The Hausdorff dimension of integral self-affine sets through fractal perturbations

Ibrahim Kirat

doi:10.1016/j.jmaa.2019.06.062

The Hausdorff dimension of integral self-affine sets through fractal perturbations

Ibrahim Kirat

Department of Mathematics Engineering

Research output: Contribution to journal › Article › peer-review

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⊂Rⁿ of integer vectors so that F=T⁻¹(F+A). In this paper, we focus on the pivotal case F⊂R² 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
Article number	123291
Journal	Journal of Mathematical Analysis and Applications
Volume	483
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2019.06.062
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
National Center for High Performance Computing	1003222014
UHeM
Deutscher Akademischer Austauschdienst

Keywords

Fractal perturbation
Hausdorff dimension
Integral self-affine sets and tiles

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10.1016/j.jmaa.2019.06.062

Cite this

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The Hausdorff dimension of integral self-affine sets through fractal perturbations

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Keywords

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