Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

TL;DR

This paper demonstrates the differentiability of the absolutely continuous invariant measure for certain nonuniformly hyperbolic unimodal maps under smooth deformations, using transfer operators and new representations.

Contribution

It introduces a novel approach to analyze the invariant measure's dependence on parameters for nonuniformly hyperbolic maps, including a new representation for Benedicks-Carleson maps.

Findings

Invariant measure depends differentiably on parameters

New representation of acim for Benedicks-Carleson maps

Solution to twisted cohomological equation is continuous

Abstract

We consider C^2 families of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of f_t 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 f_t, in terms of a single smooth function and the inverse branches of f_t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v(x)=\alpha (f (x)) - f'(x) \alpha (x) has a continuous solution \alpha, if f is Benedicks-Carleson and v is horizontal for f. In v3 we added a note containing 3 comments regarding minor typos.