Abstract
The singular behavior of functions is generally characterized by their Hölder exponent. However, we show that this exponent poorly characterizes oscillating singularities. We thus introduce a second exponent that accounts for the oscillations of a singular behavior and we give a characterization of this exponent using the wavelet transform. We then elaborate on a “grand-canonical” multifractal formalism that describes statistically the fluctuations of both the Hölder and the oscillation exponents. We prove that this formalism allows us to recover the generalized singularity spectrum of a large class of fractal functions involving oscillating singularities.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. B. Mandelbrot,The Fractal Geometry of Nature (Freeman, San Francisco, 1982).
A. Aharony and J. Feder, eds., Fractals in Physics, Essays in honour of B. B. Mandelbrot,Physica D 38 (1989), and references therein.
T. Vicskek, M. Shlesinger, and M. Matsushita, eds.,Fractals in Natural Sciences (World Scientific, Singapore, 1994), and references therein.
P. Goupillaud, A. Grossmann, and J. Morlet,Geoexploration 23:85 (1984).
A. Grossmann and J. Morlet,SIAM J. Math. Anal. 15:723 (1984); inMathematics and Physics, Lectures on Recent Results, L. Streit, ed. (World Scientific, Singapore, 1985).
J. M. Combes, A. Grossmann, and P. Tchamitchian, eds.,Wavelets (Springer, Berlin, 1989), and references therein.
Y. Meyer,Ondelettes (Hermann, Paris, 1990).
P. G. Lemarié, ed.,Les Ondelettes en 1989 (Springer, Berlin, 1990), and references therein.
Y. Meyer, ed.,Wavelets and Applications (Springer, Berlin, 1992) and references therein.
I. Daubechies,Ten Lectures on Wavelets (SIAM, Philadelphia, 1992).
M. B. Ruskai, G. Beylkin, R. Coifman, I. Daubechies, S. Mallat, Y. Meyer, and L. Raphael, eds.,Wavelets and Their Applications (Jones and Bartlett, Boston, 1992), and references therein.
C. K. Chui,An Introduction to Wavelets (Academic Press, Boston, 1992).
Y. Meyer and S. Roques, eds.,Progress in Wavelet Analysis and Applications (Editions Frontières, Gif sur Yvette, France, 1993), and references therein.
A. Arneodo, F. Argoul, E. Bacry, J. Elezgaray, and J. F. Muzy,Ondelettes, Multifractales et Turbulences: de l'ADN aux croissances cristallines (Diderot Editeur, Arts et Sciences, Paris, 1995).
G. Erlebacher, M. Y. Hussaini, and L. M. Jameson, eds.,Wavelets: Theory and Applications (Oxford University Press, Oxford, 1996), and references therein.
S. Jaffard,C. R. Acad. Sci. Paris 308 (Serie I):79 (1989).
M. Holschneider and P. Tchamitchian, inLes Ondelettes en 1989, P. G. Lemarié, ed. (Springer, Berlin, 1990), p. 102.
S. Mallat and W. L. Hwang,IEEE Trans. Information Theory 38:617 (1992).
E. Bacry, A. Arneodo, U. Frisch, Y. Gagne, and E. J. Hopfinger, inTurbulence and Coherent Structures, M. Lesieur and O. Metais, eds. (Kluwer, Dordrecht, 1991), p. 203.
M. Vergassola and U. Frisch,Physica D 54:58 (1991).
A. Arneodo, E. Bacry, and J. F. Muzy, Wavelet analysis of fractal signals: Direct determination of the singularity spectrum of fully developed turbulence data, inProceedings of the USA-French Workshop on Wavelets and Turbulence (Princeton, June 1991), to appear.
M. Vergassola, R. Benzi, L. Biferale, and D. Pissarenko,J. Phys. A 26:6093 (1993).
J. F. Muzy, E. Bacry, and A. Arneodo,Int. J. Bifurcation Chaos 4:245 (1994).
U. Frisch and G. Parisi, inTurbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, M. Ghil, R. Benzi, and G. Parisi, eds. (North-Holland, Amsterdam, 1985), p. 84.
T. C. Halsey, M. H. Jensen, L. P. Kadanoff, I. Procaccia, and B. I. Shraiman,Phys. Rev. A 33:1141 (1986).
P. Collet, J. Lebowitz, and A. Porzio,J. Stat. Phys. 47:609 (1987).
D. Rand,Ergod. Theory Dynam. Syst. 9:527 (1989).
J. F. Muzy, E. Bacry, and A. Arneodo,Phys. Rev. Lett. 67:3515 (1991); inProgress in Wavelet Analysis and Applications, Y. Meyer and S. Roques, eds. (Editions Frontières, Gif sur Yvette, France, 1993), p. 323.
E. Bacry, J. F. Muzy, and A. Arneodo,J. Stat. Phys. 70:635 (1993).
J. F. Muzy, E. Bacry, and A. Arneodo,Phys. Rev. E 47:875 (1993).
M. Holschneider,J. Stat. Phys. 50:963 (1988); Thesis, University of Aix-Marseille II (1988).
S. Jaffard, Multifractal formalism for functions: Part I and Part II,SIAM J. Math. Anal., to appear.
R. Badii, Thesis, University of Zurich (1987).
M. J. Feigenbaum,J. Stat. Phys. 46:919, 925 (1987).
T. Bohr and T. Tél, inDirections in Chaos, Vol. 2, Hao-Bai-Lin, ed. (World Scientific, Singapore, 1988).
A. Arneodo, E. Bacry, and J. F. Muzy,Physica A 213:232 (1995).
S. Jaffard and Y. Meyer, Wavelet methods for pointwise regularity and local oscillations of functions,Mem. AMS, to appear.
A. Arneodo, E. Bacry, and J. F. Muzy,Phys. Rev. Lett. 74:4823 (1995).
N. Delprat, B. Escudié, P. Guillemain, R. Kroland-Martinet, Ph. Tchamitchian, and B. Torrésani,IEEE Trans. Information Theory 38:644 (1992).
A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy,J. of Fourier Anal. & Appl. (1977), to appear.
A. Arneodo, E. Bacry, and J. F. Muzy,Europhys. Lett. 25:479 (1994).
A. Arneodo, F. Argoul, J. F. Muzy, M. Tabard, and E. Bacry,Fractals 1:629 (1993).
A. Arneodo, E. Bacry, P. V. Graves, and J. F. Muzy,Phys. Rev. Lett. 74: 3293 (1995).
A. Arneodo, inWavelets: Theory and Applications, G. Erlebacher, M. Y. Hussaini, and L. M. Jameson, eds. (Oxford University Press, Oxford, 1996), p. 349.
L. S. Young,Ergod. Theory Dynam. Syst. 2:109 (1982).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Arneodo, A., Bacry, E., Jaffard, S. et al. Oscillating singularities on cantor sets: A grand-canonical multifractal formalism. J Stat Phys 87, 179–209 (1997). https://doi.org/10.1007/BF02181485
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02181485