Linear response for intermittent maps with critical point

We consider a two-parameter family of maps $T_{\alpha ,\beta }:[0,1]\to [0,1]$ with a neutral fixed point and a non-flat critical point. Building on a cone technique due to Baladi and Todd, we show that for a class of $L^q$ observables $\phi :[0,1]\to \mathbb{R}$ the bivariate map $(\alpha ,\beta )\mapsto {\int }_0^1\phi d{\mu }_{\alpha ,\beta }$, where ${\mu }_{\alpha ,\beta }$ denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.