  In this article we show how ideas, methods and results from optimal
transportation can be used to study various aspects of the stationary
measuresof Iterated Function Systems equipped with a probability distribution.
We recover a classical existence and uniqueness result under a
contraction-on-average assumption, prove generalized moment bounds from which
tail estimates can be deduced, consider the convergence of the empirical
measure of an associated Markov chain, and prove in many cases the Lipschitz
continuity of the stationary measure when the system is perturbed, with as a
consequence a "linear response formula" at almost every parameter of the
perturbation.
