Benoit Mandelbrot







This open-ended webbook will be extended
as needs and opportunities arise.
It consists in links to the author’s
publications that concern multifractals
but not economics and finance and
are found in this home site


The first paper on random multifractals was Mandelbrot 1969b. It and other papers I wrote in the heroic nineteen-seventies were reprinted as part of my Selecta Volume N: Multifractals and 1/f Noise. Later, Frisch & Parisi 1985 proposed the term “multifractal,” and Halsey et al 1986 was an excellent and influential expository paper. Earlier papers by Besicovitch, Kolmogorov, and Yaglom faced none of the mathematical or practical issues but exerted a major influence.

This webbook is largely an organizational and taxonomic fiction, since it simply serves to bring together in orderly fashion the publications in this website that concern multifractals. Therefore, the whole reduces to a title page, a foreword, and a table of contents.



WWW M. N13. M 1969b. On intermittent free turbulence. Turbulence of Fluids and Plasmas. Polytechnic Institute of Brooklyn, April 1968. Edited by Ernst Weber. New York: Interscience.

• The geometry of turbulence. Conference on Prospects for Theoretical Turbulence Research, N. C. A. R., Boulder, Colo., June 14-20, 1974, 9-12.


WWW K & (SR). N14. M 1972i. Possible refinement of the lognormal hypothesis concerning the distribution of energy dissipation in intermittent turbulence. Statistical Models and Turbulence (La Jolla, California). (Lecture Notes in Physics 12). Edited by Murray Rosenblatt & Charles Van Atta. New York: Springer, 333-351.
[ PDF (774 KB) ]


WWW AS & SR. M 1974c. Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire, I & II. Comptes Rendus (Paris): 278A, 289-292 & 355-358.
I. [ PDF (2.13 MB) ] II. [ PDF (2.91 MB) ]

N16. English translations.
[ PDF (96 KB) ]


WWW M N18. M 1976o. Intermittent turbulence and fractal dimension: kurtosis and the spectral exponent 5/3+B. Turbulence and Navier Stokes Equations (Orsay, 1975). Edited by Roger Temam (Lecture Notes in Mathematics 565). New York: Springer, 121-145.

• Brief variant: Comment on coherent structures: Proceedings of the IUTAM Symposium on Turbulence and Chaotic Phenomena in Fluids. Edited by Tomomasa Tatsumi, Amsterdam: North-Holland, 1984, 207-208.


WWW M. 1977b. Fractals and turbulence: attractors and dispersion. Seminar on Turbulence, Berkeley 1976. Organized by Alexandre Chorin, Jerald Marsden & Stephen Smale. Edited by P. Bernard & T. Ratiu (Lecture Notes in Mathematics 615). New York: Springer, 83-93.

• Russian translation: Strannye Atraktory (=Strange Attractors). Collection of reprints edited by Yakov G. Sinai & L. P. Silnikova. Moscow: Mir Publishers, 1981, 47-57.
• Elaboration of some points: Fractals, attractors, and the fractal dimension. Bifurcation Theory and Applications in Scientific Disciplines (New York, 1977). Edited by Okan Gurel & Otto Rossler. Annals of the New York Academy of Sciences: 316, 1979, 463-464.


WWW M. M 1978h. Geometric facets of statistical physics: scaling and fractals. Statistical Physics 13, International IUPAP Conference (Haifa, 1977). Edited by D. Cabib, C.G. Kuper & I. Riess. Annals of the Israel Physical Society. Bristol: Adam Hilger. 2 (1), 225-233.


WWW M. M 1984e. Fractals in physics: squig clusters, diffusions, fractal measures and the unicity of fractal dimension. Statistical Physics 15, International IUPAP Conference (Edinburgh, 1983). Edited by David Wallace & Alistair Bruce. Journal of Statistical Physics: 34, 895-930.
[ PDF (7.79 MB) ]

• Excerpt: Each fractal set has a unique fractal dimension. Proceedings of the IUTAM Symposium on Turbulence and Chaotic Phenomena in Fluid (Kyoto, 1883). Edited by Tomomasa Tatsumi, Amsterdam: North-Holland, 1984, 203-206.
• Illustration: On the aggregative fractals called squigs, which include recursive models of polymers and of percolation clusters. Kinetics of Aggregation and Gelation (Athens, Georgia, April 1984). Edited by Fereydoon Family & David P. Landau. Amsterdam: North-Holland, 1984, 5-7.


WWW M. 1986. Fractal measures (their infinite moment sequences and dimensions) and multiplicative chaos: early works and open problems. Dimensions and Entropies in Dynamical Systems (Pecos River NM, 1985). Edited by Gottfried Mayer-Kress, New York: Springer, 19-27.

• Letter to the Editor: Multifractals and fractals. Physics Today: September 1986, 11-12.
• Multifractal measures: Book g, 84-91.


WWW M. M 1989g. Multifractal measures, especially for the geophysicist: Pure and Applied Geophysics: 131, 5-42. Also Book i.
[ PDF (6.71 MB) ]

• Brief excerpt: Annual Reviews of Materials Sciences: 19, 1989, 514-516.


WWW M. M 1989e. A class of multifractal measures with negative (latent) values for the “dimension” f(a)). Fractals’ Physical Origin and Properties (Erice, 1988). Edited by Luciano Pietronero, New York: Plenum, 3-29.

• Short version: Negative fractal dimensions and multifractals. Statistical Physics 17, International IUPAP Conference (Rio de Janeiro, 1989). Edited by Constantino Tsallis, Physica: A163, 1990, 306-315.
[ PDF (3.53 MB) ]
• Updated short version: Two meanings of multifractality, and the notion of negative fractal dimension. Chaos/Xaoc: Soviet-American Perspectives on Nonlinear Science (Woods Hole, 1989). Edited by David K. Campbell. New York: American Institute of Physics, 1990, 79-90.


WWW M. M 1990t. Limit lognormal multifractal measures. Frontiers of Physics: Landau Memorial Conference (Tel Aviv, 1988). Edited by E. A. Gotsman et al. New York: Pergamon, 309-340.


WWW M. M 1990d. New “anomalous” multiplicative multifractals: left-sided f(a) and the modeling of DLA. Condensed Matter Physics, in Honor of Cyril Domb (Bar Ilan, 1990). Physica: A168, 95-111.


WWW M. M, Carl J. G. EVERTSZ, & Yoshinari HAYAKAWA 1990. Exactly self-similar “left-sided” multifractal measures. Physical Review: A42, 1990, 4528-4536.

• Reprint combining 126 and 127: M & Carl J. G. Evertsz. Exactly self-similar multifractals with left-sided f(a). Fractals and Disordered Systems. Edited by Armin Bunde & Shlomo Havlin. New York: Springer, 323-346.


WWW M. M 1991k. Random multifractals: negative dimensions and the resulting limitations of the thermodynamic formalism. Proceedings of the Royal Society (London): A434, 79-88. Also in Turbulence and Stochastic Processes: Kolmogorov’s ideas 50 years on. Edited by Julian C. R. Hunt, O. M. Phillips, & D. Williams, London: The Royal Society.
[ PDF (2.92 MB) ]


WWW M & C22. M & Carl J. G. EVERTSZ 1991. Multifractality of the harmonic measure on fractal aggregates, and extended self-similarity. In Honor of Michael E. Fisher (Washington, 1991). Edited by Eytan Domany & David Jasnow, Physica: A177, 386-393.

• Reprint: Fractales y caos (Valencia, 1992). Edited by P. Martinez.


WWW M. Carl J. G. EVERTSZ & M 1992a. Multifractal measures. Chaos and Fractals: New Frontiers in Science, by Heinz-Otto Peitgen, Hartmut Jürgens & Dietmar Saupe. New York: Springer, 849-881.

• Reprint: Fractales y caos (Valencia, 1992). Edited by P. Martinez.
• Stand-alone reprint: Complexity vs. Simplicity (CCAST, Beijing, 1996). Edited by Hai-Cang Ren, Newark, NJ: Gordon and Breach, 1997.


WWW M. M 1992h. Plane DLA is not self-similar; is it a fractal that becomes increasingly compact as it grows? Fractals and Disordered Systems (Hamburg, 1992). Edited by Armin Bunde. Physica: A191, 95-107.


WWW M. C21. M 1993s. The Minkowski measure and multifractal anomalies in invariant measures of parabolic dynamic systems. Chaos in Australia (Sydney, 1990). Edited by Gavin Brown & Alex Opie. Singapore: World Publishing, 83-94.

• Slightly edited reprint: Fractals and Disordered Systems. Second edition. Edited by Armin Bunde & Shlomo Havlin. New York: Springer, 1995, 345-353.


WWW M. M 1995k. Negative dimensions and Hölder, multifractals and their Hölder spectra, and the role of lateral preasymptotics in science. J. P. Kahane meeting (Paris, 1993). Edited by Aline Bonami & Jacques Peyrière. The Journal of Fourier Analysis and Applications: special issue, 409-432.


WWW M. M & Rudolf H. RIEDI 1995. Multifractal formalism for infinite multinomial measures. Advances in Applied Mathematics: 16, 132-150.

• Outline: Fractals and Disordered Systems. Second edition. Edited by Armin Bunde & Shlomo Havlin. New York: Springer, 1995, 344-345.


WWW M. Stéphane JAFFARD & M 1995. Local regularity of nonsmooth wavelet expansions and application to the Polyà function. Advances in Mathematics: 120, 265-282.


WWW M. M & Rudolf H. RIEDI 1997. Inverse measures, the inversion formula, and discontinuous multifractals. Advances in Applied Mathematics: 18, 50-58.


WWW M. Rudolf H. RIEDI & M 1997. Inversion formula for continuous multifractals. Advances in Applied Mathematics: 9, 332-354.


WWW M. M & Stéphane JAFFARD 1997. Peano-Pólya motions, when time is intrinsic (uniform) or binomial (multifractal). The Mathematical Intelligencer: 19(4) 21-26.


WWW M. & P. M, Laurent CALVET, & Adlai FISHER 1997. The multifractal model of asset returns. Cowles Foundation Discussion Papers: 1164.
[ PDF (1.51 MB) ]


WWW M. & P. Laurent CALVET, Adlai FISHER, & M 1997. Large deviations and the distribution of price changes. Cowles Foundation Discussion Papers: 1165.
[ PDF (327 KB) ]


WWW M. & P. Adlai FISHER, Laurent CALVET, & M 1997. Multifractality of the Deutschmark/US Dollar exchange rates. Cowles Foundation Discussion Papers: 1166.
[ PDF (311 KB) ]


WWW M. Rudolf H. RIEDI & M 1998. Exceptions to the multifractal formalism for discontinuous measures. Mathematical Proceedings of the Cambridge Philosophical Society: 123, 133-157.


WWW M. & R. Marc-Olivier COPPENS & M 1999. Easy and natural generation of multifractals: multiplying harmonics of periodic functions. Fractals in Engineering (Delft, 1999). Edited by Jacques Lévy-Véhel, Evelyne Lutton, & Claude Tricot. New York: Springer, 113-122.
[ PDF (207.5 KB) ]


WWW M & P. M 2001a. Scaling in financial prices, I: Tails and dependence. Quantitative Finance: 1, 113-123.
[ PDF (261 KB) ]

• Reprint: Beyond Efficiency and Equilibrium. Edited by Doyne Farmer & John Geanakoplos, Oxford UK: The University Press, 2004.


WWW M & P. M 2001b. Scaling in financial prices, II: Multifractals and the star equation. Quantitative Finance: 1, 124-130.
[ PDF (108 KB) ]

• Reprint: Beyond Efficiency and Equilibrium. Edited by Doyne Farmer & John Geanakoplos, Oxford UK: The University Press, 2004.


WWW K, M & P. M 2001c. Scaling in financial prices, III: Cartoon Brownian motions in multifractal time. Quantitative Finance: 1, 427-440.
[ PDF (224 KB) ]


WWW K, M & P. M 2001d. Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance: 1, 641-649.
[ PDF (205 KB) ]


WWW M & P. M 2001e. Stochastic volatility, power-laws and long memory. Quantitative Finance: 1, 558-559.


WWW M. Julien BARRAL & M 2002. Multifractal products of cylindrical pulses. Probability Theory and Related Fields: 124, 409-430.
[ PDF (199.9 KB) ]


WWW M. M 2003f. Multifractal power-law distributions, other “anomalies,” and critical dimensions, explained by a simple example. Journal of Statistical Physics: 110, 739-777.
[ PDF (451 KB) ]


WWW K & M. Julien BARRAL, Marc-Olivier COPPENS, & M 2003. Multiperiodic multifractal martingale measures. Journal des mathématiques pures et appliquées: 82, 1555-1589.
[ PDF (1.01 MB) ]


WWW M. Julien BARRAL & M 2004a. Introduction to multifractal products of independent random functions: Fractals. Edited by Michel L. Lapidus. Providence RI: American Mathematical Society, 2004.


WWW M. Julien BARRAL & M 2004b. Non-degeneracy, moments, dimensions, and multifractal analysis for random multifractation measures. Fractals. Edited by Michel L. Lapidus. Providence RI: American Mathematical Society, 2004.