Random iteration of Möbius transformations and Furstenberg's theorem

Abstract

Let $Y_1, Y_2, \dots$ be a sequence of independent random maps, identically distributed with respect to a probability measure $\mu$ on $SL(2,R)$. A (deep) theorem of Furstenberg gives abstract conditions under which for almost every such sequence the orbit of a non-zero initial point in $R^2$ tends to infinity exponentially fast. In the present paper we translate this statement into the set-up of Möbius transformations on the upper half-plane and provide a very explicit way to determine whether or not the required conditions are satisfied.