Iteration of Möbius transformations and attractors on the real line (Q678421)

Let \(x_0\) be an arbitrary point in the complex plane. For each positive integer \(n\) the authors choose \(s_n(z)\) to be \(-5/(1+z)\) or \(-0.5/(1+z)\) with equal probability. They introduce the orbit \((z_n)^\infty_0\), where \(z_n= s_n(z_{n-1})\) for \(n\geq 1\). The authors prove that with probability one the orbit is attracted to the real axis.