Advertisement

Constructive Approximation

, Volume 43, Issue 3, pp 337–356 | Cite as

Fixed Points for the Multifractal Spectrum Map

  • Delphine Maman
  • Stéphane SeuretEmail author
Article
  • 120 Downloads

Abstract

For all continuous function g having a specific form that we call with increasing visibility, we construct a function f whose multifractal spectrum is such that \( d_f =g\circ f\). The function f is obtained as an infinite superposition of piecewise \(C^1\) functions, is also with increasing visibility, and is homogeneously multifractal; i.e., its restriction on any subinterval of \([0,1]\) has the same multifractal spectrum as the function f itself. In particular, we prove the existence of a function f which is its own multifractal spectrum; i.e., \(f=d_f\).

Keywords

Local regularity Hausdorff dimension and measure Multifractals Hölder exponent 

Mathematics Subject Classification

26A16 28A80 28C15 

References

  1. 1.
    Barral, J.: Inverse problems in multifractal analysis. Ann. ENS Paris (2015)Google Scholar
  2. 2.
    Barral, J., Durand, A., Jaffard, S., Seuret, S.: Local multifractal analysis, applications of fractals and dynamical systems in science and economics. In: Carfi, D., Lapidus, M.L., Pearse, E.J., van Frankenhuijsen, M. (eds.) Contemporary Mathematics. AMS, Providence (2013)Google Scholar
  3. 3.
    Barral, J., Fournier, N., Jaffard, S., Seuret, S.: A pure jump Markov process with a random singularity spectrum. Ann. Probab. 38(5), 1924–1946 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Beresnevich, V., Velani, S.: A mass transference principle and the Duffin–Schaeffer conjecture for Hausdorff measures. Ann. Math. (2) 164(3), 971–992 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Beresnevich, V., Velani, S.: Measure theoretic laws for lim sup sets. Mem. AMS 179, 846 (2006)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Brown, G., Michon, G., Peyrière, J.: On the multifractal analysis of measures. J. Stat. Phys. 66, 775–790 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Buczolich, Z., Seuret, S.: Measures and functions with prescribed singularity spectrum. J. Fractal Geom. 1(3), 295–333 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.LAMA (UMR 8050), UPEMLV, UPEC, CNRSUniversité Paris-EstCréteilFrance

Personalised recommendations