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 $\mathcal{A}_\alpha$ of two plane improper similitudes of the plane $f_1(z)=\alpha\overline{z}$, $f_2(z)=\alpha(\overline{z}-1)+1$, $\alpha,z\in\mathbb{C}$, $0<|\alpha|<1$. It is shown that $\mathcal{A}_\alpha$ 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 $\mathcal{A}_\alpha$ is calculated. Let $\mathcal{M}$ be the set of all values of the parameter $\alpha$ for which the attractor $\mathcal{A}_\alpha$ is connected. By analogy with Barnsley and Harington, we call $\mathcal{M}$ the Mandelbrot set. It is shown that, unlike the case of proper similitudes, the Mandelbrot set $\mathcal{M}$ for a pair of improper similitudes of the plane has a simple structure. Examples of attractors from the considered classes of IFS are presented.
Keywords:iterated function system, attractor, similitude, Mandelbrot set, Cantor set.
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