Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

[Espaces anisotropes de types Hölder et Sobolev]

Baladi, Viviane ; Tsujii, Masato

Annales de l'Institut Fourier, Tome 57 (2007), p. 127-154 / Harvested from Numdam

Résumé

Nous étudions les propriétés spectrales des opérateurs de transfert associés aux difféomorphismes $T:X\rightarrow X$ 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 $W^{p,t}(V)$, 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.

Abstract

We study spectral properties of transfer operators for diffeomorphisms $T:X\rightarrow X$ 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 $C^p(V)$ 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.

Détail

Publié le : 2007-01-01

MR 2313087 | Zbl 1138.37011 | 2 citations dans Numdam

DOI : https://doi.org/10.5802/aif.2253
Classification:  37C30,  37D20,  42B25
Mots clés: dynamique hyperbolique, opérateur de transfert, opérateur de Ruelle, spectre, Axiome A, Anosov, Perron-Frobenius, quasi-compacité

Bibliographie

[1] Avila, A.; Gouëzel, S.; Tsujii, M. Smoothness of solenoidal attractors, Discrete Cont. Dynam. Systems, Tome 15 (2006), pp. 21-35 | Article | MR 2191383 | Zbl 1106.37015

[2] Baladi, V. Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, World Scientific, Tome 16 (2000) | MR 1793194 | Zbl 1012.37015

[3] Baladi, V. Anisotropic Sobolev spaces and dynamical transfer operators: $C^{\infty }$ foliations, S.Kolyada, Y.Manin and T.Ward, Eds., Algebraic and Topological Dynamics, Contemporary Mathematics, Amer. Math. Soc. (2005), pp. 123-136 | MR 2180233 | Zbl 02236663

[4] Blank, M.; Keller, G.; Liverani, C. Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, Tome 15 (2002), pp. 1905-1973 | Article | MR 1938476 | Zbl 1021.37015

[5] Fried, D. The flat-trace asymptotics of a uniform system of contractions, Ergodic Theory Dynam. Sys., Tome 15 (1995), pp. 1061-1073 | Article | MR 1366308 | Zbl 0841.58052

[6] Fried, D. Meromorphic zeta functions for analytic flows, Comm. Math. Phys., Tome 174 (1995), pp. 161-190 | Article | MR 1372805 | Zbl 0841.58053

[7] Gouëzel, S.; Liverani, C. Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Sys., Tome 26 (2006), pp. 189-218 | Article | MR 2201945 | Zbl 05014357

[8] Gundlach, V. M.; Latushkin, Y. A sharp formula for the essential spectral radius of the Ruelle transfer operator on smooth and Hölder spaces, Ergodic Theory Dynam. Sys., Tome 23 (2003), pp. 175-191 | MR 1971201 | Zbl 02015889

[9] Hennion, H. Sur un théorème spectral et son application aux noyaux lipschitziens, Proc. Amer. Math. Soc., Tome 118 (1993), pp. 627-634 | MR 1129880 | Zbl 0772.60049

[10] Hörmander, L. The analysis of linear partial differential operators. III. Pseudo-differential operators, Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, Tome 274 (1994) | MR 1313500 | Zbl 0601.35001

[11] Kitaev, A. Yu. Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness, Nonlinearity, Tome 12 (1999), pp. 141-179 | Article | MR 1668543 | Zbl 0917.58029

[12] Paley, J. E.; Littlewood, R. Theorems on Fourier series and power series, Proc. London Math. Soc., Tome 42 (1937), pp. 52-89 | Article | Zbl 0015.25402

[13] Ruelle, D. The thermodynamic formalism for expanding maps, Comm. Math. Phys., Tome 125 (1989), pp. 239-262 | Article | MR 1016871 | Zbl 0702.58056

[14] Rugh, H. H. The correlation spectrum for hyperbolic analytic maps, Nonlinearity, Tome 5 (1992), pp. 1237-1263 | Article | MR 1192517 | Zbl 0768.58027

[15] Taylor, M. E. Pseudo differential operators, Lecture Notes in Math., Springer-Verlag, Berlin-New York, Tome 416 (1974) | MR 442523 | Zbl 0289.35001

[16] Taylor, M. E. Pseudodifferential operators and nonlinear PDE, Progress in Math., Birkhäuser, Boston, Tome 100 (1991) | MR 1121019 | Zbl 0746.35062