Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

TY - JOUR

T1 - Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

AU - Baladi, Viviane

AU - Smania, Daniel

PY - 2012

Y1 - 2012

N2 - We consider C2 families t → ft of C4 unimodal maps ft whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μt 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 μt 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 = α f - f'α has a continuous solution, if f is Benedicks-Carleson and v is horizontal for f.

AB - We consider C2 families t → ft of C4 unimodal maps ft whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure μt 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 μt 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 = α f - f'α has a continuous solution, if f is Benedicks-Carleson and v is horizontal for f.

M3 - Journal article

SN - 0012-9593

VL - 45

SP - 861

EP - 926

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

IS - 6

ER -