Linear response formula for piecewise expanding unimodal maps

The average of a smooth function φ with respect to the SRB measure μ_t of a smooth one-parameter family f_t of piecewise expanding interval maps is not always Lipschitz (Baladi 2007 Commun. Math. Phys. 275 839–59, Mazzolena 2007 Master's Thesis Rome 2, Tor Vergata). We prove that if f_t is tangent to the topological class of f, and if ∂_t f_t|_{t = 0} = X ○ f, then is differentiable at zero, and coincides with the resummation proposed (Baladi 2007) of the (a priori divergent) series given by Ruelle's conjecture. In fact, we show that t ↦ μ_t is differentiable within Radon measures. Linear response is violated if and only if f_t is transversal to the topological class of f.