Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms
Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 127-154.

We study spectral properties of transfer operators for diffeomorphisms T:XX on a Riemannian manifold X. Suppose that Ω is an isolated hyperbolic subset for T, with a compact isolating neighborhood VX. We first introduce Banach spaces of distributions supported on V, which are anisotropic versions of the usual space of C p functions C p (V) and of the generalized Sobolev spaces W p,t (V), respectively. We then show that the transfer operators associated to T and a smooth weight g extend boundedly to these spaces, and we give bounds on the essential spectral radii of such extensions in terms of hyperbolicity exponents.

Nous étudions les propriétés spectrales des opérateurs de transfert associés aux difféomorphismes T:XX sur une variété riemannienne X. Nous supposons qu’il existe un sous-ensemble hyperbolique Ω pour T, contenu dans un voisinage isolant compact V. Nous introduisons d’abord des espaces de Banach de distributions, supportées sur V, qui sont des versions anisotropes des espaces usuels de fonctions C p , d’une part, et des espaces de Sobolev généralisés W p,t (V), d’autre part. Nous montrons ensuite que les opérateurs de transfert associés à T et à une fonction poids lisse g 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.

DOI: 10.5802/aif.2253
Classification: 37C30, 37D20, 42B25
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
Baladi, Viviane 1; Tsujii, Masato 2

1 CNRS-UMR 7586 Institut de Mathématiques Jussieu 75252 Paris Cedex 05 (France)
2 Hokkaido University Department of Mathematics Sapporo, Hokkaido (Japan)
     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},
     mrnumber = {2313087},
     zbl = {1138.37011},
     language = {en},
     url = {}
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Baladi, Viviane; Tsujii, Masato. Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. Annales de l'Institut Fourier, Volume 57 (2007) no. 1, pp. 127-154. doi : 10.5802/aif.2253.

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