Linear response for intermittent maps

We consider the one parameter family $α\mapsto T_α$ ($α\in [0,1)$) of Pomeau-Manneville type interval maps $T_α(x)=x(1+2^αx^α)$ for $x \in [0,1/2)$ and $T_α(x)=2x-1$ for $x \in [1/2, 1]$, with the associated absolutely continuous invariant probability measure $μ_α$. For $α\in (0,1)$, Sarig and Gouëzel proved that the system mixes only polynomially with rate $n^{1-1/α}$ (in particular, there is no spectral gap). We show that for any $ψ\in L^q$, the map $α\to \int_0^1 ψ\, dμ_α$ 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 $α\ge 1/2$ we need the $n^{-1/α}$ decorrelation obtained by Gouëzel under additional conditions.