  We consider random dynamical systems such as groups of conformal
transformations with a probability measure, or transversaly 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 pseudo-group), or almost
surely a long composition of maps contracts exponentially a ball. We deduce
some results about the unique ergodicity.
