Regularity and linear response formula of the SRB measures for solenoidal attractors

We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$ , where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$ , $r \geq 2$ , and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$ -generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$ . When $s> {u}/{2}$ , it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.