Об одном классе самоаффинных множеств на плоскости, заданных шестью гомотетиями

Журнал Средневолжского математического общества. 2023. Т. 25. № 1. С. 519–530.

Bagaev A.

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.

Keywords: attractor, аттрактор, Lebesgue measure, Cantor set, self-affine set, homothety, iterated functions system, самоаффинное множество, гомотетия, канторово множество, система итерированных функций, мера Лебега