In this paper we show how ideas, methods and results from optimal transportation can be used to study various aspects of the stationary measures of Iterated Function Systems equipped with a probability distribution. We recover a classical existence and uniqueness result under a contraction-on-average assumption, prove generalised 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.

中文翻译：

---

迭代函数系统的最优运输和静止措施

在本文中，我们展示了如何使用最优运输的想法、方法和结果来研究配备概率分布的迭代函数系统的平稳测量的各个方面。我们在平均收缩假设下恢复经典存在性和唯一性结果，证明可以从中推导出尾估计的广义矩界限，考虑相关马尔可夫链的经验测度的收敛性，并在许多情况下证明 Lipschitz 连续性系统受到扰动时的平稳测量，因此几乎每个扰动参数都有一个“线性响应公式”。