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Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi; Daniel Smania — 2012

Annales scientifiques de l'École Normale Supérieure

We consider $C^{2}$ families $t \mapsto f_{t}$ of $C^{4}$ unimodal maps $f_{t}$ whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure $μ_{t}$ 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 $μ_{t}$ for a Benedicks-Carleson map $f_{t}$ , in terms of a single smooth function and the inverse branches...

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