Aequationes mathematicae

, Volume 86, Issue 1–2, pp 23–56 | Cite as

Dimensions of random self-affine multifractal Sierpinski sponges in \({{\mathbb{R}}^d}\)

  • Lars Olsen


In this paper we study the Hausdorff and packing dimensions and the Rényi dimensions of random self-affine multifractal Sierpinski sponges in \({{\mathbb{R}}^{d}}\).

Mathematics Subject Classification (1991)



Multifractals self-affine measures Sierpinski sponges Hausdorff dimension packing dimension Rényi dimension local dimension 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abercrombie A., Nair R.: On the Hausdorff dimension of certain self-affine sets. Studia Math. 152, 105–124 (2002)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bedford, T.: Crinkly curves, Markov partitions and box dimensions in self-similar sets. Ph.D. dissertation, University of Warwick (1984)Google Scholar
  3. 3.
    Barral J., Mensi M.: Gibbs measures on self-affine Sierpinski carpets and their singularity spectrum. Ergodic Theory Dyn. Syst. 27, 1419–1443 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Barral J., Mensi M.: Multifractal analysis of Birkhoff averages on ‘self-affine’ symbolic spaces. Nonlinearity 21, 2409–2425 (2008)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Cutler, C.D.: Measure disintegrations with respect to σ-stable monotone indices and pointwise representation of packing dimension. In: Proceedings of the 1990 Measure Theory Conference at Oberwolfach. Supplemento Ai Rendiconti del Circolo Mathematico di Palermo, Ser. II, vol. 28, pp. 319–340 (1992)Google Scholar
  6. 6.
    Falconer K.J.: The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Philos. Soc. 103, 339–350 (1988)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Falconer K.J.: Techniques in Fractal Geometry. Wiley, Chichester (1997)MATHGoogle Scholar
  8. 8.
    Falconer K.J.: Generalized dimensions of measures on self-affine sets. Nonlinearity 12, 877–891 (1999)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Feng D.-J., Wang Y.: A class of self-affine sets and self-affine measures. J. Fourier Anal. Appl. 11, 107–124 (2005)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Gui Y., Li W.: Hausdorff dimension of subsets with proportional fibre frequencies of the general Sierpinski carpet. Nonlinearity 20, 2353–2364 (2007)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Gui Y., Li W.: Hausdorff dimension of fiber-coding sub-Sierpinski carpets. J. Math. Anal. Appl. 331, 62–68 (2007)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Gui Y., Li W.: A generalized multifractal spectrum of the general Sierpinski carpets. J. Math. Anal. Appl. 348, 180–192 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Gui Y., Li W.: A random version of McMullen–Bedford general Sierpinski carpets and its application. Nonlinearity 21, 1745–1758 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gui Y., Li W.: Multiscale self-affine Sierpinski carpets. Nonlinearity 23, 495–512 (2010)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Haase, H.: A survey of the dimensions of measures. In: Bandt, C., Flachsmeyer, J., Haase, H. (eds.) Proceedings of the Conference “Topology and Measure VI”, Warnemünde, Germany, Mathematical Research, vol. 66, pp. 66–75. Topology, Measures, and Fractals, Akademie Verlag, (1992)Google Scholar
  16. 16.
    Halsey T.C., Jensen M.H., Kadanoff L.P., Procaccia I., Shraiman B.J.: Fractal measures and their singularities: The characterization of strange sets. Phys. Rev. A 33, 1141–1151 (1986)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Hutchinson J.: Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747 (1981)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Jordan T., Simon K.: Multifractal analysis of Birkhoff averages for some self-affine IFS. Dyn. Syst. 22, 469–483 (2007)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Kenyon R., Peres Y.: Measures of full dimension on affine-invariant sets. Ergodic Theory Dyn. Syst. 16, 307–323 (1996)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    King J.: The singularity spectrum for general Sierpinski carpets. Adv. Math. 116, 1–8 (1995)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Mattila, P.: Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge (1995)Google Scholar
  22. 22.
    McMullen C.: The Hausdorff dimension of general Sierpinski Carpets. Nagoya Math. J. 96, 1–9 (1984)MathSciNetMATHGoogle Scholar
  23. 23.
    Nielsen O.: The Hausdorff and packing dimensions of some sets related to Sierpinski carpets. Can. J. Math. 51, 1073–1088 (1999)MATHCrossRefGoogle Scholar
  24. 24.
    Olsen L.: Self-affine multifractal Sierpinski sponges in R d. Pac. J. Math. 183, 143–199 (1998)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Olsen L.: Symbolic and geometric local dimensions of self-affine multifractal Sierpinski Sponges in \({{\mathbb{R}}^d}\). Stoch. Dyn. 7, 37–51 (2007)MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    Olsen, L.: Random self-affine multifractal Sierpinski sponges in \({{\mathbb{R}}^d}\). Monatshefte für Mathematik 162, 89–117 (2011)Google Scholar
  27. 27.
    Peres Y.: The self-affine carpets of McMullen and Bedford have infinite Hausdorff measure. Math. Proc. Camb. Philos. Soc. 116, 513–526 (1994)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Peres, Y., Solomyak, B.: Problems on self-similar sets and self-affine sets: an update Fractal geometry and stochastics, II (Greifswald/Koserow, 1998). Progr. Probab., vol. 46 pp. 95–106. Birkhäuser, Basel, (2000)Google Scholar
  29. 29.
    Riedi, R.H.: An improved multifractal formalism and self-affine measures. Ph.D. dissertation, ETH Zurich, Diss. ETH No. 10077 (1993)Google Scholar
  30. 30.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)Google Scholar
  31. 31.
    Takahashi S.: Dimension spectra of self-affine sets. Isr. J. Math. 127, 1–17 (2002)MATHCrossRefGoogle Scholar
  32. 32.
    Young L.-S.: Dimension, entropy and Lyapunov exponents. Ergodic Theory Dyn. Syst. 2, 109–124 (1982)MATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of St AndrewsSt. AndrewsScotland

Personalised recommendations