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Anisotropic Hölder and Sobolev spaces for hyperbolic diffeomorphisms. (French. English summary) Zbl 1138.37011
Summary: We study spectral properties of transfer operators for diffeomorphisms $$T:X \to X$$ 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.

##### MSC:
 37C30 Functional analytic techniques in dynamical systems; zeta functions, (Ruelle-Frobenius) transfer operators, etc. 37D20 Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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