
Benoit Mandelbrot
RANDOM MULTIFRACTALS
VIRTUAL SELECTA
This openended 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 nineteenseventies 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

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 1420, 1974, 912.


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, 333351.
[ PDF
(774 KB) ]


71 
WWW AS & SR. M 1974c. Multiplications aléatoires
itérées et distributions invariantes par moyenne pondérée
aléatoire, I & II. Comptes Rendus (Paris): 278A,
289292 & 355358.
I. [ PDF
(2.13 MB) ] II. [ PDF
(2.91 MB) ]
• N16. English translations.
[ PDF
(96 KB) ]


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, 121145.
• Brief variant: Comment on coherent structures: Proceedings
of the IUTAM Symposium on Turbulence and Chaotic Phenomena in Fluids.
Edited by Tomomasa Tatsumi, Amsterdam: NorthHolland, 1984, 207208.


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, 8393.
• Russian translation: Strannye Atraktory (=Strange
Attractors). Collection of reprints edited by Yakov G. Sinai
& L. P. Silnikova. Moscow: Mir Publishers, 1981, 4757.
• 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, 463464.


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), 225233.


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, 895930.
[ 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: NorthHolland, 1984, 203206.
• 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:
NorthHolland, 1984, 57.


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 MayerKress,
New York: Springer, 1927.
• Letter to the Editor: Multifractals and fractals. Physics
Today: September 1986, 1112.
• Multifractal measures: Book g, 8491.


122 
WWW M. M 1989g. Multifractal measures, especially
for the geophysicist: Pure and Applied Geophysics: 131,
542. Also Book i.
[ PDF
(6.71 MB) ]
• Brief excerpt: Annual Reviews of Materials Sciences:
19, 1989, 514516.


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, 329.
• Short version: Negative fractal dimensions and multifractals.
Statistical Physics 17, International IUPAP Conference
(Rio de Janeiro, 1989). Edited by Constantino Tsallis, Physica:
A163, 1990, 306315. [ PDF
(3.53 MB) ]
• Updated short version: Two meanings of multifractality,
and the notion of negative fractal dimension. Chaos/Xaoc: SovietAmerican
Perspectives on Nonlinear Science (Woods Hole, 1989). Edited
by David K. Campbell. New York: American Institute of Physics, 1990,
7990.


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, 309340.


125 
WWW M. M 1990d. New “anomalous” multiplicative
multifractals: leftsided f(a) and the modeling of DLA. Condensed
Matter Physics, in Honor of Cyril Domb (Bar Ilan, 1990). Physica:
A168, 95111.


126 
WWW M. M, Carl J. G. EVERTSZ, & Yoshinari
HAYAKAWA 1990. Exactly selfsimilar “leftsided” multifractal
measures. Physical Review: A42, 1990,
45284536.
• Reprint combining 126 and 127: M & Carl J. G. Evertsz.
Exactly selfsimilar multifractals with leftsided f(a). Fractals
and Disordered Systems. Edited by Armin Bunde & Shlomo
Havlin. New York: Springer, 323346.


130 
WWW M. M 1991k. Random multifractals: negative
dimensions and the resulting limitations of the thermodynamic formalism.
Proceedings of the Royal Society (London): A434,
7988. 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) ]


132 
WWW M & C22. M & Carl J. G. EVERTSZ
1991. Multifractality of the harmonic measure on fractal aggregates,
and extended selfsimilarity. In Honor of Michael E. Fisher
(Washington, 1991). Edited by Eytan Domany & David Jasnow, Physica:
A177, 386393.
• Reprint: Fractales y caos (Valencia, 1992). Edited
by P. Martinez.


136 
WWW M. Carl J. G. EVERTSZ & M 1992a. Multifractal
measures. Chaos and Fractals: New Frontiers in Science,
by HeinzOtto Peitgen, Hartmut Jürgens & Dietmar Saupe.
New York: Springer, 849881.
• Reprint: Fractales y caos (Valencia, 1992). Edited
by P. Martinez.
• Standalone reprint: Complexity vs. Simplicity
(CCAST, Beijing, 1996). Edited by HaiCang Ren, Newark, NJ: Gordon
and Breach, 1997.


137 
WWW M. M 1992h. Plane DLA is not selfsimilar;
is it a fractal that becomes increasingly compact as it grows? Fractals
and Disordered Systems (Hamburg, 1992). Edited by Armin Bunde.
Physica: A191, 95107.


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, 8394.
• Slightly edited reprint: Fractals and Disordered Systems.
Second edition. Edited by Armin Bunde & Shlomo Havlin. New York:
Springer, 1995, 345353.


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, 409432.


154 
WWW M. M & Rudolf H. RIEDI 1995. Multifractal
formalism for infinite multinomial measures. Advances in Applied
Mathematics: 16, 132150.
• Outline: Fractals and Disordered Systems. Second
edition. Edited by Armin Bunde & Shlomo Havlin. New York: Springer,
1995, 344345.


158 
WWW M. Stéphane JAFFARD & M 1995. Local
regularity of nonsmooth wavelet expansions and application to the
Polyà function. Advances in Mathematics: 120,
265282.


160 
WWW M. M & Rudolf H. RIEDI 1997. Inverse measures,
the inversion formula, and discontinuous multifractals. Advances
in Applied Mathematics: 18, 5058.


161 
WWW M. Rudolf H. RIEDI & M 1997. Inversion
formula for continuous multifractals. Advances in Applied Mathematics:
9, 332354.


163 
WWW M. M & Stéphane JAFFARD 1997. PeanoPólya
motions, when time is intrinsic (uniform) or binomial (multifractal).
The Mathematical Intelligencer: 19(4) 2126.


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)
]


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) ]


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) ]


167 
WWW M. Rudolf H. RIEDI & M 1998. Exceptions
to the multifractal formalism for discontinuous measures. Mathematical
Proceedings of the Cambridge Philosophical Society: 123,
133157.


170 
WWW M. & R. MarcOlivier COPPENS & M 1999.
Easy and natural generation of multifractals: multiplying harmonics
of periodic functions. Fractals in Engineering (Delft, 1999).
Edited by Jacques LévyVéhel, Evelyne Lutton, &
Claude Tricot. New York: Springer, 113122.
[ PDF
(207.5 KB) ] 

172 
WWW M & P. M 2001a. Scaling in financial
prices, I: Tails and dependence. Quantitative Finance:
1, 113123.
[ PDF
(261 KB) ]
• Reprint: Beyond Efficiency and Equilibrium. Edited
by Doyne Farmer & John Geanakoplos, Oxford UK: The University
Press, 2004.


173 
WWW M & P. M 2001b. Scaling in financial
prices, II: Multifractals and the star equation. Quantitative
Finance: 1, 124130.
[ PDF
(108 KB) ]
• Reprint: Beyond Efficiency and Equilibrium. Edited
by Doyne Farmer & John Geanakoplos, Oxford UK: The University
Press, 2004.


174 
WWW K, M & P. M 2001c. Scaling in financial
prices, III: Cartoon Brownian motions in multifractal time. Quantitative
Finance: 1, 427440.
[ PDF
(224 KB) ]


175 
WWW K, M & P. M 2001d. Scaling in financial
prices, IV: Multifractal concentration. Quantitative Finance:
1, 641649.
[ PDF
(205 KB) ] 

176 
WWW M & P. M 2001e. Stochastic volatility,
powerlaws and long memory. Quantitative Finance: 1,
558559.


178 
WWW M. Julien BARRAL & M 2002. Multifractal
products of cylindrical pulses. Probability Theory and Related
Fields: 124, 409430.
[ PDF
(199.9 KB) ] 

179 
WWW M. M 2003f. Multifractal powerlaw distributions,
other “anomalies,” and critical dimensions, explained
by a simple example. Journal of Statistical Physics: 110,
739777.
[ PDF
(451 KB) ] 

182 
WWW K & M. Julien BARRAL, MarcOlivier COPPENS,
& M 2003. Multiperiodic multifractal martingale measures. Journal
des mathématiques pures et appliquées: 82,
15551589.
[ PDF
(1.01 MB) ]


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.


184 
WWW M. Julien BARRAL & M 2004b. Nondegeneracy,
moments, dimensions, and multifractal analysis for random multifractation
measures. Fractals. Edited by Michel L. Lapidus. Providence
RI: American Mathematical Society, 2004.

