Journal of Statistical Physics

, Volume 142, Issue 5, pp 1105–1120 | Cite as

Multifractal Analysis of Inhomogeneous Bernoulli Products

  • Athanasios BatakisEmail author
  • Benoît Testud


We are interested to the multifractal analysis of inhomogeneous Bernoulli products which are also known as coin tossing measures. We give conditions ensuring the validity of the multifractal formalism for such measures. On another hand, we show that these measures can have a dense set of phase transitions.


Hausdorff dimension Multifractal analysis Gibbs measure Phase transition 


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  1. 1.
    Batakis, A., Heurteaux, Y.: On relations between entropy and Hausdorff dimension of measures. Asian J. Math. 6(3), 399–408 (2002) MathSciNetzbMATHGoogle Scholar
  2. 2.
    BenNasr, F.: Analyse multifractale de mesures. C. R. Acad. Sci. Paris Sér. I Math. 319, 807–810 (1994) MathSciNetGoogle Scholar
  3. 3.
    BenNasr, F., Bhouri, I., Heurteaux, Y.: The validity of the multifractal formalism: results and examples. Adv. Math. 165, 264–284 (2002) MathSciNetCrossRefGoogle Scholar
  4. 4.
    Benzi, R., Paladin, G., Parisi, G., Vulpiani, A.: On the multifractal nature of fully developed turbulence and chaotic system. J. Phys. A 17, 3521–3531 (1984) MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Billingsley, P.: Ergodic Theory and Information. Wiley, New York (1965) zbMATHGoogle Scholar
  6. 6.
    Bisbas, A.: A multifractal analysis of an interesting class of measures. Colloq. Math. 69, 37–42 (1995) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Bowen, R.: Equilibrium states and the ergodic theory of Anosov diffeomorphisms. In: Lecture Notes in Mathematics, vol. 470. Springer, New York (1975) Google Scholar
  8. 8.
    Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775–790 (1992) ADSzbMATHCrossRefGoogle Scholar
  9. 9.
    Cawley, R., Mauldin, R.D.: Multifractal decompositions of Moran fractals. Adv. Math. 92, 196–236 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Collet, P.: Hausdorff dimension of singularities for invariant measures of expanding dynamical systems. In: Bamon, R. et al. (eds.) Dynamical Systems Valparaiso 1986. Lecture Notes in Mathematics, vol. 1331. Springer, Berlin (1988) CrossRefGoogle Scholar
  11. 11.
    Falconer, K.: Techniques in Fractal Geometry. Wiley, New York (1997) zbMATHGoogle Scholar
  12. 12.
    Fan, A.H.: Sur la dimension des mesures. Stud. Math. 111, 1–17 (1994) zbMATHGoogle Scholar
  13. 13.
    Feng, D.J.: Smoothness of the q spectrum of self similar measures with overlaps. J. Lond. Math. Soc. 68, 102–118 (2003) zbMATHCrossRefGoogle Scholar
  14. 14.
    Frisch, U., Parisi, G.: On the singularity structure of fully developed turbulence. In: Proc. Internat. School Phys. Enrico Fermi, pp. 84–88. U Frisch (North-Holland), Amsterdam (1985) Google Scholar
  15. 15.
    Grassberger, P.: Generalized dimension of strange attractors. Phys. Lett. A 97, 227–230 (1983) MathSciNetADSCrossRefGoogle Scholar
  16. 16.
    Grassberger, P., Procaccia, I.: Characterization of strange sets. Phys. Rev. Lett. 50, 346–349 (1983) MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Halsey, T.C., Jensen, M.H., Kadanoff, L.P., Procaccia, I., Shraiman, B.: Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986) MathSciNetADSzbMATHCrossRefGoogle Scholar
  18. 18.
    Hentschel, H., Procaccia, I.: The infinite number of generalized dimensions of fractals and strange attractors. Physica D 8, 435–444 (1983) MathSciNetADSzbMATHCrossRefGoogle Scholar
  19. 19.
    Heurteaux, Y.: Estimations de la dimension inférieure et de la dimension supérieure des mesures. Ann. Inst. H. Poincaré Probab. Stat. 34, 309–338 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Lau, K.S., Ngai, S.M.: Multifractal measures and a weak separation condition. Adv. Math. 141, 45–96 (1999) MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Mandelbrot, B.B.: Intermittent turbulence in self-similar cascades: Divergence of high moments and dimension of the carrier. J. Fluid Mech. 62, 331–358 (1974) ADSzbMATHCrossRefGoogle Scholar
  22. 22.
    Meakin, P., Conoglio, A., Stanley, H., Witten, T.: Scaling properties for the surfaces of fractal and nonfractal objects: An infinite hierarchy of critical exponents. Phys. Rev. A 34, 3325–3340 (1986) MathSciNetADSCrossRefGoogle Scholar
  23. 23.
    Michon, G.: Mesures de Gibbs sur les Cantor réguliers. Ann. Inst. H. Poincaré Phys. Théor. 58, 267–285 (1983) MathSciNetGoogle Scholar
  24. 24.
    Olsen, L.: A multifractal formalism. Adv. Math. 116, 82–196 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Rand, D.: The singularity spectrum f(α) for cookie-cutter. Ergod. Theory Dyn. Syst. 9, 527–541 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Rényi, A.: Probability Theory. North-Holland, Amsterdam (1970) Google Scholar
  27. 27.
    Riedi, R.: An improved multifractal formalism and self-similar measures. J. Math. Anal. Appl. 189, 462–490 (1995) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Ruelle, D.: Thermodynamic Formalism. Addison-Wesley, Reading (1978) zbMATHGoogle Scholar
  29. 29.
    Testud, B.: Mesures quasi-Bernoulli au sens faible: résultats et exemples. Ann. Inst. H. Poincaré Probab. Stat. 42, 1–35 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  30. 30.
    Testud, B.: Phase transitions for the multifractal analysis of self-similar measures. Nonlinearity 19, 1201–1217 (2006) MathSciNetADSzbMATHCrossRefGoogle Scholar
  31. 31.
    Ye, Y.L.: Multifractal of self-conformal measures. Nonlinearity 18, 2111–2133 (2005) MathSciNetADSzbMATHCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.MAPMO CNRS UMR 6628, Fédération Denis PoissonUniversité D’OrléansOrléans cedex 2France
  2. 2.LAMFA CNRS UMR 6140Université de Picardie Jules VerneAmiens cedexFrance

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