  We consider a Markov chain X_1, X_2, ..., X_n belonging to a class of
iterated random functions, which is "one-step contracting" with respect to some
distance d. If f is any separately Lipschitz function with respect to d, we use
a well known decomposition of S_n=f(X_1, ..., X_n) -E[f(X_1, ..., X_n)]$ into a
sum of martingale differences d_k with respect to the natural filtration F_k.
We show that each difference d_k is bounded by a random variable eta_k
independent of F_{k-1}. Using this very strong property, we obtain a large
variety of deviation inequalities for S_n, which are governed by the
distribution of the eta_k's. Finally, we give an application of these
inequalities to the Wasserstein distance between the empirical measure and the
invariant distribution of the chain.
