Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Baladi, Viviane; Smania, Daniel

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 45 (2012) no. 6, pp. 861-926.

Nous considérons des familles $t\mapsto f_t$ d’applications unimodales $C^4$, de récurrence postcritique lente, avec une dépendance $C^2$ en fonction du paramètre $t$. Nous montrons que l’unique mesure invariante ${\mu }_t$ de $f_t$ est différentiable en fonction de $t$, en tant que distribution d’ordre $1$. La preuve utilise des opérateurs de transfert sur des tours dont les bords sont mollifiés avec des fonctions de troncation lisses, pour éviter l’introduction de discontinuités artificielles. Nous donnons de plus une représentation de ${\mu }_t$ dépendant d’une unique fonction lisse et des branches inverses de $f_t$ le long de l’orbite postcritique. Nous prouvons enfin que l’équation cohomologique tordue $v=\alpha \circ f-f^{\text{'}}\alpha$ admet une solution continue $\alpha$, si $f$ est Benedicks-Carleson et $v$ est horizontal pour $f$.

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 ${\mu }_t$ 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 ${\mu }_t$ 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 $v=\alpha \circ f-f^{\text{'}}\alpha$ has a continuous solution $\alpha$, if $f$ is Benedicks-Carleson and $v$ is horizontal for $f$.

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