Hausdorff Dimension for Randomly Perturbed Self Affine Attractors Authors Thomas Jordan Mathematics Institute University of Warwick Mark Pollicott Mathematics Institute University of Warwick Károly Simon Institute of Mathematics Technical University of Budapest Article

First Online: 15 December 2006 Received: 28 February 2006 Accepted: 02 August 2006 DOI :
10.1007/s00220-006-0161-7

Cite this article as: Jordan, T., Pollicott, M. & Simon, K. Commun. Math. Phys. (2007) 270: 519. doi:10.1007/s00220-006-0161-7
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.

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.

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