We consider families of unimodal maps whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure of depends differentiably on , as a distribution of order . 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 for a Benedicks-Carleson map , in terms of a single smooth function and the inverse branches of along the postcritical orbit. Along the way, we prove that the twisted cohomological equation has a continuous solution , if is Benedicks-Carleson and is horizontal for .
Nous considérons des familles d’applications unimodales , de récurrence postcritique lente, avec une dépendance en fonction du paramètre . Nous montrons que l’unique mesure invariante de est différentiable en fonction de , en tant que distribution d’ordre . 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 dépendant d’une unique fonction lisse et des branches inverses de le long de l’orbite postcritique. Nous prouvons enfin que l’équation cohomologique tordue admet une solution continue , si est Benedicks-Carleson et est horizontal pour .
Keywords: smooth unimodal maps, linear response, Benedicks-Carleson, SRB measures, absolutely continuous invariant measures, transfer operator
Mot clés : applications unimodales lisses, réponse linéaire, Benedicks-Carleson, mesures SRB, mesures invariantes absolument continues, opérateur de transfert
@article{ASENS_2012_4_45_6_861_0, author = {Baladi, Viviane and Smania, Daniel}, title = {Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {861--926}, publisher = {Soci\'et\'e math\'ematique de France}, volume = {Ser. 4, 45}, number = {6}, year = {2012}, doi = {10.24033/asens.2179}, mrnumber = {3075107}, zbl = {1277.37045}, language = {en}, url = {http://www.numdam.org/articles/10.24033/asens.2179/} }
TY - JOUR AU - Baladi, Viviane AU - Smania, Daniel TI - Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps JO - Annales scientifiques de l'École Normale Supérieure PY - 2012 SP - 861 EP - 926 VL - 45 IS - 6 PB - Société mathématique de France UR - http://www.numdam.org/articles/10.24033/asens.2179/ DO - 10.24033/asens.2179 LA - en ID - ASENS_2012_4_45_6_861_0 ER -
%0 Journal Article %A Baladi, Viviane %A Smania, Daniel %T Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps %J Annales scientifiques de l'École Normale Supérieure %D 2012 %P 861-926 %V 45 %N 6 %I Société mathématique de France %U http://www.numdam.org/articles/10.24033/asens.2179/ %R 10.24033/asens.2179 %G en %F ASENS_2012_4_45_6_861_0
Baladi, Viviane; Smania, Daniel. Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 45 (2012) no. 6, pp. 861-926. doi : 10.24033/asens.2179. http://www.numdam.org/articles/10.24033/asens.2179/
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