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Annales scientifiques de l'ENS - Parutions - série 4, 48 (2015)

Parutions < série 4, 48

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 48, fascicule 6 (2015)

Julien Barral
Inverse problems in multifractal analysis of measures
Annales scientifiques de l'ENS 48, fascicule 6 (2015), 1457-1510

Télécharger cet article : Fichier PDF

Résumé :
Problèmes inverses dans l'analyse multifractale des mesures
Le formalisme multifractal est un cadre adapté pour décrire la distribution aux petites échelles des mesures de Borel finies positives à support compact dans R ^d, dont l'ensemble est ici noté M^+_c(R ^d). Il est dit valide pour une mesure lorsque son spectre de Hausdorff est la fonction semi-continue supérieurement obtenue comme transformée de Legendre-Fenchel concave de sa fonction d'énergie libre _; c'est le cas pour certaines classes fondamentales de mesures exactement dimensionnelles. Pour toute fonction candidate à être la fonction d'énergie libre d'un élément de M^+_c(R ^d), nous construisons une telle mesure, exactement dimensionnelle, et validant le formalisme. Ce résultat s'étend à un formalisme plus fin considérant simultanément spectres de Hausdorff et de packing. D'autre part, pour toute fonction semi-continue supérieurement candidate à être le spectre de Hausdorff inférieur d'une mesure exactement dimensionnelle, nous construisons une telle mesure.

Mots-clefs : Formalisme multifractal, analyse multifractale, dimension de Hausdorff, dimension de packing, grandes déviations, problèmes inverses.

Abstract:
Multifractal formalism is designed to describe the distribution at small scales of the elements of M^+_c(R ^d), the set of positive, finite and compactly supported Borel measures on R ^d. It is valid for such a measure when its Hausdorff spectrum is the upper semi-continuous function given by the concave Legendre-Fenchel transform of the free energy function _ associated with ; this is the case for fundamental classes of exactly dimensional measures. For any function candidate to be the free energy function of some M^+_c(R ^d), we construct such a measure, exactly dimensional, and obeying the multifractal formalism. This result is extended to a refined formalism considering jointly Hausdorff and packing spectra. Also, for any upper semi-continuous function candidate to be the lower Hausdorff spectrum of some exactly dimensional M^+_c(R ^d), we construct such a measure.

Keywords: Multifractal formalism, multifractal analysis, Hausdorff dimension, packing dimension, large deviations, inverse problems.

Class. math. : 28A78, 60F10


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

Bibliographie:

1
Arbeiter, Matthias and Patzschke, Norbert
Random self-similar multifractals
Math. Nachr. 181 (1996) 5–42
Math Reviews MR1409071 (97j:28016)
2
Attia, Najmeddine and Barral, Julien
Hausdorff and packing spectra, large deviations, and free energy for branching random walks in Rd
Comm. Math. Phys. 331 (2014) 139–187
Math Reviews MR3231998
3
Aversa, V. and Bandt, C.
The multifractal spectrum of discrete measures
Acta Univ. Carolin. Math. Phys. 31 (1990) 5–8
Math Reviews MR1101408 (92b:28008)
4
Baek, I. S. and Olsen, L. and Snigireva, N.
Divergence points of self-similar measures and packing dimension
Adv. Math. 214 (2007) 267–287
Math Reviews MR2348031 (2009a:28020)
5
Barral, Julien
Continuity of the multifractal spectrum of a random statistically self-similar measure
J. Theoret. Probab. 13 (2000) 1027–1060
Math Reviews MR1820501 (2002b:60080)
6
Barral, Julien and Feng, De-Jun
Weighted thermodynamic formalism on subshifts and applications
Asian J. Math. 16 (2012) 319–352
Math Reviews MR2916367
7
Barral, Julien and Feng, De-Jun
Multifractal formalism for almost all self-affine measures
Comm. Math. Phys. 318 (2013) 473–504
Math Reviews MR3020165
8
Barral, Julien and Mandelbrot, Benoît B.
Multifractal products of cylindrical pulses
Probab. Theory Related Fields 124 (2002) 409–430
Math Reviews MR1939653 (2004g:28005)
9
Barral, Julien and Mandelbrot, Benoît B.
Non-degeneracy, moments, dimension, and multifractal analysis for random multiplicative measures (Random multiplicative multifractal measures. II)
in Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2
Proc. Sympos. Pure Math. 72 (2004) 17–52
Math Reviews MR2112120 (2005i:28014)
10
Barral, Julien and Mensi, Mounir
Gibbs measures on self-affine Sierpiński carpets and their singularity spectrum
Ergodic Theory Dynam. Systems 27 (2007) 1419–1443
Math Reviews MR2358972 (2009c:37022)
11
Barral, Julien and Rhodes, Rémi and Vargas, Vincent
Limiting laws of supercritical branching random walks
C. R. Math. Acad. Sci. Paris 350 (2012) 535–538
Math Reviews MR2929063
12
Barral, Julien and Seuret, Stéphane
The singularity spectrum of Lévy processes in multifractal time
Adv. Math. 214 (2007) 437–468
Math Reviews MR2348038 (2008k:60086)
13
Barral, Julien and Seuret, Stéphane
The singularity spectrum of the inverse of cookie-cutters
Ergodic Theory Dynam. Systems 29 (2009) 1075–1095
Math Reviews MR2529640 (2010i:28005)
14
Barreira, Luis and Schmeling, Jörg
Sets of ``non-typical'' points have full topological entropy and full Hausdorff dimension
Israel J. Math. 116 (2000) 29–70
Math Reviews MR1759398 (2002d:37040)
15
Batakis, Athanasios and Testud, Benoît
Multifractal analysis of inhomogeneous Bernoulli products
J. Stat. Phys. 142 (2011) 1105–1120
Math Reviews MR2781722 (2012f:28008)
16
Bayart, Frédéric
Multifractal spectra of typical and prevalent measures
Nonlinearity 26 (2013) 353–367
Math Reviews MR3007893
17
Ben Nasr, Fethi
Analyse multifractale de mesures
C. R. Acad. Sci. Paris Sér. I Math. 319 (1994) 807–810
Math Reviews MR1300947 (95h:28010)
18
Ben Nasr, Fathi and Bhouri, Imen and Heurteaux, Yanick
The validity of the multifractal formalism: results and examples
Adv. Math. 165 (2002) 264–284
Math Reviews MR1887585 (2002m:28011)
19
Ben Nasr, Fathi and Peyrière, Jacques
Revisiting the multifractal analysis of measures
Rev. Mat. Iberoam. 29 (2013) 315–328
Math Reviews MR3010130
20
Billingsley, Patrick
Ergodic theory and information
John Wiley Sons, Inc., 1965
Math Reviews MR0192027 (33 \#254)
21
Brown, G. and Michon, G. and Peyrière, Jacques
On the multifractal analysis of measures
J. Statist. Phys. 66 (1992) 775–790
Math Reviews MR1151978 (93c:58120)
22
Buczolich, Zoltán and Nagy, Judit
Hölder spectrum of typical monotone continuous functions
Real Anal. Exchange 26 (2000/01) 133–156
Math Reviews MR1825500 (2002d:26012)
23
Buczolich, Zoltán and Seuret, Stéphane
Typical Borel measures on [0,1]d satisfy a multifractal formalism
Nonlinearity 23 (2010) 2905–2918
Math Reviews MR2727176 (2012a:28008)
24
Buczolich, Zoltán and Seuret, Stéphane
Measures and functions with prescribed homogeneous multifractal spectrum
J. Fractal Geom. 1 (2014) 295–333
Math Reviews MR3276837
25
Collet, P. and Lebowitz, J. L. and Porzio, Anna
The dimension spectrum of some dynamical systems
in Proceedings of the symposium on statistical mechanics of phase transitions—mathematical and physical aspects (Trebon, 1986)
J. Statist. Phys. 47 (1987) 609–644
Math Reviews MR912493 (89d:58061)
26
Cutler, Colleen D.
The Hausdorff dimension distribution of finite measures in Euclidean space
Canad. J. Math. 38 (1986) 1459–1484
Math Reviews MR873419 (88b:28013)
27
Cutler, Colleen D.
Measure disintegrations with respect to -stable monotone indices and the pointwise representation of packing dimension
Rend. Circ. Mat. Palermo (2) Suppl. 28 (1992) 319–339
Math Reviews MR1183059 (93h:28005)
28
Dembo, Amir and Zeitouni, Ofer
Large deviations techniques and applications
Springer, 1998
Math Reviews MR1619036 (99d:60030)
29
Edgar, G. A. and Mauldin, R. Daniel
Multifractal decompositions of digraph recursive fractals
Proc. London Math. Soc. 65 (1992) 604–628
Math Reviews MR1182103 (93h:28010)
30
Falconer, Kenneth
Fractal geometry
John Wiley Sons, Ltd., Chichester, 2014
Math Reviews MR3236784
31
Falconer, Kenneth
The multifractal spectrum of statistically self-similar measures
J. Theoret. Probab. 7 (1994) 681–702
Math Reviews MR1284660 (95m:60076)
32
Falconer, Kenneth
Generalized dimensions of measures on self-affine sets
Nonlinearity 12 (1999) 877–891
Math Reviews MR1709826 (2000i:28008)
33
Falconer, Kenneth
Representation of families of sets by measures, dimension spectra and Diophantine approximation
Math. Proc. Cambridge Philos. Soc. 128 (2000) 111–121
Math Reviews MR1724433 (2000j:28003)
34
Feng, De-Jun
The limited Rademacher functions and Bernoulli convolutions associated with Pisot numbers
Adv. Math. 195 (2005) 24–101
Math Reviews MR2145793 (2006b:11091)
35
Feng, De-Jun
Gibbs properties of self-conformal measures and the multifractal formalism
Ergodic Theory Dynam. Systems 27 (2007) 787–812
Math Reviews MR2322179 (2008c:37040)
36
Feng, De-Jun
Multifractal analysis of Bernoulli convolutions associated with Salem numbers
Adv. Math. 229 (2012) 3052–3077
Math Reviews MR2889154
37
Feng, De-Jun and Lau, Ka-Sing
Multifractal formalism for self-similar measures with weak separation condition
J. Math. Pures Appl. 92 (2009) 407–428
Math Reviews MR2569186 (2011b:28015)
38
Feng, De-Jun and Olivier, Eric
Multifractal analysis of weak Gibbs measures and phase transition—application to some Bernoulli convolutions
Ergodic Theory Dynam. Systems 23 (2003) 1751–1784
Math Reviews MR2032487 (2004m:37053)
39
Feng, De-Jun and Wu, Jun
The Hausdorff dimension of recurrent sets in symbolic spaces
Nonlinearity 14 (2001) 81–85
Math Reviews MR1808624 (2001m:28007)
40
Frisch, U. and Parisi, G.
Fully developped turbulence and intermittency in turbulence, and predictability in geophysical fluid dynamics and climate dynamics
in International school of Physics ``Enrico Fermi''
84–88
41
Halsey, Thomas C. and Jensen, Mogens H. and Kadanoff, Leo P. and Procaccia, Itamar and Shraiman, Boris I.
Fractal measures and their singularities: the characterization of strange sets
Phys. Rev. A 33 (1986) 1141–1151
Math Reviews MR823474 (87h:58125a)
42
Hentschel, H. G. E. and Procaccia, Itamar
The infinite number of generalized dimensions of fractals and strange attractors
Phys. D 8 (1983) 435–444
Math Reviews MR719636 (85a:58064)
43
Heurteaux, Yanick
Estimations de la dimension inférieure et de la dimension supérieure des mesures
Ann. Inst. H. Poincaré Probab. Statist. 34 (1998) 309–338
Math Reviews MR1625871 (99g:28013)
44
Heurteaux, Yanick
Dimension of measures: the probabilistic approach
Publ. Mat. 51 (2007) 243–290
Math Reviews MR2334791 (2008i:28005)
45
Holley, Richard and Waymire, Edward C.
Multifractal dimensions and scaling exponents for strongly bounded random cascades
Ann. Appl. Probab. 2 (1992) 819–845
Math Reviews MR1189419 (93k:60122)
46
Iommi, Godofredo
Multifractal analysis for countable Markov shifts
Ergodic Theory Dynam. Systems 25 (2005) 1881–1907
Math Reviews MR2183299 (2006g:37007)
47
Jaffard, Stéphane
Old friends revisited: the multifractal nature of some classical functions
J. Fourier Anal. Appl. 3 (1997) 1–22
Math Reviews MR1428813 (98b:28013)
48
Jaffard, Stéphane
The multifractal nature of Lévy processes
Probab. Theory Related Fields 114 (1999) 207–227
Math Reviews MR1701520 (2000g:60079)
49
Jordan, Thomas and Rams, Michał
Multifractal analysis of weak Gibbs measures for non-uniformly expanding C1 maps
Ergodic Theory Dynam. Systems 31 (2011) 143–164
Math Reviews MR2755925 (2012a:37057)
50
Kesseböhmer, Marc
Large deviation for weak Gibbs measures and multifractal spectra
Nonlinearity 14 (2001) 395–409
Math Reviews MR1819804 (2002a:60037)
51
King, James F.
The singularity spectrum for general Sierpiński carpets
Adv. Math. 116 (1995) 1–11
Math Reviews MR1361476 (97d:28008)
52
Lau, Ka-Sing and Ngai, Sze-Man
Multifractal measures and a weak separation condition
Adv. Math. 141 (1999) 45–96
Math Reviews MR1667146 (2000d:28010)
53
54
Ledrappier, François and Porzio, Anna
On the multifractal analysis of Bernoulli convolutions. I. Large-deviation results
J. Statist. Phys. 82 (1996) 367–395
Math Reviews MR1372657 (97b:58088)
55
Ledrappier, François and Porzio, Anna
On the multifractal analysis of Bernoulli convolutions. II. Dimensions
J. Statist. Phys. 82 (1996) 397–420
Math Reviews MR1372658 (97b:58089)
56
Lévy Véhel, Jacques and Vojak, Robert
Multifractal analysis of Choquet capacities
Adv. in Appl. Math. 20 (1998) 1–43
Math Reviews MR1488230 (98j:28003)
57
Lévy Véhel, Jacques and Tricot, Claude
On various multifractal spectra
in Fractal geometry and stochastics III
Progr. Probab. 57 (2004) 23–42
Math Reviews MR2087130 (2006e:28010)
58
Makarov, N. G.
Fine structure of harmonic measure
Algebra i Analiz 10 (1998) 1–62
Math Reviews MR1629379 (2000g:30018)
59
Mandelbrot, Benoît B. and Evertsz, C. J. G. and Hayakawa, Y.
Exactly self-similar left sided multifractal measures
Phys. Rev. A 42 (1990) 4528–4536
60
Riedi, Rudolf H. and Mandelbrot, Benoit B.
Multifractal formalism for infinite multinomial measures
Adv. in Appl. Math. 16 (1995) 132–150
Math Reviews MR1334146 (96d:28008)
61
Mattila, Pertti
Geometry of sets and measures in Euclidean spaces
Cambridge Univ. Press, Cambridge, 1995
Math Reviews MR1333890 (96h:28006)
62
Molchan, G. M.
Scaling exponents and multifractal dimensions for independent random cascades
Comm. Math. Phys. 179 (1996) 681–702
Math Reviews MR1400758 (97g:60066)
63
Ngai, Sze-Man
A dimension result arising from the Lq-spectrum of a measure
Proc. Amer. Math. Soc. 125 (1997) 2943–2951
Math Reviews MR1402878 (97m:28007)
64
Olsen, L.
A multifractal formalism
Adv. Math. 116 (1995) 82–196
Math Reviews MR1361481 (97a:28006)
65
Olsen, L.
Self-affine multifractal Sierpinski sponges in Rd
Pacific J. Math. 183 (1998) 143–199
Math Reviews MR1616626 (99c:28028)
66
Olsen, L.
Dimension inequalities of multifractal Hausdorff measures and multifractal packing measures
Math. Scand. 86 (2000) 109–129
Math Reviews MR1738518 (2001b:28009)
67
Olsen, L.
Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages
J. Math. Pures Appl. 82 (2003) 1591–1649
Math Reviews MR2025314 (2004k:37036)
68
Olsen, L. and Snigireva, N.
Multifractal spectra of in-homogenous self-similar measures
Indiana Univ. Math. J. 57 (2008) 1789–1843
Math Reviews MR2440882 (2009k:37050)
69
Patzschke, Norbert
Self-conformal multifractal measures
Adv. in Appl. Math. 19 (1997) 486–513
Math Reviews MR1479016 (99c:28020)
70
Pesin, Yakov B.
Dimension theory in dynamical systems
University of Chicago Press, Chicago, IL, 1997
Math Reviews MR1489237 (99b:58003)
71
Pesin, Yakov and Weiss, Howard
A multifractal analysis of equilibrium measures for conformal expanding maps and Moran-like geometric constructions
J. Statist. Phys. 86 (1997) 233–275
Math Reviews MR1435198 (97m:58118)
72
Peyrière, Jacques
A vectorial multifractal formalism
in Fractal geometry and applications: a jubilee of Benoît Mandelbrot, Part 2
Proc. Sympos. Pure Math. 72 (2004) 217–230
Math Reviews MR2112124 (2006k:28013)
73
Porzio, Anna
The dimension spectrum of Axiom A attractors
J. Statist. Phys. 58 (1990) 923–937
Math Reviews MR1049052 (91e:58142)
74
Rand, D. A.
The singularity spectrum f() for cookie-cutters
Ergodic Theory Dynam. Systems 9 (1989) 527–541
Math Reviews MR1016670 (90k:58115)
75
Riedi, Rudolf H.
Multifractal processes
in Theory and applications of long-range dependence
(2003) 625–716
Math Reviews MR1957510
76
Rhodes, Rémi and Vargas, Vincent
Gaussian multiplicative chaos and applications: a review
Probab. Surv. 11 (2014) 315–392
Math Reviews MR3274356
77
Rockafellar, R. Tyrrell
Convex analysis
Princeton Univ. Press, Princeton, N.J., 1970
Math Reviews MR0274683 (43 \#445)
78
Shen, Shuang
Multifractal analysis of some inhomogeneous multinomial measures with distinct analytic Olsen's b and B functions
J. Stat. Phys. 159 (2015) 1216–1235
Math Reviews MR3345417
79
Shmerkin, Pablo
A modified multifractal formalism for a class of self-similar measures with overlap
Asian J. Math. 9 (2005) 323–348
Math Reviews MR2214956 (2007b:28009)
80
Testud, Benoît
Mesures quasi-Bernoulli au sens faible: résultats et exemples
Ann. Inst. H. Poincaré Probab. Statist. 42 (2006) 1–35
Math Reviews MR2196969 (2007c:28010)
81
Tricot, Claude Jr.
Two definitions of fractional dimension
Math. Proc. Cambridge Philos. Soc. 91 (1982) 57–74
Math Reviews MR633256 (84d:28013)
82
Zhou, Xiaoyao and Chen, Ercai
Packing dimensions of the divergence points of self-similar measures with OSC
Monatsh. Math. 172 (2013) 233–246
Math Reviews MR3117190
83
Young, Lai Sang
Dimension, entropy and Lyapunov exponents
Ergodic Theory Dynam. Systems 2 (1982) 109–124
Math Reviews MR684248 (84h:58087)