Linear response formula for piecewise expanding unimodal maps
linear response | unimodal maps | SRB measure
The average $R(t)=\int \varphi\, d\mu_t$ of a smooth function $\varphi$ with respect to the SRB measure $\mu_t$ of a smooth one-parameter family $f_t$ of piecewise expanding interval maps is not always Lipschitz. We prove that if $f_t$ is tangent to the topological class of $f$ then $R(t)$ is differentiable at zero, and $R'(0)$ coincides with a resummation of the (a priori divergent) series given by Ruelle's conjecture. It is the first time that a linear response formula is obtained in a setting where structural stability does not hold.