### Abstract

In this paper we shall consider a self-affine iterated function system in 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.

Original language | English |
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Pages (from-to) | 519-544 |

Number of pages | 26 |

Journal | Communications in Mathematical Physics |

Volume | 270 |

Issue number | 2 |

DOIs | |

Publication status | Published - Mar 2007 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*270*(2), 519-544. https://doi.org/10.1007/s00220-006-0161-7

**Hausdorff dimension for randomly perturbed self affine attractors.** / Jordan, Thomas; Pollicott, Mark; Simon, K.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 270, no. 2, pp. 519-544. https://doi.org/10.1007/s00220-006-0161-7

}

TY - JOUR

T1 - Hausdorff dimension for randomly perturbed self affine attractors

AU - Jordan, Thomas

AU - Pollicott, Mark

AU - Simon, K.

PY - 2007/3

Y1 - 2007/3

N2 - In this paper we shall consider a self-affine iterated function system in ℝd, 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.

AB - In this paper we shall consider a self-affine iterated function system in ℝd, 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=33846471209&partnerID=8YFLogxK

U2 - 10.1007/s00220-006-0161-7

DO - 10.1007/s00220-006-0161-7

M3 - Article

VL - 270

SP - 519

EP - 544

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 2

ER -