Linear response for random systems with a cusp

Abstract

We study i.i.d.\ random compositions of cusp tent-like interval maps having a common cusp point and a common cusp value. For cusp exponents $-1<\beta <-\frac{1}{2}$, we prove that the associated annealed transfer operators have a spectral gap on $W^{1,1}(I)$ and $W^{2,1}(I)$. For all sufficiently small perturbations of the probability law, there is a stationary density $h_{\epsilon }\in W^{2,1}(I)$, unique among stationary densities belonging to $W^{1,1}(I)$. Moreover, the map $\epsilon \mapsto h_{\epsilon }$ is differentiable at $\epsilon =0$ in $W^{1,1}(I)$, and we obtain an explicit linear response formula.

26 pages, one figure