
Guivarc'h, Y. (1987),Remarques sur les Solutions d'une Equation Fonctionnelle Non Linéaire de Benoít Mandelbrot, Comptes Rendus (Paris)3051, 139.
Halsey, T. C., Jensen, M. H., Kadanoff, L. P., Procaccia, I., andShraiman, B. I. (1986),Fractal Measure and their Singularities: The Characterization of Strange Sets, Phys. Rev.A33, 1141.
Hentschel, H. G. E., andProcaccia, I. (1983),The Infinite Number of Generalized Dimensions of Fractals and Strange Attractors, Physica (Utrecht)8D, 435.
Huang, K.,Statistical Mechanics (J. Wiley, New York 1966).
Kahane, J. P., andPeyrière, J. (1976),Sur Certaines Martingales de B. Mandelbrot, Adv. in Math.22, 131.
Mandelbrot, B. B.,Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence, InStatistical Models and Turbulence (Lecture Note in Physics, Vol. 12), Proc. Symp. La Jolla, Calif. (eds. Rosenblatt, M., and Van Atta, C.) (Springer-Verlag, New York 1972) p. 333.
Mandelbrot, B. B. (1974),Intermittent Turbulence in Self Similar Cascades; Divergence of High Moments and Dimension of the Carrier, J. Fluid Mech62, 331; also Comptes Rendus278A, 289, 355.
Mandelbrot, B. B.,The Fractal Geometry of Nature (W. H. Freeman, New York 1982) pp. 373–381 discuss α and the “Lipshitz-Hölder heuristics”, then the “nonlacunar fractals”=multifractals.
Mandelbrot, B. B. (1984),Fractals in Physics: Squig Clusters, Diffusiions, Fractal Measures and the Unicity of Fractal Dimension J. Stat. Phys.34, 895.
Mandelbrot, B. B. (1986),Letter to the Editor: Multifractals and Fractals, Physics Today11.
Mandelbrot, B. B., (1988),An introduction to multifractal distribution functions, InFluctuations and Pattern Formation (Cargèse, 1988) (eds. Stanley, H. E., and Ostrowsky, N.) (Kluwer, Dordrecht-Boston 1988) pp. 345–360.
Mandelbrot, B. B.,Examples of multinomial multifractal measures that have negative latent values for the dimension f(α), InFractals (“Etteor Majorama” Centre for Scientific Culture, Special Seminar) (ed. Pietronero, L) (Plenum, New York 1989).
Mandelbrot, B. B. (forthcoming),Fractals and Multifractals: Noise, Turbulence and Galaxies (Selecta, Vol. 1) (Springer, New York).
Meakin, P. The growth of fractal aggregates and their fractal measures, InPhase Transitions and Critical Phenomena (eds. Domb, C., and Lebowitz, J. L.) (Academic Press London 1988)12, 335.
Meneveau, C., andSreenivasan, K. R. (1987),Simple Multifractal Cascade Model for Fully Developed Turbulence, Phys. Rev. Lett.,59, 1424.
Meneveau, C., andSreenivasan, K. R. (1989),Measurement of f(α) from Scaling of Histograms, and Applications to Dynamic Systems and Fully Developed Turbulence, Phys. Lett.A137, 103–112.
Prasad, R. P., Meneveau, C., andSreenivasan, K. R. (1988).Multifractal Nature of the Dissipation Field of Passive Scalars in Fully Turbulent Flaws, Phys. Rev. Lett.61, 74.
Volkmann, (1958),Ober Hausdorffsche Dimensionen Von Mengen, Die Durch Zifferneigenschaften Sind, Math. Zeitschrift68, 439.
Author information
Authors and Affiliations
Additional information
This text incorporatesand supersedes Mandelbrot (1988). A more detailed treatment, in preparation, will incorporateMandelbrot (1989).
Rights and permissions
About this article
Cite this article
Mandelbrot, B.B. Multifractal measures, especially for the geophysicist. PAGEOPH 131, 5–42 (1989). https://doi.org/10.1007/BF00874478
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00874478