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

Random M\"obius dynamics on the unit disc and perturbation theory for Lyapunov exponents