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Mathematics
On a class of self-affine sets on the plane given by six homotheties
A. V. Bagaev Nizhny Novgorod State Technical University
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.
Keywords:
self-affine set, homothety, Cantor set, iterated function system, attractor, Lebesgue measure.
Citation:
A. V. Bagaev, “On a class of self-affine sets on the plane given by six homotheties”, Zhurnal SVMO, 25:1 (2023), 519–530
Linking options:
https://www.mathnet.ru/eng/svmo846 https://www.mathnet.ru/eng/svmo/v25/i1/p519
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Abstract page: | 76 | Full-text PDF : | 18 | References: | 18 |
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