Abstract
In this paper, the attractors of iterated function systems (IFSs) consisting of two improper similitudes of the plane are investigated. The attractor of such IFS is either a connected or totally 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 \({{f}_{1}}(z) = \alpha \bar {z}\) , \({{f}_{2}}(z) = \alpha (\bar {z} - 1) + 1\) , \(\alpha ,z \in \mathbb{C}\) , \(0 < \left| \alpha \right| < 1\) . It is shown that \({{\mathcal{A}}_{\alpha }}\) is one of the following sets: a segment, a Cantor set in a segment, a parallelogram, or 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 Harrington, 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 plane improper similitudes has a simple structure. Examples of attractors from the considered classes of IFSs are presented.
