Rigorous Approximation of Stationary Measures for Iterated Function Systems

We study the problem of the rigorous computation of the stationary measure of an IFS described by a stochastic mixture of two or more dynamical systems which are either all uniformly expanding on the interval, either all contractive. In the expanding case, the associated transfer operators satisfy a LasotaYorke inequality, and we compute rigorously the approximations in the L1 norm. The rigorous computation requires a computer-aided proof of the contraction of the transfer operators for the maps, and we show that this property propagates to the transfer operators of the IFS. In the contractive case we perform a rigorous approximation of the stationary measure in the Wasserstein-Kantorovich distance, using the same functional analytic approach. We show that a finite computation can produce a realistic computation of all contraction rates for the whole parameter space. We conclude with a description of the implementation and numerical experiments.