Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps

Viviane Baladi; Daniel Smania

Annales scientifiques de l'École Normale Supérieure (2012)

  • Volume: 45, Issue: 6, page 861-926
  • ISSN: 0012-9593

Abstract

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We consider C 2 families t ↦ 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 of  f t along the postcritical orbit. Along the way, we prove that the twisted cohomological equation v = α ∘ f - f ' α has a continuous solution α , if f is Benedicks-Carleson and v is horizontal for  f .

How to cite

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Baladi, Viviane, and Smania, Daniel. "Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps." Annales scientifiques de l'École Normale Supérieure 45.6 (2012): 861-926. <http://eudml.org/doc/272218>.

@article{Baladi2012,
abstract = {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^\{\prime \} \alpha $ has a continuous solution $\alpha $, if $f$ is Benedicks-Carleson and $v$ is horizontal for $f$.},
author = {Baladi, Viviane, Smania, Daniel},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {smooth unimodal maps; linear response; Benedicks–Carleson; SRB measures; absolutely continuous invariant measures; transfer operator},
language = {eng},
number = {6},
pages = {861-926},
publisher = {Société mathématique de France},
title = {Linear response for smooth deformations of generic nonuniformly hyperbolic unimodal maps},
url = {http://eudml.org/doc/272218},
volume = {45},
year = {2012},
}

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
PB - Société mathématique de France
VL - 45
IS - 6
SP - 861
EP - 926
AB - 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^{\prime } \alpha $ has a continuous solution $\alpha $, if $f$ is Benedicks-Carleson and $v$ is horizontal for $f$.
LA - eng
KW - smooth unimodal maps; linear response; Benedicks–Carleson; SRB measures; absolutely continuous invariant measures; transfer operator
UR - http://eudml.org/doc/272218
ER -

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