Communications in Mathematical Physics

, Volume 270, Issue 2, pp 519–544

Hausdorff Dimension for Randomly Perturbed Self Affine Attractors



In this paper we shall consider a self-affine iterated function system in \(\mathbb{R}^d\), d ≥ 2, where we allow a small random translation at each application of the contractions. We compute the dimension of a typical attractor of the resulting random iterated function system, complementing a famous deterministic result of Falconer, which necessarily requires restrictions on the norms of the contraction. However, our result has the advantage that we do not need to impose any additional assumptions on the norms. This is of benefit in practical applications, where such perturbations would correspond to the effect of random noise. We also give analogous results for the dimension of ergodic measures (in terms of their Lyapunov dimension). Finally, we apply our method to a problem originating in the theory of fractal image compression.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Edgar G.A. (1992). Fractal dimension of self-similar sets: some examples. Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II 28: 341–358 MathSciNetGoogle Scholar
  2. 2.
    Edgar G.A. (1998). Integral, Probability and Fractal measures. Springer, Berlin-Heidelberg-NewYork MATHGoogle Scholar
  3. 3.
    Falconer K. (1988). The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103: 339–350 MathSciNetMATHGoogle Scholar
  4. 4.
    Falconer K.J. (2003). Fractal Geometry. Wiley, NewYork MATHCrossRefGoogle Scholar
  5. 5.
    Falconer K.J. (1997). Techniques in Fractal Geometry. Wiley, NewYork MATHGoogle Scholar
  6. 6.
    Fisher, Y., Dudbridge, F., Bielefeld, B.: On the Dimension of fractally encoded images. In: Fractal Image Encoding and Analysis, Proceedings of the NATO ASI (Trondheim, 1995). Berlin-Heidelberg-NewYork: Springer, 1998, pp. 89–94Google Scholar
  7. 7.
    Käenmäki A. (2004). On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2): 419–458 MathSciNetMATHGoogle Scholar
  8. 8.
    Keane M., Simon K., Solomyak B. (2003). The dimension of graph directed attractors with overlaps on the line, with an application to a problem in fractal image recognition. Fund. Math. 180(3): 279–292 MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Krengel U. (1985). Ergodic Theorems. Walter de Gruyter, Berlin MATHGoogle Scholar
  10. 10.
    Mattila P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge MATHGoogle Scholar
  11. 11.
    Mauldin R.D., Williams S.C. (1998). Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309: 811–829 CrossRefMathSciNetGoogle Scholar
  12. 12.
    McMullen C. (1984). The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96: 1–9 MathSciNetMATHGoogle Scholar
  13. 13.
    Peres Y., Solomyak B. (1996). Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3(2): 231–239 MathSciNetMATHGoogle Scholar
  14. 14.
    Peres, Y., Simon, K., Solomyak, B.: Absolute continuity for random iterated function systems with overlaps., 2005Google Scholar
  15. 15.
    Rogers C.A. (1970). Hausdorff Measures. Cambridge University Press, Cambridge MATHGoogle Scholar
  16. 16.
    Simon K., Solomyak B. (2002). On the dimension of self-similar sets. Fractals 10: 59–65 CrossRefMathSciNetMATHGoogle Scholar
  17. 17.
    Solomyak B. (1998). Measure and dimension for some fractal families. Math. Proc. Camb. Phil. Soc. 124(3): 531–546 CrossRefMathSciNetMATHGoogle Scholar
  18. 18.
    Walters P. (1982). An introduction to Ergodic Theory. Springer, Berlin-Heidelberg-NewYork MATHGoogle Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Institute of MathematicsTechnical University of BudapestBudapestHungary

Personalised recommendations