Stationary measures associated to analytic iterated function schemes

We study how the stationary measure associated to analytic contractions on the unit interval behaves under changes in the contractions and the weights. Firstly we give a simple proof of the fact that the integrals of analytic functions with respect to the stationary measure vary analytically if we perturb the contractions and the weights analytically. Secondly, we consider the special case of affine contractions and we prove a conjecture of J. Fraser in 3 on the Kantorovich–Wasserstein distance between two stationary measures associated to affine contractions on the unit interval with different rates of contraction.