# Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms

Viviane Baladi^{[1]}; Masato Tsujii^{[2]}

- [1] CNRS-UMR 7586 Institut de Mathématiques Jussieu 75252 Paris Cedex 05 (France)
- [2] Hokkaido University Department of Mathematics Sapporo, Hokkaido (Japan)

Annales de l’institut Fourier (2007)

- Volume: 57, Issue: 1, page 127-154
- ISSN: 0373-0956

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topBaladi, Viviane, and Tsujii, Masato. "Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms." Annales de l’institut Fourier 57.1 (2007): 127-154. <http://eudml.org/doc/10215>.

@article{Baladi2007,

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.},

affiliation = {CNRS-UMR 7586 Institut de Mathématiques Jussieu 75252 Paris Cedex 05 (France); Hokkaido University Department of Mathematics Sapporo, Hokkaido (Japan)},

author = {Baladi, Viviane, Tsujii, Masato},

journal = {Annales de l’institut Fourier},

keywords = {Hyperbolic dynamics; transfer operator; Ruelle operator; spectrum; axiom A; Anosov; Perron-Frobenius; quasi-compact; hyperbolic dynamics; Axiom A},

language = {eng},

number = {1},

pages = {127-154},

publisher = {Association des Annales de l’institut Fourier},

title = {Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms},

url = {http://eudml.org/doc/10215},

volume = {57},

year = {2007},

}

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

PB - Association des Annales de l’institut Fourier

VL - 57

IS - 1

SP - 127

EP - 154

AB - 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.

LA - eng

KW - Hyperbolic dynamics; transfer operator; Ruelle operator; spectrum; axiom A; Anosov; Perron-Frobenius; quasi-compact; hyperbolic dynamics; Axiom A

UR - http://eudml.org/doc/10215

ER -

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