Linear response for intermittent maps

Abstract

We consider the one parameter family α↦Tα (α∈[0,1)) of Pomeau-Manneville type interval maps Tα(x)=x(1+2αxα) for x∈[0,1/2) and Tα(x)=2x−1 for x∈[1/2,1], with the associated absolutely continuous invariant probability measure μα. For α∈(0,1), Sarig and Gouëzel proved that the system mixes only polynomially with rate n1−1/α (in particular, there is no spectral gap). We show that for any ψ∈Lq, the map α→∫10ψdμα is differentiable on [0,1−1/q), and we give a (linear response) formula for the value of the derivative. This is the first time that a linear response formula for the SRB measure is obtained in the setting of slowly mixing dynamics. Our argument shows how cone techniques can be used in this context. For α≥1/2 we need the n−1/α decorrelation obtained by Gouëzel under additional conditions.