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The Mandelbrot set for a pair of improper similitudes of the plane
https://doi.org/10.26907/0021-3446-2025-10-30-43
Abstract
In this paper, the attractors of iterated function systems (IFS) consisting of two improper similitudes of the plane are investigated. The attractor of such IFS is either a connected or completely disconnected set. Sufficient conditions are found under which the attractor of such an IFS is a connected set. For an arbitrary IFS, sufficient conditions are obtained under which its attractor is a Cantor set. The main goal of the present work is to investigate the attractor Aa of two plane improper similitudes of the plane fi(z) = az, f2(z) = a(z — 1) + 1, a, z G C, 0 < |a| < 1. It is shown that Aa is one of the following sets: a segment, a Cantor set in a segment, a parallelogram, a Cantor set in a parallelogram. The Hausdorff dimension of the attractor Aa is calculated. Let M be the set of all vaiues of the parameter a for which the attractor Aa is connected. By analogy with Barnsley and Harington, we call M the Mandelbrot set. It is shown that, unlike the case of proper similitudes, the Mandelbrot set M for a pair of improper similitudes of the plane has a simple structure. Examples of attractors from the considered classes of IFS are presented.
About the Author
A. V. Bagaev
National Research University «Higher School of Economics»
Russian Federation
Russian Federation
Andrey V. Bagaev
25/12 Bolshaya Pecherskaya str., Nizhny Novgorod, 603155
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Review
For citations:
Bagaev A.V. The Mandelbrot set for a pair of improper similitudes of the plane. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika. 2025;(10):30-43. (In Russ.) https://doi.org/10.26907/0021-3446-2025-10-30-43
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