Linear response for random dynamical systems
Linear response for random dynamical systems
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DOI：
10.1016/j.aim.2020.107011
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发表时间：
2017-10
期刊：
影响因子：
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通讯作者：
Wael Bahsoun;Marks Ruziboev;B. Saussol
中科院分区：
文献类型：
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作者：
Wael Bahsoun;Marks Ruziboev;B. Saussol
我们首次研究了根据分布P独立选择的映射的随机合成的线性响应。我们感兴趣的问题如下：当P平滑地变为Pε时，随机系统的绝对连续平稳测度如何变化？对于一类一维随机映射，我们证明了自适应随机模型关于ε的可微性，并得到了一个线性响应公式。我们的结果涵盖了其转移算子不一定允许光谱间隙的随机映射。我们将我们的结果应用于关于一致扩张圆映射、Gauss-Rényi映射(随机连分数)和Pomeau-ε映射的各种分布的IID复合。我们的结果给出了随机连分式的不变密度的精确公式，而对于Pomeau-Manneville映射，我们的结果提供了它们在某些随机扰动下的线性响应和它们在确定性扰动下的线性响应之间的精确关系。
We study for the first time linear response for random compositions of maps, chosen independently according to a distribution P. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when P changes smoothly to P ε? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to ε; moreover, we obtain a linear response formula. Our results cover random maps whose transfer operator does not necessarily admit a spectral gap. We apply our results to iid compositions, with respect to various distributions P ε, of uniformly expanding circle maps, Gauss-Rényi 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.