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

Abstract

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We study spectral properties of transfer operators for diffeomorphisms T : X → X on a Riemannian manifold X . Suppose that Ω is an isolated hyperbolic subset for T , with a compact isolating neighborhood V ⊂ 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.

How to cite

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Baladi, 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 -

References

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