## Abstract

The estimation or the computation of the Hausdorff dimension of self-affine fractals is of considerable interest. An almost-sure formula for that has been given by K.J. Falconer. However, the precise Hausdorff dimension formulas have been given only in special cases. This problem is even unsolved for a general integral self-affine set F, which is generated by an n×n integer expanding matrix T (not necessarily a similitude) and a finite set A⊂R^{n} of integer vectors so that F=T^{−1}(F+A). In this paper, we focus on the pivotal case F⊂R^{2} and show that the Hausdorff dimension of F is the limit of a monotonic sequence of McMullen-type dimensions by introducing a process, which we call the fractal perturbation (or deflection) method. In fact, the perturbation method is developed to deal with the pathological case where the characteristic polynomial of T is irreducible over Z. We also consider certain examples of exceptional self-affine fractals for which the Hausdorff dimension is less than the upper bound given by Falconer's formula. These examples show that our approach leads to highly non-trivial computation.

Original language | English |
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Article number | 123291 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 483 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 Mar 2020 |

### Bibliographical note

Publisher Copyright:© 2019 Elsevier Inc.

### Funding

The author would like to thank Professor Ka-Sing Lau for encouraging him to study the exact Hausdorff dimension of integral self-affine sets and for his constant support. The author is indebted to Dr. Ilker Kocyigit for his great help with computer programs and Dr. Abdulkerim Saracoglu for sharing his computer knowledge with me. The theoretical part (Sections 1–5 ) of this paper is related to one of the research problems proposed to DAAD under the supervision of Dr. Christoph Bandt. He also thanks DAAD for its support during his visit to Department of Mathematics, Ernst Moritz Arndt University of Greifswald. The computational work in Section 6 was supported by National Center for High Performance Computing (UHeM) of Turkey under Project Grant No. 1003222014 . Finally, he is grateful to the anonymous referee(s) for many valuable comments. The author would like to thank Professor Ka-Sing Lau for encouraging him to study the exact Hausdorff dimension of integral self-affine sets and for his constant support. The author is indebted to Dr. Ilker Kocyigit for his great help with computer programs and Dr. Abdulkerim Saracoglu for sharing his computer knowledge with me. The theoretical part (Sections 1?5) of this paper is related to one of the research problems proposed to DAAD under the supervision of Dr. Christoph Bandt. He also thanks DAAD for its support during his visit to Department of Mathematics, Ernst Moritz Arndt University of Greifswald. The computational work in Section 6 was supported by National Center for High Performance Computing (UHeM) of Turkey under Project Grant No. 1003222014. Finally, he is grateful to the anonymous referee(s) for many valuable comments.

Funders | Funder number |
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National Center for High Performance Computing | 1003222014 |

UHeM | |

Deutscher Akademischer Austauschdienst |

## Keywords

- Fractal perturbation
- Hausdorff dimension
- Integral self-affine sets and tiles