Journal of Statistical Physics

, Volume 108, Issue 3–4, pp 541–590 | Cite as

Multi-Fractal Formalism for Quasi-Self-Similar Functions

  • Jamil Aouidi
  • Mourad Ben Slimane
Article

Abstract

The study of multi-fractal functions has proved important in several domains of physics. Some physical phenomena such as fully developed turbulence or diffusion limited aggregates seem to exhibit some sort of self-similarity. The validity of the multi-fractal formalism has been proved to be valid for self-similar functions. But, multi-fractals encountered in physics or image processing are not exactly self-similar. For this reason, we extend the validity of the multi-fractal formalism for a class of some non-self-similar functions. Our functions are written as the superposition of “similar” structures at different scales, reminiscent of some possible modelization of turbulence or cascade models. Their expressions look also like wavelet decompositions. For the computation of their spectrum of singularities, it is unknown how to construct Gibbs measures. However, it suffices to use measures constructed according the Frostman's method. Besides, we compute the box dimension of the graphs.

multi-fractal formalism wavelets turbulence cascade models Gibbs measures non-self-similar functions Frostman's method box dimension 

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REFERENCES

  1. 1.
    J. Lévy Véhel and R. Riedi, Fractional Brownian Motion and Data Traffic Modeling: The Other End of the Spectrum. Fractals in Engineering, J. Lévy Véhel, E. Lutton, and C. Tricot, eds. (Springer Verlag, 1997).Google Scholar
  2. 2.
    M. Holschneider and Ph. Tchamitchan, Régularité locale de la fonction “non-differentiable” de Riemann, Lecture Notes in Math. 1438:102-124 (1990).Google Scholar
  3. 3.
    S. Jaffard, Exposants de Hölder en des points donnés et coefficients d'ondelettes, C. R. Acad. Sci. Paris Sér. I Math. 308:79-81 (1989).Google Scholar
  4. 4.
    S. Jaffard, Pointwise smoothness, two-microlocalization and wavelet coefficients, Publ. Mat. 35:155-168 (1991).Google Scholar
  5. 5.
    J. P. Kahane and J. Peyrière, Sur certaines martingales de Benoit Mandelbrot (1979).Google Scholar
  6. 6.
    B. Mandelbrot, Intermittent turbulence in selfsimilar cascades: Divergence of high moments and dimension of the carrier, J. Fluid Mech. 62:331 (1974).Google Scholar
  7. 7.
    B. Mandelbrot, Multiplications aléatoires itérées et distributions invariantes par moyenne pondérée aléatoire, C. R. Acad. Sci. Paris, Sér. I Math. 278:289-292 (1974).Google Scholar
  8. 8.
    J. Barral, Moments, continuité et analyse multifractale des martingales de Mandelbrot, Probab. Theory Related Fields 113:535-570 (1999).Google Scholar
  9. 9.
    Y. Gagne, Etude expérimentale de l'intermittence et des singularités dans le plan complexe en turbulence pleinement développée, Thèse de l'Université de Grenoble (1987).Google Scholar
  10. 10.
    A. Arneodo, E. Bacry, and J. F. Muzy, Singularity spectrum of fractal signals from wavelet analysis: Exact results, J. Statist. Phys. 70:635-674 (1993).Google Scholar
  11. 11.
    U. Frisch and G. Parisi, Fully developped turbulence and intermittency, in Proc. Internat. School Phys. Enrico Fermi (North Holland, 1985), pp. 84-88.Google Scholar
  12. 12.
    S. Jaffard, Multifractal formalism for functions. Part 1: Results valid for all functions and Part 2: Selfsimilar functions, SIAM J. Math. Anal. 28:944-998 (1997).Google Scholar
  13. 13.
    Y. Meyer, Ondelettes et opérateurs I: Ondelettes (Hermann, Paris, 1990).Google Scholar
  14. 14.
    M. Ben Slimane, Formalisme Multifractal pour quelques généralisations des fonctions autosimilaires, C. R. Acad. Sci. Paris Sér. I Math. 324:981-986 (1997).Google Scholar
  15. 15.
    M. Ben Slimane, Multifractal formalism and anisotropic selfsimilar functions, Math. Proc. Cambridge. Philos. Soc. 124:329-363 (1998).Google Scholar
  16. 16.
    M. Ben Slimane, Multifractal formalism for selfsimilar functions expanded in singular basis, Appl. Comput. Harmon. Anal. 11:387-419 (2001).Google Scholar
  17. 17.
    M. Barnsley and A. Sloan, A better way to compress images, Byte Magazine:215-223 (1988).Google Scholar
  18. 18.
    C. Meneveau and K. Srenivanasan, Phys. Rev. Lett. 59:1424 (1987).Google Scholar
  19. 19.
    A. Arneodo, A. Argoul, E. Bacry, J. F. Muzy, and M. Tabard, Golden mean arithmetic in the fractal branching of diffusion-limitted aggreagates, Technical Report, Paul-Pascal Research Center, France (1991).Google Scholar
  20. 20.
    I. Daubechies and J. C. Lagarias, On the thermodynamic formalism for functions, Rev. Math. Phys. 6:1033-1070 (1994).Google Scholar
  21. 21.
    I. Daubechies, Private communication.Google Scholar
  22. 22.
    M. Ben Slimane, Multifractal formalism for selfsimilar functions under the action of nonlinear dynamical systems, Constr. Approx. J. 15:209-240 (1999).Google Scholar
  23. 23.
    P. Collet, J. Lebowitz, and A. Porzio, The dimension specrum of some dynamical systems, J. Statist. Phys. 47:609-644 (1987).Google Scholar
  24. 24.
    L. Olsen, A multifractal formalism, Adv. Math. 116:82-196 (1995).Google Scholar
  25. 25.
    D. A. Rand, The singularity spectrum f(a) for cookie-cutters, Ergodic Theory Dynam. Systems 9:527-541 (1989).Google Scholar
  26. 26.
    J. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30:713-747 (1981).Google Scholar
  27. 27.
    G. Brown, G. Michon, and J. Peyrière, On the multifractal analysis of measures, J. Statist. Phys. 66:775-790 (1992).Google Scholar
  28. 28.
    F. Ben Nasr, Multifractal analysis of measures, C. R. Acad. Sci. Paris Sér. I Math. 319:807-810 (1994).Google Scholar
  29. 29.
    F. Ben Nasr, I. Bhouri, and Y. Heurtaux, A necessary condition and sufficient condition for a valid multifractal formalism, Prépublication d'Orsay, No. 58. To appear in Adv. Math. Google Scholar
  30. 30.
    Y. Meyer, Wavelets and Operators (Cambridge University Press, Cambridge, 1992).Google Scholar
  31. 31.
    J. Peyrière, Multifractal measures, Probabilistic and Stochastic Methods in Analysis, with Applications, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 372, pp. 175-186 (Kluwer Acad. Publ., Dordrecht, 1992).Google Scholar
  32. 32.
    P. Billingsly, Hausdorff dimension in probability theory, Illinois J. Math. 4:187-209 (1960).Google Scholar
  33. 33.
    S. Jaffard, On box dimension of graphs, C. R. Acad. Sci. Paris, Sér. I Math. 326:555-560 (1998).Google Scholar
  34. 34.
    S. Jaffard, On lacunary wavelet series, Ann. Appl. Probab. 10:313-329 (2000).Google Scholar
  35. 35.
    A. Arneodo, E. Bacry, S. Jaffard, and J. F. Muzy, Singularity spectrum of multifractal functions involving oscillating singularities, J. Fourier Anal. Appl. 4:159-174 (1998).Google Scholar
  36. 36.
    K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (Wiley, Toronto, 1990).Google Scholar
  37. 37.
    A. Arneodo, E. Bacry, and J. F. Muzy, Random cascades on wavelet dyadic trees, J. Math. Phys. 39:4142-4164 (1998).Google Scholar

Copyright information

© Plenum Publishing Corporation 2002

Authors and Affiliations

  • Jamil Aouidi
    • 1
  • Mourad Ben Slimane
    • 2
  1. 1.Département de MathématiquesFaculté des Sciences de BizerteJarzouna, BizerteTunisia
  2. 2.Département de MathématiquesFaculté des Sciences de TunisTunisTunisia

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