Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Baladi, Viviane; Tsujii, Masato

Annales de l'Institut Fourier, Tome 57 (2007) no. 1, pp. 127-154.

Nous étudions les propriétés spectrales des opérateurs de transfert associés aux difféomorphismes $T:X\to X$ sur une variété riemannienne $X$. Nous supposons qu’il existe un sous-ensemble hyperbolique $\Omega$ 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 $W^{p,t}\left(V\right)$, 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.

We study spectral properties of transfer operators for diffeomorphisms $T:X\to X$ on a Riemannian manifold $X$. Suppose that $\Omega$ is an isolated hyperbolic subset for $T$, with a compact isolating neighborhood $V\subset X$. We first introduce Banach spaces of distributions supported on $V$, which are anisotropic versions of the usual space of $C^p$ functions $C^p\left(V\right)$ and of the generalized Sobolev spaces $W^{p,t}\left(V\right)$, 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.

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

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