We study for the first time linear response for random compositions of maps, chosen independently according to a distribution $\PP$. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when $\PP$ changes smoothly to $\PP_{\eps}$? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to $\eps$; moreover, we obtain a linear response formula. We apply our results to iid compositions, with respect to various distributions $\PP_{\eps}$, of uniformly expanding circle maps, Gauss-R\'enyi maps (random continued fractions) and Pomeau-Manneville maps. Our results yield an exact formula for the invariant density of random continued fractions; while for Pomeau-Manneville maps our results provide a precise relation between their linear response under certain random perturbations and their linear response under deterministic perturbations.

中文翻译：

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随机动力系统的线性响应

我们首次研究了地图随机组合的线性响应，根据分布 $\PP$ 独立选择。我们对以下问题感兴趣：当 $\PP$ 平滑地变为 $\PP_{\eps}$ 时，随机系统的绝对连续平稳测度 (acsm) 如何变化？对于一大类一维随机映射，我们证明了 acsm 相对于 $\eps$ 的可微性；此外，我们获得了一个线性响应公式。我们将我们的结果应用于均匀扩展圆图、Gauss-R\'enyi 映射（随机连分数）和 Pomeau-Manneville 映射的各种分布 $\PP_{\eps}$ 的 iid 组合。我们的结果给出了随机连分数不变密度的精确公式；