Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps
Abstract
We consider C2 families of C4 unimodal maps ft whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of ft depends differentiably on t, as a distribution of order 1. The proof uses transfer operators on towers whose level boundaries are mollified via smooth cutoff functions, in order to avoid artificial discontinuities. We give a new representation of the acim for a Benedicks-Carleson map ft, in terms of a single smooth function and the inverse branches of ft along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=α (f (x)) - f'(x) α (x) has a continuous solution α, if f is Benedicks-Carleson and v is horizontal for f. In v3 we added a note containing 3 comments regarding minor typos.