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)
@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},
     mrnumber = {2313087},
     zbl = {1138.37011},
     language = {en},
     url = {https://aif.centre-mersenne.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  - https://aif.centre-mersenne.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 https://aif.centre-mersenne.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, Volume 57 (2007) no. 1, pp. 127-154. doi : 10.5802/aif.2253. https://aif.centre-mersenne.org/articles/10.5802/aif.2253/

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

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

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

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

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

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

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

[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., Volume 23 (2003), pp. 175-191 | MR | Zbl

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

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

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

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

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

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

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

[16] Taylor, M. E. Pseudodifferential operators and nonlinear PDE, Progress in Math., Volume 100, Birkhäuser, Boston, 1991 | MR | Zbl

Cited by Sources: