Abstract
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.
Similar content being viewed by others
References
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
Edgar G.A. (1998). Integral, Probability and Fractal measures. Springer, Berlin-Heidelberg-NewYork
Falconer K. (1988). The Hausdorff dimension of self-affine fractals. Math. Proc. Camb. Phil. Soc. 103: 339–350
Falconer K.J. (2003). Fractal Geometry. Wiley, NewYork
Falconer K.J. (1997). Techniques in Fractal Geometry. Wiley, NewYork
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. 89–94
Käenmäki A. (2004). On natural invariant measures on generalised iterated function systems. Ann. Acad. Sci. Fenn. Math. 29(2): 419–458
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
Krengel U. (1985). Ergodic Theorems. Walter de Gruyter, Berlin
Mattila P. (1995). Geometry of Sets and Measures in Euclidean Spaces. Cambridge University Press, Cambridge
Mauldin R.D., Williams S.C. (1998). Hausdorff dimension in graph directed constructions. Trans. Amer. Math. Soc. 309: 811–829
McMullen C. (1984). The Hausdorff dimension of general Sierpiński carpets. Nagoya Math. J. 96: 1–9
Peres Y., Solomyak B. (1996). Absolute continuity of Bernoulli convolutions, a simple proof. Math. Res. Lett. 3(2): 231–239
Peres, Y., Simon, K., Solomyak, B.: Absolute continuity for random iterated function systems with overlaps. http://arxiv.org/list/math.ds/0502200, 2005
Rogers C.A. (1970). Hausdorff Measures. Cambridge University Press, Cambridge
Simon K., Solomyak B. (2002). On the dimension of self-similar sets. Fractals 10: 59–65
Solomyak B. (1998). Measure and dimension for some fractal families. Math. Proc. Camb. Phil. Soc. 124(3): 531–546
Walters P. (1982). An introduction to Ergodic Theory. Springer, Berlin-Heidelberg-NewYork
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J.L. Lebowitz
Research of Jordan and Pollicott was supported by the EPSRC and the research of Simon was supported by an EU-Marie Curie grant and the OTKA Foundation #T42496.
Rights and permissions
About this article
Cite this article
Jordan, T., Pollicott, M. & Simon, K. Hausdorff Dimension for Randomly Perturbed Self Affine Attractors. Commun. Math. Phys. 270, 519–544 (2007). https://doi.org/10.1007/s00220-006-0161-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-006-0161-7