  Randomly drawn $2\times 2$ matrices induce a random dynamics on the Riemann
sphere via the M\"obius transformation. Considering a situation where this
dynamics is restricted to the unit disc and given by a random rotation
perturbed by further random terms depending on two competing small parameters,
the invariant (Furstenberg) measure of the random dynamical system is
determined. The results have applications to the perturbation theory of
Lyapunov exponents which are of relevance for one-dimensional discrete random
Schr\"odinger operators.
