Abstract
The multifractal formalism for singular measures is revisited using the wavelet transform. For Bernoulli invariant measures of some expanding Markov maps, the generalized fractal dimensions are proved to be transition points for the scaling exponents of some partition functions defined from the wavelet transform modulus maxima. The generalization of this formalism to fractal signals is established for the class of distribution functions of these singular invariant measures. It is demonstrated that the Hausdorff dimensionD(h) of the set of singularities of Hölder exponenth can be directly determined from the wavelet transform modulus maxima. The singularity spectrum so obtained is shown to be not disturbed by the presence, in the signal, of a superimposed polynomial behavior of ordern, provided one uses an analyzing wavelet that possesses at leastN>n vanishing moments. However, it is shown that aC ∞ behavior generally induces a phase transition in theD(h) singularity spectrum that somewhat masks the weakest singularities. This phase transition actually depends on the numberN of vanishing moments of the analyzing wavelet; its observation is emphasized as a reliable experimental test for the existence of nonsingular behavior in the considered signal. These theoretical results are illustrated with numerical examples. They are likely to be valid for a large class of fractal functions as suggested by recent applications to fractional Brownian motions and turbulent velocity signals.
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References
B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman,Phys. Rev. A 33:1141 (1986).
G. Paladin and A. Vulpiani,Phys. Rep. 156:148 (1987).
H. E. Stanley and N. Ostrowski, eds.,On Growth and Form: Fractal and Nonfractal Patterns in Physics (Martinus Nijhof, Dordrecht, 1986), and references therein; H. E. Stanley and N. Ostrowski, eds.,Random Fluctuations and Pattern Growth (Kluwer Academic, Dordrecht, 1988), and references therein.
L. Pietronero and E. Tosatti, eds.,Fractals in Physics (North-Holland, Amsterdam, 1986), and references therein.
A. Aharony and J. Feder, eds., Essays in honour of B. B. Mandelbrot, Fractals in Physics,Physica D 38 (1989), and references therein.
E. B. Vul, Ya. G. Sinai, and K. M. Khanin,Usp. Mat. Nauk 39:3 (1984) [J. Russ. Math. Sun. 39:1 (1984)].
R. Benzi, G. Paladin, G. Parisi, and A. Vulpiani,J. Phys. A 17:3521 (1984).
D. Rand,Ergodic Theory Dynamic Syst. 9:527 (1989).
P. Collet, J. Lebowitz, and A. Porzio,J. Stat. Phvs. 47:609 (1987).
M. J. Feigenbaum,J. Stat. Phys. 46:919, 925 (1987).
R. Badii, Thesis, University of Zurich (1987).
J. Feder,Fractals (Pergamon, New York, 1988), and references therein.
T. Vicsek,Fractal Growth Phenomena (World Scientific, Singapore, 1989), and references therein.
P. Meakin, inPhase Transition and Critical Phenomena, Vol. 12, C. Domb and J. L. Lebowitz, eds. (Academic Press, Orlando, Florida, 1988).
B. Mandelbrot,J. Fluid Mech. 62:331 (1974).
U. Frisch, P. L. Sulem, and M. Nelkin,J. Fluid Mech. 87:719 (1978).
C. Meneveau and K. R. Sreenivasan,J. Fluid Mech. 224:429 (1991).
P. Grassberger,Phys. Lett. A 97:227 (1983).
H. G. E. Hentschel and I. Procaccia,Physica D 8:435 (1983).
P. Grassberger and I. Procaccia,Physica D 13:34 (1984).
Ya. G. Sinai,Usp. Mat. Nauk 27:21 (1972) [J. Russ. Math. Surv. 166:21 (1972)].
R. Bowen,Lecture Notes in Mathematics, Vol. 470 (Springer, New York, 1975).
D. Ruelle,Statistical Mechanics (Addison-Wesley, Reading, Massachusetts, 1969);Thermodynamics Formalism (Addison-Wesley, Reading, Massachusetts, 1978).
T. Bohr and T. Tèl, inDirection in Chaos, Vol. 2, Hao Bai-Lin, ed. (World Scientific, Singapore, 1988).
U. Frisch and G. Parisi, Fully developed turbulence and intermittency, inProceedings of International School on Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi, eds. (North-Holland, Amsterdam, 1985), p. 84.
A. L. Barabási and T. Vicsek,Phys. Rev. A 44:2730 (1991).
F. Anselmet, Y. Gagne, E. J. Hopfinger, and R. A. Antonia,J. Fluid Mech. 140:63 (1984).
Y. Gagne, E. J. Hopfinger, and U. Frisch, inNew Trends in Nonlinear Dynamics and Pattern Forming Phenomena: The Geometry of Nonequilibrium, P. Huerre and P. Coullet, eds. (Plenum Press, New York, 1988).
B. Castaing, Y. Gagne, and E. J. Hopfinger,Physica D 46:177 (1990).
J. F. Muzy, E. Bacry, and A. Arnéodo,Phys. Rev. Lett. 67:3515 (1991).
A. Arnéodo, E. Bacry, and J. F. Muzy, Wavelet analysis of fractal signals: Direct determination of the singularity spectrum of fully developed turbulence data (Springer-Verlag, Berlin, 1991), to appear.
J. F. Muzy, E. Bacry, and A. Arnéodo, Multifractal formalism for self-affine signals: The structure function approach versus the wavelet transform modulus maxima method,Phys. Rev. E, to appear.
A. Grossmann and J. Morlet,SIAM J. Math. Anal. 15:723 (1984).
I. Daubechies, A. Grossmann, and Y. Meyer,J. Math. Phys. 127:1271 (1986).
J. M. Combes, A. Grossmann, and P. Tchamitchian, eds.,Wavelets (Springer-Verlag, Berlin, 1988), and references therein.
Y. Meyer,Ondelettes (Hermann, Paris, 1990).
P. G. Lemarié, ed.,Les Ondelettes en 1989 (Springer-Verlag, Berlin, 1990).
A. Arnéodo, G. Grasseau, and M. Holschneider,Phys. Rev. Lett. 61:2281 (1988); and inWavelets, J. M. Combes, A. Grossmann, and P. Tchamitchian, eds. (Springer-Verlag, Berlin, 1988), p. 182.
M. Holschneider,J. Stat. Phys. 50:963 (1988); Thesis, University of Aix-Marseille II (1988).
A. Arnéodo, F. Argoul, J. Elezgaray, and G. Grasseau, inNonlinear Dynamics, G. Turchetti, ed. (World Scientific, Singapore, 1988), p. 130.
A. Arnéodo, F. Argoul, and G. Grasseau, inLes Ondelettes en 1989, P. G. Lemarié, ed. (Springer-Verlag, Berlin, 1990), p. 125.
G. Grasseau, Thesis, University of Bordeaux (1989).
A. Arnéodo, F. Argoul, E. Bacry, J. Elezgaray, E. Freysz, G. Grasseau, J. F. Muzy, and B. Pouligny, inWavelets and Their Applications, Y. Meyer, ed. (Springer, Berlin, 1992), p. 286.
M. Holschneider and P. Tchamitchian, inLes Ondelettes en 1989, P. G. Lemarié, ed. (Springer-Verlag, Berlin, 1990), p. 102.
S. Jaffard,C. R. Acad. Sci. Paris 308:79 (1989); Sur la dimension de Hausdorff de points singuliers d'une fonction, preprint (1991).
S. Maltet and W. L. Hwang,IEEE Trans. on Information Theory 38:617 (1992).
P. Cvitanovic, inXV International Colloquium on Group Theoretical Methods in Physics, R. Gilmore, ed. (World Scientific, Singapore, 1987).
P. Grassberger, R. Badii, and A. Politi,J. Stat. Phys. 51:135 (1988).
D. Katzen and I. Procaccia,Phys. Rev. Lett. 58:1169 (1987).
T. Bohr and M. H. Jensen,Phys. Rev. A 36:4904 (1987).
A. Arnéodo, F. Argoul, E. Freysz, J. F. Muzy, and B. Pouligny, inWavelet and Their Applications, M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, eds. (Jones and Bartlett, Boston, 1991), p. 241.
J. M. Ghez and S. Vaienti,Nonlinearity 5:772; 791 (1992).
K. Falconer,The Geometry of Fractal Sets (Cambridge University Press, Cambridge, 1985).
P. Flandrin, Wavelet analysis and synthesis of fractional Brownian motion, preprint (1991).
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Bacry, E., Muzy, J.F. & Arnéodo, A. Singularity spectrum of fractal signals from wavelet analysis: Exact results. J Stat Phys 70, 635–674 (1993). https://doi.org/10.1007/BF01053588
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DOI: https://doi.org/10.1007/BF01053588