Nous étudions les propriétés spectrales des opérateurs de transfert associés aux difféomorphismes sur une variété riemannienne . Nous supposons qu’il existe un sous-ensemble hyperbolique pour , contenu dans un voisinage isolant compact . Nous introduisons d’abord des espaces de Banach de distributions, supportées sur , qui sont des versions anisotropes des espaces usuels de fonctions , d’une part, et des espaces de Sobolev généralisés , d’autre part. Nous montrons ensuite que les opérateurs de transfert associés à et à une fonction poids lisse s’étendent continûment à ces espaces, et nous donnons des bornes pour les rayons spectraux essentiels de ces extensions, en fonction d’exposants d’hyperbolicité.
We study spectral properties of transfer operators for diffeomorphisms on a Riemannian manifold . Suppose that is an isolated hyperbolic subset for , with a compact isolating neighborhood . We first introduce Banach spaces of distributions supported on , which are anisotropic versions of the usual space of functions and of the generalized Sobolev spaces , respectively. We then show that the transfer operators associated to and a smooth weight extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents.
Keywords: Hyperbolic dynamics, transfer operator, Ruelle operator, spectrum, axiom A, Anosov, Perron-Frobenius, quasi-compact
Mot clés : dynamique hyperbolique, opérateur de transfert, opérateur de Ruelle, spectre, Axiome A, Anosov, Perron-Frobenius, quasi-compacité
@article{AIF_2007__57_1_127_0, author = {Baladi, Viviane and Tsujii, Masato}, title = {Anisotropic {H\"older} and {Sobolev} spaces for hyperbolic diffeomorphisms}, journal = {Annales de l'Institut Fourier}, pages = {127--154}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {1}, year = {2007}, doi = {10.5802/aif.2253}, zbl = {1138.37011}, mrnumber = {2313087}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2253/} }
TY - JOUR AU - Baladi, Viviane AU - Tsujii, Masato TI - Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms JO - Annales de l'Institut Fourier PY - 2007 SP - 127 EP - 154 VL - 57 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2253/ DO - 10.5802/aif.2253 LA - en ID - AIF_2007__57_1_127_0 ER -
%0 Journal Article %A Baladi, Viviane %A Tsujii, Masato %T Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms %J Annales de l'Institut Fourier %D 2007 %P 127-154 %V 57 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2253/ %R 10.5802/aif.2253 %G en %F AIF_2007__57_1_127_0
Baladi, Viviane; Tsujii, Masato. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 127-154. doi : 10.5802/aif.2253. http://www.numdam.org/articles/10.5802/aif.2253/
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