The Mandelbrot Set for a Pair of Improper Similitudes of the Plane

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.