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Benoit Mandelbrot
RANDOM MULTIFRACTALS
VIRTUAL SELECTA
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
http://www.math.yale.edu/mandelbrot
FOREWORD
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.
TABLE OF CONTENTS
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53 |
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.
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64 |
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) ]
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71 |
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) ]
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80 |
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.
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81 |
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.
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83 |
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.
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102 |
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.
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114 |
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.
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122 |
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.
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123 |
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.
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124 |
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.
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125 |
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.
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126 |
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.
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130 |
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) ]
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132 |
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.
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136 |
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.
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137 |
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.
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138 |
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.
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150 |
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.
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154 |
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.
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158 |
WWW M. Stéphane JAFFARD & M 1995. Local
regularity of nonsmooth wavelet expansions and application to the
Polyà function. Advances in Mathematics: 120,
265-282.
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160 |
WWW M. M & Rudolf H. RIEDI 1997. Inverse measures,
the inversion formula, and discontinuous multifractals. Advances
in Applied Mathematics: 18, 50-58.
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161 |
WWW M. Rudolf H. RIEDI & M 1997. Inversion
formula for continuous multifractals. Advances in Applied Mathematics:
9, 332-354.
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163 |
WWW M. M & Stéphane JAFFARD 1997. Peano-Pólya
motions, when time is intrinsic (uniform) or binomial (multifractal).
The Mathematical Intelligencer: 19(4) 21-26.
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164 |
WWW M. & P. M, Laurent CALVET, & Adlai
FISHER 1997. The multifractal model of asset returns. Cowles Foundation
Discussion Papers: 1164.
[ PDF (1.51 MB)
]
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165 |
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) ]
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166 |
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) ]
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167 |
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.
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170 |
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) ] |
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172 |
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.
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173 |
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.
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174 |
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) ]
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175 |
WWW K, M & P. M 2001d. Scaling in financial
prices, IV: Multifractal concentration. Quantitative Finance:
1, 641-649.
[ PDF
(205 KB) ] |
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176 |
WWW M & P. M 2001e. Stochastic volatility,
power-laws and long memory. Quantitative Finance: 1,
558-559.
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178 |
WWW M. Julien BARRAL & M 2002. Multifractal
products of cylindrical pulses. Probability Theory and Related
Fields: 124, 409-430.
[ PDF
(199.9 KB) ] |
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179 |
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) ] |
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182 |
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) ]
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183 |
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.
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184 |
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.
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