The generalized Sierpiński problem (Q1406458)

In the fractal theory, the ``chaos game'' is well known [see \textit{H.-O. Peitgen, H. Jürgens}, and \textit{D. Saupe}, Chaos and Fractals. New frontiers of science'', New York: Springer-Verlag (1992; Zbl 0779.58004)]. The author proposes the following generalization of this classical algorithm. On the complex plane \(\mathbb{C}\) an arbitrary regular \(m\)-angular \(A_0A_1A_2\dots A_{m-1}\) is taken as well as the numerical sequence \(S = \{s_1,s_2,\dots,s_n,\dots\}\), where \(s_n\in\{0,1,2,\dots,m-1\}\), \(n\in \mathbb{N}\). Consider the sequence of points \(\{z_0,z_1,\dots,z_n,\dots\}\in \mathbb{C}\) constructed by the formula \[ z_{n+1} = A_{s_n} + \frac{z_n - A_{s_n}}{\gamma},\qquad \gamma\in [1;+\infty), \] where \(z_0\) is an arbitrary initial point. Further, the sequence \(S\) is constructed in a random way, i.e., each term takes one of the values \(\{0,1,2,\dots,m-1\}\) with the probability \(1/m\).