### Abstract

A self-affine IFS F = {f_{i} (x) = A_{i}x + t_{i}}^{i}_{i=1} is a finite list of contracting affine maps on ℝ^{d}, for some d ≥ 1. The attractor of F is (Formula presented.) where B is a sufficiently large ball centered at the origin. In most cases we cannot compute the dimension of f. However, if we add an independent additive random error to each f_{ik} in (0.1) then the dimension of this random perturbation (called almost self-affine system) is almost surely the so-called affinity dimension of theoriginal deterministic system. The dimension theory of almost self-affine sets and measures were described in Jordan et al. (Commun. Math. Phys. 270(2):519-544, 2007). The multifractal analysis of almost self-affine measures has been studied insome recent papers (Falconer, Nonlinearity 23:1047-1069, 2010; Barral and Feng, Commun. Math. Phys. 318(2):473-504, 2013). In the second part of this note I give a survey of this field but first we review some results related to the dimension theory of self-affine sets

Original language | English |
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Title of host publication | Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets |

Editors | V. Kannan, Michael F. Barnsley, Robert L. Devaney, Vinod Kumar P.B., Kenneth J. Devaney, Christoph Bandt |

Publisher | Springer New York LLC |

Pages | 103-127 |

Number of pages | 25 |

ISBN (Electronic) | 9783319081045 |

DOIs | |

Publication status | Published - Jan 1 2014 |

Event | 1st International Conference and Workshop on Fractals and Wavelets, ICFW India - Kochi, India Duration: Nov 13 2013 → Nov 16 2013 |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 92 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Other

Other | 1st International Conference and Workshop on Fractals and Wavelets, ICFW India |
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Country | India |

City | Kochi |

Period | 11/13/13 → 11/16/13 |

### Fingerprint

### Keywords

- Hausdorff dimension
- Processes in random environment
- Random fractals

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Fractals, Wavelets and their Applications - Contributions from the International Conference and Workshop on Fractals and Wavelets*(pp. 103-127). (Springer Proceedings in Mathematics and Statistics; Vol. 92). Springer New York LLC. https://doi.org/10.1007/978-3-319-08105-2__6