Random iteration of Mobius transformations and Furstenberg's theorem

Let Y-1, Yz,... be a sequence of independent random maps, identically distr ibuted with respect to a probability measure mu, on SL(2, R). A (deep) theo rem of Furstenberg gives abstract conditions under which for almost every s uch 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 Mobius transformations on the upper half-plane and provide a very explicit way to determine whether or not the required conditions are s atisfied.