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Inverse Problems in Multifractal Analysis

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Fractal Geometry and Stochastics V

Part of the book series: Progress in Probability ((PRPR,volume 70))

Abstract

We present recent results regarding the construction of positive measures with a prescribed multifractal nature, as well as their counterpart in multifractal analysis of Hölder continuous functions.

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References

  1. N. Attia, J. Barral, Hausdorff and packing spectra, large deviations, and free energy for branching random walks in \(\mathbb{R}^{d}\). Commun. Math. Phys. 331, 139–187 (2014)

    Article  Google Scholar 

  2. V. Aversa, C. Bandt, The multifractal spectrum of discrete measures. 18th winter school on abstract analysis. Acta Univ. Carolin. Math. Phys. 31(2), 5–8 (1990)

    Google Scholar 

  3. J. Barral, Inverse problems in multifractal analysis of measures. Ann. Sci. Ec. Norm. Sup. (to appear). arXiv:1311.3895v3

    Google Scholar 

  4. J. Barral, D.-J. Feng, Weighted thermodynamic formalism and applications. Asian J. Math. 16, 319–352 (2012)

    Article  Google Scholar 

  5. J. Barral, D.-J. Feng, Multifractal formalism for almost all self-affine measures. Commun. Math. Phys. 318, 473–504 (2013)

    Article  Google Scholar 

  6. J. Barral, B. Mandelbrot, Random multiplicative multifractal measures, in Fractal Geometry and Applications. Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2 (American Mathematical Society, Providence, 2004), pp. 3–90

    Google Scholar 

  7. J. Barral, M. Mensi, Gibbs measures on self-affine Sierpinski carpets and their singularity spectrum. Ergod. Theory Dyn. Syst. 27, 1419–1443 (2007)

    Article  Google Scholar 

  8. J. Barral, S. Seuret, From multifractal measures to multifractal wavelet series. J. Fourier Anal. Appl. 11, 589–614 (2005)

    Article  Google Scholar 

  9. J. Barral, S. Seuret, The singularity spectrum of Lévy processes in multifractal time. Adv. Math. 214, 437–468 (2007)

    Article  Google Scholar 

  10. A. Batakis, B. Testud, Multifractal analysis of inhomogeneous Bernoulli products. J. Stat. Phys. 142, 1105–1120 (2011)

    Article  Google Scholar 

  11. F. Bayard, Multifractal spectra of typical and prevalent measures. Nonlinearity 26, 353–367 (2013)

    Article  Google Scholar 

  12. F. Ben Nasr, Analyse multifractale de mesures, C. R. Acad. Sci. Paris, Série I 319, 807–810 (1994)

    Google Scholar 

  13. G. Brown, G. Michon, J. Peyrière, On the multifractal analysis of measures. J. Stat. Phys. 66, 775–790 (1992)

    Article  Google Scholar 

  14. Z. Buczolich, S. Seuret, Typical measures on [0, 1]d satisfy a multifractal formalism. Nonlinearity 23, 2905–2911 (2010)

    Google Scholar 

  15. Z. Buczolich, S. Seuret, Measures and functions with prescribed homogeneous spectrum. J. Fractal. Geom. 1, 295–333 (2014)

    Article  Google Scholar 

  16. G.A. Edgar, R.D. Mauldin, Multifractal decompositions of digraph recursive fractals. Proc. Lond. Math. Soc. 65, 604–628 (1992)

    Article  Google Scholar 

  17. K.J. Falconer, The multifractal spectrum of statistically self-similar measures. J. Theor. Probab. 7, 681–702 (1994)

    Article  Google Scholar 

  18. K.J. Falconer, Generalized dimensions of measures on self-affine sets. Nonlinearity 12, 877–891 (1999).

    Article  Google Scholar 

  19. K.J. Falconer, Representation of families of sets by measures, dimension spectra and Diophantine approximation. Math. Proc. Camb. Philos. Soc. 128, 111–121 (2000)

    Article  Google Scholar 

  20. K.J. Falconer, Fractal Geometry – Mathematical Foundations and Applications, 3rd ed. (Wiley, Hoboken, 2014)

    Google Scholar 

  21. A.H. Fan, Multifractal analysis of infinite products. J. Stat. Phys. 86, 1313–1336 (1997)

    Article  Google Scholar 

  22. D.-J. Feng, The limit Rademacher functions and Bernoulli convolutions associated with Pisot numbers. Adv. Math. 195, 24–101 (2005)

    Article  Google Scholar 

  23. D.-J. Feng, Gibbs properties of self-conformal measures and the multifractal formalism. Ergod. Theory Dyn. Syst. 27, 787–812 (2007)

    Article  Google Scholar 

  24. D.-J. Feng, K.S. Lau, Multifractal formalism for self-similar measures with weak separation condition. J. Math. Pures Appl. 92, 407–428 (2009)

    Article  Google Scholar 

  25. D.-J. Feng, E. Olivier, Multifractal analysis of the weak Gibbs measures and phase transition–Application to some Bernoulli convolutions. Ergod. Theory Dyn. Syst. 23, 1751–1784 (2003)

    Article  Google Scholar 

  26. U. Frisch, G. Parisi, On the singularity structure of fully developed turbulence, in Turbulence, and Predictability in Geophysical Fluid Dynamics and Climate Dynamics, ed. by M. Ghil et al. (North Holland, Amsterdam/New York, 1985), pp. 84–88

    Google Scholar 

  27. T. C. Halsey, M. H. Jensen, L.P. Kadanoff, I. Procaccia, B.I. Shraiman, Fractal measures and their singularities: the characterisation of strange sets. Phys. Rev. A 33, 1141–1151 (1986)

    Article  Google Scholar 

  28. Y. Heurteaux, Dimension of measures: the probabilistic approach. Publ. Mat. 51, 243–290 (2007)

    Article  Google Scholar 

  29. R. Holley, E.C. Waymire, Multifractal dimensions and scaling exponents for strongly bounded random fractals. Ann. Appl. Probab. 2, 819–845 (1992)

    Article  Google Scholar 

  30. G. Iommi, Multifractal Analysis for countable Markov shifts. Ergod. Theory Dyn. Syst. 25, 1881–1907 (2005)

    Article  Google Scholar 

  31. S. Jaffard, Exposants de Hölder en des points donnés et coefficients d’ondelettes. C. R. Acad. Sci. Paris Série I 308, 79–81 (1989)

    Google Scholar 

  32. S. Jaffard, Construction de fonctions multifractales ayant un spectre de singularités prescrit. C. R. Acad. Sci. Paris, Sér. I 315, 19–24 (1992)

    Google Scholar 

  33. S. Jaffard, Multifractal formalism for functions. I. Results valid for all functions. II Self-similar functions. SIAM J. Math. Anal. 28, 944–970 & 971–998 (1997)

    Google Scholar 

  34. S. Jaffard, The multifractal nature of Lévy processes. Probab. Theory Relat. Fields 114, 207–227 (1999)

    Article  Google Scholar 

  35. S. Jaffard, On the Frisch-Parisi conjecture. J. Math. Pures Appl. 79, 525–552 (2000)

    Article  Google Scholar 

  36. S. Jaffard, Wavelet techniques in multifractal analysis, in Fractal Geometry and Applications. Proceedings of Symposia in Pure Mathematics, vol. 72, Part 2 (American Mathematical Society, Providence, 2004), pp. 91–152

    Google Scholar 

  37. S. Jaffard, Y. Meyer, On the pointwise regularity of functions in critical Besov spaces. J. Funct. Anal. 175, 415–434 (2000)

    Article  Google Scholar 

  38. J.F. King, The singularity spectrum for general Sierpinski carpets. Adv. Math. 116, 1–8 (1995)

    Article  Google Scholar 

  39. K.S. Lau, S.-M. Ngai, Multifractal measures and a weak separation condition. Adv. Math. 141, 45–96 (1999)

    Article  Google Scholar 

  40. J. Lévy-Véhel, R. Vojak, Multifractal analysis of Choquet capacities. Adv. Appl. Math. 20, 1–43 (1998)

    Article  Google Scholar 

  41. J. Lévy-Véhel, C. Tricot, On various multifractal spectra, in Fractal Geometry and Stochastics III. Progress in Probability, vol. 57 (Birkhäuser, Boston, 2004), pp. 23–42

    Google Scholar 

  42. B.B. Mandelbrot, C.J.G. Evertsz, Y. Hayakawa, Exactly self-similar left sided multifractal measures. Phys. Rev. A 42, 4528–4536 (1990)

    Article  Google Scholar 

  43. B.B. Mandelbrot, R. Riedi, Multifractal formalism for infinite multinomial measures. Adv. Appl. Math. 16, 132–150 (1995)

    Article  Google Scholar 

  44. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, Cambridge, 1995)

    Book  Google Scholar 

  45. Y. Meyer, Ondelettes et Opérateurs I (Hermann, Paris, 1990)

    Google Scholar 

  46. G.M. Molchan, Scaling exponents and multifractal dimensions for independent random cascades. Commun. Math. Phys. 179, 681–702 (1996)

    Article  Google Scholar 

  47. J.F. Muzy, A. Arneodo, E. Bacry, A multifractal formalism revisited with wavelets. Int. J. Bifur. Chaos Appl. Sci. Eng. 4, 245–302 (1994)

    Article  Google Scholar 

  48. S.-M. Ngai, A dimension result arising from the L q-spectrum of a measure. Proc. Am. Math. Soc. 125, 2943–2951 (1997)

    Article  Google Scholar 

  49. L. Olsen, A multifractal formalism. Adv. Math. 116, 82–196 (1995)

    Article  Google Scholar 

  50. L. Olsen, Self-affine multifractal Sierpinski sponges in \(\mathbb{R}^{d}\). Pac. J. Math. 183, 143–199 (1998)

    Article  Google Scholar 

  51. L. Olsen, N. Snigireva, Multifractal spectra of in-homogeneous self-similar measures. Indiana Univ. Math. J. 57, 1789–1844 (2008)

    Article  Google Scholar 

  52. Y. Pesin, Dimension Theory in Dynamical Systems: Contemporary Views and Applications (University of Chicago Press, Chicago, 1997)

    Book  Google Scholar 

  53. D.A. Rand, The singularity spectrum f(α) for cookie-cutters. Ergod. Theory Dyn. Syst. 9, 527–541 (1989)

    Article  Google Scholar 

  54. R.T. Rockafellar, Convex Analysis (Princeton University Press, Princeton, 1970)

    Google Scholar 

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Authors and Affiliations

  1. Laboratoire d’Analyse, Géométrie et Applications (UMR 7539), Institut Galilée, Université Paris 13, Sorbonne-Paris-Cité, 99 avenue Jean-Baptiste Clément, 93430, Villetaneuse, France

    Julien Barral

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  1. Julien Barral
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Correspondence to Julien Barral .

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Editors and Affiliations

  1. Institut für Mathematik und Informatik, Universität Greifswald, Greifswald, Germany

    Christoph Bandt

  2. Mathematical Institute, University of St Andrews, St Andrews, United Kingdom

    Kenneth Falconer

  3. Institut für Mathematik, Friedrich-Schiller-Universität Jena, Jena, Germany

    Martina Zähle

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Barral, J. (2015). Inverse Problems in Multifractal Analysis. In: Bandt, C., Falconer, K., Zähle, M. (eds) Fractal Geometry and Stochastics V. Progress in Probability, vol 70. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-18660-3_14

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