Email this article On a class of self-affine sets on the plane given by six homotheties
- Authors: Bagaev A.V.1
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Affiliations:
- Nizhny Novgorod State Technical University n.a. R.E. Alekseev
- Issue: Vol 25, No 1 (2023)
- Pages: 519-530
- Section: Mathematics
- Submitted: 12.12.2025
- Accepted: 14.12.2025
- Published: 24.12.2025
- URL: https://bakhtiniada.ru/2079-6900/article/view/357848
- DOI: https://doi.org/10.15507/2079-6900.25.202301.519-530
- ID: 357848
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Abstract
This paper is devoted to a class of self-affine sets on the plane determined by six homotheties. Centers of these homotheties are located at the vertices of a regular hexagon P, and the homothetic coefficients belong to the interval (0, 1). One must note that equality of homothetic coefficients is not assumed. A self-affine set on the plane is a non-empty compact subset that is invariant with respect to the considered family of homotheties. The existence and uniqueness of such a set is provided by Hutchinson’s theorem. The goal of present work is to investigate the influence of homothetic coefficients on the properties of a self-affine set. To describe the set, barycentric coordinates on the plane are introduced. The conditions are found under which the self-affine set is: a) the hexagon P; b) a Cantor set in the hexagon P. The Minkowski and the Hausdorff dimensions of the indicated sets are calculated. The conditions providing vanishing Lebesgue measure of self-affine set are obtained. Examples of self-affine sets from the considered class are presented.
About the authors
Andrey V. Bagaev
Nizhny Novgorod State Technical University n.a. R.E. Alekseev
Author for correspondence.
Email: a.v.bagaev@gmail.com
ORCID iD: 0000-0001-5155-4175
Associate Professor, Department of Applied Mathematics
Russian Federation, 24 Minina St., Nizhny Novgorod 603950, RussiaReferences
- J. E. Hutchinson, "Fractals and self-similarity" , Indiana Univ. Math. J., 30:5 (1981), 713–747.
- R. M. Crownover, Introduction in fractals and chaos, Jones and Bartlett Publ., Boston, 1995, 350 p.
- K. J. Falconer, Fractal geometry: mathematical foundations and applications, Wiley, New York, 2014, 400 p.
- M. Hata, "On the structure of self-similar sets", Japan J. Appl. Math., 2 (1985), 381–414. DOI: https://doi.org/10.1007/BF03167083
- M. F. Barnsley, Fractals everywhere, Academic Press, Boston, 1988, 394 p.
- D. Broomhead, J. Montaldi, N. Sidorov, "Golden gaskets: variations on the Sierpinski", Nonlinearity, 17:4 (2004), 1455–1480. DOI: https://doi.org/10.1088/0951-7715/17/4/017
- Th. Jordan, "Dimension of fat Sierpinski gaskets", Real Anal. Exchange, 31:1 (2005), 97–110.
- A. V. Bagaev, A. V. Kiseleva, "Attractors of systems of three iterated homotheties of the Euclidean plane", XXIX Russian Scientific and Practical Conference on Graphic Information Technologies and Systems (KOGRAF-2019): Proceedings, 2019, 136–140 (In Russ.).
- A. V. Bagaev, A. V. Kiseleva, "On multidimensional analogs of the Sierpinski triangle", XXVI International Scientifc and Technical Conference "Information Systems and Technologies-2020": Proceedings, 2020, 1148–1152 (In Russ.).
- A. V. Bagaev, A. V. Kiseleva, "On the Lebesgue measure of the attractors given by homotheties with affine independent centers", XXVI International Scientifc and Technical Conference "Information Systems and Technologies-2021": Proceedings, 2021, 945–948 (In Russ.).
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