Linear response formula for piecewise expanding unimodal maps

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Abstract
The average of a smooth function with respect to the SRB measure μt of a smooth one-parameter family ft 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 ft is tangent to the topological class of f, and if ∂t ft|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 μt is differentiable within Radon measures. Linear response is violated if and only if ft is transversal to the topological class of f.