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.
@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}, volume = {57}, year = {2007}, pages = {127-154}, doi = {10.5802/aif.2253}, zbl = {1138.37011}, mrnumber = {2313087}, language = {en}, url = {http://dml.mathdoc.fr/item/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) pp. 127-154. doi : 10.5802/aif.2253. http://gdmltest.u-ga.fr/item/AIF_2007__57_1_127_0/
[1] Smoothness of solenoidal attractors, Discrete Cont. Dynam. Systems, Tome 15 (2006), pp. 21-35 | Article | MR 2191383 | Zbl 1106.37015
[2] Positive transfer operators and decay of correlations, Advanced Series in Nonlinear Dynamics, World Scientific, Tome 16 (2000) | MR 1793194 | Zbl 1012.37015
[3] Anisotropic Sobolev spaces and dynamical transfer operators: 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] Ruelle-Perron-Frobenius spectrum for Anosov maps, Nonlinearity, Tome 15 (2002), pp. 1905-1973 | Article | MR 1938476 | Zbl 1021.37015
[5] 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] Meromorphic zeta functions for analytic flows, Comm. Math. Phys., Tome 174 (1995), pp. 161-190 | Article | MR 1372805 | Zbl 0841.58053
[7] Banach spaces adapted to Anosov systems, Ergodic Theory Dynam. Sys., Tome 26 (2006), pp. 189-218 | Article | MR 2201945 | Zbl 05014357
[8] 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] 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] 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] Fredholm determinants for hyperbolic diffeomorphisms of finite smoothness, Nonlinearity, Tome 12 (1999), pp. 141-179 | Article | MR 1668543 | Zbl 0917.58029
[12] Theorems on Fourier series and power series, Proc. London Math. Soc., Tome 42 (1937), pp. 52-89 | Article | Zbl 0015.25402
[13] The thermodynamic formalism for expanding maps, Comm. Math. Phys., Tome 125 (1989), pp. 239-262 | Article | MR 1016871 | Zbl 0702.58056
[14] The correlation spectrum for hyperbolic analytic maps, Nonlinearity, Tome 5 (1992), pp. 1237-1263 | Article | MR 1192517 | Zbl 0768.58027
[15] Pseudo differential operators, Lecture Notes in Math., Springer-Verlag, Berlin-New York, Tome 416 (1974) | MR 442523 | Zbl 0289.35001
[16] Pseudodifferential operators and nonlinear PDE, Progress in Math., Birkhäuser, Boston, Tome 100 (1991) | MR 1121019 | Zbl 0746.35062