On a class of self-affine sets on the plane given by six homotheties

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.