A. V. Bagaev

Nizhny Novgorod State Technical University n.a. R.E. Alekseev (Nizhny Novgorod, Russian Federation)

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.

Key Words: self-affine set, homothety, Cantor set, iterated function system, attractor, Lebesgue measure

For citation: A. V. Bagaev. On a class of self-affine sets on the plane given by six homotheties. Zhurnal Srednevolzhskogo matematicheskogo obshchestva. 25:1(2023), 519–530. DOI: https://doi.org/10.15507/2079-6900.25.202301.519-530

Information about the author:

Andrey V. Bagaev, Associate Professor, Department of Applied Mathematics, Nizhny Novgorod State Technical University n.a. R.E. Alekseev (24 Minina St., Nizhny Novgorod 603950, Russia), Ph.D. (Phys. and Math.), ORCID: http://orcid.org/0000-0001-5155-4175, a.v.bagaev@gmail.com