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Annales scientifiques de l'ENS - Parutions - série 4, 45 (2012)

Parutions < série 4, 45

ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4 45, fascicule 6 (2012)

Viviane Baladi, Daniel Smania
Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps
Annales scientifiques de l'ENS 45, fascicule 6 (2012), 861-926

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Résumé :
Réponse linéaire pour les déformations lisses d'applications unimodales génériques non-uniformément hyperboliques
Nous considérons des familles tf_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 _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 _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=f - f' admet une solution continue , si f est Benedicks-Carleson et v est horizontal pour f.

Mots-clefs : Applications unimodales lisses, réponse linéaire, Benedicks–Carleson, mesures SRB, mesures invariantes absolument continues, opérateur de transfert.

Abstract:
We consider C^2 families tf_t of C^4 unimodal maps f_t whose critical point is slowly recurrent, and we show that the unique absolutely continuous invariant measure _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 _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=f - f' has a continuous solution , if f is Benedicks-Carleson and v is horizontal for f.

Keywords: Smooth unimodal maps, linear response, Benedicks–Carleson, SRB measures, absolutely continuous invariant measures, transfer operator

Class. math. : 37C40, 37C30, 37D25, 37E05


ISSN : 0012-9593
Publié avec le concours de : Centre National de la Recherche Scientifique

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