Linear response for random dynamical systems

Wael Bahsoun, Marks Ruziboev, Benoî t Saussol

Published 2017-10-10Version 1

We study for the first time linear response for random compositions of maps, chosen independently according to a distribution $\mathbb P$. We are interested in the following question: how does an absolutely continuous stationary measure (acsm) of a random system change when $\mathbb P$ changes smoothly to $\mathbb P_{\varepsilon}$? For a wide class of one dimensional random maps, we prove differentiability of acsm with respect to $\varepsilon$; moreover, we obtain a linear response formula. We apply our results to iid compositions of uniformly expanding circle maps, to iid compositions of the Gauss-R\'enyi maps (random continued fractions) and to iid compositions of Pomeau-Manneville maps.