Random conformal dynamical systems

We consider random dynamical systems such as groups of conformal transformations with a probability measure, or transversally conformal foliations with a Laplace operator along the leaves, in which case we consider the holonomy pseudo-group. We prove that either there exists a measure invariant under all the elements of the group (or the pseudogroup), or almost surely a long composition of maps contracts a ball exponentially. We deduce some results about the unique ergodicity.