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Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva, 2023, Volume 25, Number 1, Pages 519–530
DOI: https://doi.org/10.15507/2079-6900.25.202301.519-530
(Mi svmo846)
 

Mathematics

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

A. V. Bagaev

Nizhny Novgorod State Technical University
References:
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.
Document Type: Article
UDC: 514.7
MSC: 28A80
Language: Russian
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
Citation in format AMSBIB
\Bibitem{Bag23}
\by A.~V.~Bagaev
\paper On a class of self-affine sets on the plane given by six homotheties
\jour Zhurnal SVMO
\yr 2023
\vol 25
\issue 1
\pages 519--530
\mathnet{http://mi.mathnet.ru/svmo846}
\crossref{https://doi.org/10.15507/2079-6900.25.202301.519-530}
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    Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva
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