Random Iteration of Isometries of the Hyperbolic Plane

1999, v.5, №1, 69-88

ABSTRACT

Suppose that we start at a point $Z_0\in\overline{\C}$ in the extended complex plane and form an orbit $\{Z_n\}^\infty_0$ by random iteration in the following way. At each step $n\ge1$ we form $Z_n=f(Z_{n-1})$ where $f=f_n$ is chosen, with equal probability, as one of the functions $-5/(1+z)$ and $-0.5/(1+z)$ which map the upper half-plane onto itself preserving the hyperbolic metric. In [A. Barrlund, H. Wallin and J. Karlsson, Iteration of Mobius transformations and attractors on the real line, Comput. Math. Appl., 1997, 33, 1-12] it was proved that the orbit $\{Z_n\}^\infty_0$ tends to the extended real line $\overline{R}=R\cup\{\infty\}$ with probability 1, as $n$ tends to infinity. In this paper we study the same problem for general isometries (in the hyperbolic metric) of the upper half-plane onto itself. We prove that $Z_n \to \overline{R}$ in probability, as $n \to \infty$ and show that we do not have convergence with probability one in general. As an application we study random iteration of linear maps on $R^2$ with determinant one. The problems studied here are related to Markov chains and to the generation of fractal pictures as attractors of iterated function systems.

Keywords: random iteration,probabilistic attractor,hyperbolic isometry,Mobiustransformation,linear map