Linear response for intermittent maps with critical point

We consider a two-parameter family of maps <?CDATA Tα, β : [0,1] → [0,1]?> T α , β : [ 0 , 1 ] → [ 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 <?CDATA φ : [0,1] → R?> ϕ : [ 0 , 1 ] → R the bivariate map <?CDATA (α, β) ↦ ∫₀¹ φ \, dμα,β?> ( α , β ) ↦ ∫ 0 1 ϕ d μ α , β , where <?CDATA μα, β?> μ α , β denotes the invariant SRB measure, is differentiable in a certain parameter region, and establish a formula for its directional derivative.