Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Details Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

2010-03-29

arxiv.org/abs/1003.5592v2

Mathematics

Dynamical Systems

Typos corrected. Banach spaces (Prop 4.10, Prop 4.11, Lem 4.12, Appendix B, Section 6) cleaned up: H^1_1 Sobolev space replace

Scientific paper

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.

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